control/0000755000076500000240000000000012273411257011544 5ustar lukasstaffcontrol/COPYING0000644000076500000240000010451312273411256012602 0ustar lukasstaff GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The GNU General Public License is a free, copyleft license for software and other kinds of works. The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program--to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too. When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things. To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others. For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights. Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it. For the developers' and authors' protection, the GPL clearly explains that there is no warranty for this free software. For both users' and authors' sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions. Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users' freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users. Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free. The precise terms and conditions for copying, distribution and modification follow. TERMS AND CONDITIONS 0. Definitions. "This License" refers to version 3 of the GNU General Public License. "Copyright" also means copyright-like laws that apply to other kinds of works, such as semiconductor masks. 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If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions. When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission. 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Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10. 9. Acceptance Not Required for Having Copies. You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so. 10. Automatic Licensing of Downstream Recipients. Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License. An "entity transaction" is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party's predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts. You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it. 11. Patents. A "contributor" is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor's "contributor version". A contributor's "essential patent claims" are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, "control" includes the right to grant patent sublicenses in a manner consistent with the requirements of this License. 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If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. "Knowingly relying" means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient's use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid. If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it. A patent license is "discriminatory" if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007. Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law. 12. No Surrender of Others' Freedom. If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program. 13. Use with the GNU Affero General Public License. Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such. 14. Revised Versions of this License. The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License "or any later version" applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation. If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program. Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version. 15. Disclaimer of Warranty. THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 16. Limitation of Liability. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. 17. Interpretation of Sections 15 and 16. If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee. END OF TERMS AND CONDITIONS How to Apply These Terms to Your New Programs If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms. To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. Copyright (C) This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . Also add information on how to contact you by electronic and paper mail. If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode: Copyright (C) This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an "about box". You should also get your employer (if you work as a programmer) or school, if any, to sign a "copyright disclaimer" for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see . The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read . control/DESCRIPTION0000644000076500000240000000061212273411256013250 0ustar lukasstaffName: Control Version: 2.6.2 Date: 2014-02-02 Author: Lukas Reichlin Maintainer: Lukas Reichlin Title: Control Systems Description: Computer-Aided Control System Design (CACSD) Tools for GNU Octave, based on the proven SLICOT Library Depends: octave (>= 3.6.0) Autoload: yes License: GPLv3+ Url: http://octave.sf.net, http://www.slicot.org control/doc/0000755000076500000240000000000012273411257012311 5ustar lukasstaffcontrol/doc/control.pdf0000644000076500000240000201646312273411257014500 0ustar lukasstaff%PDF-1.5 % 1 0 obj << /Length 587 /Filter /FlateDecode >> stream xmTM@+z&?tBL$d4*.<_fW_wիrc;`GUOV&ʮ[v6W7TvbuYt/N.5=S> stream xmTM@+z&?tBL0d4*.<̿~UfW_uvc;Z̫MfG} I]/ޭmޯo⣩0^'^x]fkn{EK{*ʇupg6;ލ$4;gZ8, M[TPRJGeWxmE7 "/7j;{Yʋ"1tm|oirI ɑc׺>[TқEnn#bBSEV嶭mzsg)gR133w xAb;aGL6K&0+}&"?(Ҧa/ c,!-f3*Ix {asIC%hS7}H=ŤIY(jŧ Z4{SO5Z ekxvKǬ@2a> stream xmSn0+$z"aKU^CvF^p=!94gB˥0pދ s#P~k@hZ+vQڦ(A,Rf5Ħq8>K_X NH3$Ǟ{<0*5c~Pʯ5W42^!0^#rqxƘE3x z)cgl1BҰ?Xq!NAWA*d1)iȧΰО 9璆NVfkVaUJ?%͚5ػbTW=ј52f&p2pjV^cHMcVYxLS7E=1j g endstream endobj 6 0 obj << /Length 321 /Filter /FlateDecode >> stream xڅPN!WpO&hI=4[Z7ݚBqh4\a$Hb%'ߒ<,i`4Oy\"x(ܖO8[C8CMW +]Pҡv5TA =UEU}+wc+Jlh 29J+AEVv_CYutFX˔w!rͶ0b:a,^d+vGĦrq}zX+*P*ޟs9񅕊te`\ҦfimFI>M endstream endobj 14 0 obj << /Length 788 /Filter /FlateDecode >> stream xڍTM0Wp+/|[ժ[ꡪrk{V l3;!m̛7xÏE;v~[4-pfeâ,l\aȲǬKX DMvMyk%RP <]^yFey r`j8.$6B_6B96)¶&|ڟM% j28bq p"]#kr׬oXr3@ ioz;V-,ϳ5bvY΄zj,^8Ev() е.,O8yVtJHmt3$-p՗79N;nN0k}$5ZO ܤnot Wy.6wr ju)he@.!vԞvȕ-PI IUnޝ:wy${0:K5k[3  endstream endobj 20 0 obj << /Length 2921 /Filter /FlateDecode >> stream xڭrܸBUMwd93kS&>dr`nDle_fKǓrJ <<o_@O`ftU^v+|(U0":b[N}}4^T:V[6HjOl\ob@UQ:Ȕ < D  #t\Cz_EyXʶе5E8`m#z 0sEWSTbwnpGWQ `,mlQh֡ұfг=FަMC$%]{D#HBW۱+4:{s}KZS(::s==2pz2M*d`)kp.k0 MmǃZJkO%wS`*Q*^zʭyNZ Ba^1WCH6B)3_X$3G1(⯾NDԶh 3E$8Mp%nj~0LZQ X )?ز,Z RG8i#(Ykom4yîFXCu=A4k2NJTbOr@%aH>NX( pNx~F^~oX2{[ zFCA0{s |Xnώ\sB-40Y\gDktV};,yk F`VdUh)B3+9'=;`WEGAvjR7؋[s] dJKwϽmؒI,nGΎw`QIZOA{XSxH=jh3 eH'g)辂墌CkZBAwfalUCxj_g(hoTkHy$v AHFˆCga%HYyT6=]S]}XdCH4z|t޵Ք4&&@L Z_iӹw-л$Uq>plkC<:ؑ啓q &ˁ\)@Wddd=U ǓD%"}HFrDZJdGy'Kio3Ղ?{%t@g\%޿Ǟeąt;xզ{ZOL~1O*T%zrɦ]O~SEn [=C%=Y 63 :%&3;$XA)UBd>D1Xb\KϸR~A'O\;b$jz|:,[8߿Hש̮*tV>zM|_UT, y=?}kj4CL-B{: f~Pnl^p@N0ۺ- i<Ѭ]Z?6 ~L֞tIavg!j GO[tk]8=CՉCakI,Om\TʂZG$j,º?Kfx>}{;fU~%W1,Q.A!n>? 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{ғICH@76]eypp;8X /v; #bKjI_'¼rWO|gpciwL7 SK hldEz_ts$<8{kVw{ C ;aߌj| xiyn{/ Hϟ{ԍQ-{[ݒ7wqG endstream endobj 1110 0 obj << /Length 2441 /Filter /FlateDecode >> stream xZsܶ_!&Jgڌ,N*Oռ8~x8{d*dIƳr[Gѯ8Vt s0Si~D"yASc3Bp UO3)Կ6 9 I8> `EdnUje jPٮQW148iTC*(ă5&Pf"y&K2HQБka|*\]'QþuRSk(=3Wծ]̸SCniD]C\khPjflk d;;)Nߗ[<- 0sh^$ hQ5h(h(G|= }o8ō@l}kvT>rǘ&֥SKLB;ϵ@/\s|_ln' (J?zd'$J?3k+c+Gq+Q 8^u$zHg}*B:˳V< Nj< ZF<:TI83T&QYR;O8ISfytVC%Z8y5IsY e~l&]ĸՑhb3noK++[_0UjvÓS m(x0~a:qӲ)[gV+C[jԌTO7p`7Y<5SRX L>})p-i׈ͨ#b}dU0hnjc9A֔n X6PM tNs;N?qLavd(A/џgKx7P( m?1n䤄QxIr=o3ErF$ͪcZNZq[N *"Ȅl\?6{!V]rBs&YncCӫE?K&өtfe(jT㬻VM|S³Ew<< Ӌ_&Jj9_|w̖GC̃ӓM Qx 'e9G!Re{V=ULJaak1~g*e(wc"<N8T :Y &W yަRࣹ 6StU+հ*)N!iBn^8_IfLe+bV.R?M-Wئ_uUe^H E7 ,[STv(>Xqe;? ,ff}Io]({m)#q񙉾ᠾm~JnIĮ_4ηB}^kuWOiz̸yCaugQ8=OIJSyBx'{|")^3}8#/Zn#O#w׊5A~%Kb8O2ﱒ+khK1>C圲y]uk@,ӷ^ci>f $a](hVRgQJEE$Ad3tVnfi|ş(2q zi| ٘s~q!جR=,Ύ@MEc*?Fcj`rNߎ[iu_ t@?u<A--SY[N2Za?XuC^3`ӋдhlGG?5xoJvNw q=bhol ݚRR[܂&%)X{Quy11Favmx n+d00+vy:Hzx?}r0> stream xYK4HqS2,R{Ǚ;.ïeɶQ&X5r[C=x^H"E"?IY}>N.ؓ_q J/7ROzSJ#?\ŌZ?cޖ&&\2t7O#g4Va kά*nk]۽ydpY4M[}Y nt(o ZW`""&Qƭ>D/ cm'JLm1za!텞uD:ױB8=SIӄX *xI8DJ jªqS+XbDRy81&қm ^G1S<||LzZց zn|ulA&ʙ͜W=G>TNn2MŀXJpBLx&P{=# wg]AmWRwSȨ.S|g#L.ΰkIwjNΠ!L!听6vjНM?'(T!JR"0ًdPtuSTm2v;7EȤzjuyw~],e26 @S*z ` q)q ACV&hΔ,zrqdOo~ykv¼!7$Me Rd-{C{{< [ E:Ǝ<)sIwmlo*lv =#8"IH<.7ACv };'v,t+7ƚ>c`H1_M Vj:?q쀕!=FV^ܗ#5I5&\ Ħ܅R.}(Jg.{7+%qA| `y z v_c:QElOkXr;aU_ÚY K8-tSZ+j vlk3I&8?4DnЃۄ]!ARA4,S:%'SDP.t  5,)%ѠpOc/ڭ˳5䘋ؕ?6rpl>WWkۛK|9M16X[kW,(@ OwQd ͛?0͢>ϖ2=7kin3~P`v07\2# y·cJ(l )v=YT|462תwƶ p]0Xex &1XcH>agqY~;p%  DՐ{eKg}t?쪎S2!w=NbWot _ endstream endobj 1118 0 obj << /Length 2245 /Filter /FlateDecode >> stream xYKsܸW&  V|^YZks@V;_hCrKUen|E]*f*N/YhF5~~uX|y?HyCIݖ^m"4x6q\~wB*Rwu&+|goe$u9hF /ZwyX)S1fȨ]T\聤D#%S"s1 Tfyi X}ջ(\ZSSu$TRz9lj4#|2ism]`@vu2fyMU <0sI6W!N xM&cI6a .>+E!b|q،Ut'0(%i p{#sȿSB] Kͣ98&eʼn̤pvbi ayMCޏ5Z;7Iv3VU[ ͜>e5{+$zCoѓ>Nc)SN^z=h>qGǙƘXW4hV`K;Z$ n8ň˅x7N mmMH KMzr&Ib"IpJ='Q%s+C_bŽYnݩ ukPxǯvw×.l JakTnc/ní?nq pky9yt8aHR|ʪZ{ zSq(tuI{LԱCotI,L#)1zO 6mt;ŧX@S1bXad׵ F'M F ;%2xNX>j^bz7vve$)qJN|П'켩w]_\ .MtǗ< t.]t٢.P,olL܆m92mߜ{~&\bx}K|i, ٜ_~`V endstream endobj 1003 0 obj << /Type /ObjStm /N 100 /First 975 /Length 1662 /Filter /FlateDecode >> stream xZmk7_=H3h B^ɇ9 :G}6؞I3g^]B ZH5P͉X˂/Hȩ5(\PR 8A*# gS D&NaՉ9ej 3Ś-`,&)Pi2De0?Tbp?%p2(pn{K&mgsBBƂjc@|' % "RFD|9,1GR+6KPPY6XB˜+kȹUFWDD 1Fa^p?v/T"$arc9Hn*ۨDc  3~Ŵ iھ,1("P!y)(TT^*֦y >huqu%Kj6 Ea+E} .ZXQ#V᮵3g*Oces`nZAiiXP:*9%HO@FUAjʎK[ J8TuO E-:U5y҂Ml]Htt||xN+ԌYXA_=zԦ8,Nm5 zMi4V/^Ξ/4,>9 eK|h,חtGgNj&_o޽~N#J_A 0 LO7yJzpxxp\Ftk]䝮Mؤ3v?n|~yw75g?o|zӧOwguxsв83 B8A+1 <`CLuEvv-?m? 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ZxY@ 26>sed~#MX y]SI ʮ W*æl§} 4˴oNdA`YNW\ ,/tva-U`'j[ 6٧Զ4dQl[f}AEvfі:y:,|AE=S.챨oXmݠˢSQyX4b.p=:(8@?Ҭ殫<N!nmh%g08΃Qؠ'Cy[ 2O}|\mS*bCjs`֢'v }Wmii[i,Xt&(edӁ[|y+^y̩ȢO%t|`㕯^<y X}E\XŬbTC[| tkզ? Xţ[jy2-XY|LZUrʪ~4mU;+Vf굋JW~Y+QfO(Y!yHi /?t tê)-X=GnZ/*-Z *-Y*we5wl5|HVϏT7JNp'ѬI>`A{`OrrMơ6q=d*k9ơ6qCe*P8Tơ2qCe*P H = filt ([0, 3], [1, 4, 2]) Transfer function 'H' from input 'u1' to output ... 3 z^-1 y1: ------------------- 1 + 4 z^-1 + 2 z^-2 Sampling time: unspecified Discrete-time model. @end group @end example @seealso{tf} @end deftypefn @section frd @findex frd @deftypefn {Function File} {@var{sys} =} frd (@var{sys}) @deftypefnx {Function File} {@var{sys} =} frd (@var{sys}, @var{w}) @deftypefnx {Function File} {@var{sys} =} frd (@var{H}, @var{w}, @dots{}) @deftypefnx {Function File} {@var{sys} =} frd (@var{H}, @var{w}, @var{tsam}, @dots{}) Create or convert to frequency response data. @strong{Inputs} @table @var @item sys @acronym{LTI} model to be converted to frequency response data. If second argument @var{w} is omitted, the interesting frequency range is calculated by the zeros and poles of @var{sys}. @item H Frequency response array (p-by-m-by-lw). H(i,j,k) contains the response from input j to output i at frequency k. In the SISO case, a vector (lw-by-1) or (1-by-lw) is accepted as well. @item w Frequency vector (lw-by-1) in radian per second [rad/s]. Frequencies must be in ascending order. @item tsam Sampling time in seconds. If @var{tsam} is not specified, a continuous-time model is assumed. @item @dots{} Optional pairs of properties and values. Type @command{set (frd)} for more information. @end table @strong{Outputs} @table @var @item sys Frequency response data object. @end table @strong{Option Keys and Values} @table @var @item 'H' Frequency response array. See 'Inputs' for details. @item 'w' Frequency vector. See 'Inputs' for details. @item 'tsam' Sampling time. See 'Inputs' for details. @item 'inname' The name of the input channels in @var{sys}. Cell vector of length m containing strings. Default names are @code{@{'u1', 'u2', ...@}} @item 'outname' The name of the output channels in @var{sys}. Cell vector of length p containing strings. Default names are @code{@{'y1', 'y2', ...@}} @item 'ingroup' Struct with input group names as field names and vectors of input indices as field values. Default is an empty struct. @item 'outgroup' Struct with output group names as field names and vectors of output indices as field values. Default is an empty struct. @item 'name' String containing the name of the model. @item 'notes' String or cell of string containing comments. @item 'userdata' Any data type. @end table @seealso{dss, ss, tf} @end deftypefn @section ss @findex ss @deftypefn {Function File} {@var{sys} =} ss (@var{sys}) @deftypefnx {Function File} {@var{sys} =} ss (@var{d}) @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}) @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}) @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}, @var{d}, @dots{}) @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}, @var{d}, @var{tsam}, @dots{}) Create or convert to state-space model. @strong{Inputs} @table @var @item sys @acronym{LTI} model to be converted to state-space. @item a State matrix (n-by-n). @item b Input matrix (n-by-m). @item c Output matrix (p-by-n). If @var{c} is empty @code{[]} or not specified, an identity matrix is assumed. @item d Feedthrough matrix (p-by-m). If @var{d} is empty @code{[]} or not specified, a zero matrix is assumed. @item tsam Sampling time in seconds. If @var{tsam} is not specified, a continuous-time model is assumed. @item @dots{} Optional pairs of properties and values. Type @command{set (ss)} for more information. @end table @strong{Outputs} @table @var @item sys State-space model. @end table @strong{Option Keys and Values} @table @var @item 'a', 'b', 'c', 'd', 'e' State-space matrices. See 'Inputs' for details. @item 'stname' The name of the states in @var{sys}. Cell vector containing strings for each state. Default names are @code{@{'x1', 'x2', ...@}} @item 'scaled' Logical. If set to true, no automatic scaling is used, e.g. for frequency response plots. @item 'tsam' Sampling time. See 'Inputs' for details. @item 'inname' The name of the input channels in @var{sys}. Cell vector of length m containing strings. Default names are @code{@{'u1', 'u2', ...@}} @item 'outname' The name of the output channels in @var{sys}. Cell vector of length p containing strings. Default names are @code{@{'y1', 'y2', ...@}} @item 'ingroup' Struct with input group names as field names and vectors of input indices as field values. Default is an empty struct. @item 'outgroup' Struct with output group names as field names and vectors of output indices as field values. Default is an empty struct. @item 'name' String containing the name of the model. @item 'notes' String or cell of string containing comments. @item 'userdata' Any data type. @end table @strong{Example} @example @group octave:1> a = [1 2 3; 4 5 6; 7 8 9]; octave:2> b = [10; 11; 12]; octave:3> stname = @{"V", "A", "kJ"@}; octave:4> sys = ss (a, b, [], [], "stname", stname) @end group @end example @example @group sys.a = V A kJ V 1 2 3 A 4 5 6 kJ 7 8 9 @end group @end example @example @group sys.b = u1 V 10 A 11 kJ 12 @end group @end example @example @group sys.c = V A kJ y1 1 0 0 y2 0 1 0 y3 0 0 1 @end group @end example @example @group sys.d = u1 y1 0 y2 0 y3 0 Continuous-time model. octave:5> @end group @end example @seealso{tf, dss} @end deftypefn @section tf @findex tf @deftypefn {Function File} {@var{s} =} tf (@var{'s'}) @deftypefnx {Function File} {@var{z} =} tf (@var{'z'}, @var{tsam}) @deftypefnx {Function File} {@var{sys} =} tf (@var{sys}) @deftypefnx {Function File} {@var{sys} =} tf (@var{num}, @var{den}, @dots{}) @deftypefnx {Function File} {@var{sys} =} tf (@var{num}, @var{den}, @var{tsam}, @dots{}) Create or convert to transfer function model. @strong{Inputs} @table @var @item sys @acronym{LTI} model to be converted to transfer function. @item num Numerator or cell of numerators. Each numerator must be a row vector containing the coefficients of the polynomial in descending powers of the transfer function variable. num@{i,j@} contains the numerator polynomial from input j to output i. In the SISO case, a single vector is accepted as well. @item den Denominator or cell of denominators. Each denominator must be a row vector containing the coefficients of the polynomial in descending powers of the transfer function variable. den@{i,j@} contains the denominator polynomial from input j to output i. In the SISO case, a single vector is accepted as well. @item tsam Sampling time in seconds. If @var{tsam} is not specified, a continuous-time model is assumed. @item @dots{} Optional pairs of properties and values. Type @command{set (tf)} for more information. @end table @strong{Outputs} @table @var @item sys Transfer function model. @end table @strong{Option Keys and Values} @table @var @item 'num' Numerator. See 'Inputs' for details. @item 'den' Denominator. See 'Inputs' for details. @item 'tfvar' String containing the transfer function variable. @item 'inv' Logical. True for negative powers of the transfer function variable. @item 'tsam' Sampling time. See 'Inputs' for details. @item 'inname' The name of the input channels in @var{sys}. Cell vector of length m containing strings. Default names are @code{@{'u1', 'u2', ...@}} @item 'outname' The name of the output channels in @var{sys}. Cell vector of length p containing strings. Default names are @code{@{'y1', 'y2', ...@}} @item 'ingroup' Struct with input group names as field names and vectors of input indices as field values. Default is an empty struct. @item 'outgroup' Struct with output group names as field names and vectors of output indices as field values. Default is an empty struct. @item 'name' String containing the name of the model. @item 'notes' String or cell of string containing comments. @item 'userdata' Any data type. @end table @strong{Example} @example @group octave:1> s = tf ('s'); octave:2> G = 1/(s+1) Transfer function 'G' from input 'u1' to output ... 1 y1: ----- s + 1 Continuous-time model. @end group @end example @example @group octave:3> z = tf ('z', 0.2); octave:4> H = 0.095/(z-0.9) Transfer function 'H' from input 'u1' to output ... 0.095 y1: ------- z - 0.9 Sampling time: 0.2 s Discrete-time model. @end group @end example @example @group octave:5> num = @{[1, 5, 7], [1]; [1, 7], [1, 5, 5]@}; octave:6> den = @{[1, 5, 6], [1, 2]; [1, 8, 6], [1, 3, 2]@}; octave:7> sys = tf (num, den) @end group @end example @example @group Transfer function 'sys' from input 'u1' to output ... s^2 + 5 s + 7 y1: ------------- s^2 + 5 s + 6 s + 7 y2: ------------- s^2 + 8 s + 6 @end group @end example @example @group Transfer function 'sys' from input 'u2' to output ... 1 y1: ----- s + 2 s^2 + 5 s + 5 y2: ------------- s^2 + 3 s + 2 Continuous-time model. octave:8> @end group @end example @seealso{filt, ss, dss} @end deftypefn @section zpk @findex zpk @deftypefn {Function File} {@var{s} =} zpk (@var{"s"}) @deftypefnx {Function File} {@var{z} =} zpk (@var{"z"}, @var{tsam}) @deftypefnx {Function File} {@var{sys} =} zpk (@var{sys}) @deftypefnx {Function File} {@var{sys} =} zpk (@var{k}) @deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @dots{}) @deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @var{tsam}, @dots{}) @deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @var{tsam}, @dots{}) Create transfer function model from zero-pole-gain data. This is just a stop-gap compatibility wrapper since zpk models are not yet implemented. @strong{Inputs} @table @var @item sys @acronym{LTI} model to be converted to transfer function. @item z Cell of vectors containing the zeros for each channel. z@{i,j@} contains the zeros from input j to output i. In the SISO case, a single vector is accepted as well. @item p Cell of vectors containing the poles for each channel. p@{i,j@} contains the poles from input j to output i. In the SISO case, a single vector is accepted as well. @item k Matrix containing the gains for each channel. k(i,j) contains the gain from input j to output i. @item tsam Sampling time in seconds. If @var{tsam} is not specified, a continuous-time model is assumed. @item @dots{} Optional pairs of properties and values. Type @command{set (tf)} for more information. @end table @strong{Outputs} @table @var @item sys Transfer function model. @end table @seealso{tf, ss, dss, frd} @end deftypefn @chapter Model Data Access @section @@lti/dssdata @findex dssdata @deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}] =} dssdata (@var{sys}) @deftypefnx {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}] =} dssdata (@var{sys}, @var{[]}) Access descriptor state-space model data. Argument @var{sys} is not limited to descriptor state-space models. If @var{sys} is not a descriptor state-space model, it is converted automatically. @strong{Inputs} @table @var @item sys Any type of @acronym{LTI} model. @item [] In case @var{sys} is not a dss model (descriptor matrix @var{e} empty), @code{dssdata (sys, [])} returns the empty element @code{e = []} whereas @code{dssdata (sys)} returns the identity matrix @code{e = eye (size (a))}. @end table @strong{Outputs} @table @var @item a State matrix (n-by-n). @item b Input matrix (n-by-m). @item c Measurement matrix (p-by-n). @item d Feedthrough matrix (p-by-m). @item e Descriptor matrix (n-by-n). @item tsam Sampling time in seconds. If @var{sys} is a continuous-time model, a zero is returned. @end table @end deftypefn @section @@lti/filtdata @findex filtdata @deftypefn {Function File} {[@var{num}, @var{den}, @var{tsam}] =} filtdata (@var{sys}) @deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} filtdata (@var{sys}, @var{"vector"}) Access discrete-time transfer function data in DSP format. Argument @var{sys} is not limited to transfer function models. If @var{sys} is not a transfer function, it is converted automatically. @strong{Inputs} @table @var @item sys Any type of discrete-time @acronym{LTI} model. @item "v", "vector" For SISO models, return @var{num} and @var{den} directly as column vectors instead of cells containing a single column vector. @end table @strong{Outputs} @table @var @item num Cell of numerator(s). Each numerator is a row vector containing the coefficients of the polynomial in ascending powers of z^-1. num@{i,j@} contains the numerator polynomial from input j to output i. In the SISO case, a single vector is possible as well. @item den Cell of denominator(s). Each denominator is a row vector containing the coefficients of the polynomial in ascending powers of z^-1. den@{i,j@} contains the denominator polynomial from input j to output i. In the SISO case, a single vector is possible as well. @item tsam Sampling time in seconds. If @var{tsam} is not specified, -1 is returned. @end table @end deftypefn @section @@lti/frdata @findex frdata @deftypefn {Function File} {[@var{H}, @var{w}, @var{tsam}] =} frdata (@var{sys}) @deftypefnx {Function File} {[@var{H}, @var{w}, @var{tsam}] =} frdata (@var{sys}, @var{"vector"}) Access frequency response data. Argument @var{sys} is not limited to frequency response data objects. If @var{sys} is not a frd object, it is converted automatically. @strong{Inputs} @table @var @item sys Any type of @acronym{LTI} model. @item "v", "vector" In case @var{sys} is a SISO model, this option returns the frequency response as a column vector (lw-by-1) instead of an array (p-by-m-by-lw). @end table @strong{Outputs} @table @var @item H Frequency response array (p-by-m-by-lw). H(i,j,k) contains the response from input j to output i at frequency k. In the SISO case, a vector (lw-by-1) is possible as well. @item w Frequency vector (lw-by-1) in radian per second [rad/s]. Frequencies are in ascending order. @item tsam Sampling time in seconds. If @var{sys} is a continuous-time model, a zero is returned. @end table @end deftypefn @section @@lti/get @findex get @deftypefn {Function File} {} get (@var{sys}) @deftypefnx {Function File} {@var{value} =} get (@var{sys}, @var{"property"}) Access property values of @acronym{LTI} objects. @end deftypefn @section @@lti/set @findex set @deftypefn {Function File} {} set (@var{sys}) @deftypefnx {Function File} {} set (@var{sys}, @var{"property"}, @var{value}, @dots{}) @deftypefnx {Function File} {@var{retsys} =} set (@var{sys}, @var{"property"}, @var{value}, @dots{}) Set or modify properties of @acronym{LTI} objects. If no return argument @var{retsys} is specified, the modified @acronym{LTI} object is stored in input argument @var{sys}. @command{set} can handle multiple properties in one call: @code{set (sys, 'prop1', val1, 'prop2', val2, 'prop3', val3)}. @code{set (sys)} prints a list of the object's property names. @end deftypefn @section @@lti/ssdata @findex ssdata @deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{tsam}] =} ssdata (@var{sys}) Access state-space model data. Argument @var{sys} is not limited to state-space models. If @var{sys} is not a state-space model, it is converted automatically. @strong{Inputs} @table @var @item sys Any type of @acronym{LTI} model. @end table @strong{Outputs} @table @var @item a State matrix (n-by-n). @item b Input matrix (n-by-m). @item c Measurement matrix (p-by-n). @item d Feedthrough matrix (p-by-m). @item tsam Sampling time in seconds. If @var{sys} is a continuous-time model, a zero is returned. @end table @end deftypefn @section @@lti/tfdata @findex tfdata @deftypefn {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}) @deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}, @var{"vector"}) @deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}, @var{"tfpoly"}) Access transfer function data. Argument @var{sys} is not limited to transfer function models. If @var{sys} is not a transfer function, it is converted automatically. @strong{Inputs} @table @var @item sys Any type of @acronym{LTI} model. @item "v", "vector" For SISO models, return @var{num} and @var{den} directly as column vectors instead of cells containing a single column vector. @end table @strong{Outputs} @table @var @item num Cell of numerator(s). Each numerator is a row vector containing the coefficients of the polynomial in descending powers of the transfer function variable. num@{i,j@} contains the numerator polynomial from input j to output i. In the SISO case, a single vector is possible as well. @item den Cell of denominator(s). Each denominator is a row vector containing the coefficients of the polynomial in descending powers of the transfer function variable. den@{i,j@} contains the denominator polynomial from input j to output i. In the SISO case, a single vector is possible as well. @item tsam Sampling time in seconds. If @var{sys} is a continuous-time model, a zero is returned. @end table @end deftypefn @section @@lti/zpkdata @findex zpkdata @deftypefn {Function File} {[@var{z}, @var{p}, @var{k}, @var{tsam}] =} zpkdata (@var{sys}) @deftypefnx {Function File} {[@var{z}, @var{p}, @var{k}, @var{tsam}] =} zpkdata (@var{sys}, @var{"v"}) Access zero-pole-gain data. @strong{Inputs} @table @var @item sys Any type of @acronym{LTI} model. @item "v", "vector" For SISO models, return @var{z} and @var{p} directly as column vectors instead of cells containing a single column vector. @end table @strong{Outputs} @table @var @item z Cell of column vectors containing the zeros for each channel. z@{i,j@} contains the zeros from input j to output i. @item p Cell of column vectors containing the poles for each channel. p@{i,j@} contains the poles from input j to output i. @item k Matrix containing the gains for each channel. k(i,j) contains the gain from input j to output i. @item tsam Sampling time in seconds. If @var{sys} is a continuous-time model, a zero is returned. @end table @end deftypefn @chapter Model Conversions @section @@lti/c2d @findex c2d @deftypefn {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}) @deftypefnx {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}, @var{method}) @deftypefnx {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}, @var{'prewarp'}, @var{w0}) Convert the continuous @acronym{LTI} model into its discrete-time equivalent. @strong{Inputs} @table @var @item sys Continuous-time @acronym{LTI} model. @item tsam Sampling time in seconds. @item method Optional conversion method. If not specified, default method @var{"zoh"} is taken. @table @var @item 'zoh' Zero-order hold or matrix exponential. @item 'tustin', 'bilin' Bilinear transformation or Tustin approximation. @item 'prewarp' Bilinear transformation with pre-warping at frequency @var{w0}. @item 'matched' Matched pole/zero method. @end table @end table @strong{Outputs} @table @var @item sys Discrete-time @acronym{LTI} model. @end table @end deftypefn @section @@lti/d2c @findex d2c @deftypefn {Function File} {@var{sys} =} d2c (@var{sys}) @deftypefnx {Function File} {@var{sys} =} d2c (@var{sys}, @var{method}) @deftypefnx {Function File} {@var{sys} =} d2c (@var{sys}, @var{'prewarp'}, @var{w0}) Convert the discrete @acronym{LTI} model into its continuous-time equivalent. @strong{Inputs} @table @var @item sys Discrete-time @acronym{LTI} model. @item method Optional conversion method. If not specified, default method @var{"zoh"} is taken. @table @var @item 'zoh' Zero-order hold or matrix logarithm. @item 'tustin', 'bilin' Bilinear transformation or Tustin approximation. @item 'prewarp' Bilinear transformation with pre-warping at frequency @var{w0}. @item 'matched' Matched pole/zero method. @end table @end table @strong{Outputs} @table @var @item sys Continuous-time @acronym{LTI} model. @end table @end deftypefn @section @@lti/d2d @findex d2d @deftypefn {Function File} {@var{sys} =} d2d (@var{sys}, @var{tsam}) @deftypefnx {Function File} {@var{sys} =} d2d (@var{sys}, @var{tsam}, @var{method}) @deftypefnx {Function File} {@var{sys} =} d2d (@var{sys}, @var{tsam}, @var{'prewarp'}, @var{w0}) Resample discrete-time @acronym{LTI} model to sampling time @var{tsam}. @strong{Inputs} @table @var @item sys Discrete-time @acronym{LTI} model. @item tsam Desired sampling time in seconds. @item method Optional conversion method. If not specified, default method @var{"zoh"} is taken. @table @var @item 'zoh' Zero-order hold or matrix logarithm. @item 'tustin', 'bilin' Bilinear transformation or Tustin approximation. @item 'prewarp' Bilinear transformation with pre-warping at frequency @var{w0}. @item 'matched' Matched pole/zero method. @end table @end table @strong{Outputs} @table @var @item sys Resampled discrete-time @acronym{LTI} model with sampling time @var{tsam}. @end table @end deftypefn @section @@lti/prescale @findex prescale @deftypefn {Function File} {[@var{scaledsys}, @var{info}] =} prescale (@var{sys}) Scale state-space model. The scaled model @var{scaledsys} is equivalent to @var{sys}, but the state vector is scaled by diagonal transformation matrices in order to increase the accuracy of subsequent numerical computations. Frequency response commands perform automatic scaling unless model property @var{scaled} is set to @var{true}. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @end table @strong{Outputs} @table @var @item scaledsys Scaled state-space model. @item info Structure containing additional information. @item info.SL Left scaling factors. @code{Tl = diag (info.SL)}. @item info.SR Right scaling factors. @code{Tr = diag (info.SR)}. @end table @strong{Equations} @example @group Es = Tl * E * Tr As = Tl * A * Tr Bs = Tl * B Cs = C * Tr Ds = D @end group @end example For proper state-space models, @var{Tl} and @var{Tr} are inverse of each other. @strong{Algorithm}@* Uses SLICOT TB01ID and TG01AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.}. @end deftypefn @section @@lti/xperm @findex xperm @deftypefn {Function File} {@var{retsys} =} xperm (@var{sys}, @var{idx}) Reorder states in state-space models. @strong{Inputs} @table @var @item sys State-space model. @item idx Vector containing the state indices in the desired order. Alternatively, a cell vector containing the state names is possible as well. See @code{sys.stname}. State names only work if they were assigned explicitly before, i.e. @code{sys.stname} contains no empty strings. Note that if certain state indices of @var{sys} are missing or appear multiple times in @var{idx}, these states will be pruned or duplicated accordingly in the resulting state-space model @var{retsys}. @end table @strong{Outputs} @table @var @item retsys Resulting state-space model with states reordered according to @var{idx}. @end table @end deftypefn @chapter Model Interconnections @section @@lti/append @findex append @deftypefn {Function File} {@var{sys} =} append (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) Group @acronym{LTI} models by appending their inputs and outputs. @end deftypefn @section @@lti/blkdiag @findex blkdiag @deftypefn {Function File} {@var{sys} =} blkdiag (@var{sys1}, @var{sys2}) Block-diagonal concatenation of @acronym{LTI} models. @end deftypefn @section @@lti/connect @findex connect @deftypefn {Function File} {@var{sys} =} connect (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{inputs}, @var{outputs}) @deftypefnx {Function File} {@var{sys} =} connect (@var{sys}, @var{cm}, @var{inputs}, @var{outputs}) Name-based interconnections between the inputs and outputs of @acronym{LTI} models. @strong{Inputs} @table @var @item sys1, @dots{}, sysN @acronym{LTI} models to be connected. The properties 'inname' and 'outname' of each model should be set according to the desired input-output connections. @item inputs String or cell of strings containing the names of the inputs to be kept. The names must be part of the properties 'ingroup' or 'inname'. @item outputs String or cell of strings containing the names of the outputs to be kept. The names must be part of the properties 'outgroup' or 'outname'. @item cm Legacy connection matrix (not name-based). @end table @strong{Outputs} @table @var @item sys Resulting interconnected system with outputs @var{outputs} and inputs @var{inputs}. @end table @seealso{sumblk} @end deftypefn @section @@lti/feedback @findex feedback @deftypefn {Function File} {@var{sys} =} feedback (@var{sys1}) @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{"+"}) @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}) @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{"+"}) @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{feedin}, @var{feedout}) @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{feedin}, @var{feedout}, @var{"+"}) Feedback connection of two @acronym{LTI} models. @strong{Inputs} @table @var @item sys1 @acronym{LTI} model of forward transmission. @code{[p1, m1] = size (sys1)}. @item sys2 @acronym{LTI} model of backward transmission. If not specified, an identity matrix of appropriate size is taken. @item feedin Vector containing indices of inputs to @var{sys1} which are involved in the feedback loop. The number of @var{feedin} indices and outputs of @var{sys2} must be equal. If not specified, @code{1:m1} is taken. @item feedout Vector containing indices of outputs from @var{sys1} which are to be connected to @var{sys2}. The number of @var{feedout} indices and inputs of @var{sys2} must be equal. If not specified, @code{1:p1} is taken. @item "+" Positive feedback sign. If not specified, @var{"-"} for a negative feedback interconnection is assumed. @var{+1} and @var{-1} are possible as well, but only from the third argument onward due to ambiguity. @end table @strong{Outputs} @table @var @item sys Resulting @acronym{LTI} model. @end table @strong{Block Diagram} @example @group u + +--------+ y ------>(+)----->| sys1 |-------+-------> ^ - +--------+ | | | | +--------+ | +-------| sys2 |<------+ +--------+ @end group @end example @end deftypefn @section @@lti/lft @findex lft @deftypefn {Function File} {@var{sys} =} lft (@var{sys1}, @var{sys2}) @deftypefnx {Function File} {@var{sys} =} lft (@var{sys1}, @var{sys2}, @var{nu}, @var{ny}) Linear fractional tranformation, also known as Redheffer star product. @strong{Inputs} @table @var @item sys1 Upper @acronym{LTI} model. @item sys2 Lower @acronym{LTI} model. @item nu The last nu inputs of @var{sys1} are connected with the first nu outputs of @var{sys2}. If not specified, @code{min (m1, p2)} is taken. @item ny The last ny outputs of @var{sys1} are connected with the first ny inputs of @var{sys2}. If not specified, @code{min (p1, m2)} is taken. @end table @strong{Outputs} @table @var @item sys Resulting @acronym{LTI} model. @end table @strong{Block Diagram} @example @group .............sys.............. : +--------+ : w1 ------------>| |------------> z1 : | sys1 | : : u +---->| |-----+ y : : | +--------+ | : Lower LFT : | | : : | +--------+ | : lft (sys1, sys2) : +-----| sys2 |<----+ : : +--------+ : :............................: @end group @end example @example @group .............sys.............. : +--------+ : : u +---->| sys1 |-----+ y : : | +--------+ | : Upper LFT : | | : : | +--------+ | : lft (sys1, sys2) : +-----| |<----+ : : | sys2 | : z2 <------------| |<------------ w2 : +--------+ : :............................: @end group @end example @example @group .............sys.............. : +--------+ : w1 ------------>| |------------> z1 : | sys1 | : : u +---->| |-----+ y : : | +--------+ | : : | | : lft (sys1, sys2, nu, ny) : | +--------+ | : : +-----| |<----+ : : | sys2 | : z2 <------------| |<------------ w2 : +--------+ : :............................: @end group @end example @end deftypefn @section @@lti/mconnect @findex mconnect @deftypefn {Function File} {@var{sys} =} mconnect (@var{sys}, @var{m}) @deftypefnx {Function File} {@var{sys} =} mconnect (@var{sys}, @var{m}, @var{inputs}, @var{outputs}) Arbitrary interconnections between the inputs and outputs of an @acronym{LTI} model. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @item m Connection matrix. Each row belongs to an input and each column represents an output. @item inputs Vector of indices of those inputs which are retained. If not specified, all inputs are kept. @item outputs Vector of indices of those outputs which are retained. If not specified, all outputs are kept. @end table @strong{Outputs} @table @var @item sys Interconnected system. @end table @strong{Example} @example @group Solve the system equations of y(t) = G e(t) e(t) = u(t) + M y(t) in order to build y(t) = H u(t) The matrix M for a (p-by-m) system G has m rows and p columns (m-by-p). Example for a 3x2 system: u1 = -1*y1 + 5*y2 + 0*y3 u2 = pi*y1 + 0*y2 - 7*y3 | -1 5 0 | M = | pi 0 7 | @end group @end example @end deftypefn @section @@lti/parallel @findex parallel @deftypefn{Function File} {@var{sys} =} parallel (@var{sys1}, @var{sys2}) Parallel connection of two @acronym{LTI} systems. @strong{Block Diagram} @example @group .......................... : +--------+ : : +-->| sys1 |---+ : u : | +--------+ | + : y -------+ O---------> : | +--------+ | + : : +-->| sys2 |---+ : : +--------+ : :.........sys............: sys = parallel (sys1, sys2) @end group @end example @end deftypefn @section @@lti/series @findex series @deftypefn {Function File} {@var{sys} =} series (@var{sys1}, @var{sys2}) @deftypefnx {Function File} {@var{sys} =} series (@var{sys1}, @var{sys2}, @var{outputs1}, @var{inputs2}) Series connection of two @acronym{LTI} models. @strong{Block Diagram} @example @group ..................................... u : +--------+ y1 u2 +--------+ : y ------>| sys1 |---------->| sys2 |-------> : +--------+ +--------+ : :................sys................. sys = series (sys1, sys2) @end group @end example @example @group ..................................... : v2 +--------+ : : ---------->| | : y : +--------+ y1 u2 | sys2 |-------> u : | |---------->| | : ------>| sys1 | z1 +--------+ : : | |----------> : : +--------+ : :................sys................. outputs1 = [1] inputs2 = [2] sys = series (sys1, sys2, outputs1, inputs2) @end group @end example @end deftypefn @section sumblk @findex sumblk @deftypefn{Function File} {@var{S} =} sumblk (@var{formula}) @deftypefnx{Function File} {@var{S} =} sumblk (@var{formula}, @var{n}) Create summing junction @var{S} from string @var{formula} for name-based interconnections. @strong{Inputs} @table @var @item formula String containing the formula of the summing junction, e.g. @code{e = r - y + d} @item n Signal size. Default value is 1. @end table @strong{Outputs} @table @var @item S State-space model of the summing junction. @end table @strong{Example} @example @group octave:1> S = sumblk ('e = r - y + d') S.d = r y d e 1 -1 1 Static gain. octave:2> S = sumblk ('e = r - y + d', 2) S.d = r1 r2 y1 y2 d1 d2 e1 1 0 -1 0 1 0 e2 0 1 0 -1 0 1 Static gain. @end group @end example @seealso{connect} @end deftypefn @chapter Model Characteristics @section ctrb @findex ctrb @deftypefn {Function File} {@var{co} =} ctrb (@var{sys}) @deftypefnx {Function File} {@var{co} =} ctrb (@var{a}, @var{b}) Return controllability matrix. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item a State matrix (n-by-n). @item b Input matrix (n-by-m). @end table @strong{Outputs} @table @var @item co Controllability matrix. @end table @strong{Equation} @iftex @tex $$ C_o = [ B \ \ AB \ \ A^2B \ \ldots \ A^{n-1}B ] $$ @end tex @end iftex @ifnottex @example 2 n-1 Co = [ B AB A B ... A B ] @end example @end ifnottex @end deftypefn @section ctrbf @findex ctrbf @deftypefn{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys}) @deftypefnx{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys}, @var{tol}) @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C}) @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C}, @var{TOL}) If Co=ctrb(A,B) has rank r <= n = SIZE(A,1), then there is a similarity transformation Tc such that Tc = [t1 t2] where t1 is the controllable subspace and t2 is orthogonal to t1 @example @group Abar = Tc \ A * Tc , Bbar = Tc \ B , Cbar = C * Tc @end group @end example and the transformed system has the form @example @group | Ac A12| | Bc | Abar = |----------|, Bbar = | ---|, Cbar = [Cc | Cnc]. | 0 Anc| | 0 | @end group @end example where (Ac,Bc) is controllable, and Cc(sI-Ac)^(-1)Bc = C(sI-A)^(-1)B. and the system is stabilizable if Anc has no eigenvalues in the right half plane. The last output K is a vector of length n containing the number of controllable states. @end deftypefn @section @@lti/dcgain @findex dcgain @deftypefn {Function File} {@var{k} =} dcgain (@var{sys}) DC gain of @acronym{LTI} model. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @end table @strong{Outputs} @table @var @item k DC gain matrice. For a system with m inputs and p outputs, the array @var{k} has dimensions [p, m]. @end table @seealso{freqresp} @end deftypefn @section gram @findex gram @deftypefn {Function File} {@var{W} =} gram (@var{sys}, @var{mode}) @deftypefnx {Function File} {@var{Wc} =} gram (@var{a}, @var{b}) @code{gram (@var{sys}, "c")} returns the controllability gramian of the (continuous- or discrete-time) system @var{sys}. @code{gram (@var{sys}, "o")} returns the observability gramian of the (continuous- or discrete-time) system @var{sys}. @code{gram (@var{a}, @var{b})} returns the controllability gramian @var{Wc} of the continuous-time system @math{dx/dt = a x + b u}; i.e., @var{Wc} satisfies @math{a Wc + m Wc' + b b' = 0}. @end deftypefn @section hsvd @findex hsvd @deftypefn{Function File} {@var{hsv} =} hsvd (@var{sys}) @deftypefnx{Function File} {@var{hsv} =} hsvd (@var{sys}, @var{"offset"}, @var{offset}) @deftypefnx{Function File} {@var{hsv} =} hsvd (@var{sys}, @var{"alpha"}, @var{alpha}) Hankel singular values of the stable part of an @acronym{LTI} model. If no output arguments are given, the Hankel singular values are displayed in a plot. @strong{Algorithm}@* Uses SLICOT AB13AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section @@lti/isct @findex isct @deftypefn {Function File} {@var{bool} =} isct (@var{sys}) Determine whether @acronym{LTI} model is a continuous-time system. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @end table @strong{Outputs} @table @var @item bool = 0 @var{sys} is a discrete-time system. @item bool = 1 @var{sys} is a continuous-time system or a static gain. @end table @end deftypefn @section isctrb @findex isctrb @deftypefn {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{sys}) @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{sys}, @var{tol}) @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}) @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{e}) @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{[]}, @var{tol}) @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{e}, @var{tol}) Logical check for system controllability. For numerical reasons, @code{isctrb (sys)} should be used instead of @code{rank (ctrb (sys))}. @strong{Inputs} @table @var @item sys @acronym{LTI} model. Descriptor state-space models are possible. If @var{sys} is not a state-space model, it is converted to a minimal state-space realization, so beware of pole-zero cancellations which may lead to wrong results! @item a State matrix (n-by-n). @item b Input matrix (n-by-m). @item e Descriptor matrix (n-by-n). If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. @item tol Optional roundoff parameter. Default value is 0. @end table @strong{Outputs} @table @var @item bool = 0 System is not controllable. @item bool = 1 System is controllable. @item ncon Number of controllable states. @end table @strong{Algorithm}@* Uses SLICOT AB01OD and TG01HD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{isobsv} @end deftypefn @section isdetectable @findex isdetectable @deftypefn {Function File} {@var{bool} =} isdetectable (@var{sys}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{sys}, @var{tol}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{tol}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{tol}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{[]}, @var{dflg}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{[]}, @var{dflg}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{tol}, @var{dflg}) @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{tol}, @var{dflg}) Logical test for system detectability. All unstable modes must be observable or all unobservable states must be stable. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @item a State transition matrix. @item c Measurement matrix. @item e Descriptor matrix. If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. @item tol Optional tolerance for stability. Default value is 0. @item dflg = 0 Matrices (@var{a}, @var{c}) are part of a continuous-time system. Default Value. @item dflg = 1 Matrices (@var{a}, @var{c}) are part of a discrete-time system. @end table @strong{Outputs} @table @var @item bool = 0 System is not detectable. @item bool = 1 System is detectable. @end table @strong{Algorithm}@* Uses SLICOT AB01OD and TG01HD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} See @command{isstabilizable} for description of computational method. @seealso{isstabilizable, isstable, isctrb, isobsv} @end deftypefn @section @@lti/isdt @findex isdt @deftypefn {Function File} {@var{bool} =} isdt (@var{sys}) Determine whether @acronym{LTI} model is a discrete-time system. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @end table @strong{Outputs} @table @var @item bool = 0 @var{sys} is a continuous-time system. @item bool = 1 @var{sys} is a discrete-time system or a static gain. @end table @end deftypefn @section @@lti/isminimumphase @findex isminimumphase @deftypefn {Function File} {@var{bool} =} isminimumphase (@var{sys}) @deftypefnx {Function File} {@var{bool} =} isminimumphase (@var{sys}, @var{tol}) Determine whether @acronym{LTI} system is minimum-phase. The zeros must lie in the left complex half-plane. The name minimum-phase refers to the fact that such a system has the minimum possible phase lag for the given magnitude response |sys(jw)|. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @item tol Optional tolerance. @var{tol} must be a real-valued, non-negative scalar. Default value is 0. @end table @strong{Outputs} @table @var @item bool True if the system is minimum-phase and false otherwise. @end table @example @group real (z) < -tol*(1 + abs (z)) continuous-time abs (z) < 1 - tol discrete-time @end group @end example @end deftypefn @section isobsv @findex isobsv @deftypefn {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{sys}) @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{sys}, @var{tol}) @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}) @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{e}) @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{[]}, @var{tol}) @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{e}, @var{tol}) Logical check for system observability. For numerical reasons, @code{isobsv (sys)} should be used instead of @code{rank (obsv (sys))}. @strong{Inputs} @table @var @item sys @acronym{LTI} model. Descriptor state-space models are possible. @item a State matrix (n-by-n). @item c Measurement matrix (p-by-n). @item e Descriptor matrix (n-by-n). If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. @item tol Optional roundoff parameter. Default value is 0. @end table @strong{Outputs} @table @var @item bool = 0 System is not observable. @item bool = 1 System is observable. @item nobs Number of observable states. @end table @strong{Algorithm}@* Uses SLICOT AB01OD and TG01HD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{isctrb} @end deftypefn @section @@lti/issiso @findex issiso @deftypefn {Function File} {@var{bool} =} issiso (@var{sys}) Determine whether @acronym{LTI} model is single-input/single-output (SISO). @end deftypefn @section isstabilizable @findex isstabilizable @deftypefn {Function File} {@var{bool} =} isstabilizable (@var{sys}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{sys}, @var{tol}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{tol}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{tol}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{[]}, @var{dflg}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{[]}, @var{dflg}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{tol}, @var{dflg}) @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{tol}, @var{dflg}) Logical check for system stabilizability. All unstable modes must be controllable or all uncontrollable states must be stable. @strong{Inputs} @table @var @item sys @acronym{LTI} system. If @var{sys} is not a state-space system, it is converted to a minimal state-space realization, so beware of pole-zero cancellations which may lead to wrong results! @item a State transition matrix. @item b Input matrix. @item e Descriptor matrix. If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. @item tol Optional tolerance for stability. Default value is 0. @item dflg = 0 Matrices (@var{a}, @var{b}) are part of a continuous-time system. Default Value. @item dflg = 1 Matrices (@var{a}, @var{b}) are part of a discrete-time system. @end table @strong{Outputs} @table @var @item bool = 0 System is not stabilizable. @item bool = 1 System is stabilizable. @end table @strong{Algorithm}@* Uses SLICOT AB01OD and TG01HD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @example @group * Calculate staircase form (SLICOT AB01OD) * Extract unobservable part of state transition matrix * Calculate eigenvalues of unobservable part * Check whether real (ev) < -tol*(1 + abs (ev)) continuous-time abs (ev) < 1 - tol discrete-time @end group @end example @seealso{isdetectable, isstable, isctrb, isobsv} @end deftypefn @section @@lti/isstable @findex isstable @deftypefn {Function File} {@var{bool} =} isstable (@var{sys}) @deftypefnx {Function File} {@var{bool} =} isstable (@var{sys}, @var{tol}) Determine whether @acronym{LTI} system is stable. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @item tol Optional tolerance for stability. @var{tol} must be a real-valued, non-negative scalar. Default value is 0. @end table @strong{Outputs} @table @var @item bool True if the system is stable and false otherwise. @end table @example @group real (p) < -tol*(1 + abs (p)) continuous-time abs (p) < 1 - tol discrete-time @end group @end example @end deftypefn @section @@lti/norm @findex norm @deftypefn {Function File} {@var{gain} =} norm (@var{sys}, @var{2}) @deftypefnx {Function File} {[@var{gain}, @var{wpeak}] =} norm (@var{sys}, @var{inf}) @deftypefnx {Function File} {[@var{gain}, @var{wpeak}] =} norm (@var{sys}, @var{inf}, @var{tol}) Return H-2 or L-inf norm of @acronym{LTI} model. @strong{Algorithm}@* Uses SLICOT AB13BD and AB13DD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section obsv @findex obsv @deftypefn {Function File} {@var{ob} =} obsv (@var{sys}) @deftypefnx {Function File} {@var{ob} =} obsv (@var{a}, @var{c}) Return observability matrix. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item a State matrix (n-by-n). @item c Measurement matrix (p-by-n). @end table @strong{Outputs} @table @var @item ob Observability matrix. @end table @strong{Equation} @iftex @tex $$ O_b = \left[ \matrix{ C \cr CA \cr CA^2 \cr \vdots \cr CA^{n-1} } \right ] $$ @end tex @end iftex @ifnottex @example @group | C | | CA | Ob = | CA^2 | | ... | | CA^(n-1) | @end group @end example @end ifnottex @end deftypefn @section obsvf @findex obsvf @deftypefn{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} obsvf (@var{sys}) @deftypefnx{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} obsvf (@var{sys}, @var{tol}) @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} obsvf (@var{A}, @var{B}, @var{C}) @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} obsvf (@var{A}, @var{B}, @var{C}, @var{TOL}) If Ob=obsv(A,C) has rank r <= n = SIZE(A,1), then there is a similarity transformation Tc such that To = [t1;t2] where t1 is c and t2 is orthogonal to t1 @example @group Abar = To \ A * To , Bbar = To \ B , Cbar = C * To @end group @end example and the transformed system has the form @example @group | Ao 0 | | Bo | Abar = |----------|, Bbar = | --- |, Cbar = [Co | 0 ]. | A21 Ano| | Bno | @end group @end example where (Ao,Bo) is observable, and Co(sI-Ao)^(-1)Bo = C(sI-A)^(-1)B. And system is detectable if Ano has no eigenvalues in the right half plane. The last output K is a vector of length n containing the number of observable states. @end deftypefn @section @@lti/pole @findex pole @deftypefn {Function File} {@var{p} =} pole (@var{sys}) Compute poles of @acronym{LTI} system. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @end table @strong{Outputs} @table @var @item p Poles of @var{sys}. @end table @strong{Algorithm}@* For (descriptor) state-space models, @command{pole} relies on Octave's @command{eig}. For @acronym{SISO} transfer functions, @command{pole} uses Octave's @command{roots}. @acronym{MIMO} transfer functions are converted to a @emph{minimal} state-space representation for the computation of the poles. @end deftypefn @section pzmap @findex pzmap @deftypefn {Function File} {} pzmap (@var{sys}) @deftypefnx {Function File} {} pzmap (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx {Function File} {} pzmap (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx {Function File} {[@var{p}, @var{z}] =} pzmap (@var{sys}) Plot the poles and zeros of an LTI system in the complex plane. If no output arguments are given, the result is plotted on the screen. Otherwise, the poles and zeros are computed and returned. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item p Poles of @var{sys}. @item z Transmission zeros of @var{sys}. @end table @end deftypefn @section @@lti/size @findex size @deftypefn {Function File} {@var{nvec} =} size (@var{sys}) @deftypefnx {Function File} {@var{n} =} size (@var{sys}, @var{dim}) @deftypefnx {Function File} {[@var{p}, @var{m}] =} size (@var{sys}) @acronym{LTI} model size, i.e. number of outputs and inputs. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @item dim If given a second argument, @command{size} will return the size of the corresponding dimension. @end table @strong{Outputs} @table @var @item nvec Row vector. The first element is the number of outputs (rows) and the second element the number of inputs (columns). @item n Scalar value. The size of the dimension @var{dim}. @item p Number of outputs. @item m Number of inputs. @end table @end deftypefn @section @@lti/zero @findex zero @deftypefn {Function File} {@var{z} =} zero (@var{sys}) @deftypefnx {Function File} {@var{z} =} zero (@var{sys}, @var{type}) @deftypefnx {Function File} {[@var{z}, @var{k}, @var{info}] =} zero (@var{sys}) Compute zeros and gain of @acronym{LTI} model. By default, @command{zero} computes the invariant zeros, also known as Smith zeros. Alternatively, when called with a second input argument, @command{zero} can also compute the system zeros, transmission zeros, input decoupling zeros and output decoupling zeros. See paper [1] for an explanation of the various zero flavors as well as for further details. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item type String specifying the type of zeros: @table @var @item 'system', 's' Compute the system zeros. The system zeros include in all cases (square, non-square, degenerate or non-degenerate system) all transmission and decoupling zeros. @item 'invariant', 'inv' Compute invariant zeros. Default selection. @item 'transmission', 't' Compute transmission zeros. Transmission zeros are a subset of the invariant zeros. The transmission zeros are the zeros of the Smith-McMillan form of the transfer function matrix. @item 'input', 'inp', 'id' Compute input decoupling zeros. The input decoupling zeros are also known as the uncontrollable eigenvalues of the pair (A,B). @item 'output', 'o', 'od' Compute output decoupling zeros. The output decoupling zeros are also known as the unobservable eigenvalues of the pair (A,C). @end table @end table @strong{Outputs} @table @var @item z Depending on argument @var{type}, @var{z} contains the invariant (default), system, transmission, input decoupling or output decoupling zeros of @var{sys} as defined in [1]. @item k Gain of @acronym{SISO} system @var{sys}. For @acronym{MIMO} systems, an empty matrix @code{[]} is returned. @item info Struct containing additional information. For details, see the documentation of @acronym{SLICOT} routines @acronym{AB08ND} and @acronym{AG08BD}. @item info.rank The normal rank of the transfer function matrix (regular state-space models) or of the system pencil (descriptor state-space models). @item info.infz Contains information on the infinite elementary divisors as follows: the system has info.infz(i) infinite elementary divisors of degree i, where i=1,2,...,length(info.infz). @item info.kronr Right Kronecker (column) indices. @item info.kronl Left Kronecker (row) indices. @end table @strong{Examples} @example @group [z, k, info] = zero (sys) # invariant zeros z = zero (sys, 'system') # system zeros z = zero (sys, 'invariant') # invariant zeros z = zero (sys, 'transmission') # transmission zeros z = zero (sys, 'output') # output decoupling zeros z = zero (sys, 'input') # input decoupling zeros @end group @end example @strong{Algorithm}@* For (descriptor) state-space models, @command{zero} relies on SLICOT AB08ND and AG08BD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} For @acronym{SISO} transfer functions, @command{zero} uses Octave's @command{roots}. @acronym{MIMO} transfer functions are converted to a @emph{minimal} state-space representation for the computation of the zeros. @strong{References}@* [1] MacFarlane, A. and Karcanias, N. @cite{Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex-variable theory}. Int. J. Control, vol. 24, pp. 33-74, 1976.@* [2] Rosenbrock, H.H. @cite{Correction to 'The zeros of a system'}. Int. J. Control, vol. 20, no. 3, pp. 525-527, 1974.@* [3] Svaricek, F. @cite{Computation of the structural invariants of linear multivariable systems with an extended version of the program ZEROS}. Systems & Control Letters, vol. 6, pp. 261-266, 1985.@* [4] Emami-Naeini, A. and Van Dooren, P. @cite{Computation of zeros of linear multivariable systems}. Automatica, vol. 26, pp. 415-430, 1982.@* @end deftypefn @chapter Model Simplification @section @@lti/minreal @findex minreal @deftypefn {Function File} {@var{sys} =} minreal (@var{sys}) @deftypefnx {Function File} {@var{sys} =} minreal (@var{sys}, @var{tol}) Minimal realization or zero-pole cancellation of @acronym{LTI} models. @end deftypefn @section @@lti/sminreal @findex sminreal @deftypefn {Function File} {@var{sys} =} sminreal (@var{sys}) @deftypefnx {Function File} {@var{sys} =} sminreal (@var{sys}, @var{tol}) Perform state-space model reduction based on structure. Remove states which have no influence on the input-output behaviour. The physical meaning of the states is retained. @strong{Inputs} @table @var @item sys State-space model. @item tol Optional tolerance for controllability and observability. Entries of the state-space matrices whose moduli are less or equal to @var{tol} are assumed to be zero. Default value is 0. @end table @strong{Outputs} @table @var @item sys Reduced state-space model. @end table @seealso{minreal} @end deftypefn @chapter Time Domain Analysis @section covar @findex covar @deftypefn{Function File} {[@var{p}, @var{q}] =} covar (@var{sys}, @var{w}) Return the steady-state covariance. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item w Intensity of Gaussian white noise inputs which drive @var{sys}. @end table @strong{Outputs} @table @var @item p Output covariance. @item q State covariance. @end table @seealso{lyap, dlyap} @end deftypefn @section gensig @findex gensig @deftypefn{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}) @deftypefnx{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}, @var{tfinal}) @deftypefnx{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}, @var{tfinal}, @var{tsam}) Generate periodic signal. Useful in combination with lsim. @strong{Inputs} @table @var @item sigtype = "sin" Sine wave. @item sigtype = "cos" Cosine wave. @item sigtype = "square" Square wave. @item sigtype = "pulse" Periodic pulse. @item tau Duration of one period in seconds. @item tfinal Optional duration of the signal in seconds. Default duration is 5 periods. @item tsam Optional sampling time in seconds. Default spacing is tau/64. @end table @strong{Outputs} @table @var @item u Vector of signal values. @item t Time vector of the signal. @end table @seealso{lsim} @end deftypefn @section impulse @findex impulse @deftypefn{Function File} {} impulse (@var{sys}) @deftypefnx{Function File} {} impulse (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx{Function File} {} impulse (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx{Function File} {} impulse (@var{sys1}, @dots{}, @var{t}) @deftypefnx{Function File} {} impulse (@var{sys1}, @dots{}, @var{tfinal}) @deftypefnx{Function File} {} impulse (@var{sys1}, @dots{}, @var{tfinal}, @var{dt}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{t}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{tfinal}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{tfinal}, @var{dt}) Impulse response of @acronym{LTI} system. If no output arguments are given, the response is printed on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item t Time vector. Should be evenly spaced. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item tfinal Optional simulation horizon. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item dt Optional sampling time. Be sure to choose it small enough to capture transient phenomena. If not specified, it is calculated by the poles of the system. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item y Output response array. Has as many rows as time samples (length of t) and as many columns as outputs. @item t Time row vector. @item x State trajectories array. Has @code{length (t)} rows and as many columns as states. @end table @seealso{initial, lsim, step} @end deftypefn @section initial @findex initial @deftypefn{Function File} {} initial (@var{sys}, @var{x0}) @deftypefnx{Function File} {} initial (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{x0}) @deftypefnx{Function File} {} initial (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}, @var{x0}) @deftypefnx{Function File} {} initial (@var{sys1}, @dots{}, @var{x0}, @var{t}) @deftypefnx{Function File} {} initial (@var{sys1}, @dots{}, @var{x0}, @var{tfinal}) @deftypefnx{Function File} {} initial (@var{sys1}, @dots{}, @var{x0}, @var{tfinal}, @var{dt}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{t}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{tfinal}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{tfinal}, @var{dt}) Initial condition response of state-space model. If no output arguments are given, the response is printed on the screen. @strong{Inputs} @table @var @item sys State-space model. @item x0 Vector of initial conditions for each state. @item t Optional time vector. Should be evenly spaced. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item tfinal Optional simulation horizon. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item dt Optional sampling time. Be sure to choose it small enough to capture transient phenomena. If not specified, it is calculated by the poles of the system. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item y Output response array. Has as many rows as time samples (length of t) and as many columns as outputs. @item t Time row vector. @item x State trajectories array. Has @code{length (t)} rows and as many columns as states. @end table @strong{Example} @example @group . Continuous Time: x = A x , y = C x , x(0) = x0 Discrete Time: x[k+1] = A x[k] , y[k] = C x[k] , x[0] = x0 @end group @end example @seealso{impulse, lsim, step} @end deftypefn @section lsim @findex lsim @deftypefn{Function File} {} lsim (@var{sys}, @var{u}) @deftypefnx{Function File} {} lsim (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{u}) @deftypefnx{Function File} {} lsim (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}, @var{u}) @deftypefnx{Function File} {} lsim (@var{sys1}, @dots{}, @var{u}, @var{t}) @deftypefnx{Function File} {} lsim (@var{sys1}, @dots{}, @var{u}, @var{t}, @var{x0}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{x0}) Simulate @acronym{LTI} model response to arbitrary inputs. If no output arguments are given, the system response is plotted on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} model. System must be proper, i.e. it must not have more zeros than poles. @item u Vector or array of input signal. Needs @code{length(t)} rows and as many columns as there are inputs. If @var{sys} is a single-input system, row vectors @var{u} of length @code{length(t)} are accepted as well. @item t Time vector. Should be evenly spaced. If @var{sys} is a continuous-time system and @var{t} is a real scalar, @var{sys} is discretized with sampling time @code{tsam = t/(rows(u)-1)}. If @var{sys} is a discrete-time system and @var{t} is not specified, vector @var{t} is assumed to be @code{0 : tsam : tsam*(rows(u)-1)}. @item x0 Vector of initial conditions for each state. If not specified, a zero vector is assumed. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item y Output response array. Has as many rows as time samples (length of t) and as many columns as outputs. @item t Time row vector. It is always evenly spaced. @item x State trajectories array. Has @code{length (t)} rows and as many columns as states. @end table @seealso{impulse, initial, step} @end deftypefn @section ramp @findex ramp @deftypefn{Function File} {} ramp (@var{sys}) @deftypefnx{Function File} {} ramp (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx{Function File} {} ramp (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx{Function File} {} ramp (@var{sys1}, @dots{}, @var{t}) @deftypefnx{Function File} {} ramp (@var{sys1}, @dots{}, @var{tfinal}) @deftypefnx{Function File} {} ramp (@var{sys1}, @dots{}, @var{tfinal}, @var{dt}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}, @var{t}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}, @var{tfinal}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}, @var{tfinal}, @var{dt}) Ramp response of @acronym{LTI} system. If no output arguments are given, the response is printed on the screen. @iftex @tex $$ r(t) = t \, \cdot \, h(t) $$ @end tex @end iftex @ifnottex @example r(t) = t * h(t) @end example @end ifnottex @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item t Time vector. Should be evenly spaced. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item tfinal Optional simulation horizon. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item dt Optional sampling time. Be sure to choose it small enough to capture transient phenomena. If not specified, it is calculated by the poles of the system. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item y Output response array. Has as many rows as time samples (length of t) and as many columns as outputs. @item t Time row vector. @item x State trajectories array. Has @code{length (t)} rows and as many columns as states. @end table @seealso{impulse, initial, lsim, step} @end deftypefn @section step @findex step @deftypefn{Function File} {} step (@var{sys}) @deftypefnx{Function File} {} step (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx{Function File} {} step (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx{Function File} {} step (@var{sys1}, @dots{}, @var{t}) @deftypefnx{Function File} {} step (@var{sys1}, @dots{}, @var{tfinal}) @deftypefnx{Function File} {} step (@var{sys1}, @dots{}, @var{tfinal}, @var{dt}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{t}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{tfinal}) @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{tfinal}, @var{dt}) Step response of @acronym{LTI} system. If no output arguments are given, the response is printed on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item t Time vector. Should be evenly spaced. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item tfinal Optional simulation horizon. If not specified, it is calculated by the poles of the system to reflect adequately the response transients. @item dt Optional sampling time. Be sure to choose it small enough to capture transient phenomena. If not specified, it is calculated by the poles of the system. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item y Output response array. Has as many rows as time samples (length of t) and as many columns as outputs. @item t Time row vector. @item x State trajectories array. Has @code{length (t)} rows and as many columns as states. @end table @seealso{impulse, initial, lsim} @end deftypefn @chapter Frequency Domain Analysis @section bode @findex bode @deftypefn {Function File} {} bode (@var{sys}) @deftypefnx {Function File} {} bode (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx {Function File} {} bode (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) @deftypefnx {Function File} {} bode (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} bode (@var{sys}) @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} bode (@var{sys}, @var{w}) Bode diagram of frequency response. If no output arguments are given, the response is printed on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} system. Must be a single-input and single-output (SISO) system. @item w Optional vector of frequency values. If @var{w} is not specified, it is calculated by the zeros and poles of the system. Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, where @var{wmin} and @var{wmax} denote minimum and maximum frequencies in rad/s. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item mag Vector of magnitude. Has length of frequency vector @var{w}. @item pha Vector of phase. Has length of frequency vector @var{w}. @item w Vector of frequency values used. @end table @seealso{nichols, nyquist, sigma} @end deftypefn @section bodemag @findex bodemag @deftypefn {Function File} {} bodemag (@var{sys}) @deftypefnx {Function File} {} bodemag (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx {Function File} {} bodemag (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) @deftypefnx {Function File} {} bodemag (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx {Function File} {[@var{mag}, @var{w}] =} bodemag (@var{sys}) @deftypefnx {Function File} {[@var{mag}, @var{w}] =} bodemag (@var{sys}, @var{w}) Bode magnitude diagram of frequency response. If no output arguments are given, the response is printed on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} system. Must be a single-input and single-output (SISO) system. @item w Optional vector of frequency values. If @var{w} is not specified, it is calculated by the zeros and poles of the system. Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, where @var{wmin} and @var{wmax} denote minimum and maximum frequencies in rad/s. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item mag Vector of magnitude. Has length of frequency vector @var{w}. @item w Vector of frequency values used. @end table @seealso{bode, nichols, nyquist, sigma} @end deftypefn @section @@lti/freqresp @findex freqresp @deftypefn{Function File} {@var{H} =} freqresp (@var{sys}, @var{w}) Evaluate frequency response at given frequencies. @strong{Inputs} @table @var @item sys @acronym{LTI} system. @item w Vector of frequency values. @end table @strong{Outputs} @table @var @item H Array of frequency response. For a system with m inputs and p outputs, the array @var{H} has dimensions [p, m, length (w)]. The frequency response at the frequency w(k) is given by H(:,:,k). @end table @seealso{dcgain} @end deftypefn @section margin @findex margin @deftypefn{Function File} {[@var{gamma}, @var{phi}, @var{w_gamma}, @var{w_phi}] =} margin (@var{sys}) @deftypefnx{Function File} {[@var{gamma}, @var{phi}, @var{w_gamma}, @var{w_phi}] =} margin (@var{sys}, @var{tol}) Gain and phase margin of a system. If no output arguments are given, both gain and phase margin are plotted on a bode diagram. Otherwise, the margins and their corresponding frequencies are computed and returned. A more robust criterion to assess the stability of a feedback system is the sensitivity Ms computed by command @command{sensitivity}. @strong{Inputs} @table @var @item sys @acronym{LTI} model. Must be a single-input and single-output (SISO) system. @item tol Imaginary parts below @var{tol} are assumed to be zero. If not specified, default value @code{sqrt (eps)} is taken. @end table @strong{Outputs} @table @var @item gamma Gain margin (as gain, not dBs). @item phi Phase margin (in degrees). @item w_gamma Frequency for the gain margin (in rad/s). @item w_phi Frequency for the phase margin (in rad/s). @end table @strong{Algorithm}@* Uses command @command{roots} to calculate the frequencies @var{w_gamma}, @var{w_phi} from special polynomials created from the transfer function of @var{sys} as listed below in section @guillemetleft{}Equations@guillemetright{}. @strong{Equations} @example @group CONTINUOUS-TIME SYSTEMS Gain Margin _ _ L(jw) = L(jw) BTW: L(jw) = L(-jw) = conj (L(jw)) num(jw) num(-jw) ------- = -------- den(jw) den(-jw) num(jw) den(-jw) = num(-jw) den(jw) imag (num(jw) den(-jw)) = 0 imag (num(-jw) den(jw)) = 0 @end group @end example @example @group Phase Margin |num(jw)| |L(jw)| = |-------| = 1 |den(jw)| _ 2 2 z z = Re z + Im z num(jw) num(-jw) ------- * -------- = 1 den(jw) den(-jw) num(jw) num(-jw) - den(jw) den(-jw) = 0 real (num(jw) num(-jw) - den(jw) den(-jw)) = 0 @end group @end example @example @group DISCRETE-TIME SYSTEMS Gain Margin jwT log z L(z) = L(1/z) BTW: z = e --> w = ----- j T num(z) num(1/z) ------ = -------- den(z) den(1/z) num(z) den(1/z) - num(1/z) den(z) = 0 @end group @end example @example @group Phase Margin |num(z)| |L(z)| = |------| = 1 |den(z)| @end group @end example @example @group L(z) L(1/z) = 1 num(z) num(1/z) ------ * -------- = 1 den(z) den(1/z) num(z) num(1/z) - den(z) den(1/z) = 0 @end group @end example @example @group PS: How to get L(1/z) 4 3 2 p(z) = a z + b z + c z + d z + e -4 -3 -2 -1 p(1/z) = a z + b z + c z + d z + e -4 2 3 4 = z ( a + b z + c z + d z + e z ) 4 3 2 4 = ( e z + d z + c z + b z + a ) / ( z ) @end group @end example @seealso{sensitivity, roots} @end deftypefn @section nichols @findex nichols @deftypefn {Function File} {} nichols (@var{sys}) @deftypefnx {Function File} {} nichols (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx {Function File} {} nichols (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) @deftypefnx {Function File} {} nichols (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} nichols (@var{sys}) @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} nichols (@var{sys}, @var{w}) Nichols chart of frequency response. If no output arguments are given, the response is printed on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} system. Must be a single-input and single-output (SISO) system. @item w Optional vector of frequency values. If @var{w} is not specified, it is calculated by the zeros and poles of the system. Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, where @var{wmin} and @var{wmax} denote minimum and maximum frequencies in rad/s. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item mag Vector of magnitude. Has length of frequency vector @var{w}. @item pha Vector of phase. Has length of frequency vector @var{w}. @item w Vector of frequency values used. @end table @seealso{bode, nyquist, sigma} @end deftypefn @section nyquist @findex nyquist @deftypefn {Function File} {} nyquist (@var{sys}) @deftypefnx {Function File} {} nyquist (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx {Function File} {} nyquist (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) @deftypefnx {Function File} {} nyquist (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx {Function File} {[@var{re}, @var{im}, @var{w}] =} nyquist (@var{sys}) @deftypefnx {Function File} {[@var{re}, @var{im}, @var{w}] =} nyquist (@var{sys}, @var{w}) Nyquist diagram of frequency response. If no output arguments are given, the response is printed on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} system. Must be a single-input and single-output (SISO) system. @item w Optional vector of frequency values. If @var{w} is not specified, it is calculated by the zeros and poles of the system. Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, where @var{wmin} and @var{wmax} denote minimum and maximum frequencies in rad/s. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item re Vector of real parts. Has length of frequency vector @var{w}. @item im Vector of imaginary parts. Has length of frequency vector @var{w}. @item w Vector of frequency values used. @end table @seealso{bode, nichols, sigma} @end deftypefn @section sensitivity @findex sensitivity @deftypefn{Function File} {[@var{Ms}, @var{ws}] =} sensitivity (@var{L}) @deftypefnx{Function File} {[@var{Ms}, @var{ws}] =} sensitivity (@var{P}, @var{C}) @deftypefnx{Function File} {[@var{Ms}, @var{ws}] =} sensitivity (@var{P}, @var{C1}, @var{C2}, @dots{}) Return sensitivity margin @var{Ms}. The quantity @var{Ms} is simply the inverse of the shortest distance from the Nyquist curve to the critical point -1. Reasonable values of @var{Ms} are in the range from 1.3 to 2. @iftex @tex $$ M_s = ||S(j\omega)||_{\infty} $$ @end tex @end iftex @ifnottex @example Ms = ||S(jw)|| inf @end example @end ifnottex If no output arguments are given, the critical distance 1/Ms is plotted on a Nyquist diagram. In contrast to gain and phase margin as computed by command @command{margin}, the sensitivity @var{Ms} is a more robust criterion to assess the stability of a feedback system. @strong{Inputs} @table @var @item L Open loop transfer function. @var{L} can be any type of @acronym{LTI} system, but it must be square. @item P Plant model. Any type of @acronym{LTI} system. @item C Controller model. Any type of @acronym{LTI} system. @item C1, C2, @dots{} If several controllers are specified, command @command{sensitivity} computes the sensitivity @var{Ms} for each of them in combination with plant @var{P}. @end table @strong{Outputs} @table @var @item Ms Sensitivity margin @var{Ms} as defined in [1]. Scalar value. If several controllers are specified, @var{Ms} becomes a row vector with as many entries as controllers. @item ws The frequency [rad/s] corresponding to the sensitivity peak. Scalar value. If several controllers are specified, @var{ws} becomes a row vector with as many entries as controllers. @end table @strong{Algorithm}@* Uses SLICOT AB13DD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} to calculate the infinity norm of the sensitivity function. @strong{References}@* [1] Astr@"om, K. and H@"agglund, T. (1995) PID Controllers: Theory, Design and Tuning, Second Edition. Instrument Society of America. @end deftypefn @section sigma @findex sigma @deftypefn {Function File} {} sigma (@var{sys}) @deftypefnx {Function File} {} sigma (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) @deftypefnx {Function File} {} sigma (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) @deftypefnx {Function File} {} sigma (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) @deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}) @deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{w}) Singular values of frequency response. If no output arguments are given, the singular value plot is printed on the screen. @strong{Inputs} @table @var @item sys @acronym{LTI} system. Multiple inputs and/or outputs (MIMO systems) make practical sense. @item w Optional vector of frequency values. If @var{w} is not specified, it is calculated by the zeros and poles of the system. Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, where @var{wmin} and @var{wmax} denote minimum and maximum frequencies in rad/s. @item 'style' Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted black line. See @command{help plot} for details. @end table @strong{Outputs} @table @var @item sv Array of singular values. For a system with m inputs and p outputs, the array sv has @code{min (m, p)} rows and as many columns as frequency points @code{length (w)}. The singular values at the frequency @code{w(k)} are given by @code{sv(:,k)}. @item w Vector of frequency values used. @end table @seealso{bodemag, svd} @end deftypefn @chapter Pole Placement @section place @findex place @deftypefn {Function File} {@var{f} =} place (@var{sys}, @var{p}) @deftypefnx {Function File} {@var{f} =} place (@var{a}, @var{b}, @var{p}) @deftypefnx {Function File} {[@var{f}, @var{info}] =} place (@var{sys}, @var{p}, @var{alpha}) @deftypefnx {Function File} {[@var{f}, @var{info}] =} place (@var{a}, @var{b}, @var{p}, @var{alpha}) Pole assignment for a given matrix pair (@var{A},@var{B}) such that @code{p = eig (A-B*F)}. If parameter @var{alpha} is specified, poles with real parts (continuous-time) or moduli (discrete-time) below @var{alpha} are left untouched. @strong{Inputs} @table @var @item sys Continuous- or discrete-time @acronym{LTI} system. @item a State matrix (n-by-n) of a continuous-time system. @item b Input matrix (n-by-m) of a continuous-time system. @item p Desired eigenvalues of the closed-loop system state-matrix @var{A-B*F}. @code{length (p) <= rows (A)}. @item alpha Specifies the maximum admissible value, either for real parts or for moduli, of the eigenvalues of @var{A} which will not be modified by the eigenvalue assignment algorithm. @code{alpha >= 0} for discrete-time systems. @end table @strong{Outputs} @table @var @item f State feedback gain matrix. @item info Structure containing additional information. @item info.nfp The number of fixed poles, i.e. eigenvalues of @var{A} having real parts less than @var{alpha}, or moduli less than @var{alpha}. These eigenvalues are not modified by @command{place}. @item info.nap The number of assigned eigenvalues. @code{nap = n-nfp-nup}. @item info.nup The number of uncontrollable eigenvalues detected by the eigenvalue assignment algorithm. @item info.z The orthogonal matrix @var{z} reduces the closed-loop system state matrix @code{A + B*F} to upper real Schur form. Note the positive sign in @code{A + B*F}. @end table @strong{Note} @example Place is also suitable to design estimator gains: @group L = place (A.', C.', p).' L = place (sys.', p).' # useful for discrete-time systems @end group @end example @strong{Algorithm}@* Uses SLICOT SB01BD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section rlocus @findex rlocus @deftypefn {Function File} {} rlocus (@var{sys}) @deftypefnx {Function File} {[@var{rldata}, @var{k}] =} rlocus (@var{sys}, @var{increment}, @var{min_k}, @var{max_k}) Display root locus plot of the specified @acronym{SISO} system. @strong{Inputs} @table @var @item sys @acronym{LTI} model. Must be a single-input and single-output (SISO) system. @item increment The increment used in computing gain values. @item min_k Minimum value of @var{k}. @item max_k Maximum value of @var{k}. @end table @strong{Outputs} @table @var @item rldata Data points plotted: in column 1 real values, in column 2 the imaginary values. @item k Gains for real axis break points. @end table @strong{Block Diagram} @example @group u + +---+ +------+ y ------>(+)----->| k |----->| SISO |-------+-------> ^ - +---+ +------+ | | | +---------------------------------+ @end group @end example @end deftypefn @chapter Optimal Control @section dlqe @findex dlqe @deftypefn {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}, @var{s}) Kalman filter for discrete-time systems. @example @group x[k] = Ax[k] + Bu[k] + Gw[k] (State equation) y[k] = Cx[k] + Du[k] + v[k] (Measurement Equation) E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S @end group @end example @strong{Inputs} @table @var @item a State transition matrix of discrete-time system (n-by-n). @item g Process noise matrix of discrete-time system (n-by-g). If @var{g} is empty @code{[]}, an identity matrix is assumed. @item c Measurement matrix of discrete-time system (p-by-n). @item q Process noise covariance matrix (g-by-g). @item r Measurement noise covariance matrix (p-by-p). @item s Optional cross term covariance matrix (g-by-p), s = cov(w,v). If @var{s} is empty @code{[]} or not specified, a zero matrix is assumed. @end table @strong{Outputs} @table @var @item m Kalman filter gain matrix (n-by-p). @item p Unique stabilizing solution of the discrete-time Riccati equation (n-by-n). Symmetric matrix. @item z Error covariance (n-by-n), cov(x(k|k)-x) @item e Closed-loop poles (n-by-1). @end table @strong{Equations} @example @group x[k|k] = x[k|k-1] + M(y[k] - Cx[k|k-1] - Du[k]) x[k+1|k] = Ax[k|k] + Bu[k] for S=0 x[k+1|k] = Ax[k|k] + Bu[k] + G*S*(C*P*C' + R)^-1*(y[k] - C*x[k|k-1]) for non-zero S E = eig(A - A*M*C) for S=0 E = eig(A - A*M*C - G*S*(C*P*C' + Rv)^-1*C) for non-zero S @end group @end example @seealso{dare, care, dlqr, lqr, lqe} @end deftypefn @section dlqr @findex dlqr @deftypefn {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{sys}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{sys}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) Linear-quadratic regulator for discrete-time systems. @strong{Inputs} @table @var @item sys Continuous or discrete-time @acronym{LTI} model (p-by-m, n states). @item a State transition matrix of discrete-time system (n-by-n). @item b Input matrix of discrete-time system (n-by-m). @item q State weighting matrix (n-by-n). @item r Input weighting matrix (m-by-m). @item s Optional cross term matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. @item e Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. @end table @strong{Outputs} @table @var @item g State feedback matrix (m-by-n). @item x Unique stabilizing solution of the discrete-time Riccati equation (n-by-n). @item l Closed-loop poles (n-by-1). @end table @strong{Equations} @example @group x[k+1] = A x[k] + B u[k], x[0] = x0 inf J(x0) = SUM (x' Q x + u' R u + 2 x' S u) k=0 L = eig (A - B*G) @end group @end example @seealso{dare, care, lqr} @end deftypefn @section estim @findex estim @deftypefn {Function File} {@var{est} =} estim (@var{sys}, @var{l}) @deftypefnx {Function File} {@var{est} =} estim (@var{sys}, @var{l}, @var{sensors}, @var{known}) Return state estimator for a given estimator gain. @strong{Inputs} @table @var @item sys @acronym{LTI} model. @item l State feedback matrix. @item sensors Indices of measured output signals y from @var{sys}. If omitted, all outputs are measured. @item known Indices of known input signals u (deterministic) to @var{sys}. All other inputs to @var{sys} are assumed stochastic. If argument @var{known} is omitted, no inputs u are known. @end table @strong{Outputs} @table @var @item est State-space model of estimator. @end table @seealso{kalman, place} @end deftypefn @section kalman @findex kalman @deftypefn {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{[]}, @var{sensors}, @var{known}) @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{s}, @var{sensors}, @var{known}) Design Kalman estimator for @acronym{LTI} systems. @strong{Inputs} @table @var @item sys Nominal plant model. @item q Covariance of white process noise. @item r Covariance of white measurement noise. @item s Optional cross term covariance. Default value is 0. @item sensors Indices of measured output signals y from @var{sys}. If omitted, all outputs are measured. @item known Indices of known input signals u (deterministic) to @var{sys}. All other inputs to @var{sys} are assumed stochastic. If argument @var{known} is omitted, no inputs u are known. @end table @strong{Outputs} @table @var @item est State-space model of the Kalman estimator. @item g Estimator gain. @item x Solution of the Riccati equation. @end table @strong{Block Diagram} @example @group u +-------+ ^ +---------------------------->| |-------> y | +-------+ + y | est | ^ u ----+--->| |----->(+)------>| |-------> x | sys | ^ + +-------+ w -------->| | | +-------+ | v Q = cov (w, w') R = cov (v, v') S = cov (w, v') @end group @end example @seealso{care, dare, estim, lqr} @end deftypefn @section lqe @findex lqe @deftypefn {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{sys}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{sys}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}, @var{s}) Kalman filter for continuous-time systems. @example @group . x = Ax + Bu + Gw (State equation) y = Cx + Du + v (Measurement Equation) E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S @end group @end example @strong{Inputs} @table @var @item sys Continuous or discrete-time @acronym{LTI} model (p-by-m, n states). @item a State matrix of continuous-time system (n-by-n). @item g Process noise matrix of continuous-time system (n-by-g). If @var{g} is empty @code{[]}, an identity matrix is assumed. @item c Measurement matrix of continuous-time system (p-by-n). @item q Process noise covariance matrix (g-by-g). @item r Measurement noise covariance matrix (p-by-p). @item s Optional cross term covariance matrix (g-by-p), s = cov(w,v). If @var{s} is empty @code{[]} or not specified, a zero matrix is assumed. @end table @strong{Outputs} @table @var @item l Kalman filter gain matrix (n-by-p). @item p Unique stabilizing solution of the continuous-time Riccati equation (n-by-n). Symmetric matrix. If @var{sys} is a discrete-time model, the solution of the corresponding discrete-time Riccati equation is returned. @item e Closed-loop poles (n-by-1). @end table @strong{Equations} @example @group . x = Ax + Bu + L(y - Cx -Du) E = eig(A - L*C) @end group @end example @seealso{dare, care, dlqr, lqr, dlqe} @end deftypefn @section lqr @findex lqr @deftypefn {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{sys}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{sys}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) Linear-quadratic regulator. @strong{Inputs} @table @var @item sys Continuous or discrete-time @acronym{LTI} model (p-by-m, n states). @item a State matrix of continuous-time system (n-by-n). @item b Input matrix of continuous-time system (n-by-m). @item q State weighting matrix (n-by-n). @item r Input weighting matrix (m-by-m). @item s Optional cross term matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. @item e Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. @end table @strong{Outputs} @table @var @item g State feedback matrix (m-by-n). @item x Unique stabilizing solution of the continuous-time Riccati equation (n-by-n). @item l Closed-loop poles (n-by-1). @end table @strong{Equations} @example @group . x = A x + B u, x(0) = x0 inf J(x0) = INT (x' Q x + u' R u + 2 x' S u) dt 0 L = eig (A - B*G) @end group @end example @seealso{care, dare, dlqr} @end deftypefn @chapter Robust Control @section augw @findex augw @deftypefn{Function File} {@var{P} =} augw (@var{G}, @var{W1}, @var{W2}, @var{W3}) Extend plant for stacked S/KS/T problem. Subsequently, the robust control problem can be solved by h2syn or hinfsyn. @strong{Inputs} @table @var @item G @acronym{LTI} model of plant. @item W1 @acronym{LTI} model of performance weight. Bounds the largest singular values of sensitivity @var{S}. Model must be empty @code{[]}, SISO or of appropriate size. @item W2 @acronym{LTI} model to penalize large control inputs. Bounds the largest singular values of @var{KS}. Model must be empty @code{[]}, SISO or of appropriate size. @item W3 @acronym{LTI} model of robustness and noise sensitivity weight. Bounds the largest singular values of complementary sensitivity @var{T}. Model must be empty @code{[]}, SISO or of appropriate size. @end table All inputs must be proper/realizable. Scalars, vectors and matrices are possible instead of @acronym{LTI} models. @strong{Outputs} @table @var @item P State-space model of augmented plant. @end table @strong{Block Diagram} @example @group | W1 | -W1*G | z1 = W1 r - W1 G u | 0 | W2 | z2 = W2 u P = | 0 | W3*G | z3 = W3 G u |----+-------| | I | -G | e = r - G u @end group @end example @example @group +------+ z1 +---------------------------------------->| W1 |-----> | +------+ | +------+ z2 | +---------------------->| W2 |-----> | | +------+ r + e | +--------+ u | +--------+ y +------+ z3 ----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |-----> ^ - +--------+ +--------+ | +------+ | | +----------------------------------------+ @end group @end example @example @group +--------+ | |-----> z1 (p1x1) z1 = W1 e r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u | |-----> z3 (p3x1) z3 = W3 y u (mx1) ----->| |-----> e (px1) e = r - y +--------+ @end group @end example @example @group +--------+ r ----->| |-----> z | P(s) | u +---->| |-----+ e | +--------+ | | | | +--------+ | +-----| K(s) |<----+ +--------+ @end group @end example @strong{References}@* [1] Skogestad, S. and Postlethwaite I. (2005) @cite{Multivariable Feedback Control: Analysis and Design: Second Edition}. Wiley. @seealso{h2syn, hinfsyn, mixsyn} @end deftypefn @section fitfrd @findex fitfrd @deftypefn{Function File} {[@var{sys}, @var{n}] =} fitfrd (@var{dat}, @var{n}) @deftypefnx{Function File} {[@var{sys}, @var{n}] =} fitfrd (@var{dat}, @var{n}, @var{flag}) Fit frequency response data with a state-space system. If requested, the returned system is stable and minimum-phase. @strong{Inputs} @table @var @item dat @acronym{LTI} model containing frequency response data of a SISO system. @item n The desired order of the system to be fitted. @code{n <= length(dat.w)}. @item flag The flag controls whether the returned system is stable and minimum-phase. @table @var @item 0 The system zeros and poles are not constrained. Default value. @item 1 The system zeros and poles will have negative real parts in the continuous-time case, or moduli less than 1 in the discrete-time case. @end table @end table @strong{Outputs} @table @var @item sys State-space model of order @var{n}, fitted to frequency response data @var{dat}. @item n The order of the obtained system. The value of @var{n} could only be modified if inputs @code{n > 0} and @code{flag = 1}. @end table @strong{Algorithm}@* Uses SLICOT SB10YD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section h2syn @findex h2syn @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} h2syn (@var{P}, @var{nmeas}, @var{ncon}) @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} h2syn (@var{P}) H-2 control synthesis for @acronym{LTI} plant. @strong{Inputs} @table @var @item P Generalized plant. Must be a proper/realizable @acronym{LTI} model. If @var{P} is constructed with @command{mktito} or @command{augw}, arguments @var{nmeas} and @var{ncon} can be omitted. @item nmeas Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used to calculate the H-2 norm. @item ncon Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited by a harmonic test signal. @end table @strong{Outputs} @table @var @item K State-space model of the H-2 optimal controller. @item N State-space model of the lower LFT of @var{P} and @var{K}. @item info Structure containing additional information. @item info.gamma H-2 norm of @var{N}. @item info.rcond Vector @var{rcond} contains estimates of the reciprocal condition numbers of the matrices which are to be inverted and estimates of the reciprocal condition numbers of the Riccati equations which have to be solved during the computation of the controller @var{K}. For details, see the description of the corresponding SLICOT routine. @end table @strong{Block Diagram} @example @group gamma = min||N(K)|| N = lft (P, K) K 2 +--------+ w ----->| |-----> z | P(s) | u +---->| |-----+ v | +--------+ | | | | +--------+ | +-----| K(s) |<----+ +--------+ +--------+ w ----->| N(s) |-----> z +--------+ @end group @end example @strong{Algorithm}@* Uses SLICOT SB10HD and SB10ED by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{augw, lqr, dlqr, kalman} @end deftypefn @section hinfsyn @findex hinfsyn @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}) @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}, @dots{}) @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @dots{}) @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{opt}, @dots{}) H-infinity control synthesis for @acronym{LTI} plant. @strong{Inputs} @table @var @item P Generalized plant. Must be a proper/realizable @acronym{LTI} model. If @var{P} is constructed with @command{mktito} or @command{augw}, arguments @var{nmeas} and @var{ncon} can be omitted. @item nmeas Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used to calculate the H-infinity norm. @item ncon Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited by a harmonic test signal. @item @dots{} Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by function @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item K State-space model of the H-infinity (sub-)optimal controller. @item N State-space model of the lower LFT of @var{P} and @var{K}. @item info Structure containing additional information. @item info.gamma L-infinity norm of @var{N}. @item info.rcond Vector @var{rcond} contains estimates of the reciprocal condition numbers of the matrices which are to be inverted and estimates of the reciprocal condition numbers of the Riccati equations which have to be solved during the computation of the controller @var{K}. For details, see the description of the corresponding SLICOT routine. @end table @strong{Option Keys and Values} @table @var @item 'method' String specifying the desired kind of controller: @table @var @item 'optimal', 'opt', 'o' Compute optimal controller using gamma iteration. Default selection for compatibility reasons. @item 'suboptimal', 'sub', 's' Compute (sub-)optimal controller. For stability reasons, suboptimal controllers are to be preferred over optimal ones. @end table @item 'gmax' The maximum value of the H-infinity norm of @var{N}. It is assumed that @var{gmax} is sufficiently large so that the controller is admissible. Default value is 1e15. @item 'gmin' Initial lower bound for gamma iteration. Default value is 0. @var{gmin} is only meaningful for optimal discrete-time controllers. @item 'tolgam' Tolerance used for controlling the accuracy of @var{gamma} and its distance to the estimated minimal possible value of @var{gamma}. Default value is 0.01. If @var{tolgam} = 0, then a default value equal to @code{sqrt(eps)} is used, where @var{eps} is the relative machine precision. For suboptimal controllers, @var{tolgam} is ignored. @item 'actol' Upper bound for the poles of the closed-loop system @var{N} used for determining if it is stable. @var{actol} >= 0 for stable systems. For suboptimal controllers, @var{actol} is ignored. @end table @strong{Block Diagram} @example @group gamma = min||N(K)|| N = lft (P, K) K inf +--------+ w ----->| |-----> z | P(s) | u +---->| |-----+ v | +--------+ | | | | +--------+ | +-----| K(s) |<----+ +--------+ +--------+ w ----->| N(s) |-----> z +--------+ @end group @end example @strong{Algorithm}@* Uses SLICOT SB10FD, SB10DD and SB10AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{augw, mixsyn} @end deftypefn @section mixsyn @findex mixsyn @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} mixsyn (@var{G}, @var{W1}, @var{W2}, @var{W3}, @dots{}) Solve stacked S/KS/T H-infinity problem. Mixed-sensitivity is the name given to transfer function shaping problems in which the sensitivity function @iftex @tex $ S = (I + G K)^{-1} $ @end tex @end iftex @ifnottex @example -1 S = (I + G K) @end example @end ifnottex is shaped along with one or more other closed-loop transfer functions such as @var{K S} or the complementary sensitivity function @iftex @tex $ T = I - S = (I + G K)^{-1} G K $ @end tex @end iftex @ifnottex @example -1 T = I - S = (I + G K) @end example @end ifnottex in a typical one degree-of-freedom configuration, where @var{G} denotes the plant and @var{K} the (sub-)optimal controller to be found. The shaping of multivariable transfer functions is based on the idea that a satisfactory definition of gain (range of gain) for a matrix transfer function is given by the singular values @iftex @tex $\sigma$ @end tex @end iftex @ifnottex @end ifnottex of the transfer function. Hence the classical loop-shaping ideas of feedback design can be generalized to multivariable systems. In addition to the requirement that @var{K} stabilizes @var{G}, the closed-loop objectives are as follows [1]: @enumerate @item For @emph{disturbance rejection} make @iftex @tex $\overline{\sigma}(S)$ @end tex @end iftex @ifnottex @end ifnottex small. @item For @emph{noise attenuation} make @iftex @tex $\overline{\sigma}(T)$ @end tex @end iftex @ifnottex @end ifnottex small. @item For @emph{reference tracking} make @iftex @tex $\overline{\sigma}(T) \approx \underline{\sigma}(T) \approx 1$. @end tex @end iftex @ifnottex @end ifnottex @item For @emph{input usage (control energy) reduction} make @iftex @tex $\overline{\sigma}(K S)$ @end tex @end iftex @ifnottex @end ifnottex small. @item For @emph{robust stability} in the presence of an additive perturbation @iftex @tex $G_p = G + \Delta$, @end tex @end iftex @ifnottex @end ifnottex make @iftex @tex $\overline{\sigma}(K S)$ @end tex @end iftex @ifnottex @end ifnottex small. @item For @emph{robust stability} in the presence of a multiplicative output perturbation @iftex @tex $G_p = (I + \Delta) G$, @end tex @end iftex @ifnottex @end ifnottex make @iftex @tex $\overline{\sigma}(T)$ @end tex @end iftex @ifnottex @end ifnottex small. @end enumerate In order to find a robust controller for the so-called stacked @iftex @tex $S/KS/T \ H_{\infty}$ @end tex @end iftex @ifnottex S/KS/T H-infinity @end ifnottex problem, the user function @command{mixsyn} minimizes the following criterion @iftex @tex $$ \underset{K}{\min} || N(K) ||_{\infty}, \quad N = | W_1 S; \ W_2 K S; \ W_3 T |$$ @end tex @end iftex @ifnottex @example | W1 S | min || N(K) || N = | W2 K S | K oo | W3 T | @end example @end ifnottex @code{[K, N] = mixsyn (G, W1, W2, W3)}. The user-defined weighting functions @var{W1}, @var{W2} and @var{W3} bound the largest singular values of the closed-loop transfer functions @var{S} (for performance), @var{K S} (to penalize large inputs) and @var{T} (for robustness and to avoid sensitivity to noise), respectively [1]. A few points are to be considered when choosing the weights. The weigths @var{Wi} must all be proper and stable. Therefore if one wishes, for example, to minimize @var{S} at low frequencies by a weighting @var{W1} including integral action, @iftex @tex ${1 \over s}$ @end tex @end iftex @ifnottex @example 1 - s @end example @end ifnottex needs to be approximated by @iftex @tex ${1 \over s + \epsilon}$, where $\epsilon \ll 1$. @end tex @end iftex @ifnottex @example 1 ----- where e << 1. s + e @end example @end ifnottex Similarly one might be interested in weighting @var{K S} with a non-proper weight @var{W2} to ensure that @var{K} is small outside the system bandwidth. The trick here is to replace a non-proper term such as @iftex @tex $ 1 + \tau_1 s $ by $ {1 + \tau_1 s \over 1 + \tau_2 s} $, where @end tex @end iftex @ifnottex @example 1 + T1 s 1 + T1 s by --------, where T2 << T1. 1 + T2 s @end example @end ifnottex @iftex @tex $\tau_2 \ll \tau_1$ @end tex @end iftex [1, 2]. @strong{Inputs} @table @var @item G @acronym{LTI} model of plant. @item W1 @acronym{LTI} model of performance weight. Bounds the largest singular values of sensitivity @var{S}. Model must be empty @code{[]}, SISO or of appropriate size. @item W2 @acronym{LTI} model to penalize large control inputs. Bounds the largest singular values of @var{KS}. Model must be empty @code{[]}, SISO or of appropriate size. @item W3 @acronym{LTI} model of robustness and noise sensitivity weight. Bounds the largest singular values of complementary sensitivity @var{T}. Model must be empty @code{[]}, SISO or of appropriate size. @item @dots{} Optional arguments of @command{hinfsyn}. Type @command{help hinfsyn} for more information. @end table All inputs must be proper/realizable. Scalars, vectors and matrices are possible instead of @acronym{LTI} models. @strong{Outputs} @table @var @item K State-space model of the H-infinity (sub-)optimal controller. @item N State-space model of the lower LFT of @var{P} and @var{K}. @item info Structure containing additional information. @item info.gamma L-infinity norm of @var{N}. @item info.rcond Vector @var{rcond} contains estimates of the reciprocal condition numbers of the matrices which are to be inverted and estimates of the reciprocal condition numbers of the Riccati equations which have to be solved during the computation of the controller @var{K}. For details, see the description of the corresponding SLICOT routine. @end table @strong{Block Diagram} @example @group | W1 S | gamma = min||N(K)|| N = | W2 K S | = lft (P, K) K inf | W3 T | @end group @end example @example @group +------+ z1 +---------------------------------------->| W1 |-----> | +------+ | +------+ z2 | +---------------------->| W2 |-----> | | +------+ r + e | +--------+ u | +--------+ y +------+ z3 ----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |-----> ^ - +--------+ +--------+ | +------+ | | +----------------------------------------+ @end group @end example @example @group +--------+ | |-----> z1 (p1x1) z1 = W1 e r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u | |-----> z3 (p3x1) z3 = W3 y u (mx1) ----->| |-----> e (px1) e = r - y +--------+ @end group @end example @example @group +--------+ r ----->| |-----> z | P(s) | u +---->| |-----+ e | +--------+ | | | | +--------+ | +-----| K(s) |<----+ +--------+ @end group @end example @example @group +--------+ r ----->| N(s) |-----> z +--------+ @end group @end example @example @group Extended Plant: P = augw (G, W1, W2, W3) Controller: K = mixsyn (G, W1, W2, W3) Entire System: N = lft (P, K) Open Loop: L = G * K Closed Loop: T = feedback (L) @end group @end example @strong{Algorithm}@* Relies on functions @command{augw} and @command{hinfsyn}, which use SLICOT SB10FD, SB10DD and SB10AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @strong{References}@* [1] Skogestad, S. and Postlethwaite I. (2005) @cite{Multivariable Feedback Control: Analysis and Design: Second Edition}. Wiley, Chichester, England.@* [2] Meinsma, G. (1995) @cite{Unstable and nonproper weights in H-infinity control} Automatica, Vol. 31, No. 11, pp. 1655-1658 @seealso{hinfsyn, augw} @end deftypefn @section mktito @findex mktito @deftypefn{Function File} {@var{P} =} mktito (@var{P}, @var{nmeas}, @var{ncon}) Partition @acronym{LTI} plant @var{P} for robust controller synthesis. If a plant is partitioned this way, one can omit the inputs @var{nmeas} and @var{ncon} when calling the functions @command{hinfsyn} and @command{h2syn}. @strong{Inputs} @table @var @item P Generalized plant. @item nmeas Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used to calculate the H-2/H-infinity norm. @item ncon Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited by a harmonic test signal. @end table @strong{Outputs} @table @var @item P Partitioned plant. The input/output groups and names are overwritten with designations according to [1]. @end table @strong{Block Diagram} @example @group min||N(K)|| N = lft (P, K) K norm +--------+ w ----->| |-----> z | P(s) | u +---->| |-----+ v | +--------+ | | | | +--------+ | +-----| K(s) |<----+ +--------+ +--------+ w ----->| N(s) |-----> z +--------+ @end group @end example @strong{Reference}@* [1] Skogestad, S. and Postlethwaite, I. (2005) @cite{Multivariable Feedback Control: Analysis and Design: Second Edition}. Wiley, Chichester, England. @end deftypefn @section ncfsyn @findex ncfsyn @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} ncfsyn (@var{G}, @var{W1}, @var{W2}, @var{factor}) Loop shaping H-infinity synthesis. Compute positive feedback controller using the McFarlane/Glover loop shaping design procedure [1]. Using a precompensator @var{W1} and/or a postcompensator @var{W2}, the singular values of the nominal plant @var{G} are shaped to give a desired open-loop shape. The nominal plant @var{G} and shaping functions @var{W1}, @var{W2} are combined to form the shaped plant, @var{Gs} where @code{Gs = W2 G W1}. We assume that @var{W1} and @var{W2} are such that @var{Gs} contains no hidden modes. It is relatively easy to approximate the closed-loop requirements by the following open-loop objectives [2]: @enumerate @item For @emph{disturbance rejection} make @iftex @tex $\underline{\sigma}(W_2 G W_1)$ @end tex @end iftex @ifnottex @example @end example @end ifnottex large; valid for frequencies at which @iftex @tex $\underline{\sigma}(G_S) \gg 1$. @end tex @end iftex @ifnottex @example @end example @end ifnottex @item For @emph{noise attenuation} make @iftex @tex $\overline{\sigma}(W_2 G W_1)$ @end tex @end iftex @ifnottex @example @end example @end ifnottex small; valid for frequencies at which @iftex @tex $\overline{\sigma}(G_S) \ll 1$. @end tex @end iftex @ifnottex @example @end example @end ifnottex @item For @emph{reference tracking} make @iftex @tex $\underline{\sigma}(W_2 G W_1)$ @end tex @end iftex @ifnottex @example @end example @end ifnottex large; valid for frequencies at which @iftex @tex $\underline{\sigma}(G_S) \gg 1$. @end tex @end iftex @ifnottex @example @end example @end ifnottex @item For @emph{robust stability} to a multiplicative output perturbation @iftex @tex $G_p = (I + \Delta) G$, make @end tex @end iftex @ifnottex @example @end example @end ifnottex @iftex @tex $\overline{\sigma}(W_2 G W_1)$ @end tex @end iftex @ifnottex @example @end example @end ifnottex small; valid for frequencies at which @iftex @tex $\overline{\sigma}(G_S) \ll 1$. @end tex @end iftex @ifnottex @example . @end example @end ifnottex @end enumerate Then a stabilizing controller @var{Ks} is synthesized for shaped plant @var{Gs}. The final positive feedback controller @var{K} is then constructed by combining the @iftex @tex $H_{\infty}$ @end tex @end iftex @ifnottex @example H-infinity @end example @end ifnottex controller @var{Ks} with the shaping functions @var{W1} and @var{W2} such that @code{K = W1 Ks W2}. In [1] is stated further that the given robust stabilization objective can be interpreted as a @iftex @tex $H_{\infty}$ @end tex @end iftex @ifnottex @example H-infinity @end example @end ifnottex problem formulation of minimizing the @iftex @tex $H_{\infty}$ @end tex @end iftex @ifnottex @example H-infinity @end example @end ifnottex norm of the frequency weighted gain from disturbances on the plant input and output to the controller input and output as follows: @iftex @tex $$ \underset{K}{\min} \, || N(K) ||_{\infty}, $$ $$ N = | W_{1}^{-1}; W_2 G | \ (I - K G)^{-1} \ | W_1, \ G W_{2}^{-1} | $$ @end tex @end iftex @ifnottex @example -1 -1 -1 min || N(K) || , N = | W1 | (I - K G) | W1 G W2 | K oo | W2 G | @end example @end ifnottex @iftex @tex @end tex @end iftex @code{[K, N] = ncfsyn (G, W1, W2, f)} The function @command{ncfsyn} - the somewhat cryptic name stands for @emph{normalized coprime factorization synthesis} - allows the specification of an additional argument, factor @var{f}. Default value @code{f = 1} implies that an optimal controller is required, whereas @code{f > 1} implies that a suboptimal controller is required, achieving a performance that is @var{f} times less than optimal. @strong{Inputs} @table @var @item G @acronym{LTI} model of plant. @item W1 @acronym{LTI} model of precompensator. Model must be SISO or of appropriate size. An identity matrix is taken if @var{W1} is not specified or if an empty model @code{[]} is passed. @item W2 @acronym{LTI} model of postcompensator. Model must be SISO or of appropriate size. An identity matrix is taken if @var{W2} is not specified or if an empty model @code{[]} is passed. @item factor @code{factor = 1} implies that an optimal controller is required. @code{factor > 1} implies that a suboptimal controller is required, achieving a performance that is @var{factor} times less than optimal. Default value is 1. @end table @strong{Outputs} @table @var @item K State-space model of the H-infinity loop-shaping controller. Note that @var{K} is a @emph{positive} feedback controller. @item N State-space model of the closed loop depicted below. @item info Structure containing additional information. @item info.gamma L-infinity norm of @var{N}. @code{gamma = norm (N, inf)}. @item info.emax Nugap robustness. @code{emax = inv (gamma)}. @item info.Gs Shaped plant. @code{Gs = W2 * G * W1}. @item info.Ks Controller for shaped plant. @code{Ks = ncfsyn (Gs)}. @item info.rcond Estimates of the reciprocal condition numbers of the Riccati equations and a few other things. For details, see the description of the corresponding SLICOT routine. @end table @strong{Block Diagram of N} @example @group ^ z1 ^ z2 | | w1 + | +--------+ | +--------+ ----->(+)---+-->| Ks |----+--->(+)---->| Gs |----+ ^ + +--------+ ^ +--------+ | | w2 | | | | +-------------------------------------------------+ @end group @end example @strong{Algorithm}@* Uses SLICOT SB10ID, SB10KD and SB10ZD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @strong{References}@* [1] D. McFarlane and K. Glover, @cite{A Loop Shaping Design Procedure Using H-infinity Synthesis}, IEEE Transactions on Automatic Control, Vol. 37, No. 6, June 1992.@* [2] S. Skogestad and I. Postlethwaite, @cite{Multivariable Feedback Control: Analysis and Design: Second Edition}. Wiley, Chichester, England, 2005.@* @end deftypefn @chapter Matrix Equation Solvers @section care @findex care @deftypefn {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) Solve continuous-time algebraic Riccati equation (ARE). @strong{Inputs} @table @var @item a Real matrix (n-by-n). @item b Real matrix (n-by-m). @item q Real matrix (n-by-n). @item r Real matrix (m-by-m). @item s Optional real matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. @item e Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. @end table @strong{Outputs} @table @var @item x Unique stabilizing solution of the continuous-time Riccati equation (n-by-n). @item l Closed-loop poles (n-by-1). @item g Corresponding gain matrix (m-by-n). @end table @strong{Equations} @example @group -1 A'X + XA - XB R B'X + Q = 0 -1 A'X + XA - (XB + S) R (B'X + S') + Q = 0 -1 G = R B'X -1 G = R (B'X + S') L = eig (A - B*G) @end group @end example @example @group -1 A'XE + E'XA - E'XB R B'XE + Q = 0 -1 A'XE + E'XA - (E'XB + S) R (B'XE + S') + Q = 0 -1 G = R B'XE -1 G = R (B'XE + S) L = eig (A - B*G, E) @end group @end example @strong{Algorithm}@* Uses SLICOT SB02OD and SG02AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{dare, lqr, dlqr, kalman} @end deftypefn @section dare @findex dare @deftypefn {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}) @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) Solve discrete-time algebraic Riccati equation (ARE). @strong{Inputs} @table @var @item a Real matrix (n-by-n). @item b Real matrix (n-by-m). @item q Real matrix (n-by-n). @item r Real matrix (m-by-m). @item s Optional real matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. @item e Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. @end table @strong{Outputs} @table @var @item x Unique stabilizing solution of the discrete-time Riccati equation (n-by-n). @item l Closed-loop poles (n-by-1). @item g Corresponding gain matrix (m-by-n). @end table @strong{Equations} @example @group -1 A'XA - X - A'XB (B'XB + R) B'XA + Q = 0 -1 A'XA - X - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0 -1 G = (B'XB + R) B'XA -1 G = (B'XB + R) (B'XA + S') L = eig (A - B*G) @end group @end example @example @group -1 A'XA - E'XE - A'XB (B'XB + R) B'XA + Q = 0 -1 A'XA - E'XE - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0 -1 G = (B'XB + R) B'XA -1 G = (B'XB + R) (B'XA + S') L = eig (A - B*G, E) @end group @end example @strong{Algorithm}@* Uses SLICOT SB02OD and SG02AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{care, lqr, dlqr, kalman} @end deftypefn @section dlyap @findex dlyap @deftypefn{Function File} {@var{x} =} dlyap (@var{a}, @var{b}) @deftypefnx{Function File} {@var{x} =} dlyap (@var{a}, @var{b}, @var{c}) @deftypefnx{Function File} {@var{x} =} dlyap (@var{a}, @var{b}, @var{[]}, @var{e}) Solve discrete-time Lyapunov or Sylvester equations. @strong{Equations} @example @group AXA' - X + B = 0 (Lyapunov Equation) AXB' - X + C = 0 (Sylvester Equation) AXA' - EXE' + B = 0 (Generalized Lyapunov Equation) @end group @end example @strong{Algorithm}@* Uses SLICOT SB03MD, SB04QD and SG03AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{dlyapchol, lyap, lyapchol} @end deftypefn @section dlyapchol @findex dlyapchol @deftypefn{Function File} {@var{u} =} dlyapchol (@var{a}, @var{b}) @deftypefnx{Function File} {@var{u} =} dlyapchol (@var{a}, @var{b}, @var{e}) Compute Cholesky factor of discrete-time Lyapunov equations. @strong{Equations} @example @group A U' U A' - U' U + B B' = 0 (Lyapunov Equation) A U' U A' - E U' U E' + B B' = 0 (Generalized Lyapunov Equation) @end group @end example @strong{Algorithm}@* Uses SLICOT SB03OD and SG03BD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{dlyap, lyap, lyapchol} @end deftypefn @section lyap @findex lyap @deftypefn{Function File} {@var{x} =} lyap (@var{a}, @var{b}) @deftypefnx{Function File} {@var{x} =} lyap (@var{a}, @var{b}, @var{c}) @deftypefnx{Function File} {@var{x} =} lyap (@var{a}, @var{b}, @var{[]}, @var{e}) Solve continuous-time Lyapunov or Sylvester equations. @strong{Equations} @example @group AX + XA' + B = 0 (Lyapunov Equation) AX + XB + C = 0 (Sylvester Equation) AXE' + EXA' + B = 0 (Generalized Lyapunov Equation) @end group @end example @strong{Algorithm}@* Uses SLICOT SB03MD, SB04MD and SG03AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{lyapchol, dlyap, dlyapchol} @end deftypefn @section lyapchol @findex lyapchol @deftypefn{Function File} {@var{u} =} lyapchol (@var{a}, @var{b}) @deftypefnx{Function File} {@var{u} =} lyapchol (@var{a}, @var{b}, @var{e}) Compute Cholesky factor of continuous-time Lyapunov equations. @strong{Equations} @example @group A U' U + U' U A' + B B' = 0 (Lyapunov Equation) A U' U E' + E U' U A' + B B' = 0 (Generalized Lyapunov Equation) @end group @end example @strong{Algorithm}@* Uses SLICOT SB03OD and SG03BD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @seealso{lyap, dlyap, dlyapchol} @end deftypefn @chapter Model Reduction @section bstmodred @findex bstmodred @deftypefn{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @var{opt}, @dots{}) Model order reduction by Balanced Stochastic Truncation (BST) method. The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} approximates the one from original system @var{G}. BST is a relative error method which tries to minimize @iftex @tex $$ || G^{-1} (G-G_r) ||_{\infty} = \min $$ @end tex @end iftex @ifnottex @example -1 ||G (G-Gr)|| = min inf @end example @end ifnottex @strong{Inputs} @table @var @item G @acronym{LTI} model to be reduced. @item nr The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is chosen automatically according to the description of key @var{'order'}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Gr Reduced order state-space model. @item info Struct containing additional information. @table @var @item info.n The order of the original system @var{G}. @item info.ns The order of the @var{alpha}-stable subsystem of the original system @var{G}. @item info.hsv The Hankel singular values of the phase system corresponding to the @var{alpha}-stable part of the original system @var{G}. The @var{ns} Hankel singular values are ordered decreasingly. @item info.nu The order of the @var{alpha}-unstable subsystem of both the original system @var{G} and the reduced-order system @var{Gr}. @item info.nr The order of the obtained reduced order system @var{Gr}. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'nr' The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is the sum of NU and the number of Hankel singular values greater than @code{MAX(TOL1,NS*EPS)}; @var{nr} can be further reduced to ensure that @code{HSV(NR-NU) > HSV(NR+1-NU)}. @item 'method' Approximation method for the H-infinity norm. Valid values corresponding to this key are: @table @var @item 'sr-bta', 'b' Use the square-root Balance & Truncate method. @item 'bfsr-bta', 'f' Use the balancing-free square-root Balance & Truncate method. Default method. @item 'sr-spa', 's' Use the square-root Singular Perturbation Approximation method. @item 'bfsr-spa', 'p' Use the balancing-free square-root Singular Perturbation Approximation method. @end table @item 'alpha' Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix @var{G.A}. For a continuous-time system, ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. Default value is 0 for continuous-time systems and 1 for discrete-time systems. @item 'beta' Use @code{[G, beta*I]} as new system @var{G} to combine absolute and relative error methods. BETA > 0 specifies the absolute/relative error weighting parameter. A large positive value of BETA favours the minimization of the absolute approximation error, while a small value of BETA is appropriate for the minimization of the relative error. BETA = 0 means a pure relative error method and can be used only if rank(G.D) = rows(G.D) which means that the feedthrough matrice must not be rank-deficient. Default value is 0. @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of reduced system. For model reduction, the recommended value of @var{tol1} lies in the interval [0.00001, 0.001]. @var{tol1} < 1. If @var{tol1} <= 0 on entry, the used default value is @var{tol1} = NS*EPS, where NS is the number of ALPHA-stable eigenvalues of A and EPS is the machine precision. If @var{'order'} is specified, the value of @var{tol1} is ignored. @item 'tol2' The tolerance for determining the order of a minimal realization of the phase system (see METHOD) corresponding to the ALPHA-stable part of the given system. The recommended value is TOL2 = NS*EPS. TOL2 <= TOL1 < 1. This value is used by default if @var{'tol2'} is not specified or if TOL2 <= 0 on entry. @item 'equil', 'scale' Boolean indicating whether equilibration (scaling) should be performed on system @var{G} prior to order reduction. Default value is true if @code{G.scaled == false} and false if @code{G.scaled == true}. Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can @strong{not} be done by the equilibration option or the @command{prescale} command because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model. @end table BST is often suitable to perform model reduction in order to obtain low order design models for controller synthesis. Approximation Properties: @itemize @bullet @item Guaranteed stability of reduced models @item Approximates simultaneously gain and phase @item Preserves non-minimum phase zeros @item Guaranteed a priori error bound @iftex @tex $$ || G^{-1} (G-G_r) ||_{\infty} \leq 2 \sum_{j=r+1}^{n} {1+\sigma_j \over 1-\sigma_j} - 1 $$ @end tex @end iftex @end itemize @strong{Algorithm}@* Uses SLICOT AB09HD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section btamodred @findex btamodred @deftypefn{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @var{opt}, @dots{}) Model order reduction by frequency weighted Balanced Truncation Approximation (BTA) method. The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} approximates the one from original system @var{G}. BTA is an absolute error method which tries to minimize @iftex @tex $$ || G - G_r ||_{\infty} = \min $$ $$ || V \ (G - G_r) \ W ||_{\infty} = \min $$ @end tex @end iftex @ifnottex @example ||G-Gr|| = min inf ||V (G-Gr) W|| = min inf @end example @end ifnottex where @var{V} and @var{W} denote output and input weightings. @strong{Inputs} @table @var @item G @acronym{LTI} model to be reduced. @item nr The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is chosen automatically according to the description of key @var{'order'}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Gr Reduced order state-space model. @item info Struct containing additional information. @table @var @item info.n The order of the original system @var{G}. @item info.ns The order of the @var{alpha}-stable subsystem of the original system @var{G}. @item info.hsv The Hankel singular values of the @var{alpha}-stable part of the original system @var{G}, ordered decreasingly. @item info.nu The order of the @var{alpha}-unstable subsystem of both the original system @var{G} and the reduced-order system @var{Gr}. @item info.nr The order of the obtained reduced order system @var{Gr}. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'nr' The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is chosen automatically such that states with Hankel singular values @var{info.hsv} > @var{tol1} are retained. @item 'left', 'output' @acronym{LTI} model of the left/output frequency weighting @var{V}. Default value is an identity matrix. @item 'right', 'input' @acronym{LTI} model of the right/input frequency weighting @var{W}. Default value is an identity matrix. @item 'method' Approximation method for the L-infinity norm to be used as follows: @table @var @item 'sr', 'b' Use the square-root Balance & Truncate method. @item 'bfsr', 'f' Use the balancing-free square-root Balance & Truncate method. Default method. @end table @item 'alpha' Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix @var{G.A}. For a continuous-time system, ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. Default value is 0 for continuous-time systems and 1 for discrete-time systems. @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of the reduced model. For model reduction, the recommended value of @var{tol1} is c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. Default value is info.ns*eps*info.hsv(1). If @var{'order'} is specified, the value of @var{tol1} is ignored. @item 'tol2' The tolerance for determining the order of a minimal realization of the ALPHA-stable part of the given model. TOL2 <= TOL1. If not specified, ns*eps*info.hsv(1) is chosen. @item 'gram-ctrb' Specifies the choice of frequency-weighted controllability Grammian as follows: @table @var @item 'standard' Choice corresponding to a combination method [4] of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified combination method of [4]. @end table @item 'gram-obsv' Specifies the choice of frequency-weighted observability Grammian as follows: @table @var @item 'standard' Choice corresponding to a combination method [4] of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified combination method of [4]. @end table @item 'alpha-ctrb' Combination method parameter for defining the frequency-weighted controllability Grammian. abs(alphac) <= 1. If alphac = 0, the choice of Grammian corresponds to the method of Enns [1], while if alphac = 1, the choice of Grammian corresponds to the method of Lin and Chiu [2,3]. Default value is 0. @item 'alpha-obsv' Combination method parameter for defining the frequency-weighted observability Grammian. abs(alphao) <= 1. If alphao = 0, the choice of Grammian corresponds to the method of Enns [1], while if alphao = 1, the choice of Grammian corresponds to the method of Lin and Chiu [2,3]. Default value is 0. @item 'equil', 'scale' Boolean indicating whether equilibration (scaling) should be performed on system @var{G} prior to order reduction. This is done by state transformations. Default value is true if @code{G.scaled == false} and false if @code{G.scaled == true}. Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can @strong{not} be done by the equilibration option or the @command{prescale} command because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model. @end table Approximation Properties: @itemize @bullet @item Guaranteed stability of reduced models @item Lower guaranteed error bound @item Guaranteed a priori error bound @iftex @tex $$ \sigma_{r+1} \leq || (G-G_r) ||_{\infty} \leq 2 \sum_{j=r+1}^{n} \sigma_j $$ @end tex @end iftex @end itemize @strong{References}@* [1] Enns, D. @cite{Model reduction with balanced realizations: An error bound and a frequency weighted generalization}. Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984. [2] Lin, C.-A. and Chiu, T.-Y. @cite{Model reduction via frequency-weighted balanced realization}. Control Theory and Advanced Technology, vol. 8, pp. 341-351, 1992. [3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. @cite{New results on frequency weighted balanced reduction technique}. Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995. [4] Varga, A. and Anderson, B.D.O. @cite{Square-root balancing-free methods for the frequency-weighted balancing related model reduction}. (report in preparation) @strong{Algorithm}@* Uses SLICOT AB09ID by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section hnamodred @findex hnamodred @deftypefn{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @var{opt}, @dots{}) Model order reduction by frequency weighted optimal Hankel-norm (HNA) method. The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} approximates the one from original system @var{G}. HNA is an absolute error method which tries to minimize @iftex @tex $$ || G - G_r ||_H = \min $$ $$ || V \ (G - G_r) \ W ||_H = \min $$ @end tex @end iftex @ifnottex @example ||G-Gr|| = min H ||V (G-Gr) W|| = min H @end example @end ifnottex where @var{V} and @var{W} denote output and input weightings. @strong{Inputs} @table @var @item G @acronym{LTI} model to be reduced. @item nr The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is chosen automatically according to the description of key @var{"order"}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Gr Reduced order state-space model. @item info Struct containing additional information. @table @var @item info.n The order of the original system @var{G}. @item info.ns The order of the @var{alpha}-stable subsystem of the original system @var{G}. @item info.hsv The Hankel singular values corresponding to the projection @code{op(V)*G1*op(W)}, where G1 denotes the @var{alpha}-stable part of the original system @var{G}. The @var{ns} Hankel singular values are ordered decreasingly. @item info.nu The order of the @var{alpha}-unstable subsystem of both the original system @var{G} and the reduced-order system @var{Gr}. @item info.nr The order of the obtained reduced order system @var{Gr}. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'nr' The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is the sum of @var{info.nu} and the number of Hankel singular values greater than @code{max(tol1, ns*eps*info.hsv(1)}; @item 'method' Specifies the computational approach to be used. Valid values corresponding to this key are: @table @var @item 'descriptor' Use the inverse free descriptor system approach. @item 'standard' Use the inversion based standard approach. @item 'auto' Switch automatically to the inverse free descriptor approach in case of badly conditioned feedthrough matrices in V or W. Default method. @end table @item 'left', 'v' @acronym{LTI} model of the left/output frequency weighting. The weighting must be antistable. @iftex @math{|| V \ (G-G_r) \dots ||_H = \min} @end iftex @ifnottex @example || V (G-Gr) . || = min H @end example @end ifnottex @item 'right', 'w' @acronym{LTI} model of the right/input frequency weighting. The weighting must be antistable. @iftex @math{|| \dots (G-G_r) \ W ||_H = \min} @end iftex @ifnottex @example || . (G-Gr) W || = min H @end example @end ifnottex @item 'left-inv', 'inv-v' @acronym{LTI} model of the left/output frequency weighting. The weighting must have only antistable zeros. @iftex @math{|| inv(V) \ (G-G_r) \dots ||_H = \min} @end iftex @ifnottex @example || inv(V) (G-Gr) . || = min H @end example @end ifnottex @item 'right-inv', 'inv-w' @acronym{LTI} model of the right/input frequency weighting. The weighting must have only antistable zeros. @iftex @math{|| \dots (G-G_r) \ inv(W) ||_H = \min} @end iftex @ifnottex @example || . (G-Gr) inv(W) || = min H @end example @end ifnottex @item 'left-conj', 'conj-v' @acronym{LTI} model of the left/output frequency weighting. The weighting must be stable. @iftex @math{|| conj(V) \ (G-G_r) \dots ||_H = \min} @end iftex @ifnottex @example || V (G-Gr) . || = min H @end example @end ifnottex @item 'right-conj', 'conj-w' @acronym{LTI} model of the right/input frequency weighting. The weighting must be stable. @iftex @math{|| \dots (G-G_r) \ conj(W) ||_H = \min} @end iftex @ifnottex @example || . (G-Gr) W || = min H @end example @end ifnottex @item 'left-conj-inv', 'conj-inv-v' @acronym{LTI} model of the left/output frequency weighting. The weighting must be minimum-phase. @iftex @math{|| conj(inv(V)) \ (G-G_r) \dots ||_H = \min} @end iftex @ifnottex @example || V (G-Gr) . || = min H @end example @end ifnottex @item 'right-conj-inv', 'conj-inv-w' @acronym{LTI} model of the right/input frequency weighting. The weighting must be minimum-phase. @iftex @math{|| \dots (G-G_r) \ conj(inv(W)) ||_H = \min} @end iftex @ifnottex @example || . (G-Gr) W || = min H @end example @end ifnottex @item 'alpha' Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix @var{G.A}. For a continuous-time system, ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. Default value is 0 for continuous-time systems and 1 for discrete-time systems. @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of the reduced model. For model reduction, the recommended value of @var{tol1} is c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. @var{tol1} < 1. If @var{'order'} is specified, the value of @var{tol1} is ignored. @item 'tol2' The tolerance for determining the order of a minimal realization of the ALPHA-stable part of the given model. @var{tol2} <= @var{tol1} < 1. If not specified, ns*eps*info.hsv(1) is chosen. @item 'equil', 'scale' Boolean indicating whether equilibration (scaling) should be performed on system @var{G} prior to order reduction. Default value is true if @code{G.scaled == false} and false if @code{G.scaled == true}. Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can @strong{not} be done by the equilibration option or the @command{prescale} command because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model. @end table Approximation Properties: @itemize @bullet @item Guaranteed stability of reduced models @item Lower guaranteed error bound @item Guaranteed a priori error bound @iftex @tex $$ \sigma_{r+1} \leq || (G-G_r) ||_{\infty} \leq 2 \sum_{j=r+1}^{n} \sigma_j $$ @end tex @end iftex @end itemize @strong{Algorithm}@* Uses SLICOT AB09JD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section spamodred @findex spamodred @deftypefn{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @var{opt}, @dots{}) Model order reduction by frequency weighted Singular Perturbation Approximation (SPA). The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} approximates the one from original system @var{G}. SPA is an absolute error method which tries to minimize @iftex @tex $$ || G - G_r ||_{\infty} = \min $$ $$ || V \ (G - G_r) \ W ||_{\infty} = \min $$ @end tex @end iftex @ifnottex @example ||G-Gr|| = min inf ||V (G-Gr) W|| = min inf @end example @end ifnottex where @var{V} and @var{W} denote output and input weightings. @strong{Inputs} @table @var @item G @acronym{LTI} model to be reduced. @item nr The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is chosen automatically according to the description of key @var{'order'}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Gr Reduced order state-space model. @item info Struct containing additional information. @table @var @item info.n The order of the original system @var{G}. @item info.ns The order of the @var{alpha}-stable subsystem of the original system @var{G}. @item info.hsv The Hankel singular values of the @var{alpha}-stable part of the original system @var{G}, ordered decreasingly. @item info.nu The order of the @var{alpha}-unstable subsystem of both the original system @var{G} and the reduced-order system @var{Gr}. @item info.nr The order of the obtained reduced order system @var{Gr}. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'nr' The desired order of the resulting reduced order system @var{Gr}. If not specified, @var{nr} is chosen automatically such that states with Hankel singular values @var{info.hsv} > @var{tol1} are retained. @item 'left', 'output' @acronym{LTI} model of the left/output frequency weighting @var{V}. Default value is an identity matrix. @item 'right', 'input' @acronym{LTI} model of the right/input frequency weighting @var{W}. Default value is an identity matrix. @item 'method' Approximation method for the L-infinity norm to be used as follows: @table @var @item 'sr', 's' Use the square-root Singular Perturbation Approximation method. @item 'bfsr', 'p' Use the balancing-free square-root Singular Perturbation Approximation method. Default method. @end table @item 'alpha' Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix @var{G.A}. For a continuous-time system, ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. Default value is 0 for continuous-time systems and 1 for discrete-time systems. @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of the reduced model. For model reduction, the recommended value of @var{tol1} is c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. Default value is info.ns*eps*info.hsv(1). If @var{'order'} is specified, the value of @var{tol1} is ignored. @item 'tol2' The tolerance for determining the order of a minimal realization of the ALPHA-stable part of the given model. TOL2 <= TOL1. If not specified, ns*eps*info.hsv(1) is chosen. @item 'gram-ctrb' Specifies the choice of frequency-weighted controllability Grammian as follows: @table @var @item 'standard' Choice corresponding to a combination method [4] of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified combination method of [4]. @end table @item 'gram-obsv' Specifies the choice of frequency-weighted observability Grammian as follows: @table @var @item 'standard' Choice corresponding to a combination method [4] of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified combination method of [4]. @end table @item 'alpha-ctrb' Combination method parameter for defining the frequency-weighted controllability Grammian. abs(alphac) <= 1. If alphac = 0, the choice of Grammian corresponds to the method of Enns [1], while if alphac = 1, the choice of Grammian corresponds to the method of Lin and Chiu [2,3]. Default value is 0. @item 'alpha-obsv' Combination method parameter for defining the frequency-weighted observability Grammian. abs(alphao) <= 1. If alphao = 0, the choice of Grammian corresponds to the method of Enns [1], while if alphao = 1, the choice of Grammian corresponds to the method of Lin and Chiu [2,3]. Default value is 0. @item 'equil', 'scale' Boolean indicating whether equilibration (scaling) should be performed on system @var{G} prior to order reduction. Default value is true if @code{G.scaled == false} and false if @code{G.scaled == true}. Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can @strong{not} be done by the equilibration option or the @command{prescale} command because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model. @end table @strong{References}@* [1] Enns, D. @cite{Model reduction with balanced realizations: An error bound and a frequency weighted generalization}. Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984. [2] Lin, C.-A. and Chiu, T.-Y. @cite{Model reduction via frequency-weighted balanced realization}. Control Theory and Advanced Technology, vol. 8, pp. 341-351, 1992. [3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. @cite{New results on frequency weighted balanced reduction technique}. Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995. [4] Varga, A. and Anderson, B.D.O. @cite{Square-root balancing-free methods for the frequency-weighted balancing related model reduction}. (report in preparation) @strong{Algorithm}@* Uses SLICOT AB09ID by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @chapter Controller Reduction @section btaconred @findex btaconred @deftypefn{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @var{opt}, @dots{}) Controller reduction by frequency-weighted Balanced Truncation Approximation (BTA). Given a plant @var{G} and a stabilizing controller @var{K}, determine a reduced order controller @var{Kr} such that the closed-loop system is stable and closed-loop performance is retained. The algorithm tries to minimize the frequency-weighted error @iftex @tex $$ || V \ (K - K_r) \ W ||_{\infty} = \min $$ @end tex @end iftex @ifnottex @example ||V (K-Kr) W|| = min inf @end example @end ifnottex where @var{V} and @var{W} denote output and input weightings. @strong{Inputs} @table @var @item G @acronym{LTI} model of the plant. It has m inputs, p outputs and n states. @item K @acronym{LTI} model of the controller. It has p inputs, m outputs and nc states. @item ncr The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically according to the description of key @var{'order'}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Kr State-space model of reduced order controller. @item info Struct containing additional information. @table @var @item info.ncr The order of the obtained reduced order controller @var{Kr}. @item info.ncs The order of the alpha-stable part of original controller @var{K}. @item info.hsvc The Hankel singular values of the alpha-stable part of @var{K}. The @var{ncs} Hankel singular values are ordered decreasingly. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'ncr' The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically such that states with Hankel singular values @var{info.hsvc} > @var{tol1} are retained. @item 'method' Order reduction approach to be used as follows: @table @var @item 'sr', 'b' Use the square-root Balance & Truncate method. @item 'bfsr', 'f' Use the balancing-free square-root Balance & Truncate method. Default method. @end table @item 'weight' Specifies the type of frequency-weighting as follows: @table @var @item 'none' No weightings are used (V = I, W = I). @item 'left', 'output' Use stability enforcing left (output) weighting @iftex @tex $$ V = (I - G K)^{-1} G, \qquad W = I $$ @end tex @end iftex @ifnottex @example -1 V = (I-G*K) *G , W = I @end example @end ifnottex @item 'right', 'input' Use stability enforcing right (input) weighting @iftex @tex $$ V = I, \qquad W = (I - G K)^{-1} G $$ @end tex @end iftex @ifnottex @example -1 V = I , W = (I-G*K) *G @end example @end ifnottex @item 'both', 'performance' Use stability and performance enforcing weightings @iftex @tex $$ V = (I - G K)^{-1} G, \qquad W = (I - G K)^{-1} $$ @end tex @end iftex @ifnottex @example -1 -1 V = (I-G*K) *G , W = (I-G*K) @end example @end ifnottex Default value. @end table @item 'feedback' Specifies whether @var{K} is a positive or negative feedback controller: @table @var @item '+' Use positive feedback controller. Default value. @item '-' Use negative feedback controller. @end table @item 'alpha' Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix @var{K.A}. For a continuous-time controller, ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time controller, 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. Default value is 0 for continuous-time controllers and 1 for discrete-time controllers. @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of the reduced controller. For model reduction, the recommended value of @var{tol1} is c*info.hsvc(1), where c lies in the interval [0.00001, 0.001]. Default value is info.ncs*eps*info.hsvc(1). If @var{'order'} is specified, the value of @var{tol1} is ignored. @item 'tol2' The tolerance for determining the order of a minimal realization of the ALPHA-stable part of the given controller. TOL2 <= TOL1. If not specified, ncs*eps*info.hsvc(1) is chosen. @item 'gram-ctrb' Specifies the choice of frequency-weighted controllability Grammian as follows: @table @var @item 'standard' Choice corresponding to standard Enns' method [1]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified Enns' method of [2]. @end table @item 'gram-obsv' Specifies the choice of frequency-weighted observability Grammian as follows: @table @var @item 'standard' Choice corresponding to standard Enns' method [1]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified Enns' method of [2]. @end table @item 'equil', 'scale' Boolean indicating whether equilibration (scaling) should be performed on @var{G} and @var{K} prior to order reduction. Default value is false if both @code{G.scaled == true, K.scaled == true} and true otherwise. Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can @strong{not} be done by the equilibration option or the @command{prescale} command because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model. @end table @strong{Algorithm}@* Uses SLICOT SB16AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section cfconred @findex cfconred @deftypefn{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @var{opt}, @dots{}) Reduction of state-feedback-observer based controller by coprime factorization (CF). Given a plant @var{G}, state feedback gain @var{F} and full observer gain @var{L}, determine a reduced order controller @var{Kr}. @strong{Inputs} @table @var @item G @acronym{LTI} model of the open-loop plant (A,B,C,D). It has m inputs, p outputs and n states. @item F Stabilizing state feedback matrix (m-by-n). @item L Stabilizing observer gain matrix (n-by-p). @item ncr The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically according to the description of key @var{'order'}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Kr State-space model of reduced order controller. @item info Struct containing additional information. @table @var @item info.hsv The Hankel singular values of the extended system?!?. The @var{n} Hankel singular values are ordered decreasingly. @item info.ncr The order of the obtained reduced order controller @var{Kr}. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'ncr' The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically such that states with Hankel singular values @var{info.hsv} > @var{tol1} are retained. @item 'method' Order reduction approach to be used as follows: @table @var @item 'sr-bta', 'b' Use the square-root Balance & Truncate method. @item 'bfsr-bta', 'f' Use the balancing-free square-root Balance & Truncate method. Default method. @item 'sr-spa', 's' Use the square-root Singular Perturbation Approximation method. @item 'bfsr-spa', 'p' Use the balancing-free square-root Singular Perturbation Approximation method. @end table @item 'cf' Specifies whether left or right coprime factorization is to be used as follows: @table @var @item 'left', 'l' Use left coprime factorization. Default method. @item 'right', 'r' Use right coprime factorization. @end table @item 'feedback' Specifies whether @var{F} and @var{L} are fed back positively or negatively: @table @var @item '+' A+BK and A+LC are both Hurwitz matrices. @item '-' A-BK and A-LC are both Hurwitz matrices. Default value. @end table @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of the reduced system. For model reduction, the recommended value of @var{tol1} is c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. Default value is n*eps*info.hsv(1). If @var{'order'} is specified, the value of @var{tol1} is ignored. @item 'tol2' The tolerance for determining the order of a minimal realization of the coprime factorization controller. TOL2 <= TOL1. If not specified, n*eps*info.hsv(1) is chosen. @item 'equil', 'scale' Boolean indicating whether equilibration (scaling) should be performed on system @var{G} prior to order reduction. Default value is true if @code{G.scaled == false} and false if @code{G.scaled == true}. Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can @strong{not} be done by the equilibration option or the @command{prescale} command because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model. @end table @strong{Algorithm}@* Uses SLICOT SB16BD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section fwcfconred @findex fwcfconred @deftypefn{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @var{opt}, @dots{}) Reduction of state-feedback-observer based controller by frequency-weighted coprime factorization (FW CF). Given a plant @var{G}, state feedback gain @var{F} and full observer gain @var{L}, determine a reduced order controller @var{Kr} by using stability enforcing frequency weights. @strong{Inputs} @table @var @item G @acronym{LTI} model of the open-loop plant (A,B,C,D). It has m inputs, p outputs and n states. @item F Stabilizing state feedback matrix (m-by-n). @item L Stabilizing observer gain matrix (n-by-p). @item ncr The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically according to the description of key @var{'order'}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Kr State-space model of reduced order controller. @item info Struct containing additional information. @table @var @item info.hsv The Hankel singular values of the extended system?!?. The @var{n} Hankel singular values are ordered decreasingly. @item info.ncr The order of the obtained reduced order controller @var{Kr}. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'ncr' The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically such that states with Hankel singular values @var{info.hsv} > @var{tol1} are retained. @item 'method' Order reduction approach to be used as follows: @table @var @item 'sr', 'b' Use the square-root Balance & Truncate method. @item 'bfsr', 'f' Use the balancing-free square-root Balance & Truncate method. Default method. @end table @item 'cf' Specifies whether left or right coprime factorization is to be used as follows: @table @var @item 'left', 'l' Use left coprime factorization. @item 'right', 'r' Use right coprime factorization. Default method. @end table @item 'feedback' Specifies whether @var{F} and @var{L} are fed back positively or negatively: @table @var @item '+' A+BK and A+LC are both Hurwitz matrices. @item '-' A-BK and A-LC are both Hurwitz matrices. Default value. @end table @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of the reduced system. For model reduction, the recommended value of @var{tol1} is c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. Default value is n*eps*info.hsv(1). If @var{'order'} is specified, the value of @var{tol1} is ignored. @end table @strong{Algorithm}@* Uses SLICOT SB16CD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section spaconred @findex spaconred @deftypefn{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{ncr}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{opt}, @dots{}) @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{ncr}, @var{opt}, @dots{}) Controller reduction by frequency-weighted Singular Perturbation Approximation (SPA). Given a plant @var{G} and a stabilizing controller @var{K}, determine a reduced order controller @var{Kr} such that the closed-loop system is stable and closed-loop performance is retained. The algorithm tries to minimize the frequency-weighted error @iftex @tex $$ || V \ (K - K_r) \ W ||_{\infty} = \min $$ @end tex @end iftex @ifnottex @example ||V (K-Kr) W|| = min inf @end example @end ifnottex where @var{V} and @var{W} denote output and input weightings. @strong{Inputs} @table @var @item G @acronym{LTI} model of the plant. It has m inputs, p outputs and n states. @item K @acronym{LTI} model of the controller. It has p inputs, m outputs and nc states. @item ncr The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically according to the description of key @var{'order'}. @item @dots{} Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item Kr State-space model of reduced order controller. @item info Struct containing additional information. @table @var @item info.ncr The order of the obtained reduced order controller @var{Kr}. @item info.ncs The order of the alpha-stable part of original controller @var{K}. @item info.hsvc The Hankel singular values of the alpha-stable part of @var{K}. The @var{ncs} Hankel singular values are ordered decreasingly. @end table @end table @strong{Option Keys and Values} @table @var @item 'order', 'ncr' The desired order of the resulting reduced order controller @var{Kr}. If not specified, @var{ncr} is chosen automatically such that states with Hankel singular values @var{info.hsvc} > @var{tol1} are retained. @item 'method' Order reduction approach to be used as follows: @table @var @item 'sr', 's' Use the square-root Singular Perturbation Approximation method. @item 'bfsr', 'p' Use the balancing-free square-root Singular Perturbation Approximation method. Default method. @end table @item 'weight' Specifies the type of frequency-weighting as follows: @table @var @item 'none' No weightings are used (V = I, W = I). @item 'left', 'output' Use stability enforcing left (output) weighting @iftex @tex $$ V = (I - G K)^{-1} G, \qquad W = I $$ @end tex @end iftex @ifnottex @example -1 V = (I-G*K) *G , W = I @end example @end ifnottex @item 'right', 'input' Use stability enforcing right (input) weighting @iftex @tex $$ V = I, \qquad W = (I - G K)^{-1} G $$ @end tex @end iftex @ifnottex @example -1 V = I , W = (I-G*K) *G @end example @end ifnottex @item 'both', 'performance' Use stability and performance enforcing weightings @iftex @tex $$ V = (I - G K)^{-1} G, \qquad W = (I - G K)^{-1} $$ @end tex @end iftex @ifnottex @example -1 -1 V = (I-G*K) *G , W = (I-G*K) @end example @end ifnottex Default value. @end table @item 'feedback' Specifies whether @var{K} is a positive or negative feedback controller: @table @var @item '+' Use positive feedback controller. Default value. @item '-' Use negative feedback controller. @end table @item 'alpha' Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix @var{K.A}. For a continuous-time controller, ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time controller, 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. Default value is 0 for continuous-time controllers and 1 for discrete-time controllers. @item 'tol1' If @var{'order'} is not specified, @var{tol1} contains the tolerance for determining the order of the reduced controller. For model reduction, the recommended value of @var{tol1} is c*info.hsvc(1), where c lies in the interval [0.00001, 0.001]. Default value is info.ncs*eps*info.hsvc(1). If @var{'order'} is specified, the value of @var{tol1} is ignored. @item 'tol2' The tolerance for determining the order of a minimal realization of the ALPHA-stable part of the given controller. TOL2 <= TOL1. If not specified, ncs*eps*info.hsvc(1) is chosen. @item 'gram-ctrb' Specifies the choice of frequency-weighted controllability Grammian as follows: @table @var @item 'standard' Choice corresponding to standard Enns' method [1]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified Enns' method of [2]. @end table @item 'gram-obsv' Specifies the choice of frequency-weighted observability Grammian as follows: @table @var @item 'standard' Choice corresponding to standard Enns' method [1]. Default method. @item 'enhanced' Choice corresponding to the stability enhanced modified Enns' method of [2]. @end table @item 'equil', 'scale' Boolean indicating whether equilibration (scaling) should be performed on @var{G} and @var{K} prior to order reduction. Default value is false if both @code{G.scaled == true, K.scaled == true} and true otherwise. Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs is of utmost importance. The input and output scaling can @strong{not} be done by the equilibration option or the @command{prescale} command because these functions perform state transformations only. Furthermore, signals should not be scaled simply to a certain range. For all inputs (or outputs), a certain change should be of the same importance for the model. @end table @strong{Algorithm}@* Uses SLICOT SB16AD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @chapter Experimental Data Handling @section iddata @findex iddata @deftypefn{Function File} {@var{dat} =} iddata (@var{y}) @deftypefnx{Function File} {@var{dat} =} iddata (@var{y}, @var{u}) @deftypefnx{Function File} {@var{dat} =} iddata (@var{y}, @var{u}, @var{tsam}, @dots{}) @deftypefnx{Function File} {@var{dat} =} iddata (@var{y}, @var{u}, @var{[]}, @dots{}) Create identification dataset of output and input signals. @strong{Inputs} @table @var @item y Real matrix containing the output signal in time-domain. For a system with @var{p} outputs and @var{n} samples, @var{y} is a n-by-p matrix. For data from multiple experiments, @var{y} becomes a e-by-1 or 1-by-e cell vector of n(i)-by-p matrices, where @var{e} denotes the number of experiments and n(i) the individual number of samples for each experiment. @item u Real matrix containing the input signal in time-domain. For a system with @var{m} inputs and @var{n} samples, @var{u} is a n-by-m matrix. For data from multiple experiments, @var{u} becomes a e-by-1 or 1-by-e cell vector of n(i)-by-m matrices, where @var{e} denotes the number of experiments and n(i) the individual number of samples for each experiment. If @var{u} is not specified or an empty element @code{[]} is passed, @var{dat} becomes a time series dataset. @item tsam Sampling time. If not specified, default value -1 (unspecified) is taken. For multi-experiment data, @var{tsam} becomes a e-by-1 or 1-by-e cell vector containing individual sampling times for each experiment. If a scalar @var{tsam} is provided, then all experiments have the same sampling time. @item @dots{} Optional pairs of properties and values. @end table @strong{Outputs} @table @var @item dat iddata identification dataset. @end table @strong{Option Keys and Values} @table @var @item 'expname' The name of the experiments in @var{dat}. Cell vector of length e containing strings. Default names are @code{@{'exp1', 'exp2', ...@}} @item 'y' Output signals. See 'Inputs' for details. @item 'outname' The name of the output channels in @var{dat}. Cell vector of length p containing strings. Default names are @code{@{'y1', 'y2', ...@}} @item 'outunit' The units of the output channels in @var{dat}. Cell vector of length p containing strings. @item 'u' Input signals. See 'Inputs' for details. @item 'inname' The name of the input channels in @var{dat}. Cell vector of length m containing strings. Default names are @code{@{'u1', 'u2', ...@}} @item 'inunit' The units of the input channels in @var{dat}. Cell vector of length m containing strings. @item 'tsam' Sampling time. See 'Inputs' for details. @item 'timeunit' The units of the sampling times in @var{dat}. Cell vector of length e containing strings. @item 'name' String containing the name of the dataset. @item 'notes' String or cell of string containing comments. @item 'userdata' Any data type. @end table @end deftypefn @section @@iddata/cat @findex cat @deftypefn {Function File} {@var{dat} =} cat (@var{dim}, @var{dat1}, @var{dat2}, @dots{}) Concatenate iddata sets along dimension @var{dim}. @strong{Inputs} @table @var @item dim Dimension along which the concatenation takes place. @table @var @item 1 Concatenate samples. The samples are concatenated in the following way: @code{dat.y@{e@} = [dat1.y@{e@}; dat2.y@{e@}; @dots{}]} @code{dat.u@{e@} = [dat1.u@{e@}; dat2.u@{e@}; @dots{}]} where @var{e} denotes the experiment. The number of experiments, outputs and inputs must be equal for all datasets. Equivalent to @command{vertcat}. @item 2 Concatenate inputs and outputs. The outputs and inputs are concatenated in the following way: @code{dat.y@{e@} = [dat1.y@{e@}, dat2.y@{e@}, @dots{}]} @code{dat.u@{e@} = [dat1.u@{e@}, dat2.u@{e@}, @dots{}]} where @var{e} denotes the experiment. The number of experiments and samples must be equal for all datasets. Equivalent to @command{horzcat}. @item 3 Concatenate experiments. The experiments are concatenated in the following way: @code{dat.y = [dat1.y; dat2.y; @dots{}]} @code{dat.u = [dat1.u; dat2.u; @dots{}]} The number of outputs and inputs must be equal for all datasets. Equivalent to @command{merge}. @end table @item dat1, dat2, @dots{} iddata sets to be concatenated. @end table @strong{Outputs} @table @var @item dat Concatenated iddata set. @end table @seealso{horzcat, merge, vertcat} @end deftypefn @section @@iddata/detrend @findex detrend @deftypefn {Function File} {@var{dat} =} detrend (@var{dat}) @deftypefnx {Function File} {@var{dat} =} detrend (@var{dat}, @var{ord}) Detrend outputs and inputs of dataset @var{dat} by removing the best fit of a polynomial of order @var{ord}. If @var{ord} is not specified, default value 0 is taken. This corresponds to removing a constant. @end deftypefn @section @@iddata/diff @findex diff @deftypefn {Function File} {@var{dat} =} diff (@var{dat}) @deftypefnx {Function File} {@var{dat} =} diff (@var{dat}, @var{k}) Return @var{k}-th difference of outputs and inputs of dataset @var{dat}. If @var{k} is not specified, default value 1 is taken. @end deftypefn @section @@iddata/fft @findex fft @deftypefn {Function File} {@var{dat} =} fft (@var{dat}) @deftypefnx {Function File} {@var{dat} =} fft (@var{dat}, @var{n}) Transform iddata objects from time to frequency domain using a Fast Fourier Transform (FFT) algorithm. @strong{Inputs} @table @var @item dat iddata set containing signals in time-domain. @item n Length of the FFT transformations. If @var{n} does not match the signal length, the signals in @var{dat} are shortened or padded with zeros. @var{n} is a vector with as many elements as there are experiments in @var{dat} or a scalar with a common length for all experiments. If not specified, the signal lengths are taken as default values. @end table @strong{Outputs} @table @var @item dat iddata identification dataset in frequency-domain. In order to preserve signal power and noise level, the FFTs are normalized by dividing each transform by the square root of the signal length. The frequency values are distributed equally from 0 to the Nyquist frequency. The Nyquist frequency is only included for even signal lengths. @end table @end deftypefn @section @@iddata/filter @findex filter @deftypefn {Function File} {@var{dat} =} filter (@var{dat}, @var{sys}) @deftypefnx {Function File} {@var{dat} =} filter (@var{dat}, @var{b}, @var{a}) Filter output and input signals of dataset @var{dat}. The filter is specified either by @acronym{LTI} system @var{sys} or by transfer function polynomials @var{b} and @var{a} as described in the help text of the built-in filter command. Type @code{help filter} for more information. @strong{Inputs} @table @var @item dat iddata identification dataset containing signals in time-domain. @item sys @acronym{LTI} object containing the discrete-time filter. @item b Numerator polynomial of the discrete-time filter. Must be a row vector containing the coefficients of the polynomial in ascending powers of z^-1. @item a Denominator polynomial of the discrete-time filter. Must be a row vector containing the coefficients of the polynomial in ascending powers of z^-1. @end table @strong{Outputs} @table @var @item dat iddata identification dataset with filtered output and input signals. @end table @end deftypefn @section @@iddata/get @findex get @deftypefn {Function File} {} get (@var{dat}) @deftypefnx {Function File} {@var{value} =} get (@var{dat}, @var{"property"}) @deftypefnx {Function File} {[@var{val1}, @var{val2}, @dots{}] =} get (@var{dat}, @var{"prop1"}, @var{"prop2"}, @dots{}) Access property values of iddata objects. Type @command{get(dat)} to display a list of available properties. @end deftypefn @section @@iddata/ifft @findex ifft @deftypefn {Function File} {@var{dat} =} ifft (@var{dat}) Transform iddata objects from frequency to time domain. @strong{Inputs} @table @var @item dat iddata set containing signals in frequency domain. The frequency values must be distributed equally from 0 to the Nyquist frequency. The Nyquist frequency is only included for even signal lengths. @end table @strong{Outputs} @table @var @item dat iddata identification dataset in time domain. In order to preserve signal power and noise level, the FFTs are normalized by multiplying each transform by the square root of the signal length. @end table @end deftypefn @section @@iddata/merge @findex merge @deftypefn {Function File} {@var{dat} =} merge (@var{dat1}, @var{dat2}, @dots{}) Concatenate experiments of iddata datasets. The experiments are concatenated in the following way: @code{dat.y = [dat1.y; dat2.y; @dots{}]} @code{dat.u = [dat1.u; dat2.u; @dots{}]} The number of outputs and inputs must be equal for all datasets. @end deftypefn @section @@iddata/nkshift @findex nkshift @deftypefn {Function File} {@var{dat} =} nkshift (@var{dat}, @var{nk}) @deftypefnx {Function File} {@var{dat} =} nkshift (@var{dat}, @var{nk}, @var{'append'}) Shift input channels of dataset @var{dat} according to integer @var{nk}. A positive value of @var{nk} means that the input channels are delayed @var{nk} samples. By default, both input and output signals are shortened by @var{nk} samples. If a third argument @var{'append'} is passed, the output signals are left untouched while @var{nk} zeros are appended to the (shortened) input signals such that the number of samples in @var{dat} remains constant. @end deftypefn @section @@iddata/plot @findex plot @deftypefn {Function File} {} plot (@var{dat}) @deftypefnx {Function File} {} plot (@var{dat}, @var{exp}) Plot signals of iddata identification datasets on the screen. The signals are plotted experiment-wise, either in time- or frequency-domain. For multi-experiment datasets, press any key to switch to the next experiment. If the plot of a single experiment should be saved by the @command{print} command, use @code{plot(dat,exp)}, where @var{exp} denotes the desired experiment. @end deftypefn @section @@iddata/resample @findex resample @deftypefn {Function File} {@var{dat} =} resample (@var{dat}, @var{p}, @var{q}) @deftypefnx {Function File} {@var{dat} =} resample (@var{dat}, @var{p}, @var{q}, @var{n}) @deftypefnx {Function File} {@var{dat} =} resample (@var{dat}, @var{p}, @var{q}, @var{h}) Change the sample rate of the output and input signals in dataset @var{dat} by a factor of @code{p/q}. This is performed using a polyphase algorithm. The anti-aliasing @acronym{FIR} filter can be specified as follows: Either by order @var{n} (scalar) with default value 0. The band edges are then chosen automatically. Or by impulse response @var{h} (vector). Requires the signal package to be installed. @strong{Algorithm}@* Uses commands @command{fir1} and @command{resample} from the signal package. @strong{References}@* [1] J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th ed., Prentice Hall, 2007. Chap. 6 [2] A. V. Oppenheim, R. W. Schafer and J. R. Buck, Discrete-time signal processing, Signal processing series, Prentice-Hall, 1999 @end deftypefn @section @@iddata/set @findex set @deftypefn {Function File} {} set (@var{dat}) @deftypefnx {Function File} {} set (@var{dat}, @var{"property"}, @var{value}, @dots{}) @deftypefnx {Function File} {@var{dat} =} set (@var{dat}, @var{"property"}, @var{value}, @dots{}) Set or modify properties of iddata objects. If no return argument @var{dat} is specified, the modified @acronym{LTI} object is stored in input argument @var{dat}. @command{set} can handle multiple properties in one call: @code{set (dat, 'prop1', val1, 'prop2', val2, 'prop3', val3)}. @code{set (dat)} prints a list of the object's property names. @end deftypefn @section @@iddata/size @findex size @deftypefn {Function File} {@var{nvec} =} size (@var{dat}) @deftypefnx {Function File} {@var{ndim} =} size (@var{dat}, @var{dim}) @deftypefnx {Function File} {[@var{n}, @var{p}, @var{m}, @var{e}] =} size (@var{dat}) Return dimensions of iddata set @var{dat}. @strong{Inputs} @table @var @item dat iddata set. @item dim If given a second argument, @command{size} will return the size of the corresponding dimension. @end table @strong{Outputs} @table @var @item nvec Row vector. The first element is the total number of samples (rows of dat.y and dat.u). The second element is the number of outputs (columns of dat.y) and the third element the number of inputs (columns of dat.u). The fourth element is the number of experiments. @item ndim Scalar value. The size of the dimension @var{dim}. @item n Row vector containing the number of samples of each experiment. @item p Number of outputs. @item m Number of inputs. @item e Number of experiments. @end table @end deftypefn @chapter System Identification @section arx @findex arx @deftypefn {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{n}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{n}, @var{opt}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{opt}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{'na'}, @var{na}, @var{'nb'}, @var{nb}) Estimate ARX model using QR factorization. @iftex @tex $$ A(q) \, y(t) = B(q) \, u(t) \, + \, e(t) $$ @end tex @end iftex @ifnottex @example A(q) y(t) = B(q) u(t) + e(t) @end example @end ifnottex @strong{Inputs} @table @var @item dat iddata identification dataset containing the measurements, i.e. time-domain signals. @item n The desired order of the resulting model @var{sys}. @item @dots{} Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item sys Discrete-time transfer function model. If the second output argument @var{x0} is returned, @var{sys} becomes a state-space model. @item x0 Initial state vector. If @var{dat} is a multi-experiment dataset, @var{x0} becomes a cell vector containing an initial state vector for each experiment. @end table @strong{Option Keys and Values} @table @var @item 'na' Order of the polynomial A(q) and number of poles. @item 'nb' Order of the polynomial B(q)+1 and number of zeros+1. @var{nb} <= @var{na}. @item 'nk' Input-output delay specified as number of sampling instants. Scalar positive integer. This corresponds to a call to command @command{nkshift}, followed by padding the B polynomial with @var{nk} leading zeros. @end table @strong{Algorithm}@* Uses the formulae given in [1] on pages 318-319, 'Solving for the LS Estimate by QR Factorization'. For the initial conditions, SLICOT IB01CD is used by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @strong{References}@* [1] Ljung, L. (1999) @cite{System Identification: Theory for the User: Second Edition}. Prentice Hall, New Jersey, USA. @end deftypefn @section moen4 @findex moen4 @deftypefn {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @var{n}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @var{opt}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @var{n}, @var{opt}, @dots{}) Estimate state-space model using combined subspace method: @acronym{MOESP} algorithm for finding the matrices A and C, and @acronym{N4SID} algorithm for finding the matrices B and D. If no output arguments are given, the singular values are plotted on the screen in order to estimate the system order. @strong{Inputs} @table @var @item dat iddata set containing the measurements, i.e. time-domain signals. @item n The desired order of the resulting state-space system @var{sys}. If not specified, @var{n} is chosen automatically according to the singular values and tolerances. @item @dots{} Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item sys Discrete-time state-space model. @item x0 Initial state vector. If @var{dat} is a multi-experiment dataset, @var{x0} becomes a cell vector containing an initial state vector for each experiment. @item info Struct containing additional information. @table @var @item info.K Kalman gain matrix. @item info.Q State covariance matrix. @item info.Ry Output covariance matrix. @item info.S State-output cross-covariance matrix. @item info.L Noise variance matrix factor. LL'=Ry. @end table @end table @strong{Option Keys and Values} @table @var @item 'n' The desired order of the resulting state-space system @var{sys}. @var{s} > @var{n} > 0. @item 's' The number of block rows @var{s} in the input and output block Hankel matrices to be processed. @var{s} > 0. In the MOESP theory, @var{s} should be larger than @var{n}, the estimated dimension of state vector. @item 'alg', 'algorithm' Specifies the algorithm for computing the triangular factor R, as follows: @table @var @item 'C' Cholesky algorithm applied to the correlation matrix of the input-output data. Default method. @item 'F' Fast QR algorithm. @item 'Q' QR algorithm applied to the concatenated block Hankel matrices. @end table @item 'tol' Absolute tolerance used for determining an estimate of the system order. If @var{tol} >= 0, the estimate is indicated by the index of the last singular value greater than or equal to @var{tol}. (Singular values less than @var{tol} are considered as zero.) When @var{tol} = 0, an internally computed default value, @var{tol} = @var{s}*@var{eps}*SV(1), is used, where SV(1) is the maximal singular value, and @var{eps} is the relative machine precision. When @var{tol} < 0, the estimate is indicated by the index of the singular value that has the largest logarithmic gap to its successor. Default value is 0. @item 'rcond' The tolerance to be used for estimating the rank of matrices. If the user sets @var{rcond} > 0, the given value of @var{rcond} is used as a lower bound for the reciprocal condition number; an m-by-n matrix whose estimated condition number is less than 1/@var{rcond} is considered to be of full rank. If the user sets @var{rcond} <= 0, then an implicitly computed, default tolerance, defined by @var{rcond} = m*n*@var{eps}, is used instead, where @var{eps} is the relative machine precision. Default value is 0. @item 'confirm' Specifies whether or not the user's confirmation of the system order estimate is desired, as follows: @table @var @item true User's confirmation. @item false No confirmation. Default value. @end table @item 'noiseinput' The desired type of noise input channels. @table @var @item 'n' No error inputs. Default value. @iftex @tex $$ x_{k+1} = A x_k + B u_k $$ $$ y_k = C x_k + D u_k $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] y[k] = C x[k] + D u[k] @end example @end ifnottex @item 'e' Return @var{sys} as a (p-by-m+p) state-space model with both measured input channels u and noise channels e with covariance matrix @var{Ry}. @iftex @tex $$ x_{k+1} = A x_k + B u_k + K e_k $$ $$ y_k = C x_k + D u_k + e_k $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] + K e[k] y[k] = C x[k] + D u[k] + e[k] @end example @end ifnottex @item 'v' Return @var{sys} as a (p-by-m+p) state-space model with both measured input channels u and white noise channels v with identity covariance matrix. @iftex @tex $$ x_{k+1} = A x_k + B u_k + K L v_k $$ $$ y_k = C x_k + D u_k + L v_k $$ $$ e = L v, \ L L^T = R_y $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] + K L v[k] y[k] = C x[k] + D u[k] + L v[k] e = L v, L L' = Ry @end example @end ifnottex @item 'k' Return @var{sys} as a Kalman predictor for simulation. @iftex @tex $$ \widehat{x}_{k+1} = A \widehat{x}_k + B u_k + K (y_k - \widehat{y}_k) $$ $$ \widehat{y}_k = C \widehat{x}_k + D u_k $$ @end tex @end iftex @ifnottex @example ^ ^ ^ x[k+1] = A x[k] + B u[k] + K(y[k] - y[k]) ^ ^ y[k] = C x[k] + D u[k] @end example @end ifnottex @iftex @tex $$ \widehat{x}_{k+1} = (A-KC) \widehat{x}_k + (B-KD) u_k + K y_k $$ $$ \widehat{y}_k = C \widehat{x}_k + D u_k + 0 y_k $$ @end tex @end iftex @ifnottex @example ^ ^ x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k] ^ ^ y[k] = C x[k] + D u[k] + 0 y[k] @end example @end ifnottex @end table @end table @strong{Algorithm}@* Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section moesp @findex moesp @deftypefn {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @var{n}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @var{opt}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @var{n}, @var{opt}, @dots{}) Estimate state-space model using @acronym{MOESP} algorithm. @acronym{MOESP}: Multivariable Output Error State sPace. If no output arguments are given, the singular values are plotted on the screen in order to estimate the system order. @strong{Inputs} @table @var @item dat iddata set containing the measurements, i.e. time-domain signals. @item n The desired order of the resulting state-space system @var{sys}. If not specified, @var{n} is chosen automatically according to the singular values and tolerances. @item @dots{} Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item sys Discrete-time state-space model. @item x0 Initial state vector. If @var{dat} is a multi-experiment dataset, @var{x0} becomes a cell vector containing an initial state vector for each experiment. @item info Struct containing additional information. @table @var @item info.K Kalman gain matrix. @item info.Q State covariance matrix. @item info.Ry Output covariance matrix. @item info.S State-output cross-covariance matrix. @item info.L Noise variance matrix factor. LL'=Ry. @end table @end table @strong{Option Keys and Values} @table @var @item 'n' The desired order of the resulting state-space system @var{sys}. @var{s} > @var{n} > 0. @item 's' The number of block rows @var{s} in the input and output block Hankel matrices to be processed. @var{s} > 0. In the MOESP theory, @var{s} should be larger than @var{n}, the estimated dimension of state vector. @item 'alg', 'algorithm' Specifies the algorithm for computing the triangular factor R, as follows: @table @var @item 'C' Cholesky algorithm applied to the correlation matrix of the input-output data. Default method. @item 'F' Fast QR algorithm. @item 'Q' QR algorithm applied to the concatenated block Hankel matrices. @end table @item 'tol' Absolute tolerance used for determining an estimate of the system order. If @var{tol} >= 0, the estimate is indicated by the index of the last singular value greater than or equal to @var{tol}. (Singular values less than @var{tol} are considered as zero.) When @var{tol} = 0, an internally computed default value, @var{tol} = @var{s}*@var{eps}*SV(1), is used, where SV(1) is the maximal singular value, and @var{eps} is the relative machine precision. When @var{tol} < 0, the estimate is indicated by the index of the singular value that has the largest logarithmic gap to its successor. Default value is 0. @item 'rcond' The tolerance to be used for estimating the rank of matrices. If the user sets @var{rcond} > 0, the given value of @var{rcond} is used as a lower bound for the reciprocal condition number; an m-by-n matrix whose estimated condition number is less than 1/@var{rcond} is considered to be of full rank. If the user sets @var{rcond} <= 0, then an implicitly computed, default tolerance, defined by @var{rcond} = m*n*@var{eps}, is used instead, where @var{eps} is the relative machine precision. Default value is 0. @item 'confirm' Specifies whether or not the user's confirmation of the system order estimate is desired, as follows: @table @var @item true User's confirmation. @item false No confirmation. Default value. @end table @item 'noiseinput' The desired type of noise input channels. @table @var @item 'n' No error inputs. Default value. @iftex @tex $$ x_{k+1} = A x_k + B u_k $$ $$ y_k = C x_k + D u_k $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] y[k] = C x[k] + D u[k] @end example @end ifnottex @item 'e' Return @var{sys} as a (p-by-m+p) state-space model with both measured input channels u and noise channels e with covariance matrix @var{Ry}. @iftex @tex $$ x_{k+1} = A x_k + B u_k + K e_k $$ $$ y_k = C x_k + D u_k + e_k $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] + K e[k] y[k] = C x[k] + D u[k] + e[k] @end example @end ifnottex @item 'v' Return @var{sys} as a (p-by-m+p) state-space model with both measured input channels u and white noise channels v with identity covariance matrix. @iftex @tex $$ x_{k+1} = A x_k + B u_k + K L v_k $$ $$ y_k = C x_k + D u_k + L v_k $$ $$ e = L v, \ L L^T = R_y $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] + K L v[k] y[k] = C x[k] + D u[k] + L v[k] e = L v, L L' = Ry @end example @end ifnottex @item 'k' Return @var{sys} as a Kalman predictor for simulation. @iftex @tex $$ \widehat{x}_{k+1} = A \widehat{x}_k + B u_k + K (y_k - \widehat{y}_k) $$ $$ \widehat{y}_k = C \widehat{x}_k + D u_k $$ @end tex @end iftex @ifnottex @example ^ ^ ^ x[k+1] = A x[k] + B u[k] + K(y[k] - y[k]) ^ ^ y[k] = C x[k] + D u[k] @end example @end ifnottex @iftex @tex $$ \widehat{x}_{k+1} = (A-KC) \widehat{x}_k + (B-KD) u_k + K y_k $$ $$ \widehat{y}_k = C \widehat{x}_k + D u_k + 0 y_k $$ @end tex @end iftex @ifnottex @example ^ ^ x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k] ^ ^ y[k] = C x[k] + D u[k] + 0 y[k] @end example @end ifnottex @end table @end table @strong{Algorithm}@* Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @section n4sid @findex n4sid @deftypefn {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @var{n}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @var{opt}, @dots{}) @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @var{n}, @var{opt}, @dots{}) Estimate state-space model using @acronym{N4SID} algorithm. @acronym{N4SID}: Numerical algorithm for Subspace State Space System IDentification. If no output arguments are given, the singular values are plotted on the screen in order to estimate the system order. @strong{Inputs} @table @var @item dat iddata set containing the measurements, i.e. time-domain signals. @item n The desired order of the resulting state-space system @var{sys}. If not specified, @var{n} is chosen automatically according to the singular values and tolerances. @item @dots{} Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. @item opt Optional struct with keys as field names. Struct @var{opt} can be created directly or by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. @end table @strong{Outputs} @table @var @item sys Discrete-time state-space model. @item x0 Initial state vector. If @var{dat} is a multi-experiment dataset, @var{x0} becomes a cell vector containing an initial state vector for each experiment. @item info Struct containing additional information. @table @var @item info.K Kalman gain matrix. @item info.Q State covariance matrix. @item info.Ry Output covariance matrix. @item info.S State-output cross-covariance matrix. @item info.L Noise variance matrix factor. LL'=Ry. @end table @end table @strong{Option Keys and Values} @table @var @item 'n' The desired order of the resulting state-space system @var{sys}. @var{s} > @var{n} > 0. @item 's' The number of block rows @var{s} in the input and output block Hankel matrices to be processed. @var{s} > 0. In the MOESP theory, @var{s} should be larger than @var{n}, the estimated dimension of state vector. @item 'alg', 'algorithm' Specifies the algorithm for computing the triangular factor R, as follows: @table @var @item 'C' Cholesky algorithm applied to the correlation matrix of the input-output data. Default method. @item 'F' Fast QR algorithm. @item 'Q' QR algorithm applied to the concatenated block Hankel matrices. @end table @item 'tol' Absolute tolerance used for determining an estimate of the system order. If @var{tol} >= 0, the estimate is indicated by the index of the last singular value greater than or equal to @var{tol}. (Singular values less than @var{tol} are considered as zero.) When @var{tol} = 0, an internally computed default value, @var{tol} = @var{s}*@var{eps}*SV(1), is used, where SV(1) is the maximal singular value, and @var{eps} is the relative machine precision. When @var{tol} < 0, the estimate is indicated by the index of the singular value that has the largest logarithmic gap to its successor. Default value is 0. @item 'rcond' The tolerance to be used for estimating the rank of matrices. If the user sets @var{rcond} > 0, the given value of @var{rcond} is used as a lower bound for the reciprocal condition number; an m-by-n matrix whose estimated condition number is less than 1/@var{rcond} is considered to be of full rank. If the user sets @var{rcond} <= 0, then an implicitly computed, default tolerance, defined by @var{rcond} = m*n*@var{eps}, is used instead, where @var{eps} is the relative machine precision. Default value is 0. @item 'confirm' Specifies whether or not the user's confirmation of the system order estimate is desired, as follows: @table @var @item true User's confirmation. @item false No confirmation. Default value. @end table @item 'noiseinput' The desired type of noise input channels. @table @var @item 'n' No error inputs. Default value. @iftex @tex $$ x_{k+1} = A x_k + B u_k $$ $$ y_k = C x_k + D u_k $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] y[k] = C x[k] + D u[k] @end example @end ifnottex @item 'e' Return @var{sys} as a (p-by-m+p) state-space model with both measured input channels u and noise channels e with covariance matrix @var{Ry}. @iftex @tex $$ x_{k+1} = A x_k + B u_k + K e_k $$ $$ y_k = C x_k + D u_k + e_k $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] + K e[k] y[k] = C x[k] + D u[k] + e[k] @end example @end ifnottex @item 'v' Return @var{sys} as a (p-by-m+p) state-space model with both measured input channels u and white noise channels v with identity covariance matrix. @iftex @tex $$ x_{k+1} = A x_k + B u_k + K L v_k $$ $$ y_k = C x_k + D u_k + L v_k $$ $$ e = L v, \ L L^T = R_y $$ @end tex @end iftex @ifnottex @example x[k+1] = A x[k] + B u[k] + K L v[k] y[k] = C x[k] + D u[k] + L v[k] e = L v, L L' = Ry @end example @end ifnottex @item 'k' Return @var{sys} as a Kalman predictor for simulation. @iftex @tex $$ \widehat{x}_{k+1} = A \widehat{x}_k + B u_k + K (y_k - \widehat{y}_k) $$ $$ \widehat{y}_k = C \widehat{x}_k + D u_k $$ @end tex @end iftex @ifnottex @example ^ ^ ^ x[k+1] = A x[k] + B u[k] + K(y[k] - y[k]) ^ ^ y[k] = C x[k] + D u[k] @end example @end ifnottex @iftex @tex $$ \widehat{x}_{k+1} = (A-KC) \widehat{x}_k + (B-KD) u_k + K y_k $$ $$ \widehat{y}_k = C \widehat{x}_k + D u_k + 0 y_k $$ @end tex @end iftex @ifnottex @example ^ ^ x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k] ^ ^ y[k] = C x[k] + D u[k] + 0 y[k] @end example @end ifnottex @end table @end table @strong{Algorithm}@* Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of @uref{http://www.slicot.org, NICONET e.V.} @end deftypefn @chapter Overloaded LTI Operators @section @@lti/ctranspose @findex ctranspose Conjugate transpose or pertransposition of @acronym{LTI} objects. Used by Octave for "sys'". For a transfer-function matrix G, G' denotes the conjugate of G given by G.'(-s) for a continuous-time system or G.'(1/z) for a discrete-time system. The frequency response of the pertransposition of G is the Hermitian (conjugate) transpose of G(jw), i.e. freqresp (G', w) = freqresp (G, w)'. @strong{WARNING:} Do @strong{NOT} use this for dual problems, use the transpose "sys.'" (note the dot) instead. @section @@lti/end @findex end End indexing for @acronym{LTI} objects. Used by Octave for "sys(1:end, end-1)". @section @@lti/horzcat @findex horzcat Horizontal concatenation of @acronym{LTI} objects. If necessary, object conversion is done by sys_group. Used by Octave for "[sys1, sys2]". @section @@lti/inv @findex inv Inversion of @acronym{LTI} objects. @section @@lti/minus @findex minus Binary subtraction of @acronym{LTI} objects. If necessary, object conversion is done by sys_group. Used by Octave for "sys1 - sys2". @section @@lti/mldivide @findex mldivide Matrix left division of @acronym{LTI} objects. If necessary, object conversion is done by sys_group in mtimes. Used by Octave for "sys1 \ sys2". @section @@lti/mpower @findex mpower Matrix power of @acronym{LTI} objects. The exponent must be an integer. Used by Octave for "sys^int". @section @@lti/mrdivide @findex mrdivide Matrix right division of @acronym{LTI} objects. If necessary, object conversion is done by sys_group in mtimes. Used by Octave for "sys1 / sys2". @section @@lti/mtimes @findex mtimes Matrix multiplication of @acronym{LTI} objects. If necessary, object conversion is done by sys_group. Used by Octave for "sys1 * sys2". @section @@lti/plus @findex plus Binary addition of @acronym{LTI} objects. If necessary, object conversion is done by sys_group. Used by Octave for "sys1 + sys2". Operation is also known as "parallel connection". @section @@lti/subsasgn @findex subsasgn Subscripted assignment for @acronym{LTI} objects. Used by Octave for "sys.property = value". @section @@lti/subsref @findex subsref Subscripted reference for @acronym{LTI} objects. Used by Octave for "sys = sys(2:4, :)" or "val = sys.prop". @section @@lti/transpose @findex transpose Transpose of @acronym{LTI} objects. Used by Octave for "sys.'". Useful for dual problems, i.e. controllability and observability or designing estimator gains with @command{lqr} and @command{place}. @section @@lti/uminus @findex uminus Unary minus of @acronym{LTI} object. Used by Octave for "-sys". @section @@lti/uplus @findex uplus Unary plus of @acronym{LTI} object. Used by Octave for "+sys". @section @@lti/vertcat @findex vertcat Vertical concatenation of @acronym{LTI} objects. If necessary, object conversion is done by sys_group. Used by Octave for "[sys1; sys2]". @chapter Overloaded IDDATA Operators @section @@iddata/end @findex end End indexing for @acronym{IDDATA} objects. Used by Octave for "dat(1:end)". @section @@iddata/horzcat @findex horzcat @deftypefn {Function File} {@var{dat} =} horzcat (@var{dat1}, @var{dat2}, @dots{}) Horizontal concatenation of iddata datasets. The outputs and inputs are concatenated in the following way: @code{dat.y@{e@} = [dat1.y@{e@}, dat2.y@{e@}, @dots{}]} @code{dat.u@{e@} = [dat1.u@{e@}, dat2.u@{e@}, @dots{}]} where @var{e} denotes the experiment. The number of experiments and samples must be equal for all datasets. @end deftypefn @section @@iddata/subsasgn @findex subsasgn Subscripted assignment for iddata objects. Used by Octave for "dat.property = value". @section @@iddata/subsref @findex subsref Subscripted reference for iddata objects. Used by Octave for "dat = dat(2:4, :)" or "val = dat.prop". @section @@iddata/vertcat @findex vertcat @deftypefn {Function File} {@var{dat} =} vertcat (@var{dat1}, @var{dat2}, @dots{}) Vertical concatenation of iddata datasets. The samples are concatenated in the following way: @code{dat.y@{e@} = [dat1.y@{e@}; dat2.y@{e@}; @dots{}]} @code{dat.u@{e@} = [dat1.u@{e@}; dat2.u@{e@}; @dots{}]} where @var{e} denotes the experiment. The number of experiments, outputs and inputs must be equal for all datasets. @end deftypefn @chapter Miscellaneous @section db2mag @findex db2mag @deftypefn {Function File} {@var{mag} =} db2mag (@var{db}) Convert Decibels (dB) to Magnitude. @strong{Inputs} @table @var @item db Decibel (dB) value(s). Both real-valued scalars and matrices are accepted. @end table @strong{Outputs} @table @var @item mag Magnitude value(s). @end table @seealso{mag2db} @end deftypefn @section mag2db @findex mag2db @deftypefn {Function File} {@var{db} =} mag2db (@var{mag}) Convert Magnitude to Decibels (dB). @strong{Inputs} @table @var @item mag Magnitude value(s). Both real-valued scalars and matrices are accepted. @end table @strong{Outputs} @table @var @item db Decibel (dB) value(s). @end table @seealso{db2mag} @end deftypefn @section options @findex options @deftypefn{Function File} {@var{opt} =} options (@var{'key1'}, @var{value1}, @var{'key2'}, @var{value2}, @dots{}) Create options struct @var{opt} from a number of key and value pairs. For use with order reduction and system identification functions. Option structs are a way to avoid typing the same key and value pairs over and over again. @strong{Inputs} @table @var @item key, property The name of the property. @item value The value of the property. @end table @strong{Outputs} @table @var @item opt Struct with fields for each key. @end table @strong{Example} @example @group octave:1> opt = options ("method", "spa", "tol", 1e-6) opt = scalar structure containing the fields: method = spa tol = 1.0000e-06 @end group @end example @example @group octave:2> save filename opt octave:3> # save the struct 'opt' to file 'filename' for later use octave:4> load filename octave:5> # load struct 'opt' from file 'filename' @end group @end example @end deftypefn @section strseq @findex strseq @deftypefn {Function File} {@var{strvec} =} strseq (@var{str}, @var{idx}) Return a cell vector of indexed strings by appending the indices @var{idx} to the string @var{str}. @example strseq ("x", 1:3) = @{"x1"; "x2"; "x3"@} strseq ("u", [1, 2, 5]) = @{"u1"; "u2"; "u5"@} @end example @end deftypefn @section test_control @findex test_control @deftypefn {Script File} {} test_control Execute all available tests at once. The Octave control package is based on the @uref{http://www.slicot.org, @acronym{SLICOT}} library. @acronym{SLICOT} needs @acronym{BLAS} and @acronym{LAPACK} libraries which are also prerequisites for Octave itself. In case of failing tests, it is highly recommended to use @uref{http://www.netlib.org/blas/, Netlib's reference @acronym{BLAS}} and @uref{http://www.netlib.org/lapack/, @acronym{LAPACK}} for building Octave. Using @acronym{ATLAS} may lead to sign changes in some entries of the state-space matrices. In general, these sign changes are not 'wrong' and can be regarded as the result of state transformations. Such state transformations (but not input/output transformations) have no influence on the input-output behaviour of the system. For better numerics, the control package uses such transformations by default when calculating the frequency responses and a few other things. However, arguments like the Hankel singular Values (@acronym{HSV}) must not change. Differing @acronym{HSV}s and failing algorithms are known for using Framework Accelerate from Mac OS X 10.7. @end deftypefn @section thiran @findex thiran @deftypefn {Function File} {@var{sys} =} thiran (@var{tau}, @var{tsam}) Approximation of continuous-time delay using a discrete-time allpass Thiran filter. Thiran filters can approximate continuous-time delays that are non-integer multiples of the sampling time (fractional delays). This approximation gives a better matching of the phase shift between the continuous- and the discrete-time system. If there is no fractional part in the delay, then the standard discrete-time delay representation is used. @strong{Inputs} @table @var @item tau A continuous-time delay, given in time units (seconds). @item tsam The sampling time of the resulting Thiran filter. @end table @strong{Outputs} @table @var @item sys Transfer function model of the resulting filter. The order of the filter is determined automatically. @end table @strong{Example} @example @group octave:1> sys = thiran (1.33, 0.5) Transfer function 'sys' from input 'u1' to output ... 0.003859 z^3 - 0.03947 z^2 + 0.2787 z + 1 y1: ----------------------------------------- z^3 + 0.2787 z^2 - 0.03947 z + 0.003859 Sampling time: 0.5 s Discrete-time model. @end group @end example @example @group octave:2> sys = thiran (1, 0.5) Transfer function 'sys' from input 'u1' to output ... 1 y1: --- z^2 Sampling time: 0.5 s Discrete-time model. @end group @end example @seealso{absorbdelay, pade} @end deftypefn @section BMWengine @findex BMWengine @deftypefn{Function File} {@var{sys} =} BMWengine () @deftypefnx{Function File} {@var{sys} =} BMWengine (@var{"scaled"}) @deftypefnx{Function File} {@var{sys} =} BMWengine (@var{"unscaled"}) Model of the BMW 4-cylinder engine at ETH Zurich's control laboratory. @example @group OPERATING POINT Drosselklappenstellung alpha_DK = 10.3 Grad Saugrohrdruck p_s = 0.48 bar Motordrehzahl n = 860 U/min Lambda-Messwert lambda = 1.000 Relativer Wandfilminhalt nu = 1 @end group @end example @example @group INPUTS U_1 Sollsignal Drosselklappenstellung [Grad] U_2 Relative Einspritzmenge [-] U_3 Zuendzeitpunkt [Grad KW] M_L Lastdrehmoment [Nm] @end group @end example @example @group STATES X_1 Drosselklappenstellung [Grad] X_2 Saugrohrdruck [bar] X_3 Motordrehzahl [U/min] X_4 Messwert Lamba-Sonde [-] X_5 Relativer Wandfilminhalt [-] @end group @end example @example @group OUTPUTS Y_1 Motordrehzahl [U/min] Y_2 Messwert Lambda-Sonde [-] @end group @end example @example @group SCALING U_1N, X_1N 1 Grad U_2N, X_4N, X_5N, Y_2N 0.05 U_3N 1.6 Grad KW X_2N 0.05 bar X_3N, Y_1N 200 U/min @end group @end example @end deftypefn @section Boeing707 @findex Boeing707 @deftypefn {Function File} {@var{sys} =} Boeing707 () Creates a linearized state-space model of a Boeing 707-321 aircraft at @var{v}=80 m/s @iftex @tex ($M = 0.26$, $G_{a0} = -3^{\circ}$, ${\alpha}_0 = 4^{\circ}$, ${\kappa}= 50^{\circ}$). @end tex @end iftex @ifnottex (@var{M} = 0.26, @var{Ga0} = -3 deg, @var{alpha0} = 4 deg, @var{kappa} = 50 deg). @end ifnottex System inputs: (1) thrust and (2) elevator angle. System outputs: (1) airspeed and (2) pitch angle. @strong{Reference}: R. Brockhaus: @cite{Flugregelung} (Flight Control), Springer, 1994. @end deftypefn @section WestlandLynx @findex WestlandLynx @deftypefn{Function File} {@var{sys} =} WestlandLynx () Model of the Westland Lynx Helicopter about hover. @example @group INPUTS main rotor collective longitudinal cyclic lateral cyclic tail rotor collective @end group @end example @example @group STATES pitch attitude theta [rad] roll attitude phi [rad] roll rate (body-axis) p [rad/s] pitch rate (body-axis) q [rad/s] yaw rate xi [rad/s] forward velocity v_x [ft/s] lateral velocity v_y [ft/s] vertical velocity v_z [ft/s] @end group @end example @example @group OUTPUTS heave velocity H_dot [ft/s] pitch attitude theta [rad] roll attitude phi [rad] heading rate psi_dot [rad/s] roll rate p [rad/s] pitch rate q [rad/s] @end group @end example @strong{References}@* [1] Skogestad, S. and Postlethwaite I. (2005) @cite{Multivariable Feedback Control: Analysis and Design: Second Edition}. Wiley. @url{http://www.nt.ntnu.no/users/skoge/book/2nd_edition/matlab_m/matfiles.html} @end deftypefn control/doc/gpl.texi0000644000076500000240000010433012273411257013767 0ustar lukasstaff@node Copying @appendix GNU General Public License @cindex warranty @cindex copyright @center Version 3, 29 June 2007 @display Copyright @copyright{} 2007 Free Software Foundation, Inc. @url{http://fsf.org/} Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. @end display @heading Preamble The GNU General Public License is a free, copyleft license for software and other kinds of works. The licenses for most software and other practical works are designed to take away your freedom to share and change the works. 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For more information on this, and how to apply and follow the GNU GPL, see @url{http://www.gnu.org/licenses/}. The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read @url{http://www.gnu.org/philosophy/why-not-lgpl.html}. control/doc/octave_forge_specific_api.txt0000644000076500000240000000147612273411256020222 0ustar lukasstaffisstable (sys, tol) # feedback (sys) # equivalent to feedback (sys, eye (size (sys))) feedback (sys, "+") # equivalent to feedback (sys, eye (size (sys)), +1) H = sys(w) # equivalent to H = freqresp (sys, w) where w is a frequency vector place (sys, p, alpha) # If parameter alpha is specified, poles with real parts place (a, b, p, alpha) # (continuous time) or moduli (discrete time) below alpha # are left untouched. sminreal (sys, tol) # Optional tolerance for controllability and observability. # Entries of the state-space matrices whose moduli are less or # equal to TOL are assumed to be zero. Default value is 0. frd (sys) # calculate interesting frequency range by the zeros and poles of SYS.control/doc/references.txt0000644000076500000240000000207612273411256015177 0ustar lukasstaffREFERENCES ********** German ****** Geering, H.P. Regelungstechnik Springer 2004 Meyer, M. Signalverarbeitung Vieweg & Teubner 2009 English ******* Anderson, B.D.O. and Moore, J.B. Optimal Control Linear Quadratic Methods Dover Publications 1990 Astrm, K. and Hgglund, T. PID Controllers Theory, Design and Tuning Second Edition Instrument Society of America 1995 Doyle, J.C., Francis, B.A. and Tannenbaum, A.R. Feedback Control Theory Dover Publications 1992 Geering, H.P. Optimal Control with Engineering Applications Springer 2007 Gu, D.W., Petkov, P.Hr. and Konstantinov, M.M. Robust Control with MATLAB Springer 2005 Guzzella, L. Analysis and Design of SISO Control Systems VDF Hochschulverlag ETH Zurich 2007 Kailath, T. Linear Systems Prentice Hall 1980 Leigh, J.R. Applied Digital Control Theory, Design and Implementation Second Edition Dover Publications 1992 Ljung, L. System Identification Theory for the User Second Edition Prentice Hall 1999 Skogestad, S. and Postlethwaite, I. Multivariable Feedback Control Analysis and Design Second Edition Wiley 2005 control/doc/slicot_readme.txt0000644000076500000240000000531612273411256015670 0ustar lukasstaffImportant ********* If you encounter any bugs in the octave control package, please DONT bother the great people behind SLICOT. Most likely it's a problem of the control package, not the proven SLICOT library. Therefore, please send any reports to the octave mailing list and the maintainer of the control package. See the file packinfo/DESCRIPTION for the current maintainer of the control package and his mail address. Acknowledgments *************** LTI SYNCOPE - LTI SYstems aNd COntrol PackagE - is ever-increasingly based on SLICOT subroutines by courtesy of the Numerics In COntrol NETwork NICONET e.V. . Abstract ******** SLICOT is a Fortran 77 subroutine library for systems and control, distributed under GPL, developed by NICONET e.V., and available at . Commercial use requires specialized license models. SLICOT Readme ************* SLICOT - Subroutine Library In COntrol Theory - is a general purpose basic mathematical library for control theoretical computations. The library provides tools to perform essential system analysis and synthesis tasks. The main emphasis in SLICOT is on numerical reliability of implemented algorithms and the numerical robustness and efficiency of routines. Providing algorithmic flexibility and the use of rigorous implementation and documentation standards are other SLICOT features. The SLICOT Library is available as standard Fortran 77 code in double precision. Each user-callable subroutine for control computations is accompanied by an example program which illustrates the use of the subroutine and can act as a template for the user's own routines. The SLICOT Library is organized by chapters, sections and subsections. The following chapters are currently included: A : Analysis Routines B : Benchmark and Test Problems D : Data Analysis F : Filtering I : Identification M : Mathematical Routines N : Nonlinear Systems (not yet available, except for some auxiliary routines for Wiener systems) S : Synthesis Routines T : Transformation Routines U : Utility Routines The SLICOT Library is built on LAPACK (Linear Algebra PACKage) and BLAS (Basic Linear Algebra Subprograms) collections. Therefore, these packages should be available on the platform used. Basic References: 1. P. Benner, V. Mehrmann, V. Sima, S. Van Huffel, and A. Varga, "SLICOT - A Subroutine Library in Systems and Control Theory", Applied and Computational Control, Signals, and Circuits (Birkhauser), Vol. 1, Ch. 10, pp. 505-546, 1999. 2. S. Van Huffel, V. Sima, A. Varga, S. Hammarling, and F. Delebecque, "Development of High Performance Numerical Software for Control", IEEE Control Systems Magazine, Vol. 24, Nr. 1, Feb., pp. 60-76, 2004. control/INDEX0000644000076500000240000000415412273411256012341 0ustar lukasstaffcontrol >> Control Theory Examples MDSSystem optiPID Anderson Madievski Linear Time Invariant Models dss filt frd ss tf zpk Model Data Access @lti/dssdata @lti/filtdata @lti/frdata @lti/get @lti/set @lti/ssdata @lti/tfdata @lti/zpkdata Model Conversions @lti/c2d @lti/d2c @lti/d2d @lti/prescale @lti/xperm Model Interconnections @lti/append @lti/blkdiag @lti/connect @lti/feedback @lti/lft @lti/mconnect @lti/parallel @lti/series sumblk Model Characteristics ctrb ctrbf @lti/dcgain gram hsvd @lti/isct isctrb isdetectable @lti/isdt @lti/isminimumphase isobsv @lti/issiso isstabilizable @lti/isstable @lti/norm obsv obsvf @lti/pole pzmap @lti/size @lti/zero Model Simplification @lti/minreal @lti/sminreal Time Domain Analysis covar gensig impulse initial lsim ramp step Frequency Domain Analysis bode bodemag @lti/freqresp margin nichols nyquist sensitivity sigma Pole Placement place rlocus Optimal Control dlqe dlqr estim kalman lqe lqr Robust Control augw fitfrd h2syn hinfsyn mixsyn mktito ncfsyn Matrix Equation Solvers care dare dlyap dlyapchol lyap lyapchol Model Reduction bstmodred btamodred hnamodred spamodred Controller Reduction btaconred cfconred fwcfconred spaconred Experimental Data Handling iddata @iddata/cat @iddata/detrend @iddata/diff @iddata/fft @iddata/filter @iddata/get @iddata/ifft @iddata/merge @iddata/nkshift @iddata/plot @iddata/resample @iddata/set @iddata/size System Identification arx moen4 moesp n4sid Overloaded LTI Operators @lti/ctranspose @lti/end @lti/horzcat @lti/inv @lti/minus @lti/mldivide @lti/mpower @lti/mrdivide @lti/mtimes @lti/plus @lti/subsasgn @lti/subsref @lti/transpose @lti/uminus @lti/uplus @lti/vertcat Overloaded IDDATA Operators @iddata/end @iddata/horzcat @iddata/subsasgn @iddata/subsref @iddata/vertcat Miscellaneous db2mag mag2db options strseq test_control thiran BMWengine Boeing707 WestlandLynx control/inst/0000755000076500000240000000000012273411257012521 5ustar lukasstaffcontrol/inst/@frd/0000755000076500000240000000000012273411256013373 5ustar lukasstaffcontrol/inst/@frd/__c2d__.m0000644000076500000240000000211612273411256015015 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert the continuous FRD object into its discrete-time equivalent. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function sys = __c2d__ (sys, tsam, method = "zoh") error ("frd: c2d: conversion not possible"); ## NOTE: changing just the sampling time wouldn't make sense here endfunction control/inst/@frd/__ctranspose__.m0000644000076500000240000000210412273411256016523 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Conjugate transpose of FRD objects. ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function sys = __ctranspose__ (sys) [p, m, l] = size (sys.H); H = mat2cell (sys.H, p, m, ones (1, l))(:); H = cellfun (@ctranspose, H, "uniformoutput", false); sys.H = cat (3, H{:}); endfunction control/inst/@frd/__d2c__.m0000644000076500000240000000212012273411256015010 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert the discrete FRD object into its continuous-time equivalent. ## Author: Lukas Reichlin ## Created: September 2011 ## Version: 0.1 function sys = __d2c__ (sys, tsam, method = "zoh") error ("frd: d2c: conversion not possible"); ## NOTE: changing just the sampling time wouldn't make sense here endfunction control/inst/@frd/__freqresp__.m0000644000076500000240000000307212273411256016176 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Frequency response of FRD models :-) ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.3 function H = __freqresp__ (sys, w, cellflag = false) [H, w_sys, tsam] = frdata (sys, "array"); if (! isempty (w)) # freqresp (frdsys, w), sigma (frdsys, w), ... tol = sqrt (eps); w_idx = arrayfun (@(x) find (abs (w_sys - x) < tol), w, "uniformoutput", false); w_idx = vertcat (w_idx{:}); ## NOTE: There are problems when cellfun uses "uniformoutput", true ## and find returns an empty matrix, if (length (w_idx) != numel (w)) error ("frd: freqresp: some frequencies are not within tolerance %g", tol); endif H = H(:, :, w_idx); endif if (cellflag) [p, m, l] = size (H); H = mat2cell (H, p, m, ones (1, l))(:); endif endfunction control/inst/@frd/__get__.m0000644000076500000240000000225612273411256015131 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Access property values of FRD objects. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function val = __get__ (sys, prop) switch (prop) # {, } case {"h", "r", "resp", "response"} val = sys.H; case {"w", "f", "freq", "frequency"} val = sys.w; otherwise error ("frd: get: invalid property name '%s'", prop); endswitch endfunction control/inst/@frd/__minreal__.m0000644000076500000240000000176012273411256016000 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Minimal realization of FRD objects. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function sys = __minreal__ (sys, tol) warning ("frd: minreal: frequency responses are always minimal"); endfunction control/inst/@frd/__pole__.m0000644000076500000240000000170212273411256015304 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Poles of FRD object. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function pol = __pole__ (sys) error ("frd: pole: this is not possible"); endfunction control/inst/@frd/__property_names__.m0000644000076500000240000000311312273411256017412 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This program is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{sys}) ## @deftypefnx {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{sys}, @var{"specific"}) ## Return the list of properties as well as the assignable values for a frd object sys. ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [props, vals] = __property_names__ (sys, flg) ## cell vector of tf-specific properties props = {"H"; "w"}; ## cell vector of frd-specific assignable values vals = {"p-by-m-by-l array of complex frequency responses"; "l-by-1 vector of real frequencies (l = length (w))"}; if (nargin == 1) [ltiprops, ltivals] = __property_names__ (sys.lti); props = [props; ltiprops]; vals = [vals; ltivals]; endif endfunction control/inst/@frd/__set__.m0000644000076500000240000000252612273411256015145 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Set or modify properties of FRD objects. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function sys = __set__ (sys, prop, val) switch (prop) # {, } case {"h", "r", "resp", "response"} val = __adjust_frd_data__ (val, sys.w); __frd_dim__ (val, sys.w); sys.H = val; case {"w", "f", "freq", "frequency"} [~, val] = __adjust_frd_data__ (sys.H, val); __frd_dim__ (sys.H, val); sys.w = val; otherwise error ("frd: set: invalid property name '%s'", prop); endswitch endfunction control/inst/@frd/__sys2ss__.m0000644000076500000240000000207212273411256015614 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## FRD to SS conversion. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [retsys, retlti] = __sys2ss__ (sys) error ("frd: frd2ss: system identification not implemented yet"); retsys = dss (a, b, c, d, e); retlti = sys.lti; # preserve lti properties endfunction control/inst/@frd/__sys2tf__.m0000644000076500000240000000220712273411256015600 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## FRD to TF conversion. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [retsys, retlti] = __sys2tf__ (sys) error ("frd: frd2tf: system identification not implemented yet"); retsys = tf (num, den, get (sys, "tsam")); # tsam needed to set appropriate tfvar retlti = sys.lti; # preserve lti properties endfunction control/inst/@frd/__sys_connect__.m0000644000076500000240000000314612273411256016700 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{retsys} =} __sys_connect__ (@var{sys}, @var{M}) ## This function is part of the Model Abstraction Layer. No argument checking. ## For internal use only. ## @example ## @group ## Problem: Solve the system equations of ## Y(s) = G(s) E(s) ## E(s) = U(s) + M Y(s) ## in order to build ## Y(s) = H(s) U(s) ## Solution: ## Y(s) = G(s) [U(s) + M Y(s)] ## Y(s) = G(s) U(s) + G(s) M Y(s) ## Y(s) = [I - G(s) M]^-1 G(s) U(s) ## \_______ _______/ ## H(s) ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function sys = __sys_connect__ (sys, M) [p, m, l] = size (sys.H); I = eye (p); H = mat2cell (sys.H, p, m, ones (1, l))(:); H = cellfun (@(x) (I - x*M) \ x, H, "uniformoutput", false); sys.H = cat (3, H{:}); endfunction control/inst/@frd/__sys_data__.m0000644000076500000240000000171712273411256016162 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Used by frdata instead of multiple get calls. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [H, w] = __sys_data__ (sys) H = sys.H; w = sys.w; endfunction control/inst/@frd/__sys_group__.m0000644000076500000240000000440612273411256016403 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Block diagonal concatenation of two FRD models. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.2 function retsys = __sys_group__ (sys1, sys2) if (! isa (sys1, "frd")) sys1 = frd (sys1, sys2.w); endif if (! isa (sys2, "frd")) sys2 = frd (sys2, sys1.w); endif retsys = frd (); retsys.lti = __lti_group__ (sys1.lti, sys2.lti); lw1 = length (sys1.w); lw2 = length (sys2.w); [p1, m1, l1] = size (sys1.H); [p2, m2, l2] = size (sys2.H); ## TODO: tolerances for frequencies, i.e. don't check for equality ## find intersection of frequency vectors if (lw1 == lw2 && all (sys1.w == sys2.w)) # identical frequency vectors retsys.w = sys1.w; H1 = sys1.H; H2 = sys2.H; else # differing frequency vectors ## find common frequencies retsys.w = w = intersect (sys1.w, sys2.w); ## indices of common frequencies w1_idx = arrayfun (@(x) find (sys1.w == x), w); w2_idx = arrayfun (@(x) find (sys2.w == x), w); ## extract common responses H1 = sys1.H(:, :, w1_idx); H2 = sys2.H(:, :, w2_idx); endif ## block-diagonal concatenation lw = length (retsys.w); z12 = zeros (p1, m2); z21 = zeros (p2, m1); H1 = mat2cell (H1, p1, m1, ones (1, lw))(:); H2 = mat2cell (H2, p2, m2, ones (1, lw))(:); H = cellfun (@(x, y) [x, z12; z21, y], H1, H2, "uniformoutput", false); retsys.H = cat (3, H{:}); endfunction control/inst/@frd/__sys_inverse__.m0000644000076500000240000000207012273411256016715 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Inversion of FRD objects. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function sys = __sys_inverse__ (sys) [p, m, l] = size (sys.H); H = mat2cell (sys.H, p, m, ones (1, l))(:); H = cellfun (@inv, H, "uniformoutput", false); sys.H = cat (3, H{:}); endfunction control/inst/@frd/__sys_prune__.m0000644000076500000240000000223112273411256016372 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Submodel extraction and reordering for FRD objects. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function sys = __sys_prune__ (sys, out_idx, in_idx, w_idx = ":") [sys.lti, out_idx, in_idx] = __lti_prune__ (sys.lti, out_idx, in_idx); sys.H = sys.H(out_idx, in_idx, w_idx); endfunction control/inst/@frd/__transpose__.m0000644000076500000240000000207412273411256016366 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Transpose of FRD objects. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function sys = __transpose__ (sys) [p, m, l] = size (sys.H); H = mat2cell (sys.H, p, m, ones (1, l))(:); H = cellfun (@transpose, H, "uniformoutput", false); sys.H = cat (3, H{:}); endfunction control/inst/@frd/__zero__.m0000644000076500000240000000173512273411256015332 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Transmission zeros of FRD object. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [zer, gain, info] = __zero__ (sys) error ("frd: zero: this is not possible"); endfunction control/inst/@frd/display.m0000644000076500000240000000710612273411256015222 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Display routine for FRD objects. ## Author: Lukas Reichlin ## Created: February 2010 ## Version: 0.2 function display (sys) sysname = inputname (1); [inname, outname, tsam] = __lti_data__ (sys.lti); [inname, m] = __labels__ (inname, "u"); [outname, p] = __labels__ (outname, "y"); w = __freq2str__ (sys.w); disp (""); for k = 1 : m disp (["Frequency response '", sysname, "' from input '", inname{k}, "' to output ..."]); disp (""); __disp_resp__ (sys.H(:,k,:), w, outname); endfor display (sys.lti); # display sampling time if (tsam == -2) disp ("Static gain."); elseif (tsam == 0) disp ("Continuous-time frequency response."); else disp ("Discrete-time frequency response."); endif endfunction function __disp_resp__ (H, w, outname) p = rows (H); # number of outputs len = size (H, 3); # number of frequencies H = mat2cell (H, ones (1, p), 1, len)(:); H = cellfun (@__resp2str__, H, outname, "uniformoutput", false); tsize = terminal_size (); col_freq = columns (w); col_resp = cellfun (@columns, H); col_max = tsize(2) - col_freq; width = cumsum (col_resp); start = 0; stop = col_max; while (start < width(end)) idx = find (width > start & width <= stop); disp ([w, H{idx}]); disp (""); start = width(idx(end)); stop = start + col_max; endwhile ## FIXME: Handle case where tsize(2) is not enough ## to display frequencies and one output. endfunction function str = __freq2str__ (w, title = "w [rad/s]") len = rows (w); str = __vec2str__ (w); line = repmat ("-", 1, max (columns (str), columns (title))); str = strvcat (title, line, str); space = repmat (" ", len+2, 1); str = [space, str]; endfunction function str = __resp2str__ (H, outname) H = H(:); len = length (H); real_str = __vec2str__ (real (H)); im = imag (H); if (any (im)) imag_str = __vec2str__ (abs (im), "i"); sign_str = repmat (" + ", len, 1); neg = im < 0; sign_str(neg, 2) = "-"; str = [real_str, sign_str, imag_str]; else str = real_str; endif line = repmat ("-", 1, max (columns (str), columns (outname))); str = strvcat (outname, line, str); space = repmat (" ", len+2, 1); str = [space, str]; endfunction function str = __vec2str__ (vec, post) vec = vec(:); tmp = isfinite (vec); tmp = abs (vec(tmp & vec != 0)); if (isempty (tmp) || min (tmp) < 1e-3 || max (tmp) > 1e4) str = arrayfun (@(x) sprintf ("%.3e", x), vec, "uniformoutput", false); elseif (all (floor (tmp) == tmp)) str = arrayfun (@(x) sprintf ("%d", x), vec, "uniformoutput", false); else str = arrayfun (@(x) sprintf ("%.4f", x), vec, "uniformoutput", false); endif str = strjust (char (str), "right"); if (nargin > 1) str = [str, repmat(post, length (vec), 1)]; endif endfunction control/inst/@frd/frd.m0000644000076500000240000001410712273411256014327 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} frd (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} frd (@var{sys}, @var{w}) ## @deftypefnx {Function File} {@var{sys} =} frd (@var{H}, @var{w}, @dots{}) ## @deftypefnx {Function File} {@var{sys} =} frd (@var{H}, @var{w}, @var{tsam}, @dots{}) ## Create or convert to frequency response data. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model to be converted to frequency response data. ## If second argument @var{w} is omitted, the interesting ## frequency range is calculated by the zeros and poles of @var{sys}. ## @item H ## Frequency response array (p-by-m-by-lw). H(i,j,k) contains the ## response from input j to output i at frequency k. In the SISO case, ## a vector (lw-by-1) or (1-by-lw) is accepted as well. ## @item w ## Frequency vector (lw-by-1) in radian per second [rad/s]. ## Frequencies must be in ascending order. ## @item tsam ## Sampling time in seconds. If @var{tsam} is not specified, ## a continuous-time model is assumed. ## @item @dots{} ## Optional pairs of properties and values. ## Type @command{set (frd)} for more information. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Frequency response data object. ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'H' ## Frequency response array. See 'Inputs' for details. ## ## @item 'w' ## Frequency vector. See 'Inputs' for details. ## ## @item 'tsam' ## Sampling time. See 'Inputs' for details. ## ## @item 'inname' ## The name of the input channels in @var{sys}. ## Cell vector of length m containing strings. ## Default names are @code{@{'u1', 'u2', ...@}} ## ## @item 'outname' ## The name of the output channels in @var{sys}. ## Cell vector of length p containing strings. ## Default names are @code{@{'y1', 'y2', ...@}} ## ## @item 'ingroup' ## Struct with input group names as field names and ## vectors of input indices as field values. ## Default is an empty struct. ## ## @item 'outgroup' ## Struct with output group names as field names and ## vectors of output indices as field values. ## Default is an empty struct. ## ## @item 'name' ## String containing the name of the model. ## ## @item 'notes' ## String or cell of string containing comments. ## ## @item 'userdata' ## Any data type. ## @end table ## ## @seealso{dss, ss, tf} ## @end deftypefn ## Author: Lukas Reichlin ## Created: February 2010 ## Version: 0.1 function sys = frd (H = [], w = [], varargin) ## NOTE: * There's no such thing as a static gain ## because FRD objects are measurements, ## not models. ## * If something like sys1 = frd (5) existed, ## it would cause troubles in cases like ## sys2 = ss (...), sys = sys1 * sys2 ## because sys2 needs to be converted to FRD, ## but sys1 contains no valid frequencies. ## * However, things like frd (ss (5)) should ## be possible. ## model precedence: frd > ss > zpk > tf > double superiorto ("ss", "zpk", "tf", "double"); argc = 0; # initialize argument count switch (nargin) case 0 # empty object sys = frd () tsam = -2; # undefined sampling time case 1 if (isa (H, "frd")) # already in frd form sys = frd (frdsys) sys = H; return; # nothing more to do here elseif (isa (H, "lti")) # another lti object sys = frd (sys) [sys, alti] = __sys2frd__ (H); sys.lti = alti; # preserve lti properties return; # nothing more to do here else # sys = frd (H) *must* fail print_usage (); endif case 2 if (isa (H, "lti")) # another lti object sys = frd (sys, w) [sys, alti] = __sys2frd__ (H, w); sys.lti = alti; # preserve lti properties return; # nothing more to do here else # sys = frd (H, w) tsam = 0; # continuous-time endif otherwise # default case argc = numel (varargin); # number of additional arguments after H and w if (issample (varargin{1}, -10)) # sys = frd (H, w, tsam, "prop1", val1, ...) tsam = varargin{1}; # sampling time, could be 0 as well argc--; # tsam is not a property-value pair if (argc > 0) # if there are any properties and values ... varargin = varargin(2:end); # remove tsam from property-value list endif else # sys = frd (H, w, "prop1", val1, ...) tsam = 0; # continuous-time endif endswitch [H, w] = __adjust_frd_data__ (H, w); [p, m] = __frd_dim__ (H, w); # determine number of outputs and inputs frdata = struct ("H", H, "w", w); # struct for frd-specific data ltisys = lti (p, m, tsam); # parent class for general lti data sys = class (frdata, "frd", ltisys); # create frd object if (argc > 0) # if there are any properties and values, ... sys = set (sys, varargin{:}); # use the general set function endif endfunction control/inst/@iddata/0000755000076500000240000000000012273411257014047 5ustar lukasstaffcontrol/inst/@iddata/__property_names__.m0000644000076500000240000000400212273411256020063 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{dat}) ## Return the list of properties as well as the assignable values for an iddata set. ## @end deftypefn ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.1 function [props, vals] = __property_names__ (dat) [n, p, m, e] = size (dat); ## cell vector of iddata-specific properties props = {"y"; "outname"; "outunit"; "u"; "inname"; "inunit"; "tsam"; "timeunit"; "expname"; "name"; "notes"; "userdata"}; ## cell vector of lti-specific assignable values vals = {sprintf("(%dx1) cell vector of (nx%d) matrices", e, p); sprintf("(%dx1) cell vector of strings", p); sprintf("(%dx1) cell vector of strings", p); sprintf("(%dx1) cell vector of (nx%d) matrices", e, m); sprintf("(%dx1) cell vector of strings", m); sprintf("(%dx1) cell vector of strings", m); sprintf("(%dx1) cell vector of scalars", e); "string"; sprintf("(%dx1) cell vector of strings", e); "string"; "string or cell of strings"; "any data type"}; endfunctioncontrol/inst/@iddata/cat.m0000644000076500000240000001600512273411256014775 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} cat (@var{dim}, @var{dat1}, @var{dat2}, @dots{}) ## Concatenate iddata sets along dimension @var{dim}. ## ## @strong{Inputs} ## @table @var ## @item dim ## Dimension along which the concatenation takes place. ## @table @var ## @item 1 ## Concatenate samples. ## The samples are concatenated in the following way: ## @code{dat.y@{e@} = [dat1.y@{e@}; dat2.y@{e@}; @dots{}]} ## @code{dat.u@{e@} = [dat1.u@{e@}; dat2.u@{e@}; @dots{}]} ## where @var{e} denotes the experiment. ## The number of experiments, outputs and inputs must be equal for all datasets. ## Equivalent to @command{vertcat}. ## ## @item 2 ## Concatenate inputs and outputs. ## The outputs and inputs are concatenated in the following way: ## @code{dat.y@{e@} = [dat1.y@{e@}, dat2.y@{e@}, @dots{}]} ## @code{dat.u@{e@} = [dat1.u@{e@}, dat2.u@{e@}, @dots{}]} ## where @var{e} denotes the experiment. ## The number of experiments and samples must be equal for all datasets. ## Equivalent to @command{horzcat}. ## ## @item 3 ## Concatenate experiments. ## The experiments are concatenated in the following way: ## @code{dat.y = [dat1.y; dat2.y; @dots{}]} ## @code{dat.u = [dat1.u; dat2.u; @dots{}]} ## The number of outputs and inputs must be equal for all datasets. ## Equivalent to @command{merge}. ## @end table ## ## @item dat1, dat2, @dots{} ## iddata sets to be concatenated. ## @end table ## ## @strong{Outputs} ## @table @var ## @item dat ## Concatenated iddata set. ## @end table ## ## @seealso{horzcat, merge, vertcat} ## @end deftypefn ## Author: Lukas Reichlin ## Created: March 2012 ## Version: 0.1 function dat = cat (dim, varargin) ## I think this code is pretty elegant because it works for ## any number of arguments and without a single for-loop :-) ## if this overloaded cat method is called, it is guaranteed that ## * nargin > 0 ## * at least one argument is an iddata object if (! is_real_scalar (dim)) print_usage (); endif ## store all datasets in a single struct 'tmp' ## tmp is not a valid iddata set anymore, ## but it doesn't matter, we want just a ## temporary struct containing all the data tmp = cellfun (@iddata, varargin); [n, p, m, e] = cellfun (@size, varargin, "uniformoutput", false); ## TODO: dat = iddata (ones (100, 3)); ## dat = cat (1, dat, zeros (4, 3), dat) ## default values for metadata ## some of them are overwritten in the switch statement below tsam = tmp(1).tsam; expname = tmp(1).expname; outname = tmp(1).outname; outunit = tmp(1).outunit; inname = tmp(1).inname; inunit = tmp(1).inunit; check_domain (tmp, e); switch (dim) case 1 # vertcat - catenate samples check_experiments (tmp, e); check_outputs (tmp, p); check_inputs (tmp, m); y = cellfun (@vertcat, tmp.y, "uniformoutput", false); u = cellfun (@vertcat, tmp.u, "uniformoutput", false); ## note that this also works for time series (u = {}) case 2 # horzcat - catenate channels check_experiments (tmp, e); check_samples (n); y = cellfun (@horzcat, tmp.y, "uniformoutput", false); u = cellfun (@horzcat, tmp.u, "uniformoutput", false); outname = vertcat (tmp.outname); outunit = vertcat (tmp.outunit); inname = vertcat (tmp.inname); inunit = vertcat (tmp.inunit); case 3 # merge - catenate experiments check_outputs (tmp, p); check_inputs (tmp, m); y = vertcat (tmp.y); u = vertcat (tmp.u); tsam = vertcat (tmp.tsam); expname = vertcat (tmp.expname); otherwise error ("iddata: cat: '%d' is an invalid dimension", dim); endswitch dat = iddata (y, u, tsam); ## copy metadata dat.expname = expname; dat.outname = outname; dat.outunit = outunit; dat.inname = inname; dat.inunit = inunit; % TODO: handle w endfunction function check_experiments (tmp, e) if (numel (e) > 1 && ! isequal (e{:})) # isequal doesn't work with less than 2 arguments error ("iddata: cat: number of experiments don't match [%s]", ... num2str (cell2mat (e), "%d ")); endif if (! compare_strings (tmp.expname)) warning ("iddata: cat: experiment names don't match") endif if (numel (e) > 1 && ! isequal (tmp.tsam)) warning ("iddata: cat: sampling times don't match"); endif endfunction function check_outputs (tmp, p) if (numel (p) > 1 && ! isequal (p{:})) error ("iddata: cat: number of outputs don't match [%s]", ... num2str (cell2mat (p), "%d ")); endif if (! compare_strings (tmp.outname)) warning ("iddata: cat: output names don't match") endif if (! compare_strings (tmp.outunit)) warning ("iddata: cat: output units don't match") endif endfunction function check_inputs (tmp, m) if (numel (m) > 1 && ! isequal (m{:})) error ("iddata: cat: number of inputs don't match [%s]", ... num2str (cell2mat (m), "%d ")); endif if (! compare_strings (tmp.inname)) warning ("iddata: cat: input names don't match") endif if (! compare_strings (tmp.inunit)) warning ("iddata: cat: input units don't match") endif endfunction function check_samples (n) if (numel (n) > 1 && ! isequal (n{:})) error ("iddata: cat: number of samples don't match %s", ... mat2str (vertcat (n{:}), 10)); endif endfunction function check_domain (tmp, e) if (numel (e) > 1 && ! isequal (tmp.timedomain)) # isequal doesn't work with less than 2 arguments error ("iddata: cat: can't mix time- and frequency-domain datasets"); endif endfunction ## kind of strcmp for more than two arguments ## return true if all cells of strings are equal ## and false otherwise function bool = compare_strings (str, varargin) if (nargin > 1) ## compare n-th string of first cell with n-th string of remaining cells tmp = cellfun (@strcmp, {str}, varargin, "uniformoutput", false); ## check whether all strings of each pair are equal tmp = cellfun (@all, tmp); ## check whether all pairs are equal bool = all (tmp); else ## one or no cell at all is always equal to itself bool = true; endif endfunction %!error (cat (1, iddata (1, 1), iddata ({2, 3}, {2, 3}))); control/inst/@iddata/detrend.m0000644000076500000240000000440612273411256015655 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} detrend (@var{dat}) ## @deftypefnx {Function File} {@var{dat} =} detrend (@var{dat}, @var{ord}) ## Detrend outputs and inputs of dataset @var{dat} by ## removing the best fit of a polynomial of order @var{ord}. ## If @var{ord} is not specified, default value 0 is taken. ## This corresponds to removing a constant. ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function dat = detrend (dat, ord = 0) if (nargin > 2) # no need to test nargin == 0, this is handled by built-in detrend print_usage (); endif if ((! is_real_scalar (ord) || fix (ord) != ord) && ! ischar (ord)) # chars are handled by built-in detrend error ("iddata: detrend: second argument must be a positve integer"); endif [n, p, m] = size (dat); dat.y = cellfun (@detrend, dat.y, {ord}, "uniformoutput", false); dat.u = cellfun (@detrend, dat.u, {ord}, "uniformoutput", false); ## if a MIMO experiment has only 1 sample, detrend works ## row-wisely instead of column-wisely ## therefore we set these experiments to zero idx = (n == 1); dat.y(idx) = zeros (1, p); dat.u(idx) = zeros (1, m); endfunction %!shared DATD, Z %! DAT = iddata ({[(1:10).', (1:2:20).'], [(10:-1:1).', (20:-2:1).']}, {[(41:50).', (46:55).'], [(61:70).', (-66:-1:-75).']}); %! DATD = detrend (DAT, "linear"); %! Z = zeros (10, 2); %!assert (DATD.y{1}, Z, 1e-10); %!assert (DATD.y{2}, Z, 1e-10); %!assert (DATD.u{1}, Z, 1e-10); %!assert (DATD.u{2}, Z, 1e-10); control/inst/@iddata/diff.m0000644000076500000240000000262712273411256015143 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} diff (@var{dat}) ## @deftypefnx {Function File} {@var{dat} =} diff (@var{dat}, @var{k}) ## Return @var{k}-th difference of outputs and inputs of dataset @var{dat}. ## If @var{k} is not specified, default value 1 is taken. ## @end deftypefn ## Author: Lukas Reichlin ## Created: March 2012 ## Version: 0.1 function dat = diff (dat, k = 1) if (nargin > 2) # no need to test nargin == 0, this is handled by built-in diff print_usage (); endif dat.y = cellfun (@diff, dat.y, {k}, {1}, "uniformoutput", false); dat.u = cellfun (@diff, dat.u, {k}, {1}, "uniformoutput", false); endfunction control/inst/@iddata/display.m0000644000076500000240000000574512273411256015704 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Display routine for iddata objects. ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.1 function display (dat) datname = inputname (1); [outname, p] = __labels__ (dat.outname, "y"); [inname, m] = __labels__ (dat.inname, "u"); [expname, e] = __labels__ (dat.expname, "exp"); [n, p, m, e] = size (dat); if (dat.timedomain) domain = "Time"; sf = "Samples"; else domain = "Frequency"; sf = "Frequencies"; endif str = [domain, " domain dataset '", datname, "' containing ", num2str(sum(n)), " ", lower(sf)]; disp (""); disp (str); disp (""); disp (__horzcat__ (__col2str__ (expname, "Experiment"), ... __vec2str__ (n, sf), ... __vec2str__ (cell2mat (dat.tsam), "Sampling Interval"))); disp (""); disp (__horzcat__ (__col2str__ (outname, "Outputs"), ... __col2str__ (dat.outunit, "Unit (if specified)"))); disp (""); disp (__horzcat__ (__col2str__ (inname, "Inputs"), ... __col2str__ (dat.inunit, "Unit (if specified)"))); disp (""); endfunction function str = __horzcat__ (col, varargin) len = rows (col); sp2 = repmat (" ", len, 1); sp4 = repmat (" ", len, 1); str = [sp2, col]; for k = 2 : nargin str = [str, sp4, varargin{k-1}]; endfor endfunction function str = __col2str__ (col, title) len = rows (col); str = strjust (strvcat (col), "left"); if (columns (str) == 0) str = repmat (" ", len, 1); endif line = repmat ("-", 1, max (columns (str), columns (title))); str = strvcat (title, line, str); endfunction function str = __vec2str__ (vec, title) vec = vec(:); tmp = isfinite (vec); tmp = abs (vec(tmp & vec != 0)); if (isempty (tmp) || min (tmp) < 1e-3 || max (tmp) > 1e4) str = arrayfun (@(x) sprintf ("%.3e", x), vec, "uniformoutput", false); elseif (all (floor (tmp) == tmp)) str = arrayfun (@(x) sprintf ("%d", x), vec, "uniformoutput", false); else str = arrayfun (@(x) sprintf ("%.4f", x), vec, "uniformoutput", false); endif str = strjust (char (str), "right"); line = repmat ("-", 1, max (columns (str), columns (title))); %str = strvcat (title, str) str = strvcat (title, line, str); endfunction control/inst/@iddata/end.m0000644000076500000240000000302212273411257014770 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## End indexing for @acronym{IDDATA} objects. ## Used by Octave for "dat(1:end)". ## Author: Lukas Reichlin ## Created: December 2013 ## Version: 0.1 function ret = end (dat, k, n) if (n > 4) error ("iddata: end: require at most 4 indices in the expression"); endif switch (k) case 1 # selecting samples ret = size (dat, 1); if (numel (ret) != 1 && ! isequal (num2cell (ret){:})) error ("iddata: end: for multi-experiment datasets, require equal number of samples when selecting samples with 'end'"); endif ret = ret(1); case {2, 3, 4} # selecting outputs, inputs or experiments ret = size (dat, k); otherwise error ("iddata: end: invalid expression index k = %d", k); endswitch endfunction control/inst/@iddata/fft.m0000644000076500000240000000662012273411257015010 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} fft (@var{dat}) ## @deftypefnx {Function File} {@var{dat} =} fft (@var{dat}, @var{n}) ## Transform iddata objects from time to frequency domain ## using a Fast Fourier Transform (FFT) algorithm. ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata set containing signals in time-domain. ## @item n ## Length of the FFT transformations. If @var{n} does not match ## the signal length, the signals in @var{dat} are shortened or ## padded with zeros. @var{n} is a vector with as many elements ## as there are experiments in @var{dat} or a scalar with a common ## length for all experiments. ## If not specified, the signal lengths are taken as default values. ## @end table ## ## @strong{Outputs} ## @table @var ## @item dat ## iddata identification dataset in frequency-domain. ## In order to preserve signal power and noise level, ## the FFTs are normalized by dividing each transform ## by the square root of the signal length. ## The frequency values are distributed equally from 0 ## to the Nyquist frequency. The Nyquist frequency is ## only included for even signal lengths. ## @end table ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function dat = fft (dat, n = []) if (nargin > 2 || ! isa (dat, "iddata")) # no need to test nargin == 0, this is handled by built-in fft print_usage (); endif if (! dat.timedomain) return; endif [x, ~, ~, e] = size (dat); if (isempty (n)) # default case, n not specified n = num2cell (x(:)); elseif (is_real_vector (n) && length (n) == e && fix (n) == n) # individual n for each experiment n = num2cell (n(:)); elseif (is_real_scalar (n) && fix (n) == n) # common n for all experiments n = num2cell (repmat (n, e, 1)); else error ("iddata: fft: second argument invalid"); endif dat.y = cellfun (@(y, n) fft (y, n, 1)(1:fix(n/2)+1, :) / sqrt (n), dat.y, n, "uniformoutput", false); dat.u = cellfun (@(u, n) fft (u, n, 1)(1:fix(n/2)+1, :) / sqrt (n), dat.u, n, "uniformoutput", false); ## fft (x, n, dim=1) because x could be a row vector (n=1) dat.w = cellfun (@(n, tsam) (0:fix(n/2)).' * (2*pi/abs(tsam)/n), n, dat.tsam, "uniformoutput", false); ## abs(tsam) because of -1 for undefined sampling times dat.timedomain = false; endfunction %!shared DATD, Y, U %! Y = 1:10; %! U = 20:-2:1; %! W = warning ("query", "iddata:transpose"); %! warning ("off", W.identifier); %! DAT = iddata (Y, U); %! DATD = fft (DAT); %! warning (W.identifier, W.state); %!assert (DATD.y{1}, Y, 1e-10); %!assert (DATD.u{1}, U, 1e-10); control/inst/@iddata/filter.m0000644000076500000240000000715712273411257015524 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} filter (@var{dat}, @var{sys}) ## @deftypefnx {Function File} {@var{dat} =} filter (@var{dat}, @var{b}, @var{a}) ## Filter output and input signals of dataset @var{dat}. ## The filter is specified either by @acronym{LTI} system @var{sys} ## or by transfer function polynomials @var{b} and @var{a} as described ## in the help text of the built-in filter command. Type @code{help filter} ## for more information. ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata identification dataset containing signals in time-domain. ## @item sys ## @acronym{LTI} object containing the discrete-time filter. ## @item b ## Numerator polynomial of the discrete-time filter. ## Must be a row vector containing the coefficients ## of the polynomial in ascending powers of z^-1. ## @item a ## Denominator polynomial of the discrete-time filter. ## Must be a row vector containing the coefficients ## of the polynomial in ascending powers of z^-1. ## @end table ## ## @strong{Outputs} ## @table @var ## @item dat ## iddata identification dataset with filtered ## output and input signals. ## @end table ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: August 2012 ## Version: 0.1 function dat = filter (dat, b, a = [], si = []) if (nargin < 2 || nargin > 4) print_usage (); endif if (! isa (dat, "iddata")) # there's at least one iddata set, but not as the first argument error ("iddata: filter: first argument must be an iddata set"); endif if (! dat.timedomain) error ("iddata: filter: require iddata set in time-domain"); endif if (isa (b, "lti")) # filter (dat, sys) if (nargin > 3) # filter (dat, sys, si) has at most 3 inputs print_usage (); endif if (! issiso (b)) error ("iddata: filter: second argument must be a SISO LTI system"); endif si = a; # filter (dat, sys, si) if (isct (b)) # sys is continuous-time b = c2d (b, dat.tsam{1}); # does this discretization/tsam make sense? endif [b, a] = filtdata (b, "vector"); # convert LTI system to transfer function elseif (nargin < 3) print_usage (); endif ## use Octave's filter function for each experiment ## the fifth argument '1' specifies the dimension in case of datasets with only 1 sample dat.y = cellfun (@(y) filter (b, a, y, si, 1), dat.y, "uniformoutput", false); dat.u = cellfun (@(u) filter (b, a, u, si, 1), dat.u, "uniformoutput", false); endfunction ## TODO: adapt test %!shared DATD, Z %! DAT = iddata ({[(1:10).', (1:2:20).'], [(10:-1:1).', (20:-2:1).']}, {[(41:50).', (46:55).'], [(61:70).', (-66:-1:-75).']}); %! DATD = detrend (DAT, "linear"); %! Z = zeros (10, 2); %!assert (DATD.y{1}, Z, 1e-10); %!assert (DATD.y{2}, Z, 1e-10); %!assert (DATD.u{1}, Z, 1e-10); %!assert (DATD.u{2}, Z, 1e-10); control/inst/@iddata/get.m0000644000076500000240000000535112273411257015010 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} get (@var{dat}) ## @deftypefnx {Function File} {@var{value} =} get (@var{dat}, @var{"property"}) ## @deftypefnx {Function File} {[@var{val1}, @var{val2}, @dots{}] =} get (@var{dat}, @var{"prop1"}, @var{"prop2"}, @dots{}) ## Access property values of iddata objects. ## Type @command{get(dat)} to display a list of available properties. ## @end deftypefn ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.1 function varargout = get (dat, varargin) if (nargin == 1) [props, vals] = __property_names__ (dat); nrows = numel (props); str = strjust (strvcat (props), "right"); str = horzcat (repmat (" ", nrows, 1), str, repmat (": ", nrows, 1), strvcat (vals)); disp (str); else for k = 1 : (nargin-1) prop = lower (varargin{k}); switch (prop) case {"y", "outdata", "outputdata", "outd", "outputd"} val = dat.y; case {"u", "indata", "inputdata", "ind", "inputd"} val = dat.u; case {"outname", "outputname", "outn", "outputn"} val = dat.outname; case {"inname", "inputname", "inn", "inputn"} val = dat.inname; case {"outunit", "outputunit", "outu", "outputu"} val = dat.outunit; case {"inunit", "inputunit", "inu", "inputu"} val = dat.inunit; case {"tsam", "ts"} val = dat.tsam; case {"timeunit", "timeu"} val = dat.timeunit case {"expname", "experimentname", "expn", "experimentn"} val = dat.expname; case "name" val = dat.name; case "notes" val = dat.notes; case "userdata" val = dat.userdata; case {"domain", "timedomain"} val = dat.timedomain; case {"w", "frequency", "samplinginstants"} val = dat.w; otherwise error ("iddata: get: invalid property name '%s'", varargin{k}); endswitch varargout{k} = val; endfor endif endfunctioncontrol/inst/@iddata/horzcat.m0000644000076500000240000000253612273411257015705 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} horzcat (@var{dat1}, @var{dat2}, @dots{}) ## Horizontal concatenation of iddata datasets. ## The outputs and inputs are concatenated in the following way: ## @code{dat.y@{e@} = [dat1.y@{e@}, dat2.y@{e@}, @dots{}]} ## @code{dat.u@{e@} = [dat1.u@{e@}, dat2.u@{e@}, @dots{}]} ## where @var{e} denotes the experiment. ## The number of experiments and samples must be equal for all datasets. ## @end deftypefn ## Author: Lukas Reichlin ## Created: March 2012 ## Version: 0.1 function dat = horzcat (varargin) dat = cat (2, varargin{:}); endfunction control/inst/@iddata/iddata.m0000644000076500000240000001171012273411257015453 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{dat} =} iddata (@var{y}) ## @deftypefnx{Function File} {@var{dat} =} iddata (@var{y}, @var{u}) ## @deftypefnx{Function File} {@var{dat} =} iddata (@var{y}, @var{u}, @var{tsam}, @dots{}) ## @deftypefnx{Function File} {@var{dat} =} iddata (@var{y}, @var{u}, @var{[]}, @dots{}) ## Create identification dataset of output and input signals. ## ## @strong{Inputs} ## @table @var ## @item y ## Real matrix containing the output signal in time-domain. ## For a system with @var{p} outputs and @var{n} samples, ## @var{y} is a n-by-p matrix. ## For data from multiple experiments, @var{y} becomes a ## e-by-1 or 1-by-e cell vector of n(i)-by-p matrices, ## where @var{e} denotes the number of experiments ## and n(i) the individual number of samples for each experiment. ## @item u ## Real matrix containing the input signal in time-domain. ## For a system with @var{m} inputs and @var{n} samples, ## @var{u} is a n-by-m matrix. ## For data from multiple experiments, @var{u} becomes a ## e-by-1 or 1-by-e cell vector of n(i)-by-m matrices, ## where @var{e} denotes the number of experiments ## and n(i) the individual number of samples for each experiment. ## If @var{u} is not specified or an empty element @code{[]} is passed, ## @var{dat} becomes a time series dataset. ## @item tsam ## Sampling time. If not specified, default value -1 (unspecified) is taken. ## For multi-experiment data, @var{tsam} becomes a ## e-by-1 or 1-by-e cell vector containing individual ## sampling times for each experiment. If a scalar @var{tsam} ## is provided, then all experiments have the same sampling time. ## @item @dots{} ## Optional pairs of properties and values. ## @end table ## ## @strong{Outputs} ## @table @var ## @item dat ## iddata identification dataset. ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'expname' ## The name of the experiments in @var{dat}. ## Cell vector of length e containing strings. ## Default names are @code{@{'exp1', 'exp2', ...@}} ## ## @item 'y' ## Output signals. See 'Inputs' for details. ## ## @item 'outname' ## The name of the output channels in @var{dat}. ## Cell vector of length p containing strings. ## Default names are @code{@{'y1', 'y2', ...@}} ## ## @item 'outunit' ## The units of the output channels in @var{dat}. ## Cell vector of length p containing strings. ## ## @item 'u' ## Input signals. See 'Inputs' for details. ## ## @item 'inname' ## The name of the input channels in @var{dat}. ## Cell vector of length m containing strings. ## Default names are @code{@{'u1', 'u2', ...@}} ## ## @item 'inunit' ## The units of the input channels in @var{dat}. ## Cell vector of length m containing strings. ## ## @item 'tsam' ## Sampling time. See 'Inputs' for details. ## ## @item 'timeunit' ## The units of the sampling times in @var{dat}. ## Cell vector of length e containing strings. ## ## @item 'name' ## String containing the name of the dataset. ## ## @item 'notes' ## String or cell of string containing comments. ## ## @item 'userdata' ## Any data type. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2011 ## Version: 0.1 function dat = iddata (y = {}, u = {}, tsam = {}, varargin) if (nargin == 1 && isa (y, "iddata")) dat = y; return; elseif (nargin < 1) print_usage (); endif [y, u] = __adjust_iddata__ (y, u); [p, m, e] = __iddata_dim__ (y, u); tsam = __adjust_iddata_tsam__ (tsam, e); outname = repmat ({""}, p, 1); inname = repmat ({""}, m, 1); expname = repmat ({""}, e, 1); dat = struct ("y", {y}, "outname", {outname}, "outunit", {outname}, "u", {u}, "inname", {inname}, "inunit", {inname}, "tsam", {tsam}, "timeunit", {""}, "timedomain", true, "w", {{}}, "expname", {expname}, "name", "", "notes", {{}}, "userdata", []); dat = class (dat, "iddata"); if (nargin > 3) dat = set (dat, varargin{:}); endif if (dat.timedomain && ! is_real_matrix (dat.y{:}, dat.u{:})) error ("iddata: require real-valued input and output signals for time domain datasets"); endif endfunction %!error (iddata); %!error (iddata ((1:10).', (1:11).')); %!warning (iddata (1:10)); %!warning (iddata (1:10, 1:10)); control/inst/@iddata/ifft.m0000644000076500000240000000563312273411257015164 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} ifft (@var{dat}) ## Transform iddata objects from frequency to time domain. ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata set containing signals in frequency domain. ## The frequency values must be distributed equally from 0 ## to the Nyquist frequency. The Nyquist frequency is ## only included for even signal lengths. ## @end table ## ## @strong{Outputs} ## @table @var ## @item dat ## iddata identification dataset in time domain. ## In order to preserve signal power and noise level, ## the FFTs are normalized by multiplying each transform ## by the square root of the signal length. ## @end table ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function dat = ifft (dat) if (nargin > 1) # no need to test nargin == 0, this is handled by built-in ifft print_usage (); endif if (dat.timedomain) # dat is always an iddata set, otherwise built-in ifft would be called return; endif if (any (cellfun (@(w) w(1) >= eps, dat.w))) error ("iddata: ifft: first frequency must be zero"); endif if (any (cellfun (@(w) any (abs (diff (w, 2)) > 1e-4*w(2:end-1)), dat.w))) error ("iddata: ifft: require linearly spaced frequency vectors"); endif [x, ~, ~, e] = size (dat); x = x(:); n = num2cell (x); %nconj = num2cell (x - ! rem (x, 2)); nconj = num2cell (x - rem (x, 2)); dat.y = cellfun (@(y, n, nconj) real (ifft ([y; conj(y(nconj:-1:2, :))], [], 1)) * sqrt (n+nconj), dat.y, n, nconj, "uniformoutput", false); dat.u = cellfun (@(u, n, nconj) real (ifft ([u; conj(u(nconj:-1:2, :))], [], 1)) * sqrt (n+nconj), dat.u, n, nconj, "uniformoutput", false); ## ifft (x, n, dim=1) because x could be a row vector (n=1) % dat.w = cellfun (@(n, tsam) (0:fix(n/2)).' * (2*pi/abs(tsam)/n), n, dat.tsam, "uniformoutput", false); dat.w = {}; % dat.w = repmat ({[]}, e, 1); ??? ## abs(tsam) because of -1 for undefined sampling times dat.timedomain = true; endfunction %!shared DATD, Y, U %! Y = 1:10; %! U = 20:-2:1; %! DAT = iddata (Y, U); %! DATD = fft (DAT); %!assert (DATD.y{1}, Y, 1e-10); %!assert (DATD.u{1}, U, 1e-10); control/inst/@iddata/merge.m0000644000076500000240000000240612273411257015326 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} merge (@var{dat1}, @var{dat2}, @dots{}) ## Concatenate experiments of iddata datasets. ## The experiments are concatenated in the following way: ## @code{dat.y = [dat1.y; dat2.y; @dots{}]} ## @code{dat.u = [dat1.u; dat2.u; @dots{}]} ## The number of outputs and inputs must be equal for all datasets. ## @end deftypefn ## Author: Lukas Reichlin ## Created: March 2012 ## Version: 0.1 function dat = merge (varargin) dat = cat (3, varargin{:}); endfunction control/inst/@iddata/nkshift.m0000644000076500000240000000474012273411257015700 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} nkshift (@var{dat}, @var{nk}) ## @deftypefnx {Function File} {@var{dat} =} nkshift (@var{dat}, @var{nk}, @var{'append'}) ## Shift input channels of dataset @var{dat} according to integer @var{nk}. ## A positive value of @var{nk} means that the input channels are delayed ## @var{nk} samples. By default, both input and output signals are shortened ## by @var{nk} samples. ## If a third argument @var{'append'} is passed, the output signals are left ## untouched while @var{nk} zeros are appended to the (shortened) input signals ## such that the number of samples in @var{dat} remains constant. ## @end deftypefn ## Author: Lukas Reichlin ## Created: July 2012 ## Version: 0.1 function dat = nkshift (dat, nk = 0) if (nargin > 3) print_usage (); endif if (! is_real_scalar (nk)) error ("iddata: nkshift: 'nk' must be a scalar integer"); endif ## TODO: - nk per inputs ## - frequency-domain data snk = sign (nk); nk = abs (nk); if (nargin == 2) # default: shortening y and u by nk if (snk >= 0) dat.y = cellfun (@(y) y(nk+1:end, :), dat.y, "uniformoutput", false); dat.u = cellfun (@(u) u(1:end-nk, :), dat.u, "uniformoutput", false); else dat.y = cellfun (@(y) y(1:end-nk, :), dat.y, "uniformoutput", false); dat.u = cellfun (@(u) u(nk+1:end, :), dat.u, "uniformoutput", false); endif else # append: keep y, padding u with nk zeros [~, ~, m] = size (dat); if (snk >= 0) dat.u = cellfun (@(u) [zeros(nk, m), u(1:end-nk, :)], dat.u, "uniformoutput", false); else dat.u = cellfun (@(u) [u(nk+1:end, :), zeros(nk, m)], dat.u, "uniformoutput", false); endif endif endfunction control/inst/@iddata/plot.m0000644000076500000240000000711712273411257015211 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} plot (@var{dat}) ## @deftypefnx {Function File} {} plot (@var{dat}, @var{exp}) ## Plot signals of iddata identification datasets on the screen. ## The signals are plotted experiment-wise, either in time- or ## frequency-domain. For multi-experiment datasets, ## press any key to switch to the next experiment. ## If the plot of a single experiment should be saved by the ## @command{print} command, use @code{plot(dat,exp)}, ## where @var{exp} denotes the desired experiment. ## @end deftypefn ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.2 function plot (dat, exp = ":") if (nargin > 2) # nargin == 0 is handled by built-in plot print_usage (); endif if (nargin == 2 && ! is_real_vector (exp)) error ("iddata: plot: second argument must be a vector of indices"); endif expname = __labels__ (dat.expname, "exp"); expname = expname(exp); idx = substruct ("()", {":", ":", ":", exp}); dat = subsref (dat, idx); [n, p, m, e] = size (dat); if (dat.timedomain) if (m == 0) # time series for k = 1 : e if (k > 1) pause endif plot (dat.y{k}) title (expname{k}) legend (__labels__ (dat.outname, "y"){:}) xlabel ("Time") ylabel ("Output Signal") endfor else # inputs present for k = 1 : e if (k > 1) pause endif subplot (2, 1, 1) plot (dat.y{k}) title (expname{k}) legend (__labels__ (dat.outname, "y"){:}) ylabel ("Output Signal") subplot (2, 1, 2) stairs (dat.u{k}) legend (__labels__ (dat.inname, "u"){:}) xlabel ("Time") ylabel ("Input Signal") endfor endif else # frequency domain if (m == 0) # time series for k = 1 : e if (k > 1) pause endif bar (dat.w{k}, 20*log10 (abs (dat.y{k}))) xlim ([dat.w{k}(1), dat.w{k}(end)]) title (expname{k}) legend (__labels__ (dat.outname, "y"){:}) xlabel ("Frequency") ylabel ("Output Magnitude [dB]") endfor else # inputs present for k = 1 : e if (k > 1) pause endif subplot (2, 1, 1) bar (dat.w{k}, 20*log10 (abs (dat.y{k}))) xlim ([dat.w{k}(1), dat.w{k}(end)]) title (expname{k}) legend (__labels__ (dat.outname, "y"){:}) ylabel ("Output Magnitude [dB]") subplot (2, 1, 2) bar (dat.w{k}, 20*log10(abs (dat.u{k}))) xlim ([dat.w{k}(1), dat.w{k}(end)]) legend (__labels__ (dat.inname, "u"){:}) xlabel ("Frequency") ylabel ("Input Magnitude [dB]") endfor endif endif ## TODO: think about the 20*log10 and the bars in general endfunction control/inst/@iddata/resample.m0000644000076500000240000000567012273411257016045 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} resample (@var{dat}, @var{p}, @var{q}) ## @deftypefnx {Function File} {@var{dat} =} resample (@var{dat}, @var{p}, @var{q}, @var{n}) ## @deftypefnx {Function File} {@var{dat} =} resample (@var{dat}, @var{p}, @var{q}, @var{h}) ## Change the sample rate of the output and input signals in dataset @var{dat} ## by a factor of @code{p/q}. This is performed using a polyphase algorithm. ## The anti-aliasing @acronym{FIR} filter can be specified as follows: ## Either by order @var{n} (scalar) with default value 0. The band edges ## are then chosen automatically. Or by impulse response @var{h} (vector). ## Requires the signal package to be installed. ## ## @strong{Algorithm}@* ## Uses commands @command{fir1} and @command{resample} ## from the signal package. ## ## @strong{References}@* ## [1] J. G. Proakis and D. G. Manolakis, ## Digital Signal Processing: Principles, Algorithms, and Applications, ## 4th ed., Prentice Hall, 2007. Chap. 6 ## ## [2] A. V. Oppenheim, R. W. Schafer and J. R. Buck, ## Discrete-time signal processing, Signal processing series, ## Prentice-Hall, 1999 ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: June 2012 ## Version: 0.3 function dat = resample (dat, p, q, n = 0) if (nargin < 3 || nargin > 4) print_usage (); endif ## requires signal package try pkg load signal; catch error ("iddata: resample: please install signal package to proceed"); end_try_catch if (is_real_scalar (n)) # fourth scalar argument n is the order of the anti-aliasing filter h = fir1 (n, 1/q); elseif (is_real_vector (n)) # fourth vector argument is the (impulse response of the) anti-aliasing filter h = n; else error ("iddata: resample: fourth argument invalid"); endif dat.y = cellfun (@resample, dat.y, {p}, {q}, {h}, "uniformoutput", false); dat.u = cellfun (@resample, dat.u, {p}, {q}, {h}, "uniformoutput", false); dat.tsam = cellfun (@sampling_time, dat.tsam, {p}, {q}, "uniformoutput", false); endfunction function tsam = sampling_time (tsam, p, q) if (issample (tsam, 1)) # unspecified sampling times (-1) are left untouched tsam = (tsam*q)/p; endif endfunction control/inst/@iddata/set.m0000644000076500000240000001234312273411257015023 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} set (@var{dat}) ## @deftypefnx {Function File} {} set (@var{dat}, @var{"property"}, @var{value}, @dots{}) ## @deftypefnx {Function File} {@var{dat} =} set (@var{dat}, @var{"property"}, @var{value}, @dots{}) ## Set or modify properties of iddata objects. ## If no return argument @var{dat} is specified, the modified @acronym{LTI} object is stored ## in input argument @var{dat}. @command{set} can handle multiple properties in one call: ## @code{set (dat, 'prop1', val1, 'prop2', val2, 'prop3', val3)}. ## @code{set (dat)} prints a list of the object's property names. ## @end deftypefn ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.1 function retdat = set (dat, varargin) if (nargin == 1) # set (dat), dat = set (dat) [props, vals] = __property_names__ (dat); nrows = numel (props); str = strjust (strvcat (props), "right"); str = horzcat (repmat (" ", nrows, 1), str, repmat (": ", nrows, 1), strvcat (vals)); disp (str); if (nargout != 0) # function dat = set (dat, varargin) retdat = dat; # would lead to unwanted output when using endif # set (dat) else # set (dat, "prop1", val1, ...), dat = set (dat, "prop1", val1, ...) if (rem (nargin-1, 2)) error ("iddata: set: properties and values must come in pairs"); endif [n, p, m, e] = size (dat); for k = 1 : 2 : (nargin-1) prop = lower (varargin{k}); val = varargin{k+1}; switch (prop) case {"y", "outdata", "outputdata", "outd", "outputd"} val = __adjust_iddata__ (val, dat.u); [pval, ~, eval] = __iddata_dim__ (val, dat.u); if (pval != p) error ("iddata: set: argument has %d instead of %d outputs", pval, p); endif if (eval != e) # iddata_dim is not sufficient if dat.u = [] error ("iddata: set: argument has %d instead of %d experiments", eval, e); endif dat.y = val; case {"u", "indata", "inputdata", "ind", "inputd"} [~, val] = __adjust_iddata__ (dat.y, val); [~, mval] = __iddata_dim__ (dat.y, val); if (mval != m) error ("iddata: set: argument has %d instead of %d inputs", mval, m); endif dat.u = val; case {"outname", "outputname", "outn", "outputn"} dat.outname = __adjust_labels__ (val, p); case {"inname", "inputname", "inn", "inputn"} dat.inname = __adjust_labels__ (val, m); case {"outunit", "outputunit", "outu", "outputu"} dat.outunit = __adjust_labels__ (val, p); case {"inunit", "inputunit", "inu", "inputu"} dat.inunit = __adjust_labels__ (val, m); case {"timeunit", "timeu"} if (ischar (val)) dat.timeunit = val; else error ("iddata: set: property 'timeunit' requires a string"); endif case {"expname", "experimentname", "expn", "experimentn"} dat.expname = __adjust_labels__ (val, e); case {"tsam", "ts"} dat.tsam = __adjust_iddata_tsam__ (val, e); case {"w", "frequency"} if (! iscell (val)) val = {val}; endif if (any (cellfun (@(w) ! isempty (w) && (! is_real_vector (w) || any (w < 0) ... || ! issorted (w) || w(1) > w(end) ... || length (unique (w)) != length (w)), val))) error ("iddata: set: w must be a vector of positive real values in ascending order"); endif dat.w = val; dat.timedomain = false; case "name" if (ischar (val)) sys.name = val; else error ("iddata: set: property 'name' requires a string"); endif case "notes" if (iscellstr (val)) sys.notes = val; elseif (ischar (val)) sys.notes = {val}; else error ("lti: set: property 'notes' requires string or cell of strings"); endif case "userdata" sys.userdata = val; otherwise error ("iddata: set: invalid property name '%s'", varargin{k}); endswitch endfor if (nargout == 0) # set (dat, "prop1", val1, ...) assignin ("caller", inputname (1), dat); else # dat = set (dat, "prop1", val1, ...) retdat = dat; endif endif endfunctioncontrol/inst/@iddata/size.m0000644000076500000240000000657312273411257015212 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{nvec} =} size (@var{dat}) ## @deftypefnx {Function File} {@var{ndim} =} size (@var{dat}, @var{dim}) ## @deftypefnx {Function File} {[@var{n}, @var{p}, @var{m}, @var{e}] =} size (@var{dat}) ## Return dimensions of iddata set @var{dat}. ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata set. ## @item dim ## If given a second argument, @command{size} will return the size of the ## corresponding dimension. ## @end table ## ## @strong{Outputs} ## @table @var ## @item nvec ## Row vector. The first element is the total number of samples (rows of dat.y and dat.u). ## The second element is the number of outputs (columns of dat.y) and the third element ## the number of inputs (columns of dat.u). The fourth element is the number of experiments. ## @item ndim ## Scalar value. The size of the dimension @var{dim}. ## @item n ## Row vector containing the number of samples of each experiment. ## @item p ## Number of outputs. ## @item m ## Number of inputs. ## @item e ## Number of experiments. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2011 ## Version: 0.1 function [x, p, m, e] = size (dat, dim = 0) if (nargin > 2) print_usage (); endif n = cellfun (@rows, dat.y).'; # number of samples p = numel (dat.outname); # number of output channels m = numel (dat.inname); # number of input channels e = numel (dat.y); # number of experiments switch (dim) case 0 # ... size (dat) switch (nargout) case 0 # size (dat) stry = stru = stre = ""; if (p != 1) stry = "s"; endif if (m != 1) stru = "s"; endif if (e != 1) stre = "s"; endif printf ("IDDATA set with [%s] samples, %d output%s, %d input%s and %d experiment%s.\n", ... num2str (n, "%d "), p, stry, m, stru, e, stre); case 1 # x = size (dat) x = [sum(n), p, m, e]; case {2, 3, 4} # [n, p, m, e] = size (dat) x = n; otherwise # more than 4 return values print_usage (); endswitch case 1 # nvec = size (dat, 1) x = n; case 2 # p = size (dat, 2) x = p; case 3 # m = size (dat, 3) x = m; case 4 # e = size (dat, 4) x = e; otherwise # invalid dimension print_usage (); endswitch endfunction control/inst/@iddata/subsasgn.m0000644000076500000240000000370212273411257016054 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Subscripted assignment for iddata objects. ## Used by Octave for "dat.property = value". ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.1 function dat = subsasgn (dat, idx, val) switch (idx(1).type) case "()" # dat(...) = val if (length (idx(1).subs) == 1 && isa (val, "iddata")) # dat(x) = dat, required by cat for ... dat(idx.subs{:}) = val; # dat = cellfun (@iddata, varargin) else # dat(...) = val, general case error ("iddata: subsasgn type not implemented yet"); endif case "." # dat.y... = val if (length (idx) == 1) # dat.y = val dat = set (dat, idx.subs, val); else # dat.y(...) = val, dat.expname{3} = val key = idx(1).subs; dat = set (dat, key, subsasgn (get (dat, key), idx(2:end), val)); endif otherwise error ("iddata: subsasgn: invalid subscripted assignment type"); endswitch endfunctioncontrol/inst/@iddata/subsref.m0000644000076500000240000000520212273411257015675 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Subscripted reference for iddata objects. ## Used by Octave for "dat = dat(2:4, :)" or "val = dat.prop". ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.3 function a = subsref (a, s) if (numel (s) == 0) return; endif switch (s(1).type) case "()" idx = s(1).subs; if (numel (idx) > 4) error ("iddata: subsref: need four or less indices"); else a = __dat_prune__ (a, idx{:}); endif case "." fld = s(1).subs; a = get (a, fld); otherwise error ("iddata: subsref: invalid subscript type"); endswitch a = subsref (a, s(2:end)); endfunction function dat = __dat_prune__ (dat, spl_idx = ":", out_idx = ":", in_idx = ":", exp_idx = ":") out_idx = __handle_idx__ (dat.outname, out_idx, "outname"); in_idx = __handle_idx__ (dat.inname, in_idx, "inname"); exp_idx = __handle_idx__ (dat.expname, exp_idx, "expname"); dat.y = dat.y(exp_idx); dat.y = cellfun (@(y) y(spl_idx, out_idx), dat.y, "uniformoutput", false); dat.outname = dat.outname(out_idx); dat.outunit = dat.outunit(out_idx); if (! isempty (dat.u)) dat.u = dat.u(exp_idx); dat.u = cellfun (@(u) u(spl_idx, in_idx), dat.u, "uniformoutput", false); dat.inname = dat.inname(in_idx); dat.inunit = dat.inunit(in_idx); endif dat.expname = dat.expname(exp_idx); dat.tsam = dat.tsam(exp_idx); endfunction function idx = __handle_idx__ (name, idx, id) if (ischar (idx) && ! strcmp (idx, ":")) idx = {idx}; endif if (iscell (idx)) idx = cellfun (@(x) __str2idx__ (name, x, id), idx); endif endfunction function idx = __str2idx__ (name, str, id) tmp = strcmp (name, str)(:); switch (nnz (tmp)) case 1 idx = find (tmp); case 0 error ("iddata: %s '%s' not found", id, str); otherwise error ("iddata: %s '%s' is ambiguous", id, str); endswitch endfunction control/inst/@iddata/vertcat.m0000644000076500000240000000253112273411257015676 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{dat} =} vertcat (@var{dat1}, @var{dat2}, @dots{}) ## Vertical concatenation of iddata datasets. ## The samples are concatenated in the following way: ## @code{dat.y@{e@} = [dat1.y@{e@}; dat2.y@{e@}; @dots{}]} ## @code{dat.u@{e@} = [dat1.u@{e@}; dat2.u@{e@}; @dots{}]} ## where @var{e} denotes the experiment. ## The number of experiments, outputs and inputs must be equal for all datasets. ## @end deftypefn ## Author: Lukas Reichlin ## Created: March 2012 ## Version: 0.1 function dat = vertcat (varargin) dat = cat (1, varargin{:}); endfunction control/inst/@lti/0000755000076500000240000000000012273411257013411 5ustar lukasstaffcontrol/inst/@lti/__lti_data__.m0000644000076500000240000000202512273411257016143 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Used by display routines instead of multiple get calls. ## Author: Lukas Reichlin ## Created: September 2010 ## Version: 0.1 function [inname, outname, tsam] = __lti_data__ (sys) inname = sys.inname; outname = sys.outname; tsam = sys.tsam; endfunction control/inst/@lti/__lti_group__.m0000644000076500000240000000536412273411257016377 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Block diagonal concatenation of two LTI models. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function retlti = __lti_group__ (lti1, lti2) retlti = lti (); retlti.inname = [lti1.inname; lti2.inname]; retlti.outname = [lti1.outname; lti2.outname]; if (nfields (lti1.ingroup) || nfields (lti2.ingroup)) m1 = numel (lti1.inname); lti2_ingroup = structfun (@(x) x + m1, lti2.ingroup, "uniformoutput", false); retlti.ingroup = __merge_struct__ (lti1.ingroup, lti2_ingroup, "in"); endif if (nfields (lti1.outgroup) || nfields (lti2.outgroup)) p1 = numel (lti1.outname); lti2_outgroup = structfun (@(x) x + p1, lti2.outgroup, "uniformoutput", false); retlti.outgroup = __merge_struct__ (lti1.outgroup, lti2_outgroup, "out"); endif if (lti1.tsam == lti2.tsam) retlti.tsam = lti1.tsam; elseif (lti1.tsam == -2) retlti.tsam = lti2.tsam; elseif (lti2.tsam == -2) retlti.tsam = lti1.tsam; elseif (lti1.tsam == -1 && lti2.tsam > 0) retlti.tsam = lti2.tsam; elseif (lti2.tsam == -1 && lti1.tsam > 0) retlti.tsam = lti1.tsam; else error ("lti_group: systems must have identical sampling times"); endif endfunction function ret = __merge_struct__ (a, b, iostr) ## FIXME: this is too complicated; ## isn't there a simple function for this task? ## bug #40224: orderfields (struct ()) errors out in Octave 3.6.4 ## therefore use nfields to check for empty structs if (nfields (a)) a = orderfields (a); endif if (nfields (b)) b = orderfields (b); endif fa = fieldnames (a); fb = fieldnames (b); [fi, ia, ib] = intersect (fa, fb); ca = struct2cell (a); cb = struct2cell (b); for k = numel (fi) : -1 : 1 ca{ia(k)} = vertcat (ca{ia(k)}(:), cb{ib(k)}(:)); fb(ib(k)) = []; cb(ib(k)) = []; endfor ret = cell2struct ([ca; cb], [fa; fb]); endfunction control/inst/@lti/__lti_prune__.m0000644000076500000240000000524112273411257016366 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Submodel extraction and reordering for LTI objects. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function [lti, out_idx, in_idx] = __lti_prune__ (lti, out_idx, in_idx) if (ischar (out_idx) && ! strcmp (out_idx, ":")) # sys("grp", :) out_idx = {out_idx}; endif if (ischar (in_idx) && ! strcmp (in_idx, ":")) # sys(:, "grp") in_idx = {in_idx}; endif if (iscell (out_idx)) # sys({"grp1", "grp2"}, :) tmp = cellfun (@(x) __str2idx__ (lti.outgroup, lti.outname, x, "out"), out_idx, "uniformoutput", false); out_idx = vertcat (tmp{:}); endif if (iscell (in_idx)) # sys(:, {"grp1", "grp2"}) tmp = cellfun (@(x) __str2idx__ (lti.ingroup, lti.inname, x, "in"), in_idx, "uniformoutput", false); in_idx = vertcat (tmp{:}); endif if (nfields (lti.outgroup)) p = numel (lti.outname); # get size before pruning outnames! [lti.outgroup, empty] = structfun (@(x) __group_prune__ (x, out_idx, p), lti.outgroup, "uniformoutput", false); empty = cell2mat (struct2cell (empty)); fields = fieldnames (lti.outgroup); lti.outgroup = rmfield (lti.outgroup, fields(empty)); endif if (nfields (lti.ingroup)) m = numel (lti.inname); [lti.ingroup, empty] = structfun (@(x) __group_prune__ (x, in_idx, m), lti.ingroup, "uniformoutput", false); empty = cell2mat (struct2cell (empty)); fields = fieldnames (lti.ingroup); lti.ingroup = rmfield (lti.ingroup, fields(empty)); endif lti.outname = lti.outname(out_idx); lti.inname = lti.inname(in_idx); endfunction function [group, empty] = __group_prune__ (group, idx, n) lg = length (group); group = sparse (group, 1:lg, 1, n, lg); group = group(idx, :); [group, ~] = find (group); empty = isempty (group); endfunction control/inst/@lti/__property_names__.m0000644000076500000240000000322312273411257017432 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{sys}) ## Return the list of properties as well as the assignable values for an LTI object sys. ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function [props, vals] = __property_names__ (sys) ## cell vector of lti-specific properties props = {"tsam"; "inname"; "outname"; "ingroup"; "outgroup"; "name"; "notes"; "userdata"}; ## cell vector of lti-specific assignable values vals = {"scalar (sample time in seconds)"; "m-by-1 cell vector of strings"; "p-by-1 cell vector of strings"; "struct with indices as fields"; "struct with indices as fields"; "string"; "string or cell of strings"; "any data type"}; endfunction control/inst/@lti/append.m0000644000076500000240000000227212273411257015041 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} append (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## Group @acronym{LTI} models by appending their inputs and outputs. ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function sys = append (varargin) sys = varargin{1}; if (nargin > 1) for k = 2 : nargin sys = __sys_group__ (sys, varargin{k}); endfor endif endfunction control/inst/@lti/blkdiag.m0000644000076500000240000000207112273411257015164 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} blkdiag (@var{sys1}, @var{sys2}) ## Block-diagonal concatenation of @acronym{LTI} models. ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function sys = blkdiag (varargin) sys = append (varargin{:}); endfunction control/inst/@lti/c2d.m0000644000076500000240000001164012273411257014241 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}) ## @deftypefnx {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}, @var{method}) ## @deftypefnx {Function File} {@var{sys} =} c2d (@var{sys}, @var{tsam}, @var{'prewarp'}, @var{w0}) ## Convert the continuous @acronym{LTI} model into its discrete-time equivalent. ## ## @strong{Inputs} ## @table @var ## @item sys ## Continuous-time @acronym{LTI} model. ## @item tsam ## Sampling time in seconds. ## @item method ## Optional conversion method. If not specified, default method @var{"zoh"} ## is taken. ## @table @var ## @item 'zoh' ## Zero-order hold or matrix exponential. ## @item 'tustin', 'bilin' ## Bilinear transformation or Tustin approximation. ## @item 'prewarp' ## Bilinear transformation with pre-warping at frequency @var{w0}. ## @item 'matched' ## Matched pole/zero method. ## @end table ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Discrete-time @acronym{LTI} model. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function sys = c2d (sys, tsam, method = "std", w0 = 0) if (nargin < 2 || nargin > 4) print_usage (); endif if (! isa (sys, "lti")) error ("c2d: first argument is not an LTI model"); endif if (isdt (sys)) error ("c2d: system is already discrete-time"); endif if (! issample (tsam)) error ("c2d: second argument is not a valid sample time"); endif if (! ischar (method)) error ("c2d: third argument is not a string"); endif if (! issample (w0, 0)) error ("c2d: fourth argument is not a valid pre-warping frequency"); endif sys = __c2d__ (sys, tsam, lower (method), w0); sys.tsam = tsam; endfunction ## bilinear transformation ## using oct-file directly %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ].'; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ].'; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ].'; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ].'; %! %! [Ao, Bo, Co, Do] = __sl_ab04md__ (A, B, C, D, 1.0, 1.0, false); %! %! Ae = [ -1.0000 -4.0000 %! -4.0000 -1.0000 ]; %! %! Be = [ 2.8284 0.0000 %! 0.0000 -2.8284 ]; %! %! Ce = [ 0.0000 2.8284 %! -2.8284 0.0000 ]; %! %! De = [ -1.0000 0.0000 %! 0.0000 -3.0000 ]; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); ## bilinear transformation ## user function %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ].'; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ].'; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ].'; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ].'; %! %! [Ao, Bo, Co, Do] = ssdata (c2d (ss (A, B, C, D), 2, "tustin")); %! %! Ae = [ -1.0000 -4.0000 %! -4.0000 -1.0000 ]; %! %! Be = [ 2.8284 0.0000 %! 0.0000 -2.8284 ]; %! %! Ce = [ 0.0000 2.8284 %! -2.8284 0.0000 ]; %! %! De = [ -1.0000 0.0000 %! 0.0000 -3.0000 ]; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); ## bilinear transformation ## both directions %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ]; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ]; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ]; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ]; %! %! [Ao, Bo, Co, Do] = ssdata (d2c (c2d (ss (A, B, C, D), 2, "tustin"), "tustin")); %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [A, B; C, D]; %! %!assert (Mo, Me, 1e-4); ## zero-order hold ## both directions %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ]; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ]; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ]; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ]; %! %! [Ao, Bo, Co, Do] = ssdata (d2c (c2d (ss (A, B, C, D), 2, "zoh"), "zoh")); %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [A, B; C, D]; %! %!assert (Mo, Me, 1e-4); ## bilinear transformation with pre-warping ## both directions %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ]; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ]; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ]; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ]; %! %! [Ao, Bo, Co, Do] = ssdata (d2c (c2d (ss (A, B, C, D), 2, "prewarp", 1000), "prewarp", 1000)); %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [A, B; C, D]; %! %!assert (Mo, Me, 1e-4); control/inst/@lti/connect.m0000644000076500000240000001045112273411257015221 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} connect (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{inputs}, @var{outputs}) ## @deftypefnx {Function File} {@var{sys} =} connect (@var{sys}, @var{cm}, @var{inputs}, @var{outputs}) ## Name-based interconnections between the inputs and outputs of @acronym{LTI} models. ## ## @strong{Inputs} ## @table @var ## @item sys1, @dots{}, sysN ## @acronym{LTI} models to be connected. The properties 'inname' and 'outname' ## of each model should be set according to the desired input-output connections. ## @item inputs ## String or cell of strings containing the names of the inputs to be kept. ## The names must be part of the properties 'ingroup' or 'inname'. ## @item outputs ## String or cell of strings containing the names of the outputs to be kept. ## The names must be part of the properties 'outgroup' or 'outname'. ## @item cm ## Legacy connection matrix (not name-based). ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Resulting interconnected system with outputs @var{outputs} and ## inputs @var{inputs}. ## @end table ## ## @seealso{sumblk} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function sys = connect (varargin) if (nargin < 2) print_usage (); endif if (is_real_matrix (varargin{2})) # connect (sys, cm, in_idx, out_idx) if (nargin != 4) print_usage (); endif sys = varargin{1}; cm = varargin{2}; in_idx = varargin{3}; out_idx = varargin{4}; [p, m] = size (sys); [cmrows, cmcols] = size (cm); ## TODO: proper argument checking ## TODO: replace nested for-if statement ## if (! is_real_matrix (cm)) ## error ("connect: second argument must be a matrix with real-valued coefficients"); ## endif M = zeros (m, p); in = cm(:, 1); out = cm(:, 2:cmcols); for a = 1 : cmrows for b = 1 : cmcols-1 if (out(a, b) != 0) M(in(a, 1), abs (out(a, b))) = sign (out(a, b)); endif endfor endfor sys = __sys_connect__ (sys, M); sys = __sys_prune__ (sys, out_idx, in_idx); else # connect (sys1, sys2, ..., sysN, in_idx, out_idx) lti_idx = cellfun (@isa, varargin, {"lti"}); sys = append (varargin{lti_idx}); io_idx = ! lti_idx; if (nnz (io_idx) != 2) error ("connect: require arguments 'inputs' and 'outputs'"); endif in_idx = varargin(io_idx){1}; out_idx = varargin(io_idx){2}; inname = sys.inname; outname = sys.outname; ioname = intersect (inname, outname); tmp = cellfun (@(x) find (strcmp (inname, x)(:)), ioname, "uniformoutput", false); inputs = vertcat (tmp{:}); # there could be more than one input with the same name ## FIXME: sys_prune will error out if names in out_idx and in_idx are not unique ## the dark side handles cases with common in_idx names [p, m] = size (sys); M = zeros (m, p); for k = 1 : length (inputs) outputs = strcmp (outname, inname(inputs(k))); M(inputs(k), :) = outputs; endfor sys = __sys_connect__ (sys, M); sys = __sys_prune__ (sys, out_idx, in_idx); if (isa (sys, "ss")) sys = sminreal (sys); endif endif endfunction %!shared T, Texp %! P = Boeing707; %! I = ss (-eye (2)); %! I.inname = P.outname; %! I.outname = P.inname; %! T = connect (P, I, P.inname, P.outname); %! Texp = feedback (P); %!assert (T.a, Texp.a, 1e-4); %!assert (T.b, Texp.b, 1e-4); %!assert (T.c, Texp.c, 1e-4); %!assert (T.d, Texp.d, 1e-4); control/inst/@lti/ctranspose.m0000644000076500000240000000325412273411257015754 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Conjugate transpose or pertransposition of @acronym{LTI} objects. ## Used by Octave for "sys'". ## For a transfer-function matrix G, G' denotes the conjugate ## of G given by G.'(-s) for a continuous-time system or G.'(1/z) ## for a discrete-time system. ## The frequency response of the pertransposition of G is the ## Hermitian (conjugate) transpose of G(jw), i.e. ## freqresp (G', w) = freqresp (G, w)'. ## @strong{WARNING:} Do @strong{NOT} use this for dual problems, ## use the transpose "sys.'" (note the dot) instead. ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function sys = ctranspose (sys) if (nargin != 1) # prevent sys = ctranspose (sys1, sys2, sys3, ...) error ("lti: ctranspose: this is an unary operator"); endif [p, m] = size (sys); ct = isct (sys); sys = __ctranspose__ (sys, ct); sys.inname = repmat ({""}, p, 1); sys.outname = repmat ({""}, m, 1); endfunction control/inst/@lti/d2c.m0000644000076500000240000000675412273411257014253 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} d2c (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} d2c (@var{sys}, @var{method}) ## @deftypefnx {Function File} {@var{sys} =} d2c (@var{sys}, @var{'prewarp'}, @var{w0}) ## Convert the discrete @acronym{LTI} model into its continuous-time equivalent. ## ## @strong{Inputs} ## @table @var ## @item sys ## Discrete-time @acronym{LTI} model. ## @item method ## Optional conversion method. If not specified, default method @var{"zoh"} ## is taken. ## @table @var ## @item 'zoh' ## Zero-order hold or matrix logarithm. ## @item 'tustin', 'bilin' ## Bilinear transformation or Tustin approximation. ## @item 'prewarp' ## Bilinear transformation with pre-warping at frequency @var{w0}. ## @item 'matched' ## Matched pole/zero method. ## @end table ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Continuous-time @acronym{LTI} model. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2011 ## Version: 0.2 function sys = d2c (sys, method = "std", w0 = 0) if (nargin == 0 || nargin > 3) print_usage (); endif if (! isa (sys, "lti")) error ("d2c: first argument is not an LTI model"); endif if (isct (sys)) error ("d2c: system is already continuous-time"); endif if (! ischar (method)) error ("d2c: second argument is not a string"); endif if (! issample (w0, 0)) error ("d2c: third argument is not a valid pre-warping frequency"); endif sys = __d2c__ (sys, sys.tsam, lower (method), w0); sys.tsam = 0; endfunction ## bilinear transformation ## both directions %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ]; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ]; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ]; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ]; %! %! [Ao, Bo, Co, Do] = ssdata (d2c (c2d (ss (A, B, C, D), 2, "tustin"), "tustin")); %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [A, B; C, D]; %! %!assert (Mo, Me, 1e-4); ## zero-order hold ## both directions %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ]; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ]; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ]; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ]; %! %! [Ao, Bo, Co, Do] = ssdata (d2c (c2d (ss (A, B, C, D), 2, "zoh"), "zoh")); %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [A, B; C, D]; %! %!assert (Mo, Me, 1e-4); ## bilinear transformation with pre-warping ## both directions %!shared Mo, Me %! A = [ 1.0 0.5 %! 0.5 1.0 ]; %! %! B = [ 0.0 -1.0 %! 1.0 0.0 ]; %! %! C = [ -1.0 0.0 %! 0.0 1.0 ]; %! %! D = [ 1.0 0.0 %! 0.0 -1.0 ]; %! %! [Ao, Bo, Co, Do] = ssdata (d2c (c2d (ss (A, B, C, D), 2, "prewarp", 1000), "prewarp", 1000)); %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [A, B; C, D]; %! %!assert (Mo, Me, 1e-4); control/inst/@lti/d2d.m0000644000076500000240000000460212273411257014242 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} d2d (@var{sys}, @var{tsam}) ## @deftypefnx {Function File} {@var{sys} =} d2d (@var{sys}, @var{tsam}, @var{method}) ## @deftypefnx {Function File} {@var{sys} =} d2d (@var{sys}, @var{tsam}, @var{'prewarp'}, @var{w0}) ## Resample discrete-time @acronym{LTI} model to sampling time @var{tsam}. ## ## @strong{Inputs} ## @table @var ## @item sys ## Discrete-time @acronym{LTI} model. ## @item tsam ## Desired sampling time in seconds. ## @item method ## Optional conversion method. If not specified, default method @var{"zoh"} ## is taken. ## @table @var ## @item 'zoh' ## Zero-order hold or matrix logarithm. ## @item 'tustin', 'bilin' ## Bilinear transformation or Tustin approximation. ## @item 'prewarp' ## Bilinear transformation with pre-warping at frequency @var{w0}. ## @item 'matched' ## Matched pole/zero method. ## @end table ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Resampled discrete-time @acronym{LTI} model with sampling time @var{tsam}. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2013 ## Version: 0.1 function sys = d2d (sys, tsam, method = "std", w0 = 0) if (nargin < 2) print_usage (); endif tmp = d2c (sys, method, w0); sys = c2d (tmp, tsam, method, w0); endfunction %!shared num, den, z, p, k %! H1 = zpk (0.7, 0.5, 1, 0.1); %! H2 = d2d (H1, 0.05); %! H3 = d2d (H2, 0.1); %! [num, den] = tfdata (H2, "vector"); %! [z, p, k] = zpkdata (H3, "vector"); %!assert (num, [1.00000 -0.82426], 1e-4); %!assert (den, [1.00000 -0.70711], 1e-4); %!assert (z, 0.7, 1e-4); %!assert (p, 0.5, 1e-4); %!assert (k, 1.0, 1e-4); control/inst/@lti/dcgain.m0000644000076500000240000000257012273411257015020 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{k} =} dcgain (@var{sys}) ## DC gain of @acronym{LTI} model. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @end table ## ## @strong{Outputs} ## @table @var ## @item k ## DC gain matrice. For a system with m inputs and p outputs, the array @var{k} ## has dimensions [p, m]. ## @end table ## ## @seealso{freqresp} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function gain = dcgain (sys) if (nargin != 1) # sys is always an LTI model print_usage (); endif gain = __freqresp__ (sys, 0); endfunction control/inst/@lti/display.m0000644000076500000240000000301412273411257015232 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Display routine for LTI objects. Called by its child classes. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function display (sys) if (nfields (sys.ingroup) > 0) __disp_group__ (sys.ingroup, "Input"); endif if (nfields (sys.outgroup) > 0) __disp_group__ (sys.outgroup, "Output"); endif if (sys.tsam > 0) disp (sprintf ("Sampling time: %g s", sys.tsam)); elseif (sys.tsam == -1) disp ("Sampling time: unspecified"); endif endfunction function __disp_group__ (group, io) name = fieldnames (group); idx = struct2cell (group); cellfun (@(name, idx) printf ("%s group '%s' = %s\n", io, name, mat2str (idx(:).')), ... name, idx, "uniformoutput", false); endfunction control/inst/@lti/dss.m0000644000076500000240000000243512273411257014364 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert to descriptor state-space model. ## Since it is impossible to call a function inside @NAME without any ## argument of class NAME (except the constructor @NAME/NAME.m), ## the "constructor" for dss models is located outside folder @ss. ## Author: Lukas Reichlin ## Created: September 2010 ## Version: 0.1 function sys = dss (sys) if (nargin != 1) print_usage (); endif if (! isa (sys, "ss")) sys = ss (sys); endif a = __sys_data__ (sys); sys = __set__ (sys, "e", eye (size (a))); endfunction control/inst/@lti/dssdata.m0000644000076500000240000000473512273411257015223 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}] =} dssdata (@var{sys}) ## @deftypefnx {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}] =} dssdata (@var{sys}, @var{[]}) ## Access descriptor state-space model data. ## Argument @var{sys} is not limited to descriptor state-space models. ## If @var{sys} is not a descriptor state-space model, it is converted automatically. ## ## @strong{Inputs} ## @table @var ## @item sys ## Any type of @acronym{LTI} model. ## @item [] ## In case @var{sys} is not a dss model (descriptor matrix @var{e} empty), ## @code{dssdata (sys, [])} returns the empty element @code{e = []} whereas ## @code{dssdata (sys)} returns the identity matrix @code{e = eye (size (a))}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item a ## State matrix (n-by-n). ## @item b ## Input matrix (n-by-m). ## @item c ## Measurement matrix (p-by-n). ## @item d ## Feedthrough matrix (p-by-m). ## @item e ## Descriptor matrix (n-by-n). ## @item tsam ## Sampling time in seconds. If @var{sys} is a continuous-time model, ## a zero is returned. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2010 ## Version: 0.2 function [a, b, c, d, e, tsam, scaled] = dssdata (sys, flg = 0) ## NOTE: In case sys is not a dss model (matrice e empty), ## dssdata (sys, []) returns e = [] whereas ## dssdata (sys) returns e = eye (size (a)) if (nargin > 2) print_usage (); endif if (! isa (sys, "ss")) sys = ss (sys); endif [a, b, c, d, e, ~, scaled] = __sys_data__ (sys); if (isempty (e) && ! isempty (flg)) e = eye (size (a)); # return eye for ss models endif tsam = sys.tsam; endfunction control/inst/@lti/eig.m0000644000076500000240000000201712273411257014333 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{p} =} eig (@var{sys}) ## Compute poles of @acronym{LTI} system. ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function pol = eig (varargin) pol = pole (varargin{:}); endfunction control/inst/@lti/end.m0000644000076500000240000000233212273411257014335 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## End indexing for @acronym{LTI} objects. ## Used by Octave for "sys(1:end, end-1)". ## Author: Lukas Reichlin ## Created: December 2013 ## Version: 0.1 function ret = end (sys, k, n) if (n != 2) error ("lti: end: require 2 indices in the expression"); endif [p, m] = size (sys); switch (k) case 1 ret = p; case 2 ret = m; otherwise error ("lti: end: invalid expression index k = %d", k); endswitch endfunction control/inst/@lti/feedback.m0000644000076500000240000002075212273411257015321 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} feedback (@var{sys1}) ## @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{"+"}) ## @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}) ## @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{"+"}) ## @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{feedin}, @var{feedout}) ## @deftypefnx {Function File} {@var{sys} =} feedback (@var{sys1}, @var{sys2}, @var{feedin}, @var{feedout}, @var{"+"}) ## Feedback connection of two @acronym{LTI} models. ## ## @strong{Inputs} ## @table @var ## @item sys1 ## @acronym{LTI} model of forward transmission. @code{[p1, m1] = size (sys1)}. ## @item sys2 ## @acronym{LTI} model of backward transmission. ## If not specified, an identity matrix of appropriate size is taken. ## @item feedin ## Vector containing indices of inputs to @var{sys1} which are involved in the feedback loop. ## The number of @var{feedin} indices and outputs of @var{sys2} must be equal. ## If not specified, @code{1:m1} is taken. ## @item feedout ## Vector containing indices of outputs from @var{sys1} which are to be connected to @var{sys2}. ## The number of @var{feedout} indices and inputs of @var{sys2} must be equal. ## If not specified, @code{1:p1} is taken. ## @item "+" ## Positive feedback sign. If not specified, @var{"-"} for a negative feedback interconnection ## is assumed. @var{+1} and @var{-1} are possible as well, but only from the third argument ## onward due to ambiguity. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Resulting @acronym{LTI} model. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## u + +--------+ y ## ------>(+)----->| sys1 |-------+-------> ## ^ - +--------+ | ## | | ## | +--------+ | ## +-------| sys2 |<------+ ## +--------+ ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.7 function sys = feedback (sys1, sys2, feedin, feedout, fbsign = -1) [p1, m1] = size (sys1); switch (nargin) case 1 # sys = feedback (sys) if (p1 != m1) error ("feedback: argument must be a square system"); endif sys2 = eye (p1); feedin = 1 : m1; feedout = 1 : p1; case 2 if (ischar (sys2)) # sys = feedback (sys, "+") if (p1 != m1) error ("feedback: first argument must be a square system"); endif fbsign = __check_fbsign__ (sys2); sys2 = eye (p1); endif # sys = feedback (sys1, sys2) feedin = 1 : m1; feedout = 1 : p1; case 3 # sys = feedback (sys1, sys2, "+") fbsign = __check_fbsign__ (feedin); feedin = 1 : m1; feedout = 1 : p1; case 4 # sys = feedback (sys1, sys2, feedin, feedout) ## nothing needs to be done here ## case 4 required to prevent "otherwise" case 5 # sys = feedback (sys1, sys2, feedin, feedout, "+") fbsign = __check_fbsign__ (fbsign); otherwise print_usage (); endswitch if (ischar (feedin)) feedin = {feedin}; endif if (ischar (feedout)) feedout = {feedout}; endif if (iscell (feedin)) tmp = cellfun (@(x) __str2idx__ (sys1.ingroup, sys1.inname, x, "in"), feedin, "uniformoutput", false); feedin = vertcat (tmp{:}); endif if (iscell (feedout)) tmp = cellfun (@(x) __str2idx__ (sys1.outgroup, sys1.outname, x, "out"), feedout, "uniformoutput", false); feedout = vertcat (tmp{:}); endif if (! is_real_vector (feedin) || ! isequal (feedin, abs (fix (feedin)))) error ("feedback: require 'feedin' to be a vector of integers"); endif if (! is_real_vector (feedout) || ! isequal (feedout, abs (fix (feedout)))) error ("feedback: require 'feedout' to be a vector of integers"); endif [p2, m2] = size (sys2); l_feedin = length (feedin); l_feedout = length (feedout); if (l_feedin != p2) error ("feedback: feedin indices: %d, outputs sys2: %d", l_feedin, p2); endif if (l_feedout != m2) error ("feedback: feedout indices: %d, inputs sys2: %d", l_feedout, m2); endif if (any (feedin > m1 | feedin < 1)) error ("feedback: range of feedin indices exceeds dimensions of sys1"); endif if (any (feedout > p1 | feedout < 1)) error ("feedback: range of feedout indices exceeds dimensions of sys1"); endif M11 = zeros (m1, p1); M22 = zeros (m2, p2); M12 = full (sparse (feedin, 1:l_feedin, fbsign, m1, p2)); M21 = full (sparse (1:l_feedout, feedout, 1, m2, p1)); ## NOTE: for-loops do NOT the same as ## M12(feedin, 1:l_feedin) = fbsign; ## M21(1:l_feedout, feedout) = 1; ## ## M12 = zeros (m1, p2); ## M21 = zeros (m2, p1); ## ## for k = 1 : l_feedin ## M12(feedin(k), k) = fbsign; ## endfor ## ## for k = 1 : l_feedout ## M21(k, feedout(k)) = 1; ## endfor M = [M11, M12; M21, M22]; in_idx = 1 : m1; out_idx = 1 : p1; sys = __sys_group__ (sys1, sys2); sys = __sys_connect__ (sys, M); sys = __sys_prune__ (sys, out_idx, in_idx); endfunction function fbsign = __check_fbsign__ (fbsign) if (is_real_scalar (fbsign)) fbsign = sign (fbsign); elseif (ischar (fbsign)) if (strcmp (fbsign, "+")) fbsign = +1; elseif (strcmp (fbsign, "-")) fbsign = -1; else error ("feedback: invalid feedback sign string"); endif else error ("feedback: invalid feedback sign type"); endif endfunction ## Feedback inter-connection of two systems in state-space form ## Test from SLICOT AB05ND %!shared M, Me %! A1 = [ 1.0 0.0 -1.0 %! 0.0 -1.0 1.0 %! 1.0 1.0 2.0 ]; %! %! B1 = [ 1.0 1.0 0.0 %! 2.0 0.0 1.0 ].'; %! %! C1 = [ 3.0 -2.0 1.0 %! 0.0 1.0 0.0 ]; %! %! D1 = [ 1.0 0.0 %! 0.0 1.0 ]; %! %! A2 = [-3.0 0.0 0.0 %! 1.0 0.0 1.0 %! 0.0 -1.0 2.0 ]; %! %! B2 = [ 0.0 -1.0 0.0 %! 1.0 0.0 2.0 ].'; %! %! C2 = [ 1.0 1.0 0.0 %! 1.0 1.0 -1.0 ]; %! %! D2 = [ 1.0 1.0 %! 0.0 1.0 ]; %! %! sys1 = ss (A1, B1, C1, D1); %! sys2 = ss (A2, B2, C2, D2); %! sys = feedback (sys1, sys2); %! [A, B, C, D] = ssdata (sys); %! M = [A, B; C, D]; %! %! Ae = [-0.5000 -0.2500 -1.5000 -1.2500 -1.2500 0.7500 %! -1.5000 -0.2500 0.5000 -0.2500 -0.2500 -0.2500 %! 1.0000 0.5000 2.0000 -0.5000 -0.5000 0.5000 %! 0.0000 0.5000 0.0000 -3.5000 -0.5000 0.5000 %! -1.5000 1.2500 -0.5000 1.2500 0.2500 1.2500 %! 0.0000 1.0000 0.0000 -1.0000 -2.0000 3.0000 ]; %! %! Be = [ 0.5000 0.7500 %! 0.5000 -0.2500 %! 0.0000 0.5000 %! 0.0000 0.5000 %! -0.5000 0.2500 %! 0.0000 1.0000 ]; %! %! Ce = [ 1.5000 -1.2500 0.5000 -0.2500 -0.2500 -0.2500 %! 0.0000 0.5000 0.0000 -0.5000 -0.5000 0.5000 ]; %! %! De = [ 0.5000 -0.2500 %! 0.0000 0.5000 ]; %! %! Me = [Ae, Be; Ce, De]; %! %!assert (M, Me, 1e-4); ## sensitivity function ## Note the correct physical meaning of the states. ## Test would fail on a commercial octave clone ## because of wrong signs of matrices B and C. ## NOTE: Don't use T = I - S for complementary sensitivity, ## use T = feedback (L) instead! %!shared S1, S2 %! P = ss (-2, 3, 4, 5); # meaningless numbers %! C = ss (-1, 1, 1, 0); # ditto %! L = P * C; %! I = eye (size (L)); %! S1 = feedback (I, L*-I, "+"); # draw a block diagram for better understanding %! S2 = inv (I + L); %!assert (S1.a, S2.a, 1e-4); %!assert (S1.b, S2.b, 1e-4); %!assert (S1.c, S2.c, 1e-4); %!assert (S1.d, S2.d, 1e-4); control/inst/@lti/filtdata.m0000644000076500000240000000621312273411257015361 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{num}, @var{den}, @var{tsam}] =} filtdata (@var{sys}) ## @deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} filtdata (@var{sys}, @var{"vector"}) ## Access discrete-time transfer function data in DSP format. ## Argument @var{sys} is not limited to transfer function models. ## If @var{sys} is not a transfer function, it is converted automatically. ## ## @strong{Inputs} ## @table @var ## @item sys ## Any type of discrete-time @acronym{LTI} model. ## @item "v", "vector" ## For SISO models, return @var{num} and @var{den} directly as column vectors ## instead of cells containing a single column vector. ## @end table ## ## @strong{Outputs} ## @table @var ## @item num ## Cell of numerator(s). Each numerator is a row vector ## containing the coefficients of the polynomial in ascending powers of z^-1. ## num@{i,j@} contains the numerator polynomial from input j to output i. ## In the SISO case, a single vector is possible as well. ## @item den ## Cell of denominator(s). Each denominator is a row vector ## containing the coefficients of the polynomial in ascending powers of z^-1. ## den@{i,j@} contains the denominator polynomial from input j to output i. ## In the SISO case, a single vector is possible as well. ## @item tsam ## Sampling time in seconds. If @var{tsam} is not specified, -1 is returned. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function [num, den, tsam] = filtdata (sys, rtype = "cell") if (nargin > 2) print_usage (); endif if (! isdt (sys)) error ("lti: filtdata: require discrete-time system"); endif [num, den, tsam] = tfdata (sys); ## make numerator and denominator polynomials equally long ## by adding leading zeros lnum = cellfun (@length, num, "uniformoutput", false); lden = cellfun (@length, den, "uniformoutput", false); lmax = cellfun (@max, lnum, lden, "uniformoutput", false); num = cellfun (@prepad, num, lmax, "uniformoutput", false); den = cellfun (@prepad, den, lmax, "uniformoutput", false); ## remove trailing zeros ## such that polynomials are as short as possible num = cellfun (@__remove_trailing_zeros__, num, "uniformoutput", false); den = cellfun (@__remove_trailing_zeros__, den, "uniformoutput", false); if (strncmpi (rtype, "v", 1) && issiso (sys)) num = num{1}; den = den{1}; endif endfunction control/inst/@lti/frdata.m0000644000076500000240000000426012273411257015032 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{H}, @var{w}, @var{tsam}] =} frdata (@var{sys}) ## @deftypefnx {Function File} {[@var{H}, @var{w}, @var{tsam}] =} frdata (@var{sys}, @var{"vector"}) ## Access frequency response data. ## Argument @var{sys} is not limited to frequency response data objects. ## If @var{sys} is not a frd object, it is converted automatically. ## ## @strong{Inputs} ## @table @var ## @item sys ## Any type of @acronym{LTI} model. ## @item "v", "vector" ## In case @var{sys} is a SISO model, this option returns the frequency response ## as a column vector (lw-by-1) instead of an array (p-by-m-by-lw). ## @end table ## ## @strong{Outputs} ## @table @var ## @item H ## Frequency response array (p-by-m-by-lw). H(i,j,k) contains the ## response from input j to output i at frequency k. In the SISO case, ## a vector (lw-by-1) is possible as well. ## @item w ## Frequency vector (lw-by-1) in radian per second [rad/s]. ## Frequencies are in ascending order. ## @item tsam ## Sampling time in seconds. If @var{sys} is a continuous-time model, ## a zero is returned. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.3 function [H, w, tsam] = frdata (sys, rtype = "array") if (! isa (sys, "frd")) sys = frd (sys); endif [H, w] = __sys_data__ (sys); tsam = sys.tsam; if (strncmpi (rtype, "v", 1) && issiso (sys)) H = reshape (H, [], 1); endif endfunction control/inst/@lti/freqresp.m0000644000076500000240000000330612273411257015420 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{H} =} freqresp (@var{sys}, @var{w}) ## Evaluate frequency response at given frequencies. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @item w ## Vector of frequency values. ## @end table ## ## @strong{Outputs} ## @table @var ## @item H ## Array of frequency response. For a system with m inputs and p outputs, the array @var{H} ## has dimensions [p, m, length (w)]. ## The frequency response at the frequency w(k) is given by H(:,:,k). ## @end table ## ## @seealso{dcgain} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function H = freqresp (sys, w) if (nargin != 2) # case freqresp () not possible print_usage (); endif if (! is_real_vector (w)) # catches freqresp (sys, sys) and freqresp (w, sys) as well error ("freqresp: second argument must be a real vector"); endif H = __freqresp__ (sys, w); endfunction control/inst/@lti/get.m0000644000076500000240000000406712273411257014355 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} get (@var{sys}) ## @deftypefnx {Function File} {@var{value} =} get (@var{sys}, @var{"property"}) ## Access property values of @acronym{LTI} objects. ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function varargout = get (sys, varargin) if (nargin == 1) [props, vals] = __property_names__ (sys); nrows = numel (props); str = strjust (strvcat (props), "right"); str = horzcat (repmat (" ", nrows, 1), str, repmat (": ", nrows, 1), strvcat (vals)); disp (str); else for k = 1 : (nargin-1) prop = lower (varargin{k}); switch (prop) case {"inname", "inputname", "inn", "inputn"} val = sys.inname; case {"outname", "outputname", "outn", "outputn"} val = sys.outname; case {"ingroup", "inputgroup", "ing", "inputg"} val = sys.ingroup; case {"outgroup", "outputgroup", "outg", "outputg"} val = sys.outgroup; case {"tsam", "ts"} val = sys.tsam; case "name" val = sys.name; case "notes" val = sys.notes; case "userdata" val = sys.userdata; otherwise val = __get__ (sys, prop); endswitch varargout{k} = val; endfor endif endfunction control/inst/@lti/horzcat.m0000644000076500000240000000603312273411257015243 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Horizontal concatenation of @acronym{LTI} objects. If necessary, object conversion ## is done by sys_group. Used by Octave for "[sys1, sys2]". ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function sys = horzcat (sys, varargin) for k = 1 : (nargin-1) sys1 = sys; sys2 = varargin{k}; [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (p1 != p2) error ("lti: horzcat: number of system outputs incompatible: [(%dx%d), (%dx%d)]", p1, m1, p2, m2); endif sys = __sys_group__ (sys1, sys2); out_scl = [eye(p1), eye(p2)]; sys = out_scl * sys; endfor endfunction ## Rowwise concatenation of two systems in state-space form ## Test from SLICOT AB05OD %!shared M, Me %! A1 = [ 1.0 0.0 -1.0 %! 0.0 -1.0 1.0 %! 1.0 1.0 2.0 ]; %! %! B1 = [ 1.0 1.0 0.0 %! 2.0 0.0 1.0 ].'; %! %! C1 = [ 3.0 -2.0 1.0 %! 0.0 1.0 0.0 ]; %! %! D1 = [ 1.0 0.0 %! 0.0 1.0 ]; %! %! A2 = [-3.0 0.0 0.0 %! 1.0 0.0 1.0 %! 0.0 -1.0 2.0 ]; %! %! B2 = [ 0.0 -1.0 0.0 %! 1.0 0.0 2.0 ].'; %! %! C2 = [ 1.0 1.0 0.0 %! 1.0 1.0 -1.0 ]; %! %! D2 = [ 1.0 1.0 %! 0.0 1.0 ]; %! %! sys1 = ss (A1, B1, C1, D1); %! sys2 = ss (A2, B2, C2, D2); %! sys = [sys1, sys2]; %! [A, B, C, D] = ssdata (sys); %! M = [A, B; C, D]; %! %! Ae = [ 1.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 %! 0.0000 -1.0000 1.0000 0.0000 0.0000 0.0000 %! 1.0000 1.0000 2.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 -3.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 1.0000 0.0000 1.0000 %! 0.0000 0.0000 0.0000 0.0000 -1.0000 2.0000 ]; %! %! Be = [ 1.0000 2.0000 0.0000 0.0000 %! 1.0000 0.0000 0.0000 0.0000 %! 0.0000 1.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 1.0000 %! 0.0000 0.0000 -1.0000 0.0000 %! 0.0000 0.0000 0.0000 2.0000 ]; %! %! Ce = [ 3.0000 -2.0000 1.0000 1.0000 1.0000 0.0000 %! 0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000 ]; %! %! De = [ 1.0000 0.0000 1.0000 1.0000 %! 0.0000 1.0000 0.0000 1.0000 ]; %! %! Me = [Ae, Be; Ce, De]; %! %!assert (M, Me, 1e-4); control/inst/@lti/inv.m0000644000076500000240000000415112273411257014364 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Inversion of @acronym{LTI} objects. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function retsys = inv (sys) if (nargin != 1) # prevent sys = inv (sys1, sys2, sys3, ...) error ("lti: inv: this is an unary operator"); endif [p, m] = size (sys); if (p != m) error ("lti: inv: system must be square"); endif retsys = __sys_inverse__ (sys); ## TODO: handle i/o names endfunction ## inverse of state-space models ## test from SLICOT AB07ND ## result differs intentionally from a commercial ## implementation of an octave-like language %!shared M, Me %! A = [ 1.0 2.0 0.0 %! 4.0 -1.0 0.0 %! 0.0 0.0 1.0 ]; %! %! B = [ 1.0 0.0 %! 0.0 1.0 %! 1.0 0.0 ]; %! %! C = [ 0.0 1.0 -1.0 %! 0.0 0.0 1.0 ]; %! %! D = [ 4.0 0.0 %! 0.0 1.0 ]; %! %! sys = ss (A, B, C, D); %! sysinv = inv (sys); %! [Ai, Bi, Ci, Di] = ssdata (sysinv); %! M = [Ai, Bi; Ci, Di]; %! %! Ae = [ 1.0000 1.7500 0.2500 %! 4.0000 -1.0000 -1.0000 %! 0.0000 -0.2500 1.2500 ]; %! %! Be = [-0.2500 0.0000 %! 0.0000 -1.0000 %! -0.2500 0.0000 ]; %! %! Ce = [ 0.0000 0.2500 -0.2500 %! 0.0000 0.0000 1.0000 ]; %! %! De = [ 0.2500 0.0000 %! 0.0000 1.0000 ]; %! %! Me = [Ae, Be; Ce, De]; # Me = [Ae, -Be; -Ce, De]; %! %!assert (M, Me, 1e-4); control/inst/@lti/isct.m0000644000076500000240000000261312273411257014533 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} isct (@var{sys}) ## Determine whether @acronym{LTI} model is a continuous-time system. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool = 0 ## @var{sys} is a discrete-time system. ## @item bool = 1 ## @var{sys} is a continuous-time system or a static gain. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function bool = isct (ltisys) if (nargin != 1) print_usage (); endif bool = (ltisys.tsam == 0 || ltisys.tsam == -2); endfunction control/inst/@lti/isdt.m0000644000076500000240000000256412273411257014541 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} isdt (@var{sys}) ## Determine whether @acronym{LTI} model is a discrete-time system. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool = 0 ## @var{sys} is a continuous-time system. ## @item bool = 1 ## @var{sys} is a discrete-time system or a static gain. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function bool = isdt (ltisys) if (nargin != 1) print_usage (); endif bool = (ltisys.tsam != 0); endfunction control/inst/@lti/isminimumphase.m0000644000076500000240000000364512273411257016627 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} isminimumphase (@var{sys}) ## @deftypefnx {Function File} {@var{bool} =} isminimumphase (@var{sys}, @var{tol}) ## Determine whether @acronym{LTI} system is minimum-phase. ## The zeros must lie in the left complex half-plane. ## The name minimum-phase refers to the fact that such a system has the ## minimum possible phase lag for the given magnitude response |sys(jw)|. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @item tol ## Optional tolerance. ## @var{tol} must be a real-valued, non-negative scalar. ## Default value is 0. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool ## True if the system is minimum-phase and false otherwise. ## @end table ## ## @example ## @group ## real (z) < -tol*(1 + abs (z)) continuous-time ## abs (z) < 1 - tol discrete-time ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2011 ## Version: 0.1 function bool = isminimumphase (sys, tol = 0) if (nargin > 2) print_usage (); endif z = zero (sys); ct = isct (sys); bool = __is_stable__ (z, ct, tol); endfunction control/inst/@lti/issiso.m0000644000076500000240000000215712273411257015105 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} issiso (@var{sys}) ## Determine whether @acronym{LTI} model is single-input/single-output (SISO). ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function bool = issiso (sys) if (nargin != 1) print_usage (); endif bool = all (size (sys) == 1); endfunction control/inst/@lti/isstable.m0000644000076500000240000000331712273411257015401 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} isstable (@var{sys}) ## @deftypefnx {Function File} {@var{bool} =} isstable (@var{sys}, @var{tol}) ## Determine whether @acronym{LTI} system is stable. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @item tol ## Optional tolerance for stability. ## @var{tol} must be a real-valued, non-negative scalar. ## Default value is 0. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool ## True if the system is stable and false otherwise. ## @end table ## ## @example ## @group ## real (p) < -tol*(1 + abs (p)) continuous-time ## abs (p) < 1 - tol discrete-time ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function bool = isstable (sys, tol = 0) if (nargin > 2) print_usage (); endif pol = pole (sys); ct = isct (sys); bool = __is_stable__ (pol, ct, tol); endfunction control/inst/@lti/lft.m0000644000076500000240000001131512273411257014355 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} lft (@var{sys1}, @var{sys2}) ## @deftypefnx {Function File} {@var{sys} =} lft (@var{sys1}, @var{sys2}, @var{nu}, @var{ny}) ## Linear fractional tranformation, also known as Redheffer star product. ## ## @strong{Inputs} ## @table @var ## @item sys1 ## Upper @acronym{LTI} model. ## @item sys2 ## Lower @acronym{LTI} model. ## @item nu ## The last nu inputs of @var{sys1} are connected with the first nu outputs of @var{sys2}. ## If not specified, @code{min (m1, p2)} is taken. ## @item ny ## The last ny outputs of @var{sys1} are connected with the first ny inputs of @var{sys2}. ## If not specified, @code{min (p1, m2)} is taken. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Resulting @acronym{LTI} model. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## .............sys.............. ## : +--------+ : ## w1 ------------>| |------------> z1 ## : | sys1 | : ## : u +---->| |-----+ y : ## : | +--------+ | : Lower LFT ## : | | : ## : | +--------+ | : lft (sys1, sys2) ## : +-----| sys2 |<----+ : ## : +--------+ : ## :............................: ## @end group ## @end example ## @example ## @group ## .............sys.............. ## : +--------+ : ## : u +---->| sys1 |-----+ y : ## : | +--------+ | : Upper LFT ## : | | : ## : | +--------+ | : lft (sys1, sys2) ## : +-----| |<----+ : ## : | sys2 | : ## z2 <------------| |<------------ w2 ## : +--------+ : ## :............................: ## @end group ## @end example ## @example ## @group ## .............sys.............. ## : +--------+ : ## w1 ------------>| |------------> z1 ## : | sys1 | : ## : u +---->| |-----+ y : ## : | +--------+ | : ## : | | : lft (sys1, sys2, nu, ny) ## : | +--------+ | : ## : +-----| |<----+ : ## : | sys2 | : ## z2 <------------| |<------------ w2 ## : +--------+ : ## :............................: ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function sys = lft (sys1, sys2, nu, ny) if (nargin != 2 && nargin != 4) print_usage (); endif ## object conversion done by sys_group if necessary [p1, m1] = size (sys1); [p2, m2] = size (sys2); nu_max = min (m1, p2); ny_max = min (m2, p1); if (nargin == 2) # sys = lft (sys1, sys2) nu = nu_max; ny = ny_max; else # sys = lft (sys1, sys2, nu, ny) if (! is_real_scalar (nu) || nu < 0) error ("lft: argument 'nu' must be a positive integer"); endif if (! is_real_scalar (ny) || ny < 0) error ("lft: argument 'ny' must be a positive integer"); endif if (nu > nu_max) error ("lft: argument 'nu' (%d) must be at most %d", nu, nu_max); endif if (ny > ny_max) error ("lft: argument 'ny' (%d) must be at most %d", ny, ny_max); endif endif M11 = zeros (m1, p1); M12 = [zeros(m1-nu, p2); eye(nu), zeros(nu, p2-nu)]; M21 = [zeros(ny, p1-ny), eye(ny); zeros(m2-ny, p1)]; M22 = zeros (m2, p2); M = [M11, M12; M21, M22]; in_idx = [1 : (m1-nu), m1 + (ny+1 : m2)]; out_idx = [1 : (p1-ny), p1 + (nu+1 : p2)]; sys = __sys_group__ (sys1, sys2); sys = __sys_connect__ (sys, M); sys = __sys_prune__ (sys, out_idx, in_idx); [p, m] = size (sys); if (m == 0) warning ("lft: resulting system has no inputs"); endif if (p == 0) warning ("lft: resulting system has no outputs"); endif endfunction control/inst/@lti/lti.m0000644000076500000240000000252712273411257014365 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Constructor for LTI objects. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function ltisys = lti (p = 0, m = 0, tsam = -2) inname = repmat ({""}, m, 1); outname = repmat ({""}, p, 1); ltisys = struct ("tsam", tsam, "inname", {inname}, "outname", {outname}, "ingroup", struct (), "outgroup", struct (), "name", "", "notes", {{}}, "userdata", []); ltisys = class (ltisys, "lti"); endfunction control/inst/@lti/mconnect.m0000644000076500000240000000503212273411257015375 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} mconnect (@var{sys}, @var{m}) ## @deftypefnx {Function File} {@var{sys} =} mconnect (@var{sys}, @var{m}, @var{inputs}, @var{outputs}) ## Arbitrary interconnections between the inputs and outputs of an @acronym{LTI} model. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @item m ## Connection matrix. Each row belongs to an input and each column represents an output. ## @item inputs ## Vector of indices of those inputs which are retained. If not specified, all inputs are kept. ## @item outputs ## Vector of indices of those outputs which are retained. If not specified, all outputs are kept. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Interconnected system. ## @end table ## ## @strong{Example} ## @example ## @group ## Solve the system equations of ## y(t) = G e(t) ## e(t) = u(t) + M y(t) ## in order to build ## y(t) = H u(t) ## The matrix M for a (p-by-m) system G ## has m rows and p columns (m-by-p). ## ## Example for a 3x2 system: ## u1 = -1*y1 + 5*y2 + 0*y3 ## u2 = pi*y1 + 0*y2 - 7*y3 ## ## | -1 5 0 | ## M = | pi 0 7 | ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function sys = mconnect (sys, M, in_idx, out_idx = ":") if (nargin < 2 || nargin > 4) print_usage (); endif [p, m] = size (sys); [mrows, mcols] = size (M); if (p != mcols || m != mrows) error ("mconnect: second argument must be a (%dx%d) matrix", m, p); endif if (! is_real_matrix (M)) error ("mconnect: second argument must be a matrix with real-valued coefficients"); endif sys = __sys_connect__ (sys, M); if (nargin > 2) sys = __sys_prune__ (sys, out_idx, in_idx); endif endfunction control/inst/@lti/minreal.m0000644000076500000240000001334312273411257015222 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} minreal (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} minreal (@var{sys}, @var{tol}) ## Minimal realization or zero-pole cancellation of @acronym{LTI} models. ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function sys = minreal (sys, tol = "def") if (nargin > 2) # nargin == 0 not possible because minreal is inside @lti print_usage (); endif if (! is_real_scalar (tol) && tol != "def") error ("minreal: second argument must be a real scalar"); endif sys = __minreal__ (sys, tol); endfunction ## ss: minreal (SLICOT TB01PD) %!shared C, D %! %! A = ss (-2, 3, 4, 5); %! B = A / A; %! C = minreal (B, 1e-15); %! D = ss (1); %! %!assert (C.a, D.a); %!assert (C.b, D.b); %!assert (C.c, D.c); %!assert (C.d, D.d); %!shared M, Me %! A = [ 1.0 2.0 0.0 %! 4.0 -1.0 0.0 %! 0.0 0.0 1.0 ]; %! %! B = [ 1.0 %! 0.0 %! 1.0 ]; %! %! C = [ 0.0 1.0 -1.0 %! 0.0 0.0 1.0 ]; %! %! D = zeros (2, 1); %! %! sys = ss (A, B, C, D, "scaled", true); %! sysmin = minreal (sys, 0.0); %! [Ar, Br, Cr, Dr] = ssdata (sysmin); %! M = [Ar, Br; Cr, Dr]; %! %! Ae = [ 1.0000 -1.4142 1.4142 %! -2.8284 0.0000 1.0000 %! 2.8284 1.0000 0.0000 ]; %! %! Be = [-1.0000 %! 0.7071 %! 0.7071 ]; %! %! Ce = [ 0.0000 0.0000 -1.4142 %! 0.0000 0.7071 0.7071 ]; %! %! De = zeros (2, 1); %! %! Me = [Ae, Be; Ce, De]; %! %!assert (M, Me, 1e-4); ## dss: minreal (SLICOT TG01JD) ## FIXME: Test fails with larger ldwork in sltg01jd.cc %!shared Ar, Br, Cr, Dr, Er, Ae, Be, Ce, De, Ee %! A = [ -2 -3 0 0 0 0 0 0 0 %! 1 0 0 0 0 0 0 0 0 %! 0 0 -2 -3 0 0 0 0 0 %! 0 0 1 0 0 0 0 0 0 %! 0 0 0 0 1 0 0 0 0 %! 0 0 0 0 0 1 0 0 0 %! 0 0 0 0 0 0 1 0 0 %! 0 0 0 0 0 0 0 1 0 %! 0 0 0 0 0 0 0 0 1 ]; %! %! E = [ 1 0 0 0 0 0 0 0 0 %! 0 1 0 0 0 0 0 0 0 %! 0 0 1 0 0 0 0 0 0 %! 0 0 0 1 0 0 0 0 0 %! 0 0 0 0 0 0 0 0 0 %! 0 0 0 0 1 0 0 0 0 %! 0 0 0 0 0 0 0 0 0 %! 0 0 0 0 0 0 1 0 0 %! 0 0 0 0 0 0 0 1 0 ]; %! %! B = [ 1 0 %! 0 0 %! 0 1 %! 0 0 %! -1 0 %! 0 0 %! 0 -1 %! 0 0 %! 0 0 ]; %! %! C = [ 1 0 1 -3 0 1 0 2 0 %! 0 1 1 3 0 1 0 0 1 ]; %! %! D = zeros (2, 2); %! %! sys = dss (A, B, C, D, E, "scaled", true); %! sysmin = minreal (sys, 0.0); %! [Ar, Br, Cr, Dr, Er] = dssdata (sysmin); %! %! Ae = [ 1.0000 -0.0393 -0.0980 -0.1066 0.0781 -0.2330 0.0777 %! 0.0000 1.0312 0.2717 0.2609 -0.1533 0.6758 -0.3553 %! 0.0000 0.0000 1.3887 0.6699 -0.4281 1.6389 -0.7615 %! 0.0000 0.0000 0.0000 -1.2147 0.2423 -0.9792 0.4788 %! 0.0000 0.0000 0.0000 0.0000 -1.0545 0.5035 -0.2788 %! 0.0000 0.0000 0.0000 0.0000 0.0000 1.6355 -0.4323 %! 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 ]; %! %! Ee = [ 0.4100 0.2590 0.5080 -0.3109 0.0705 0.1429 -0.1477 %! -0.7629 -0.3464 0.0992 -0.3007 0.0619 0.2483 -0.0152 %! 0.1120 -0.2124 -0.4184 -0.1288 0.0569 -0.4213 -0.6182 %! 0.0000 0.1122 -0.0039 0.2771 -0.0758 0.0975 0.3923 %! 0.0000 0.0000 0.3708 -0.4290 0.1006 0.1402 -0.2699 %! 0.0000 0.0000 0.0000 0.0000 0.9458 -0.2211 0.2378 %! 0.0000 0.0000 0.0000 0.5711 0.2648 0.5948 -0.5000 ]; %! %! Be = [ -0.5597 0.2363 %! -0.4843 -0.0498 %! -0.4727 -0.1491 %! 0.1802 1.1574 %! 0.5995 0.1556 %! -0.1729 -0.3999 %! 0.0000 0.2500 ]; %! %! Ce = [ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 3.1524 -1.7500 ]; %! %! De = zeros (2, 2); %! %!assert (Ar, Ae, 1e-4); %!assert (Br, Be, 1e-4); %!assert (Cr, Ce, 1e-4); %!assert (Dr, De, 1e-4); %!assert (Er, Ee, 1e-4); ## tf: minreal %!shared a, b, c, d %! s = tf ("s"); %! G1 = (s+1)*s*5/(s+1)/(s^2+s+1); %! G2 = tf ([1, 1, 1], [2, 2, 2]); %! G1min = minreal (G1); %! G2min = minreal (G2); %! a = G1min.num{1, 1}; %! b = G1min.den{1, 1}; %! c = G2min.num{1, 1}; %! d = G2min.den{1, 1}; %!assert (a, [5, 0], 1e-4); %!assert (b, [1, 1, 1], 1e-4); %!assert (c, 0.5, 1e-4); %!assert (d, 1, 1e-4); control/inst/@lti/minus.m0000644000076500000240000000273112273411257014725 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Binary subtraction of @acronym{LTI} objects. If necessary, object conversion ## is done by sys_group. Used by Octave for "sys1 - sys2". ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function sys = minus (sys1, sys2) if (nargin != 2) # prevent sys = minus (sys1, sys2, sys3, ...) error ("lti: minus: this is a binary operator"); endif [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (p1 != p2 || m1 != m2) error ("lti: minus: system dimensions incompatible: (%dx%d) - (%dx%d)", p1, m1, p2, m2); endif sys = __sys_group__ (sys1, sys2); in_scl = [eye(m1); eye(m2)]; out_scl = [eye(p1), -eye(p2)]; sys = out_scl * sys * in_scl; endfunction control/inst/@lti/mldivide.m0000644000076500000240000000266412273411257015374 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Matrix left division of @acronym{LTI} objects. If necessary, object conversion ## is done by sys_group in mtimes. Used by Octave for "sys1 \\ sys2". ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function sys = mldivide (sys1, sys2) if (nargin != 2) # prevent sys = mldivide (sys1, sys2, sys3, ...) error ("lti: mldivide: this is a binary operator"); endif sys1 = inv (sys1); # let octave decide which inv() it uses [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (m2 != p1) error ("lti: mldivide: system dimensions incompatible: (%dx%d) \ (%dx%d)", p1, m1, p2, m2); endif sys = sys1 * sys2; endfunction control/inst/@lti/mpower.m0000644000076500000240000000324212273411257015101 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Matrix power of @acronym{LTI} objects. The exponent must be an integer. ## Used by Octave for "sys^int". ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function retsys = mpower (sys, e) if (nargin != 2) # prevent sys = mpower (a, b, c, ...) error ("lti: mpower: this is a binary operator"); endif if (! is_real_scalar (e) || e != round (e)) error ("lti: mpower: exponent must be an integer"); endif [p, m] = size (sys); if (p != m) error ("lti: mpower: system must be square"); endif ex = round (abs (e)); # make sure ex is a positive integer switch (sign (e)) case -1 # lti^-ex sys = inv (sys); retsys = sys; case 0 # lti^0 retsys = eye (p); return; case 1 # lti^ex retsys = sys; endswitch for k = 2 : ex retsys = retsys * sys; endfor endfunction control/inst/@lti/mrdivide.m0000644000076500000240000000266412273411257015402 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Matrix right division of @acronym{LTI} objects. If necessary, object conversion ## is done by sys_group in mtimes. Used by Octave for "sys1 / sys2". ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function sys = mrdivide (sys1, sys2) if (nargin != 2) # prevent sys = mrdivide (sys1, sys2, sys3, ...) error ("lti: mrdivide: this is a binary operator"); endif sys2 = inv (sys2); # let octave decide which inv() it uses [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (m2 != p1) error ("lti: mrdivide: system dimensions incompatible: (%dx%d) / (%dx%d)", p1, m1, p2, m2); endif sys = sys1 * sys2; endfunction control/inst/@lti/mtimes.m0000644000076500000240000001002212273411257015060 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Matrix multiplication of @acronym{LTI} objects. If necessary, object conversion ## is done by sys_group. Used by Octave for "sys1 * sys2". ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function sys = mtimes (sys2, sys1) if (nargin != 2) # prevent sys = mtimes (sys1, sys2, sys3, ...) error ("lti: mtimes: this is a binary operator"); endif [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (m2 != p1) error ("lti: mtimes: system dimensions incompatible: (%dx%d) * (%dx%d)", p2, m2, p1, m1); endif M22 = zeros (m2, p2); M21 = eye (m2, p1); M12 = zeros (m1, p2); M11 = zeros (m1, p1); M = [M22, M21; M12, M11]; out_idx = 1 : p2; in_idx = m2 + (1 : m1); sys = __sys_group__ (sys2, sys1); sys = __sys_connect__ (sys, M); sys = __sys_prune__ (sys, out_idx, in_idx); endfunction ## Alternative code: consistency vs. compatibility #{ M11 = zeros (m1, p1); M12 = zeros (m1, p2); M21 = eye (m2, p1); M22 = zeros (m2, p2); M = [M11, M12; M21, M22]; out_idx = p1 + (1 : p2); in_idx = 1 : m1; sys = __sys_group__ (sys1, sys2); #} ## Don't forget to adapt @tf/__sys_connect__.m draft code ## mtimes %!shared sysmat, sysmat_exp %! sys1 = ss ([0, 1; -3, -2], [0; 1], [-5, 1], [2]); %! sys2 = ss ([-10], [1], [-40], [5]); %! sys3 = sys2 * sys1; %! [A, B, C, D] = ssdata (sys3); %! sysmat = [A, B; C, D]; %! A_exp = [ -10 -5 1 %! 0 0 1 %! 0 -3 -2 ]; %! B_exp = [ 2 %! 0 %! 1 ]; %! C_exp = [ -40 -25 5 ]; %! D_exp = [ 10 ]; %! sysmat_exp = [A_exp, B_exp; C_exp, D_exp]; %!assert (sysmat, sysmat_exp) ## Cascade inter-connection of two systems in state-space form ## Test from SLICOT AB05MD ## TODO: order of united state vector: consistency vs. compatibility? #%!shared M, Me #%! A1 = [ 1.0 0.0 -1.0 #%! 0.0 -1.0 1.0 #%! 1.0 1.0 2.0 ]; #%! #%! B1 = [ 1.0 1.0 0.0 #%! 2.0 0.0 1.0 ].'; #%! #%! C1 = [ 3.0 -2.0 1.0 #%! 0.0 1.0 0.0 ]; #%! #%! D1 = [ 1.0 0.0 #%! 0.0 1.0 ]; #%! #%! A2 = [-3.0 0.0 0.0 #%! 1.0 0.0 1.0 #%! 0.0 -1.0 2.0 ]; #%! #%! B2 = [ 0.0 -1.0 0.0 #%! 1.0 0.0 2.0 ].'; #%! #%! C2 = [ 1.0 1.0 0.0 #%! 1.0 1.0 -1.0 ]; #%! #%! D2 = [ 1.0 1.0 #%! 0.0 1.0 ]; #%! #%! sys1 = ss (A1, B1, C1, D1); #%! sys2 = ss (A2, B2, C2, D2); #%! sys = sys2 * sys1; #%! [A, B, C, D] = ssdata (sys); #%! M = [A, B; C, D]; #%! #%! Ae = [ 1.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 #%! 0.0000 -1.0000 1.0000 0.0000 0.0000 0.0000 #%! 1.0000 1.0000 2.0000 0.0000 0.0000 0.0000 #%! 0.0000 1.0000 0.0000 -3.0000 0.0000 0.0000 #%! -3.0000 2.0000 -1.0000 1.0000 0.0000 1.0000 #%! 0.0000 2.0000 0.0000 0.0000 -1.0000 2.0000 ]; #%! #%! Be = [ 1.0000 2.0000 #%! 1.0000 0.0000 #%! 0.0000 1.0000 #%! 0.0000 1.0000 #%! -1.0000 0.0000 #%! 0.0000 2.0000 ]; #%! #%! Ce = [ 3.0000 -1.0000 1.0000 1.0000 1.0000 0.0000 #%! 0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000 ]; #%! #%! De = [ 1.0000 1.0000 #%! 0.0000 1.0000 ]; #%! #%! Me = [Ae, Be; Ce, De]; #%! #%!assert (M, Me, 1e-4); control/inst/@lti/norm.m0000644000076500000240000000701512273411257014545 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{gain} =} norm (@var{sys}, @var{2}) ## @deftypefnx {Function File} {[@var{gain}, @var{wpeak}] =} norm (@var{sys}, @var{inf}) ## @deftypefnx {Function File} {[@var{gain}, @var{wpeak}] =} norm (@var{sys}, @var{inf}, @var{tol}) ## Return H-2 or L-inf norm of @acronym{LTI} model. ## ## @strong{Algorithm}@* ## Uses SLICOT AB13BD and AB13DD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.5 function [gain, varargout] = norm (sys, ntype = "2", tol = 0.01) if (nargin > 3) # norm () is caught by built-in function print_usage (); endif if (is_real_scalar (ntype)) if (ntype == 2) ntype = "2"; elseif (isinf (ntype)) ntype = "inf"; else error ("lti: norm: invalid norm type"); endif elseif (ischar (ntype)) ntype = lower (ntype); else error ("lti: norm: invalid norm type"); endif switch (ntype) case "2" gain = h2norm (sys); case "inf" [gain, varargout{1}] = linfnorm (sys, tol); otherwise error ("lti: norm: invalid norm type"); endswitch endfunction function gain = h2norm (sys) if (isstable (sys)) [a, b, c, d] = ssdata (sys); discrete = ! isct (sys); if (! discrete && any (d(:))) # continuous and non-zero feedthrough gain = inf; else gain = __sl_ab13bd__ (a, b, c, d, discrete); endif else gain = inf; endif endfunction function [gain, wpeak] = linfnorm (sys, tol = 0.01) [a, b, c, d, e, tsam, scaled] = dssdata (sys, []); discrete = ! isct (sys); tol = max (tol, 100*eps); if (isempty (e)) [fpeak, gpeak] = __sl_ab13dd__ (a, a, b, c, d, discrete, false, tol, scaled); # TODO: avoid dummy argument else if (rcond (e) < eps) gain = inf; wpeak = inf; return; else [fpeak, gpeak] = __sl_ab13dd__ (a, e, b, c, d, discrete, true, tol, scaled); endif endif if (fpeak(2) > 0) if (discrete) wpeak = fpeak(1) / abs (tsam); # tsam could be -1 else wpeak = fpeak(1); endif else wpeak = inf; endif if (gpeak(2) > 0) gain = gpeak(1); else gain = inf; endif endfunction ## norm ct %!shared H2, Hinf %! sys = ss (-1, 1, 1, 0); %! H2 = norm (sys, 2); %! Hinf = norm (sys, inf); %!assert (H2, 0.7071, 1.5e-5); %!assert (Hinf, 1, 5e-4); ## norm dt %!shared H2, Hinf %! a = [ 2.417 -1.002 0.5488 %! 2 0 0 %! 0 0.5 0 ]; %! b = [ 1 %! 0 %! 0 ]; %! c = [-0.424 0.436 -0.4552 ]; %! d = [ 1 ]; %! sys = ss (a, b, c, d, 0.1); %! H2 = norm (sys, 2); %! Hinf = norm (sys, inf); %!assert (H2, 1.2527, 1.5e-5); %!assert (Hinf, 2.7, 0.1); control/inst/@lti/parallel.m0000644000076500000240000000321212273411257015361 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{sys} =} parallel (@var{sys1}, @var{sys2}) ## Parallel connection of two @acronym{LTI} systems. ## ## @strong{Block Diagram} ## @example ## @group ## .......................... ## : +--------+ : ## : +-->| sys1 |---+ : ## u : | +--------+ | + : y ## -------+ O---------> ## : | +--------+ | + : ## : +-->| sys2 |---+ : ## : +--------+ : ## :.........sys............: ## ## sys = parallel (sys1, sys2) ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function sys = parallel (sys1, sys2) if (nargin == 2) sys = sys1 + sys2; ## elseif (nargin == 6) ## TODO: implement "complicated" case sys = parallel (sys1, sys2, in1, in2, out1, out2) else print_usage (); endif endfunction control/inst/@lti/plus.m0000644000076500000240000000577312273411257014566 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Binary addition of @acronym{LTI} objects. If necessary, object conversion ## is done by sys_group. Used by Octave for "sys1 + sys2". ## Operation is also known as "parallel connection". ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function sys = plus (sys1, sys2) if (nargin != 2) # prevent sys = plus (sys1, sys2, sys3, ...) error ("lti: plus: this is a binary operator"); endif [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (p1 != p2 || m1 != m2) error ("lti: plus: system dimensions incompatible: (%dx%d) + (%dx%d)", p1, m1, p2, m2); endif sys = __sys_group__ (sys1, sys2); in_scl = [eye(m1); eye(m2)]; out_scl = [eye(p1), eye(p2)]; sys = out_scl * sys * in_scl; endfunction ## Parallel inter-connection of two systems in state-space form ## Test from SLICOT AB05PD %!shared M, Me %! A1 = [ 1.0 0.0 -1.0 %! 0.0 -1.0 1.0 %! 1.0 1.0 2.0 ]; %! %! B1 = [ 1.0 1.0 0.0 %! 2.0 0.0 1.0 ].'; %! %! C1 = [ 3.0 -2.0 1.0 %! 0.0 1.0 0.0 ]; %! %! D1 = [ 1.0 0.0 %! 0.0 1.0 ]; %! %! A2 = [-3.0 0.0 0.0 %! 1.0 0.0 1.0 %! 0.0 -1.0 2.0 ]; %! %! B2 = [ 0.0 -1.0 0.0 %! 1.0 0.0 2.0 ].'; %! %! C2 = [ 1.0 1.0 0.0 %! 1.0 1.0 -1.0 ]; %! %! D2 = [ 1.0 1.0 %! 0.0 1.0 ]; %! %! sys1 = ss (A1, B1, C1, D1); %! sys2 = ss (A2, B2, C2, D2); %! sys = sys1 + sys2; %! [A, B, C, D] = ssdata (sys); %! M = [A, B; C, D]; %! %! Ae = [ 1.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 %! 0.0000 -1.0000 1.0000 0.0000 0.0000 0.0000 %! 1.0000 1.0000 2.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 -3.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 1.0000 0.0000 1.0000 %! 0.0000 0.0000 0.0000 0.0000 -1.0000 2.0000 ]; %! %! Be = [ 1.0000 2.0000 %! 1.0000 0.0000 %! 0.0000 1.0000 %! 0.0000 1.0000 %! -1.0000 0.0000 %! 0.0000 2.0000 ]; %! %! Ce = [ 3.0000 -2.0000 1.0000 1.0000 1.0000 0.0000 %! 0.0000 1.0000 0.0000 1.0000 1.0000 -1.0000 ]; %! %! De = [ 2.0000 1.0000 %! 0.0000 2.0000 ]; %! %! Me = [Ae, Be; Ce, De]; %! %!assert (M, Me, 1e-4); control/inst/@lti/pole.m0000644000076500000240000000310412273411257014524 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{p} =} pole (@var{sys}) ## Compute poles of @acronym{LTI} system. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @end table ## ## @strong{Outputs} ## @table @var ## @item p ## Poles of @var{sys}. ## @end table ## ## @strong{Algorithm}@* ## For (descriptor) state-space models, @command{pole} ## relies on Octave's @command{eig}. ## For @acronym{SISO} transfer functions, @command{pole} ## uses Octave's @command{roots}. ## @acronym{MIMO} transfer functions are converted to ## a @emph{minimal} state-space representation for the ## computation of the poles. ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function pol = pole (sys) if (nargin > 1) print_usage (); endif pol = __pole__ (sys); endfunction control/inst/@lti/prescale.m0000644000076500000240000002502212273411257015366 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{scaledsys}, @var{info}] =} prescale (@var{sys}) ## Scale state-space model. The scaled model @var{scaledsys} is equivalent to ## @var{sys}, but the state vector is scaled by diagonal transformation matrices ## in order to increase the accuracy of subsequent numerical computations. ## Frequency response commands perform automatic scaling unless model property ## @var{scaled} is set to @var{true}. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @end table ## ## @strong{Outputs} ## @table @var ## @item scaledsys ## Scaled state-space model. ## @item info ## Structure containing additional information. ## @item info.SL ## Left scaling factors. @code{Tl = diag (info.SL)}. ## @item info.SR ## Right scaling factors. @code{Tr = diag (info.SR)}. ## @end table ## ## @strong{Equations} ## @example ## @group ## Es = Tl * E * Tr ## As = Tl * A * Tr ## Bs = Tl * B ## Cs = C * Tr ## Ds = D ## @end group ## @end example ## ## For proper state-space models, @var{Tl} and @var{Tr} are inverse of each other. ## ## @strong{Algorithm}@* ## Uses SLICOT TB01ID and TG01AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.}. ## @end deftypefn ## Author: Lukas Reichlin ## Created: June 2011 ## Version: 0.2 function [retsys, varargout] = prescale (sys) if (nargin != 1) print_usage (); endif if (! isa (sys, "ss")) warning ("prescale: system not in state-space form"); sys = ss (sys); endif [retsys, lscale, rscale] = __prescale__ (sys); if (nargout > 1) varargout{1} = struct ("SL", lscale, "SR", rscale); endif endfunction ## Scaling of state-space models, direct access to TB01ID %!shared Ao, Bo, Co, SCALEo, MAXREDo, Ae, Be, Ce, SCALEe, MAXREDe %! A = [ 0.0 1.0000e+000 0.0 0.0 0.0 %! -1.5800e+006 -1.2570e+003 0.0 0.0 0.0 %! 3.5410e+014 0.0 -1.4340e+003 0.0 -5.3300e+011 %! 0.0 0.0 0.0 0.0 1.0000e+000 %! 0.0 0.0 0.0 -1.8630e+004 -1.4820e+000 ]; %! %! B = [ 0.0 0.0 %! 1.1030e+002 0.0 %! 0.0 0.0 %! 0.0 0.0 %! 0.0 8.3330e-003 ]; %! %! C = [ 1.0000e+000 0.0 0.0 0.0 0.0 %! 0.0 0.0 1.0000e+000 0.0 0.0 %! 0.0 0.0 0.0 1.0000e+000 0.0 %! 6.6640e-001 0.0 -6.2000e-013 0.0 0.0 %! 0.0 0.0 -1.0000e-003 1.8960e+006 1.5080e+002 ]; %! %! MAXRED = 0.0; %! %! [Ao, Bo, Co, MAXREDo, SCALEo] = __sl_tb01id__ (A, B, C, MAXRED); %! %! Ae = [ 0.0000000D+00 0.1000000D+05 0.0000000D+00 0.0000000D+00 0.0000000D+00 %! -0.1580000D+03 -0.1257000D+04 0.0000000D+00 0.0000000D+00 0.0000000D+00 %! 0.3541000D+05 0.0000000D+00 -0.1434000D+04 0.0000000D+00 -0.5330000D+03 %! 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D+03 %! 0.0000000D+00 0.0000000D+00 0.0000000D+00 -0.1863000D+03 -0.1482000D+01 ]; %! %! Be = [ 0.0000000D+00 0.0000000D+00 %! 0.1103000D+04 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.8333000D+02 ]; %! %! Ce = [ 0.1000000D-04 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 0.1000000D+06 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D-05 0.0000000D+00 %! 0.6664000D-05 0.0000000D+00 -0.6200000D-07 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 -0.1000000D+03 0.1896000D+01 0.1508000D-01 ]; %! %! SCALEe = [0.1000000D-04 0.1000000D+00 0.1000000D+06 0.1000000D-05 0.1000000D-03 ]; %! %! MAXREDe = 0.3488E+10; %! %!assert (Ao, Ae, 1e-4); %!assert (Bo, Be, 1e-4); %!assert (Co, Ce, 1e-4); %!assert (MAXREDo, MAXREDe, 1e6); %!assert (SCALEo, SCALEe.', 1e-4); ## Scaling of descriptor state-space models, direct access to TG01AD %!shared Ao, Eo, Bo, Co, LSCALEo, RSCALEo, Ae, Ee, Be, Ce, LSCALEe, RSCALEe %! A = [ -1 0 0 0.003 %! 0 0 0.1000 0.02 %! 100 10 0 0.4 %! 0 0 0 0.0]; %! %! E = [ 1 0.2 0 0.0 %! 0 1 0 0.01 %! 300 90 6 0.3 %! 0 0 20 0.0]; %! %! B = [ 10 0 %! 0 0 %! 0 1000 %! 10000 10000]; %! %! C = [ -0.1 0.0 0.001 0.0 %! 0.0 0.01 -0.001 0.0001]; %! %! TRESH = 0.0; %! %! [Ao, Eo, Bo, Co, LSCALEo, RSCALEo] = __sl_tg01ad__ (A, E, B, C, TRESH); %! %! Ae = [ -1.0000 0.0000 0.0000 0.3000 %! 0.0000 0.0000 1.0000 2.0000 %! 1.0000 0.1000 0.0000 0.4000 %! 0.0000 0.0000 0.0000 0.0000]; %! %! Ee = [ 1.0000 0.2000 0.0000 0.0000 %! 0.0000 1.0000 0.0000 1.0000 %! 3.0000 0.9000 0.6000 0.3000 %! 0.0000 0.0000 0.2000 0.0000 ]; %! %! Be = [100.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 100.0000 %! 100.0000 100.0000 ]; %! %! Ce = [ -0.0100 0.0000 0.0010 0.0000 %! 0.0000 0.0010 -0.0010 0.0010]; %! %! LSCALEe = [ 10.0000 10.0000 0.1000 0.0100 ]; %! %! RSCALEe = [ 0.1000 0.1000 1.0000 10.0000 ]; %! %!assert (Ao, Ae, 1e-4); %!assert (Eo, Ee, 1e-4); %!assert (Bo, Be, 1e-4); %!assert (Co, Ce, 1e-4); %!assert (LSCALEo, LSCALEe.', 1e-4); %!assert (RSCALEo, RSCALEe.', 1e-4); ## Scaling of state-space models, user function %!shared Ao, Bo, Co, INFOo, Ae, Be, Ce, SCALEe %! A = [ 0.0 1.0000e+000 0.0 0.0 0.0 %! -1.5800e+006 -1.2570e+003 0.0 0.0 0.0 %! 3.5410e+014 0.0 -1.4340e+003 0.0 -5.3300e+011 %! 0.0 0.0 0.0 0.0 1.0000e+000 %! 0.0 0.0 0.0 -1.8630e+004 -1.4820e+000 ]; %! %! B = [ 0.0 0.0 %! 1.1030e+002 0.0 %! 0.0 0.0 %! 0.0 0.0 %! 0.0 8.3330e-003 ]; %! %! C = [ 1.0000e+000 0.0 0.0 0.0 0.0 %! 0.0 0.0 1.0000e+000 0.0 0.0 %! 0.0 0.0 0.0 1.0000e+000 0.0 %! 6.6640e-001 0.0 -6.2000e-013 0.0 0.0 %! 0.0 0.0 -1.0000e-003 1.8960e+006 1.5080e+002 ]; %! %! SYS = ss (A, B, C); %! %! [SYSo, INFOo] = prescale (SYS); %! %! [Ao, Bo, Co] = ssdata (SYSo); %! %! Ae = [ 0.0000000D+00 0.1000000D+05 0.0000000D+00 0.0000000D+00 0.0000000D+00 %! -0.1580000D+03 -0.1257000D+04 0.0000000D+00 0.0000000D+00 0.0000000D+00 %! 0.3541000D+05 0.0000000D+00 -0.1434000D+04 0.0000000D+00 -0.5330000D+03 %! 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D+03 %! 0.0000000D+00 0.0000000D+00 0.0000000D+00 -0.1863000D+03 -0.1482000D+01 ]; %! %! Be = [ 0.0000000D+00 0.0000000D+00 %! 0.1103000D+04 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.8333000D+02 ]; %! %! Ce = [ 0.1000000D-04 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 0.1000000D+06 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 0.0000000D+00 0.1000000D-05 0.0000000D+00 %! 0.6664000D-05 0.0000000D+00 -0.6200000D-07 0.0000000D+00 0.0000000D+00 %! 0.0000000D+00 0.0000000D+00 -0.1000000D+03 0.1896000D+01 0.1508000D-01 ]; %! %! SCALEe = [0.1000000D-04 0.1000000D+00 0.1000000D+06 0.1000000D-05 0.1000000D-03 ]; %! %!assert (Ao, Ae, 1e-4); %!assert (Bo, Be, 1e-4); %!assert (Co, Ce, 1e-4); %!assert (INFOo.SL.^-1, SCALEe.', 1e-4); %!assert (INFOo.SR, SCALEe.', 1e-4); ## Scaling of descriptor state-space models, user function %!shared Ao, Eo, Bo, Co, INFOo, Ae, Ee, Be, Ce, LSCALEe, RSCALEe %! A = [ -1 0 0 0.003 %! 0 0 0.1000 0.02 %! 100 10 0 0.4 %! 0 0 0 0.0]; %! %! E = [ 1 0.2 0 0.0 %! 0 1 0 0.01 %! 300 90 6 0.3 %! 0 0 20 0.0]; %! %! B = [ 10 0 %! 0 0 %! 0 1000 %! 10000 10000]; %! %! C = [ -0.1 0.0 0.001 0.0 %! 0.0 0.01 -0.001 0.0001]; %! %! SYS = dss (A, B, C, [], E); %! %! [SYSo, INFOo] = prescale (SYS); %! %! [Ao, Bo, Co, ~, Eo] = dssdata (SYSo); %! %! Ae = [ -1.0000 0.0000 0.0000 0.3000 %! 0.0000 0.0000 1.0000 2.0000 %! 1.0000 0.1000 0.0000 0.4000 %! 0.0000 0.0000 0.0000 0.0000]; %! %! Ee = [ 1.0000 0.2000 0.0000 0.0000 %! 0.0000 1.0000 0.0000 1.0000 %! 3.0000 0.9000 0.6000 0.3000 %! 0.0000 0.0000 0.2000 0.0000 ]; %! %! Be = [100.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 100.0000 %! 100.0000 100.0000 ]; %! %! Ce = [ -0.0100 0.0000 0.0010 0.0000 %! 0.0000 0.0010 -0.0010 0.0010]; %! %! LSCALEe = [ 10.0000 10.0000 0.1000 0.0100 ]; %! %! RSCALEe = [ 0.1000 0.1000 1.0000 10.0000 ]; %! %!assert (Ao, Ae, 1e-4); %!assert (Eo, Ee, 1e-4); %!assert (Bo, Be, 1e-4); %!assert (Co, Ce, 1e-4); %!assert (INFOo.SL, LSCALEe.', 1e-4); %!assert (INFOo.SR, RSCALEe.', 1e-4); control/inst/@lti/series.m0000644000076500000240000000672212273411257015070 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} series (@var{sys1}, @var{sys2}) ## @deftypefnx {Function File} {@var{sys} =} series (@var{sys1}, @var{sys2}, @var{outputs1}, @var{inputs2}) ## Series connection of two @acronym{LTI} models. ## ## @strong{Block Diagram} ## @example ## @group ## ..................................... ## u : +--------+ y1 u2 +--------+ : y ## ------>| sys1 |---------->| sys2 |-------> ## : +--------+ +--------+ : ## :................sys................. ## ## sys = series (sys1, sys2) ## @end group ## @end example ## @example ## @group ## ..................................... ## : v2 +--------+ : ## : ---------->| | : y ## : +--------+ y1 u2 | sys2 |-------> ## u : | |---------->| | : ## ------>| sys1 | z1 +--------+ : ## : | |----------> : ## : +--------+ : ## :................sys................. ## ## outputs1 = [1] ## inputs2 = [2] ## sys = series (sys1, sys2, outputs1, inputs2) ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function sys = series (sys1, sys2, out1, in2) if (nargin == 2) sys = sys2 * sys1; elseif (nargin == 4) [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (! is_real_vector (out1)) error ("series: argument 3 (outputs1) invalid"); endif if (! is_real_vector (in2)) error ("series: argument 4 (inputs2) invalid"); endif l_out1 = length (out1); l_in2 = length (in2); if (l_out1 > p1) error ("series: 'outputs1' has too many indices for 'sys1'"); endif if (l_in2 > m2) error ("series: 'inputs2' has too many indices for 'sys2'"); endif if (l_out1 != l_in2) error ("series: number of 'outputs1' and 'inputs2' indices must be equal"); endif if (any (out1 > m1 | out1 < 1)) error ("series: range of 'outputs1' indices exceeds dimensions of 'sys1'"); endif if (any (in2 > p1 | in2 < 1)) error ("series: range of 'inputs2' indices exceeds dimensions of 'sys2'"); endif out_scl = full (sparse (1:l_out1, out1, 1, l_out1, p1)); in_scl = full (sparse (in2, 1:l_out1, 1, m2, l_in2)); ## NOTE: for-loop does NOT the same as ## out_scl(1:l_out1, out1) = 1; ## in_scl(in2, 1:l_out1) = 1; ## ## out_scl = zeros (l_out1, p1); ## in_scl = zeros (m2, l_in2); ## ## for k = 1 : l_out1 ## out_scl(k, out1(k)) = 1; ## in_scl(in2(k), k) = 1; ## endfor scl = in_scl * out_scl; sys = sys2 * scl * sys1; else print_usage (); endif endfunction control/inst/@lti/set.m0000644000076500000240000001101712273411257014362 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} set (@var{sys}) ## @deftypefnx {Function File} {} set (@var{sys}, @var{"property"}, @var{value}, @dots{}) ## @deftypefnx {Function File} {@var{retsys} =} set (@var{sys}, @var{"property"}, @var{value}, @dots{}) ## Set or modify properties of @acronym{LTI} objects. ## If no return argument @var{retsys} is specified, the modified @acronym{LTI} object is stored ## in input argument @var{sys}. @command{set} can handle multiple properties in one call: ## @code{set (sys, 'prop1', val1, 'prop2', val2, 'prop3', val3)}. ## @code{set (sys)} prints a list of the object's property names. ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.5 function retsys = set (sys, varargin) if (nargin == 1) # set (sys), sys = set (sys) [props, vals] = __property_names__ (sys); nrows = numel (props); str = strjust (strvcat (props), "right"); str = horzcat (repmat (" ", nrows, 1), str, repmat (": ", nrows, 1), strvcat (vals)); disp (str); if (nargout != 0) # function sys = set (sys, varargin) retsys = sys; # would lead to unwanted output when using endif # set (sys) else # set (sys, "prop1", val1, ...), sys = set (sys, "prop1", val1, ...) if (rem (nargin-1, 2)) error ("lti: set: properties and values must come in pairs"); endif [p, m] = size (sys); for k = 1 : 2 : (nargin-1) prop = lower (varargin{k}); val = varargin{k+1}; switch (prop) case {"inname", "inputname", "inn", "inputn"} sys.inname = __adjust_labels__ (val, m); case {"outname", "outputname", "outn", "outputn"} sys.outname = __adjust_labels__ (val, p); case {"tsam", "ts"} if (issample (val, -1)) sys.tsam = val; ## TODO: display hint for 'd2d' to resample model? else error ("lti: set: invalid sampling time"); endif case {"ingroup", "inputgroup", "ing", "inputg"} if (isstruct (val) && all (structfun (@(x) is_group_idx (x, m), val))) empty = structfun (@isempty, val); fields = fieldnames (val); sys.ingroup = rmfield (val, fields(empty)); else error ("lti: set: property 'ingroup' requires a struct containing valid input indices in the range [1, %d]", m); endif case {"outgroup", "outputgroup", "outg", "outputg"} if (isstruct (val) && all (structfun (@(x) is_group_idx (x, p), val))) empty = structfun (@isempty, val); fields = fieldnames (val); sys.outgroup = rmfield (val, fields(empty)); else error ("lti: set: property 'outgroup' requires a struct containing valid output indices in the range [1, %d]", p); endif case "name" if (ischar (val)) sys.name = val; else error ("lti: set: property 'name' requires a string"); endif case "notes" if (iscellstr (val)) sys.notes = val; elseif (ischar (val)) sys.notes = {val}; else error ("lti: set: property 'notes' requires string or cell of strings"); endif case "userdata" sys.userdata = val; otherwise sys = __set__ (sys, prop, val); endswitch endfor if (nargout == 0) # set (sys, "prop1", val1, ...) assignin ("caller", inputname (1), sys); else # sys = set (sys, "prop1", val1, ...) retsys = sys; endif endif endfunction function bool = is_group_idx (idx, n) bool = (isempty (idx) || (is_real_vector (idx) && all (idx > 0) && all (idx <= n) && all (abs (fix (idx)) == idx))); endfunction control/inst/@lti/size.m0000644000076500000240000000474512273411257014553 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{nvec} =} size (@var{sys}) ## @deftypefnx {Function File} {@var{n} =} size (@var{sys}, @var{dim}) ## @deftypefnx {Function File} {[@var{p}, @var{m}] =} size (@var{sys}) ## @acronym{LTI} model size, i.e. number of outputs and inputs. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @item dim ## If given a second argument, @command{size} will return the size of the ## corresponding dimension. ## @end table ## ## @strong{Outputs} ## @table @var ## @item nvec ## Row vector. The first element is the number of outputs (rows) and the second ## element the number of inputs (columns). ## @item n ## Scalar value. The size of the dimension @var{dim}. ## @item p ## Number of outputs. ## @item m ## Number of inputs. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function [n, varargout] = size (sys, dim = 0) if (nargin > 2) print_usage (); endif p = numel (sys.outname); # WARNING: system matrices may change without m = numel (sys.inname); # being noticed by the i/o names! switch (dim) case 0 switch (nargout) case 0 if (p == 1) stry = ""; else stry = "s"; endif if (m == 1) stru = ""; else stru = "s"; endif disp (sprintf ("LTI model with %d output%s and %d input%s.", p, stry, m, stru)); case 1 n = [p, m]; case 2 n = p; varargout{1} = m; otherwise print_usage (); endswitch case 1 n = p; case 2 n = m; otherwise print_usage (); endswitch endfunction control/inst/@lti/size_equal.m0000644000076500000240000000244012273411257015730 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} size_equal (@var{a}, @var{b}, @dots{}) ## Return true if @acronym{LTI} models (and matrices) @var{a}, @var{b}, @dots{} ## are of equal size and false otherwise. ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2012 ## Version: 0.1 function bool = size_equal (varargin) s = cellfun (@size, varargin, "uniformoutput", false); bool = (nargin == 1 || isequal (s{:})); # isequal errors out with only 1 argument, nargin==0 handled by built-in size_equal endfunction control/inst/@lti/sminreal.m0000644000076500000240000000740312273411257015405 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} sminreal (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} sminreal (@var{sys}, @var{tol}) ## Perform state-space model reduction based on structure. ## Remove states which have no influence on the input-output behaviour. ## The physical meaning of the states is retained. ## ## @strong{Inputs} ## @table @var ## @item sys ## State-space model. ## @item tol ## Optional tolerance for controllability and observability. ## Entries of the state-space matrices whose moduli are less or equal to @var{tol} ## are assumed to be zero. Default value is 0. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Reduced state-space model. ## @end table ## ## @seealso{minreal} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.4 function sys = sminreal (sys, tol = 0) if (nargin > 2) # sminreal () not possible (inside @lti) print_usage (); endif if (! isa (sys, "ss")) warning ("sminreal: system not in state-space form"); sys = ss (sys); # needed by __sys_prune__ endif if (! (is_real_scalar (tol) && tol >= 0)) error ("sminreal: second argument is not a valid tolerance"); endif [a, b, c, d, e] = dssdata (sys, []); a = abs (a) > tol; b = abs (b) > tol; c = abs (c) > tol; if (! isempty (e)) e = abs (e) > tol; a = a | e; endif co_idx = __controllable_states__ (a, b); ob_idx = __controllable_states__ (a.', c.'); st_idx = intersect (co_idx, ob_idx); sys = __sys_prune__ (sys, ":", ":", st_idx); endfunction function c_idx = __controllable_states__ (a, b) n = rows (a); # number of states a = a & ! eye (n); # set diagonal entries to zero c_vec = any (b, 2); # states directly controllable c_idx = find (c_vec); # indices of directly controllable states c_idx_new = 0; # any vector of length > 0 possible while (all (length (c_idx) != [0, n]) && length(c_idx_new) != 0) u_idx = find (! c_vec); # indices of uncontrollable states #{ ## debug code a(u_idx, :) repmat (c_vec.', length (u_idx), 1) a(u_idx, :) & repmat (c_vec.', length (u_idx), 1) any (a(u_idx, :) & repmat (c_vec.', length (u_idx), 1), 2) find (any (a(u_idx, :) & repmat (c_vec.', length (u_idx), 1), 2)) #} c_idx_new = u_idx (find (any (a(u_idx, :) & repmat (c_vec.', length (u_idx), 1), 2))); c_idx = union (c_idx, c_idx_new); c_vec(c_idx_new) = 1; endwhile endfunction ## ss: sminreal %!shared B, C %! %! A = ss (-2, 3, 4, 5); %! B = A / A; %! C = sminreal (B); # no states should be removed %! %!assert (C.a, B.a); %!assert (C.b, B.b); %!assert (C.c, B.c); %!assert (C.d, B.d); %!shared A, B, D, E %! %! A = ss (-1, 1, 1, 0); %! B = ss (-2, 3, 4, 5); %! C = [A, B]; %! D = sminreal (C(:, 1)); %! E = sminreal (C(:, 2)); %! %!assert (D.a, A.a); %!assert (D.b, A.b); %!assert (D.c, A.c); %!assert (D.d, A.d); %!assert (E.a, B.a); %!assert (E.b, B.b); %!assert (E.c, B.c); %!assert (E.d, B.d); control/inst/@lti/ssdata.m0000644000076500000240000000340712273411257015052 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}, @var{tsam}] =} ssdata (@var{sys}) ## Access state-space model data. ## Argument @var{sys} is not limited to state-space models. ## If @var{sys} is not a state-space model, it is converted automatically. ## ## @strong{Inputs} ## @table @var ## @item sys ## Any type of @acronym{LTI} model. ## @end table ## ## @strong{Outputs} ## @table @var ## @item a ## State matrix (n-by-n). ## @item b ## Input matrix (n-by-m). ## @item c ## Measurement matrix (p-by-n). ## @item d ## Feedthrough matrix (p-by-m). ## @item tsam ## Sampling time in seconds. If @var{sys} is a continuous-time model, ## a zero is returned. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.4 function [a, b, c, d, tsam, scaled] = ssdata (sys) if (! isa (sys, "ss")) sys = ss (sys); endif [a, b, c, d, e, ~, scaled] = __sys_data__ (sys); [a, b, c, d, e] = __dss2ss__ (a, b, c, d, e); tsam = sys.tsam; endfunction control/inst/@lti/subsasgn.m0000644000076500000240000000266612273411257015426 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Subscripted assignment for @acronym{LTI} objects. ## Used by Octave for "sys.property = value". ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function sys = subsasgn (sys, idx, val) ## TODO: enable *more* stuff like sys.a(2, 1:3) = [4, 5, 6] ## warning ("lti: subsasgn: do not use subsasgn for development"); switch (idx(1).type) case "." if (length (idx) == 1) sys = set (sys, idx.subs, val); else prop = idx(1).subs; sys = set (sys, prop, subsasgn (get (sys, prop), idx(2:end), val)); endif otherwise error ("lti: subsasgn: invalid subscripted assignment type"); endswitch endfunction control/inst/@lti/subsref.m0000644000076500000240000000325512273411257015245 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Subscripted reference for @acronym{LTI} objects. ## Used by Octave for "sys = sys(2:4, :)" or "val = sys.prop". ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.4 function a = subsref (a, s) if (numel (s) == 0) return; endif switch (s(1).type) case "()" idx = s(1).subs; if (numel (idx) == 2) a = __sys_prune__ (a, idx{1}, idx{2}); elseif (numel (idx) == 1) a = __freqresp__ (a, idx{1}); else error ("lti: subsref: need one or two indices"); endif case "." fld = s(1).subs; a = get (a, fld); ## warning ("lti: subsref: do not use subsref for development"); otherwise error ("lti: subsref: invalid subscript type"); endswitch a = subsref (a, s(2:end)); endfunction ## lti: subsref %!shared a %! s = tf ("s"); %! G = (s+1)*s*5/(s+1)/(s^2+s+1); %! a = G(1,1).num{1,1}(1); %!assert (a, 5, 1e-4); control/inst/@lti/tfdata.m0000644000076500000240000000544312273411257015040 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}) ## @deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}, @var{"vector"}) ## @deftypefnx {Function File} {[@var{num}, @var{den}, @var{tsam}] =} tfdata (@var{sys}, @var{"tfpoly"}) ## Access transfer function data. ## Argument @var{sys} is not limited to transfer function models. ## If @var{sys} is not a transfer function, it is converted automatically. ## ## @strong{Inputs} ## @table @var ## @item sys ## Any type of @acronym{LTI} model. ## @item "v", "vector" ## For SISO models, return @var{num} and @var{den} directly as column vectors ## instead of cells containing a single column vector. ## @end table ## ## @strong{Outputs} ## @table @var ## @item num ## Cell of numerator(s). Each numerator is a row vector ## containing the coefficients of the polynomial in descending powers of ## the transfer function variable. ## num@{i,j@} contains the numerator polynomial from input j to output i. ## In the SISO case, a single vector is possible as well. ## @item den ## Cell of denominator(s). Each denominator is a row vector ## containing the coefficients of the polynomial in descending powers of ## the transfer function variable. ## den@{i,j@} contains the denominator polynomial from input j to output i. ## In the SISO case, a single vector is possible as well. ## @item tsam ## Sampling time in seconds. If @var{sys} is a continuous-time model, ## a zero is returned. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.5 function [num, den, tsam] = tfdata (sys, rtype = "cell") if (! isa (sys, "tf")) sys = tf (sys); endif [num, den] = __sys_data__ (sys); tsam = sys.tsam; if (! strncmpi (rtype, "t", 1)) # != tfpoly num = cellfun (@get, num, "uniformoutput", false); den = cellfun (@get, den, "uniformoutput", false); endif if (strncmpi (rtype, "v", 1) && issiso (sys)) # == vector num = num{1}; den = den{1}; endif endfunction control/inst/@lti/transpose.m0000644000076500000240000000252612273411257015612 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Transpose of @acronym{LTI} objects. Used by Octave for "sys.'". ## Useful for dual problems, i.e. controllability and observability ## or designing estimator gains with @command{lqr} and @command{place}. ## Author: Lukas Reichlin ## Created: February 2010 ## Version: 0.1 function sys = transpose (sys) if (nargin != 1) # prevent sys = transpose (sys1, sys2, sys3, ...) error ("lti: transpose: this is an unary operator"); endif [p, m] = size (sys); sys = __transpose__ (sys); sys.inname = repmat ({""}, p, 1); sys.outname = repmat ({""}, m, 1); endfunction control/inst/@lti/uminus.m0000644000076500000240000000221212273411257015104 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Unary minus of @acronym{LTI} object. Used by Octave for "-sys". ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function sys = uminus (sys) if (nargin != 1) # prevent sys = uminus (sys1, sys2, sys3, ...) error ("lti: uminus: this is an unary operator"); endif [p, m] = size (sys); out_scl = - eye (p); sys = out_scl * sys; endfunction control/inst/@lti/uplus.m0000644000076500000240000000207112273411257014737 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Unary plus of @acronym{LTI} object. Used by Octave for "+sys". ## Author: Lukas Reichlin ## Created: June 2012 ## Version: 0.1 function sys = uplus (sys) if (nargin != 1) # prevent sys = uplus (sys1, sys2, sys3, ...) error ("lti: uplus: this is an unary operator"); endif endfunction control/inst/@lti/vertcat.m0000644000076500000240000000262412273411257015243 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Vertical concatenation of @acronym{LTI} objects. If necessary, object conversion ## is done by sys_group. Used by Octave for "[sys1; sys2]". ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function sys = vertcat (sys, varargin) for k = 1 : (nargin-1) sys1 = sys; sys2 = varargin{k}; [p1, m1] = size (sys1); [p2, m2] = size (sys2); if (m1 != m2) error ("lti: vertcat: number of system inputs incompatible: [(%dx%d); (%dx%d)]", p1, m1, p2, m2); endif sys = __sys_group__ (sys1, sys2); in_scl = [eye(m1); eye(m2)]; sys = sys * in_scl; endfor endfunction control/inst/@lti/xperm.m0000644000076500000240000000370312273411257014725 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{retsys} =} xperm (@var{sys}, @var{idx}) ## Reorder states in state-space models. ## ## @strong{Inputs} ## @table @var ## @item sys ## State-space model. ## @item idx ## Vector containing the state indices in the desired order. ## Alternatively, a cell vector containing the state names ## is possible as well. See @code{sys.stname}. State names ## only work if they were assigned explicitly before, i.e. ## @code{sys.stname} contains no empty strings. ## Note that if certain state indices of @var{sys} are ## missing or appear multiple times in @var{idx}, these ## states will be pruned or duplicated accordingly in the ## resulting state-space model @var{retsys}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item retsys ## Resulting state-space model with states reordered according to @var{idx}. ## @end table ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.2 function sys = xperm (sys, st_idx) if (nargin != 2) print_usage (); endif if (! isa (sys, "ss")) warning ("xperm: system not in state-space form"); sys = ss (sys); endif sys = __sys_prune__ (sys, ":", ":", st_idx); endfunction control/inst/@lti/zero.m0000644000076500000240000003562212273411257014556 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## Copyright (C) 2011 Ferdinand Svaricek, UniBw Munich. ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{z} =} zero (@var{sys}) ## @deftypefnx {Function File} {@var{z} =} zero (@var{sys}, @var{type}) ## @deftypefnx {Function File} {[@var{z}, @var{k}, @var{info}] =} zero (@var{sys}) ## Compute zeros and gain of @acronym{LTI} model. ## By default, @command{zero} computes the invariant zeros, ## also known as Smith zeros. Alternatively, when called with ## a second input argument, @command{zero} can also compute ## the system zeros, transmission zeros, input decoupling zeros ## and output decoupling zeros. See paper [1] for an explanation ## of the various zero flavors as well as for further details. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item type ## String specifying the type of zeros: ## @table @var ## @item 'system', 's' ## Compute the system zeros. ## The system zeros include in all cases ## (square, non-square, degenerate or non-degenerate system) ## all transmission and decoupling zeros. ## @item 'invariant', 'inv' ## Compute invariant zeros. Default selection. ## @item 'transmission', 't' ## Compute transmission zeros. Transmission zeros ## are a subset of the invariant zeros. ## The transmission zeros are the zeros of the ## Smith-McMillan form of the transfer function matrix. ## @item 'input', 'inp', 'id' ## Compute input decoupling zeros. The input decoupling zeros are ## also known as the uncontrollable eigenvalues of the pair (A,B). ## @item 'output', 'o', 'od' ## Compute output decoupling zeros. The output decoupling zeros are ## also known as the unobservable eigenvalues of the pair (A,C). ## @end table ## @end table ## ## @strong{Outputs} ## @table @var ## @item z ## Depending on argument @var{type}, @var{z} contains the ## invariant (default), system, transmission, input decoupling ## or output decoupling zeros of @var{sys} as defined in [1]. ## @item k ## Gain of @acronym{SISO} system @var{sys}. For @acronym{MIMO} ## systems, an empty matrix @code{[]} is returned. ## @item info ## Struct containing additional information. For details, ## see the documentation of @acronym{SLICOT} routines ## @acronym{AB08ND} and @acronym{AG08BD}. ## @item info.rank ## The normal rank of the transfer function matrix (regular state-space models) ## or of the system pencil (descriptor state-space models). ## @item info.infz ## Contains information on the infinite elementary divisors as follows: ## the system has info.infz(i) infinite elementary divisors of degree i, ## where i=1,2,...,length(info.infz). ## @item info.kronr ## Right Kronecker (column) indices. ## @item info.kronl ## Left Kronecker (row) indices. ## @end table ## ## @strong{Examples} ## @example ## @group ## [z, k, info] = zero (sys) # invariant zeros ## z = zero (sys, 'system') # system zeros ## z = zero (sys, 'invariant') # invariant zeros ## z = zero (sys, 'transmission') # transmission zeros ## z = zero (sys, 'output') # output decoupling zeros ## z = zero (sys, 'input') # input decoupling zeros ## @end group ## @end example ## ## @strong{Algorithm}@* ## For (descriptor) state-space models, @command{zero} ## relies on SLICOT AB08ND and AG08BD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## For @acronym{SISO} transfer functions, @command{zero} ## uses Octave's @command{roots}. ## @acronym{MIMO} transfer functions are converted to ## a @emph{minimal} state-space representation for the ## computation of the zeros. ## ## @strong{References}@* ## [1] MacFarlane, A. and Karcanias, N. ## @cite{Poles and zeros of linear multivariable systems: ## a survey of the algebraic, geometric and complex-variable ## theory}. Int. J. Control, vol. 24, pp. 33-74, 1976.@* ## [2] Rosenbrock, H.H. ## @cite{Correction to 'The zeros of a system'}. ## Int. J. Control, vol. 20, no. 3, pp. 525-527, 1974.@* ## [3] Svaricek, F. ## @cite{Computation of the structural invariants of linear ## multivariable systems with an extended version of the ## program ZEROS}. ## Systems & Control Letters, vol. 6, pp. 261-266, 1985.@* ## [4] Emami-Naeini, A. and Van Dooren, P. ## @cite{Computation of zeros of linear multivariable systems}. ## Automatica, vol. 26, pp. 415-430, 1982.@* ## ## @end deftypefn ## TODO: write a short summary about the characteristics of the ## various zero flavors and add it to the docstring. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function [zer, gain, info] = zero (sys, type = "invariant") if (nargin > 2) print_usage (); endif if (strncmpi (type, "invariant", 3)) # invariant zeros, default [zer, gain, info] = __zero__ (sys, nargout); elseif (strncmpi (type, "transmission", 1)) # transmission zeros [zer, gain, info] = zero (minreal (sys)); elseif (strncmpi (type, "input", 3) || strncmpi (type, "id", 2)) # input decoupling zeros [a, b, c, d, e, tsam] = dssdata (sys, []); tmp = dss (a, b, zeros (0, columns (a)), zeros (0, columns (b)), e, tsam); [zer, gain, info] = zero (tmp); elseif (strncmpi (type, "output", 1)) # output decoupling zeros [a, b, c, d, e, tsam] = dssdata (sys, []); tmp = dss (a, zeros (rows (a), 0), c, zeros (rows (c), 0), e, tsam); [zer, gain, info] = zero (tmp); elseif (strncmpi (type, "system", 1)) # system zeros [zer, gain, info] = __szero__ (sys); else error ("zero: type '%s' invalid", type); endif endfunction ## Function for computing the system zeros. ## Adapted from Ferdinand Svaricek's szero.m function [z, gain, info] = __szero__ (sys) ## TODO: support descriptor state-space models ## with singular 'E' matrices [a, b, c, d] = ssdata (sys); [pp, mm] = size (sys); nn = rows (a); ## Tolerance for intersection of zeros Zeps = 10 * sqrt ((nn+pp)*(nn+mm)) * eps * norm (a,'fro'); [z, gain, info] = zero (ss (a, b, c, d)); # zero (sys) lets descriptor test fail Rank = info.rank; ## System is not degenerated and square if (Rank == 0 || (Rank == min(pp,mm) && mm == pp)) return; endif ## System (A,B,C,D) is degenerated and/or non-square z = []; ## Computation of the greatest common divisor of all minors of the ## Rosenbrock system matrix that have the following form ## ## 1, 2, ..., n, n+i_1, n+i_2, ..., n+i_k ## P ## 1, 2, ..., n, n+j_1, n+j_2, ..., n+j_k ## ## with k = Rank. NKP = nchoosek (1:pp, Rank); [IP, JP] = size (NKP); NKM = nchoosek (1:mm, Rank); [IM, JM] = size (NKM); for i = 1:IP for j = 1:JP k = NKP(i,j); C1(j,:) = c(k,:); # Build C of dimension (Rank x n) endfor for ii = 1:IM for jj = 1:JM k = NKM(ii,jj); B1(:,jj) = b(:,k); # Build B of dimension (n x Rank) endfor [z1, ~, info1] = zero (ss (a, B1, C1, zeros (Rank, Rank))); rank1 = info1.rank; if (rank1 == Rank) if (isempty (z1)) z = z1; # Subsystem has no zeros -> system has no system zeros return; else if (isempty (z)) z = z1; # Zeros of the first subsystem else # Compute intersection of z and z1 with tolerance Zeps z2 = []; for ii=1:length(z) for jj=1:length(z1) if (abs (z(ii)-z1(jj)) < Zeps) z2(end+1) = z(ii); z1(jj) = []; break; endif endfor endfor z = z2; # System zeros are the common zeros of all subsystems endif endif endif endfor endfor endfunction ## Invariant zeros of state-space models ## ## Results from the "Dark Side" 7.5 and 7.8 ## ## -13.2759 ## 12.5774 ## -0.0155 ## ## Results from Scilab 5.2.0b1 (trzeros) ## ## - 13.275931 ## 12.577369 ## - 0.0155265 ## %!shared z, z_exp %! A = [ -0.7 -0.0458 -12.2 0 %! 0 -0.014 -0.2904 -0.562 %! 1 -0.0057 -1.4 0 %! 1 0 0 0 ]; %! %! B = [ -19.1 -3.1 %! -0.0119 -0.0096 %! -0.14 -0.72 %! 0 0 ]; %! %! C = [ 0 0 -1 1 %! 0 0 0.733 0 ]; %! %! D = [ 0 0 %! 0.0768 0.1134 ]; %! %! sys = ss (A, B, C, D, "scaled", true); %! z = sort (zero (sys)); %! %! z_exp = sort ([-13.2759; 12.5774; -0.0155]); %! %!assert (z, z_exp, 1e-4); ## Invariant zeros of regular state-space models %!shared z, z_exp, info, rank_exp, infz_exp, kronr_exp, kronl_exp %! A = [ 1.0 0.0 0.0 0.0 0.0 0.0 %! 0.0 1.0 0.0 0.0 0.0 0.0 %! 0.0 0.0 3.0 0.0 0.0 0.0 %! 0.0 0.0 0.0 -4.0 0.0 0.0 %! 0.0 0.0 0.0 0.0 -1.0 0.0 %! 0.0 0.0 0.0 0.0 0.0 3.0 ]; %! %! B = [ 0.0 -1.0 %! -1.0 0.0 %! 1.0 -1.0 %! 0.0 0.0 %! 0.0 1.0 %! -1.0 -1.0 ]; %! %! C = [ 1.0 0.0 0.0 1.0 0.0 0.0 %! 0.0 1.0 0.0 1.0 0.0 1.0 %! 0.0 0.0 1.0 0.0 0.0 1.0 ]; %! %! D = [ 0.0 0.0 %! 0.0 0.0 %! 0.0 0.0 ]; %! %! sys = ss (A, B, C, D, "scaled", true); %! [z, ~, info] = zero (sys); %! %! z_exp = [ 2.0000 %! -1.0000 ]; %! %! rank_exp = 2; %! infz_exp = 2; %! kronr_exp = zeros (1, 0); %! kronl_exp = 2; %! %!assert (z, z_exp, 1e-4); %!assert (info.rank, rank_exp); %!assert (info.infz, infz_exp); %!assert (info.kronr, kronr_exp); %!assert (info.kronl, kronl_exp); ## Invariant zeros of descriptor state-space models %!shared z, z_exp, info, rank_exp, infz_exp, kronr_exp, kronl_exp %! A = [ 1 0 0 0 0 0 0 0 0 %! 0 1 0 0 0 0 0 0 0 %! 0 0 1 0 0 0 0 0 0 %! 0 0 0 1 0 0 0 0 0 %! 0 0 0 0 1 0 0 0 0 %! 0 0 0 0 0 1 0 0 0 %! 0 0 0 0 0 0 1 0 0 %! 0 0 0 0 0 0 0 1 0 %! 0 0 0 0 0 0 0 0 1 ]; %! %! E = [ 0 0 0 0 0 0 0 0 0 %! 1 0 0 0 0 0 0 0 0 %! 0 1 0 0 0 0 0 0 0 %! 0 0 0 0 0 0 0 0 0 %! 0 0 0 1 0 0 0 0 0 %! 0 0 0 0 1 0 0 0 0 %! 0 0 0 0 0 0 0 0 0 %! 0 0 0 0 0 0 1 0 0 %! 0 0 0 0 0 0 0 1 0 ]; %! %! B = [ -1 0 0 %! 0 0 0 %! 0 0 0 %! 0 -1 0 %! 0 0 0 %! 0 0 0 %! 0 0 -1 %! 0 0 0 %! 0 0 0 ]; %! %! C = [ 0 1 1 0 3 4 0 0 2 %! 0 1 0 0 4 0 0 2 0 %! 0 0 1 0 -1 4 0 -2 2 ]; %! %! D = [ 1 2 -2 %! 0 -1 -2 %! 0 0 0 ]; %! %! sys = dss (A, B, C, D, E, "scaled", true); %! [z, ~, info] = zero (sys); %! %! z_exp = 1; %! %! rank_exp = 11; %! infz_exp = [0, 1]; %! kronr_exp = 2; %! kronl_exp = 1; %! %!assert (z, z_exp, 1e-4); %!assert (info.rank, rank_exp); %!assert (info.infz, infz_exp); %!assert (info.kronr, kronr_exp); %!assert (info.kronl, kronl_exp); ## Gain of descriptor state-space models %!shared p, pi, z, zi, k, ki, p_tf, pi_tf, z_tf, zi_tf, k_tf, ki_tf %! P = ss (-2, 3, 4, 5); %! Pi = inv (P); %! %! p = pole (P); %! [z, k] = zero (P); %! %! pi = pole (Pi); %! [zi, ki] = zero (Pi); %! %! P_tf = tf (P); %! Pi_tf = tf (Pi); %! %! p_tf = pole (P_tf); %! [z_tf, k_tf] = zero (P_tf); %! %! pi_tf = pole (Pi_tf); %! [zi_tf, ki_tf] = zero (Pi_tf); %! %!assert (p, zi, 1e-4); %!assert (z, pi, 1e-4); %!assert (k, inv (ki), 1e-4); %!assert (p_tf, zi_tf, 1e-4); %!assert (z_tf, pi_tf, 1e-4); %!assert (k_tf, inv (ki_tf), 1e-4); ## Example taken from Paper [1] ## Regular state-space system %!shared z_inv, z_tra, z_inp, z_out, z_sys, z_inv_e, z_tra_e, z_inp_e, z_out_e, z_sys_e %! A = diag ([1, 1, 3, -4, -1, 3]); %! %! B = [ 0, -1 %! -1, 0 %! 1, -1 %! 0, 0 %! 0, 1 %! -1, -1 ]; %! %! C = [ 1, 0, 0, 1, 0, 0 %! 0, 1, 0, 1, 0, 1 %! 0, 0, 1, 0, 0, 1 ]; %! %! D = zeros (3, 2); %! %! SYS = ss (A, B, C, D); %! %! z_inv = zero (SYS); %! z_tra = zero (SYS, "transmission"); %! z_inp = zero (SYS, "input decoupling"); %! z_out = zero (SYS, "output decoupling"); %! z_sys = zero (SYS, "system"); %! %! z_inv_e = [2; -1]; %! z_tra_e = [2]; %! z_inp_e = [-4]; %! z_out_e = [-1]; %! z_sys_e = [-4, -1, 2]; %! %!assert (z_inv, z_inv_e, 1e-4); %!assert (z_tra, z_tra_e, 1e-4); %!assert (z_inp, z_inp_e, 1e-4); %!assert (z_out, z_out_e, 1e-4); %!assert (z_sys, z_sys_e, 1e-4); ## Example taken from Paper [1] ## Well, this is not exactly a descriptor state-space model, ## but it is the best thing I have right now and it is better ## than no test at all. The routine for the system zeros works ## only for descriptor state-space models with regular 'E' matrices. %!shared z_inv, z_tra, z_inp, z_out, z_sys, z_inv_e, z_tra_e, z_inp_e, z_out_e, z_sys_e %! A = diag ([1, 1, 3, -4, -1, 3]); %! %! B = [ 0, -1 %! -1, 0 %! 1, -1 %! 0, 0 %! 0, 1 %! -1, -1 ]; %! %! C = [ 1, 0, 0, 1, 0, 0 %! 0, 1, 0, 1, 0, 1 %! 0, 0, 1, 0, 0, 1 ]; %! %! D = zeros (3, 2); %! %! E = eye (6); %! %! SYS = dss (A, B, C, D, E); %! %! z_inv = zero (SYS); %! z_tra = zero (SYS, "transmission"); %! z_inp = zero (SYS, "input decoupling"); %! z_out = zero (SYS, "output decoupling"); %! z_sys = zero (SYS, "system"); %! %! z_inv_e = [2; -1]; %! z_tra_e = [2]; %! z_inp_e = [-4]; %! z_out_e = [-1]; %! z_sys_e = [-4, -1, 2]; %! %!assert (z_inv, z_inv_e, 1e-4); %!assert (z_tra, z_tra_e, 1e-4); %!assert (z_inp, z_inp_e, 1e-4); %!assert (z_out, z_out_e, 1e-4); %!assert (z_sys, z_sys_e, 1e-4); control/inst/@lti/zpkdata.m0000644000076500000240000000427212273411257015232 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{z}, @var{p}, @var{k}, @var{tsam}] =} zpkdata (@var{sys}) ## @deftypefnx {Function File} {[@var{z}, @var{p}, @var{k}, @var{tsam}] =} zpkdata (@var{sys}, @var{"v"}) ## Access zero-pole-gain data. ## ## @strong{Inputs} ## @table @var ## @item sys ## Any type of @acronym{LTI} model. ## @item "v", "vector" ## For SISO models, return @var{z} and @var{p} directly as column vectors ## instead of cells containing a single column vector. ## @end table ## ## @strong{Outputs} ## @table @var ## @item z ## Cell of column vectors containing the zeros for each channel. ## z@{i,j@} contains the zeros from input j to output i. ## @item p ## Cell of column vectors containing the poles for each channel. ## p@{i,j@} contains the poles from input j to output i. ## @item k ## Matrix containing the gains for each channel. ## k(i,j) contains the gain from input j to output i. ## @item tsam ## Sampling time in seconds. If @var{sys} is a continuous-time model, ## a zero is returned. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2011 ## Version: 0.1 function [z, p, k, tsam] = zpkdata (sys, rtype = "cell") [num, den, tsam] = tfdata (sys); z = cellfun (@roots, num, "uniformoutput", false); p = cellfun (@roots, den, "uniformoutput", false); k = cellfun (@(n,d) n(1)/d(1), num, den); if (strncmpi (rtype, "v", 1) && issiso (sys)) z = z{1}; p = p{1}; endif endfunction control/inst/@ss/0000755000076500000240000000000012273411257013246 5ustar lukasstaffcontrol/inst/@ss/__c2d__.m0000644000076500000240000000414012273411257014667 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert the continuous SS model into its discrete-time equivalent. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.4 function sys = __c2d__ (sys, tsam, method = "zoh", w0 = 0) switch (method(1)) case {"z", "s"} # {"zoh", "std"} [sys.a, sys.b, sys.c, sys.d, sys.e] = __dss2ss__ (sys.a, sys.b, sys.c, sys.d, sys.e); [n, m] = size (sys.b); # n: states, m: inputs ## TODO: use SLICOT MB05OD tmp = expm ([sys.a*tsam, sys.b*tsam; zeros(m, n+m)]); sys.a = tmp (1:n, 1:n); # F sys.b = tmp (1:n, n+(1:m)); # G case {"t", "b", "p"} # {"tustin", "bilin", "prewarp"} if (method(1) == "p") # prewarping beta = w0 / tan (w0*tsam/2); else beta = 2/tsam; endif if (isempty (sys.e)) [sys.a, sys.b, sys.c, sys.d] = __sl_ab04md__ (sys.a, sys.b, sys.c, sys.d, 1, beta, false); else [sys.a, sys.b, sys.c, sys.d, sys.e] = __dss_bilin__ (sys.a, sys.b, sys.c, sys.d, sys.e, beta, false); endif case "m" # "matched" tmp = ss (c2d (zpk (sys), tsam, method)); sys.e = tmp.e; sys.a = tmp.a; sys.b = tmp.b; sys.c = tmp.c; sys.d = tmp.d; otherwise error ("ss: c2d: '%s' is an invalid or missing method", method); endswitch endfunction control/inst/@ss/__ctranspose__.m0000644000076500000240000000273312273411257016406 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Conjugate transpose of SS models. ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function sys = __ctranspose__ (sys, ct) a = sys.a; b = sys.b; c = sys.c; d = sys.d; e = sys.e; if (ct) # continuous-time sys.a = -a.'; sys.b = -c.'; sys.c = b.'; sys.d = d.'; sys.e = e.'; sys.stname = repmat ({""}, rows (a), 1); else # discrete-time [n, m] = size (b); p = rows (c); if (isempty (e)) e = eye (n); endif sys.a = blkdiag (e.', eye (p)); sys.b = [zeros(n, p); -eye(p)]; sys.c = [b.', zeros(m, p)]; sys.d = d.'; sys.e = [a.', c.'; zeros(p, n+p)]; sys.stname = repmat ({""}, n+p, 1); endif endfunction control/inst/@ss/__d2c__.m0000644000076500000240000000425612273411257014677 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert the discrete SS model into its continuous-time equivalent. ## Author: Lukas Reichlin ## Created: September 2011 ## Version: 0.3 function sys = __d2c__ (sys, tsam, method = "zoh", w0 = 0) switch (method(1)) case {"z", "s"} # {"zoh", "std"} [sys.a, sys.b, sys.c, sys.d, sys.e] = __dss2ss__ (sys.a, sys.b, sys.c, sys.d, sys.e); [n, m] = size (sys.b); # n: states, m: inputs tmp = logm ([sys.a, sys.b; zeros(m,n), eye(m)]) / tsam; if (norm (imag (tmp), inf) > sqrt (eps)) warning ("ss: d2c: possibly inaccurate results"); endif sys.a = real (tmp(1:n, 1:n)); sys.b = real (tmp(1:n, n+1:n+m)); case {"t", "b", "p"} # {"tustin", "bilin", "prewarp"} if (method(1) == "p") # prewarping beta = w0 / tan (w0*tsam/2); else beta = 2/tsam; endif if (isempty (sys.e)) [sys.a, sys.b, sys.c, sys.d] = __sl_ab04md__ (sys.a, sys.b, sys.c, sys.d, 1, beta, true); else [sys.a, sys.b, sys.c, sys.d, sys.e] = __dss_bilin__ (sys.a, sys.b, sys.c, sys.d, sys.e, beta, true); endif case "m" # "matched" tmp = ss (d2c (zpk (sys), method)); sys.e = tmp.e; sys.a = tmp.a; sys.b = tmp.b; sys.c = tmp.c; sys.d = tmp.d; otherwise error ("ss: d2c: '%s' is an invalid or missing method", method); endswitch endfunction control/inst/@ss/__freqresp__.m0000644000076500000240000000243412273411257016052 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Frequency response of SS models. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.6 function H = __freqresp__ (sys, w, cellflag = false) if (sys.scaled == false) sys = prescale (sys); endif [a, b, c, d, e, tsam] = dssdata (sys); if (isct (sys)) # continuous system s = i * w; else # discrete system s = exp (i * w * abs (tsam)); endif H = arrayfun (@(x) c/(x*e - a)*b + d, s, "uniformoutput", false); if (! cellflag) H = cat (3, H{:}); endif endfunction control/inst/@ss/__get__.m0000644000076500000240000000251612273411257015003 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Access property values of SS objects. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function val = __get__ (sys, prop) switch (prop) # {, } case "a" val = sys.a; case "b" val = sys.b; case "c" val = sys.c; case "d" val = sys.d; case "e" val = sys.e; case {"stname", "statename", "stn", "staten"} val = sys.stname; case "scaled" val = sys.scaled; otherwise error ("ss: get: invalid property name '%s'", prop); endswitch endfunction control/inst/@ss/__minreal__.m0000644000076500000240000000273612273411257015657 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Minimal realization of SS models. The physical meaning of states is lost. ## Uses SLICOT TB01PD and TG01JD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function retsys = __minreal__ (sys, tol) if (tol == "def") tol = 0; elseif (tol > 1) error ("ss: minreal: require tol <= 1"); endif if (isempty (sys.e)) [a, b, c] = __sl_tb01pd__ (sys.a, sys.b, sys.c, tol, sys.scaled); retsys = ss (a, b, c, sys.d); else [a, e, b, c] = __sl_tg01jd__ (sys.a, sys.e, sys.b, sys.c, tol, sys.scaled, 0, 0); retsys = dss (a, b, c, sys.d, e); endif retsys.lti = sys.lti; # retain i/o names and tsam endfunction control/inst/@ss/__pole__.m0000644000076500000240000000235312273411257015162 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Poles of SS object. ## Special thanks to Peter Benner for his advice. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function pol = __pole__ (sys) if (isempty (sys.e)) pol = eig (sys.a); else pol = eig (sys.a, sys.e); tol = norm ([sys.a, sys.e], 2); idx = find (abs (pol) < tol/eps); pol = pol(idx); endif endfunction ## sys = ss (-2, 3, 4, 0) ## sysi = inv (sys) # singular e ## p = pole (sysi) ## infinite poles correct? control/inst/@ss/__prescale__.m0000644000076500000240000000264612273411257016026 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Prescale state-space model. ## Uses SLICOT TB01ID and TG01AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## Author: Lukas Reichlin ## Created: June 2011 ## Version: 0.1 function [retsys, lscale, rscale] = __prescale__ (sys, optarg = 0.0) if (isempty (sys.e)) [a, b, c, ~, scale] = __sl_tb01id__ (sys.a, sys.b, sys.c, optarg); retsys = ss (a, b, c, sys.d); lscale = scale.^-1; rscale = scale; else [a, e, b, c, lscale, rscale] = __sl_tg01ad__ (sys.a, sys.e, sys.b, sys.c, optarg); retsys = dss (a, b, c, sys.d, e); endif retsys.scaled = true; retsys.lti = sys.lti; # retain i/o names and tsam endfunction control/inst/@ss/__property_names__.m0000644000076500000240000000355012273411257017272 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{sys}) ## @deftypefnx {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{sys}, @var{"specific"}) ## Return the list of properties as well as the assignable values for a ss object sys. ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function [props, vals] = __property_names__ (sys, flg) ## cell vector of ss-specific properties props = {"a"; "b"; "c"; "d"; "e"; "stname"; "scaled"}; ## cell vector of ss-specific assignable values vals = {"n-by-n matrix (n = number of states)"; "n-by-m matrix (m = number of inputs)"; "p-by-n matrix (p = number of outputs)"; "p-by-m matrix"; "n-by-n matrix"; "n-by-1 cell vector of strings"; "scalar logical value"}; if (nargin == 1) [ltiprops, ltivals] = __property_names__ (sys.lti); props = [props; ltiprops]; vals = [vals; ltivals]; endif endfunction control/inst/@ss/__set__.m0000644000076500000240000000352712273411257015022 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Set or modify properties of SS objects. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function sys = __set__ (sys, prop, val) switch (prop) # {, } case "a" __ss_dim__ (val, sys.b, sys.c, sys.d); sys.a = val; case "b" __ss_dim__ (sys.a, val, sys.c, sys.d); sys.b = val; case "c" __ss_dim__ (sys.a, sys.b, val, sys.d); sys.c = val; case "d" __ss_dim__ (sys.a, sys.b, sys.c, val); sys.d = val; case "e" if (isempty (val)) sys.e = []; # avoid [](nx0) or [](0xn) else __ss_dim__ (sys.a, sys.b, sys.c, sys.d, val); sys.e = val; endif case {"stname", "statename", "stn", "staten"} n = rows (sys.a); sys.stname = __adjust_labels__ (val, n); case "scaled" if (isscalar (val)) sys.scaled = logical (val); else error ("ss: set: property 'scaled' must be a logical value"); endif otherwise error ("ss: set: invalid property name '%s'", prop); endswitch endfunction control/inst/@ss/__sys2frd__.m0000644000076500000240000000220612273411257015614 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## SS to FRD conversion. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [retsys, retlti] = __sys2frd__ (sys, w = []) if (isempty (w)) # case sys = frd (sys) w = __frequency_vector__ (sys); endif H = __freqresp__ (sys, w); retsys = frd (H, w); # tsam is set below retlti = sys.lti; # preserve lti properties endfunction control/inst/@ss/__sys2tf__.m0000644000076500000240000000560112273411257015454 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## SS to TF conversion. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.4 function [retsys, retlti] = __sys2tf__ (sys) sg_flag = false; # static gain flag try [a, b, c, d, tsam, scaled] = ssdata (sys); # system could be a descriptor model if (tsam == -2 || isempty (a)) # static gain sg_flag = true; else [num, den, ign, igd, md, p, m] = __sl_tb04bd__ (a, b, c, d, scaled); num = reshape (num, md, p, m); den = reshape (den, md, p, m); num = mat2cell (num, md, ones(1,p), ones(1,m)); den = mat2cell (den, md, ones(1,p), ones(1,m)); num = squeeze (num); den = squeeze (den); ign = mat2cell (ign, ones(1,p), ones(1,m)); igd = mat2cell (igd, ones(1,p), ones(1,m)); num = cellfun (@(x, y) x(1:y+1), num, ign, "uniformoutput", false); den = cellfun (@(x, y) x(1:y+1), den, igd, "uniformoutput", false); endif catch ## sys.e was probably singular, therefore ssdata failed. if (issiso (sys)) [num, den] = __siso_ss2tf__ (sys); else [p, m] = size (sys); num = cell (p, m); den = cell (p, m); for i = 1 : p for j = 1 : m idx = substruct ("()", {i, j}); tmp = subsref (sys, idx); # extract siso model tmp = minreal (tmp); # irreducible descriptor representation [n, d] = __siso_ss2tf__ (tmp); num(i, j) = n; den(i, j) = d; endfor endfor endif tsam = get (sys, "tsam"); end_try_catch if (sg_flag) retsys = tf (d); else retsys = tf (num, den, tsam); # tsam needed to set appropriate tfvar endif retlti = sys.lti; # preserve lti properties endfunction function [num, den] = __siso_ss2tf__ (sys) if (isempty (sys.a)) # static gain num = sys.d; den = 1; else # default case [zer, gain] = zero (sys); pol = pole (sys); num = gain * real (poly (zer)); den = real (poly (pol)); endif endfunction control/inst/@ss/__sys_connect__.m0000644000076500000240000000466612273411257016563 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{retsys} =} __sys_connect__ (@var{sys}, @var{M}) ## This function is part of the Model Abstraction Layer. No argument checking. ## For internal use only. ## @example ## @group ## Problem: Solve the system equations of ## . ## E x(t) = A x(t) + B e(t) ## ## y(t) = C x(t) + D e(t) ## ## e(t) = u(t) + M y(t) ## ## in order to build ## . ## K x(t) = F x(t) + G u(t) ## ## y(t) = H x(t) + J u(t) ## ## Solution: Laplace Transformation ## E s X(s) = A X(s) + B U(s) + B M Y(s) [1] ## ## Y(s) = C X(s) + D U(s) + D M Y(s) [2] ## ## solve [2] for Y(s) ## Y(s) = [I - D M]^(-1) C X(s) + [I - D M]^(-1) D U(s) ## ## substitute Z = [I - D M]^(-1) ## Y(s) = Z C X(s) + Z D U(s) [3] ## ## insert [3] in [1], solve for X(s) ## X(s) = [s E - (A + B M Z C)]^(-1) (B + B M Z D) U(s) [4] ## ## inserting [4] in [3] finally yields ## Y(s) = Z C [s E - (A + B M Z C)]^(-1) (B + B M Z D) U(s) + Z D U(s) ## \ / | \_____ _____/ \_____ _____/ \ / ## H K F G J ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function sys = __sys_connect__ (sys, m) a = sys.a; b = sys.b; c = sys.c; d = sys.d; z = eye (rows (d)) - d*m; if (rcond (z) < eps) # check for singularity error ("ss: sys_connect: (I - D*M) singular"); endif z = inv (z); sys.a = a + b*m*z*c; # F sys.b = b + b*m*z*d; # G sys.c = z*c; # H sys.d = z*d; # J ## sys.e remains constant: [] for ss models, e for dss models endfunction control/inst/@ss/__sys_data__.m0000644000076500000240000000207712273411257016035 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Used by ssdata instead of multiple get calls. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.3 function [a, b, c, d, e, stname, scaled] = __sys_data__ (sys) a = sys.a; b = sys.b; c = sys.c; d = sys.d; e = sys.e; stname = sys.stname; scaled = sys.scaled; endfunction control/inst/@ss/__sys_group__.m0000644000076500000240000000361612273411257016260 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Block diagonal concatenation of two SS models. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function retsys = __sys_group__ (sys1, sys2) if (! isa (sys1, "ss")) sys1 = ss (sys1); endif if (! isa (sys2, "ss")) sys2 = ss (sys2); endif retsys = ss (); retsys.lti = __lti_group__ (sys1.lti, sys2.lti); n1 = rows (sys1.a); n2 = rows (sys2.a); [p1, m1] = size (sys1.d); [p2, m2] = size (sys2.d); retsys.a = [sys1.a, zeros(n1,n2); zeros(n2,n1), sys2.a]; retsys.b = [sys1.b, zeros(n1,m2); zeros(n2,m1), sys2.b]; retsys.c = [sys1.c, zeros(p1,n2); zeros(p2,n1), sys2.c]; retsys.d = [sys1.d, zeros(p1,m2); zeros(p2,m1), sys2.d]; e1 = ! isempty (sys1.e); e2 = ! isempty (sys2.e); if (e1 || e2) if (e1 && e2) retsys.e = [sys1.e, zeros(n1,n2); zeros(n2,n1), sys2.e]; elseif (e1) retsys.e = [sys1.e, zeros(n1,n2); zeros(n2,n1), eye(n2)]; else retsys.e = [eye(n1), zeros(n1,n2); zeros(n2,n1), sys2.e]; endif endif retsys.stname = [sys1.stname; sys2.stname]; endfunction control/inst/@ss/__sys_inverse__.m0000644000076500000240000000311412273411257016570 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Inversion of SS models. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function sys = __sys_inverse__ (sys) a = sys.a; b = sys.b; c = sys.c; d = sys.d; e = sys.e; if (! isempty (e) || rcond (d) < eps) # dss or strictly proper ss n = rows (a); m = columns (b); # p = m (square system) if (isempty (e)) # avoid testing twice? e = eye (n); endif sys.a = [a, b; c, d]; sys.b = [zeros(n, m); -eye(m)]; sys.c = [zeros(m, n), eye(m)]; sys.d = zeros (m); sys.e = [e, zeros(n, m); zeros(m, n+m)]; sys.stname = repmat ({""}, n+m, 1); else # proper ss bid = b / d; sys.a = a - bid * c; sys.b = -bid; sys.c = d \ c; sys.d = inv (d); endif endfunction control/inst/@ss/__sys_prune__.m0000644000076500000240000000346312273411257016255 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Submodel extraction and reordering for SS objects. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.3 function sys = __sys_prune__ (sys, out_idx, in_idx, st_idx = ":") [sys.lti, out_idx, in_idx] = __lti_prune__ (sys.lti, out_idx, in_idx); if (ischar (st_idx) && ! strcmp (st_idx, ":")) st_idx = {st_idx}; endif if (iscell (st_idx)) st_idx = cellfun (@(x) __str2idx__ (sys.stname, x), st_idx); endif sys.a = sys.a(st_idx, st_idx); sys.b = sys.b(st_idx, in_idx); sys.c = sys.c(out_idx, st_idx); sys.d = sys.d(out_idx, in_idx); if (! isempty (sys.e)) sys.e = sys.e(st_idx, st_idx); endif sys.stname = sys.stname(st_idx); endfunction function idx = __str2idx__ (name, str) tmp = strcmp (name, str)(:); switch (nnz (tmp)) case 1 idx = find (tmp); case 0 error ("ss: state name '%s' not found", str); otherwise error ("ss: state name '%s' is ambiguous", str); endswitch endfunction control/inst/@ss/__transpose__.m0000644000076500000240000000213112273411257016233 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Transpose of SS models. ## Author: Lukas Reichlin ## Created: February 2010 ## Version: 0.2 function sys = __transpose__ (sys) a = sys.a; b = sys.b; c = sys.c; d = sys.d; e = sys.e; sys.stname = repmat ({""}, rows (a), 1); sys.a = a.'; sys.b = c.'; sys.c = b.'; sys.d = d.'; sys.e = e.'; endfunction control/inst/@ss/__zero__.m0000644000076500000240000000307112273411257015200 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Invariant zeros of SS object. ## Uses SLICOT AB08ND and AG08BD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.4 function [zer, gain, info] = __zero__ (sys, argc) if (isempty (sys.e)) [zer, gain, rank, infz, kronr, kronl] = __sl_ab08nd__ (sys.a, sys.b, sys.c, sys.d, sys.scaled); else [zer, rank, infz, kronr, kronl] = __sl_ag08bd__ (sys.a, sys.e, sys.b, sys.c, sys.d, sys.scaled); if (argc > 1 && issiso (sys)) pol = pole (sys); gain = __sl_tg04bx__ (sys.a, sys.e, sys.b, sys.c, sys.d, ... real (pol), imag (pol), real (zer), imag (zer)); else gain = []; endif endif info = struct ("rank", rank, "infz", infz, "kronr", kronr, "kronl", kronl); endfunction control/inst/@ss/display.m0000644000076500000240000000530612273411257015075 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Display routine for SS objects. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function display (sys) sysname = inputname (1); [inname, outname, tsam] = __lti_data__ (sys.lti); stname = sys.stname; inname = __labels__ (inname, "u"); outname = __labels__ (outname, "y"); stname = __labels__ (stname, "x"); disp (""); if (! isempty (sys.e)) __disp_mat__ (sys.e, [sysname, ".e"], stname, stname); endif if (! isempty (sys.a)) __disp_mat__ (sys.a, [sysname, ".a"], stname, stname); __disp_mat__ (sys.b, [sysname, ".b"], stname, inname); __disp_mat__ (sys.c, [sysname, ".c"], outname, stname); endif __disp_mat__ (sys.d, [sysname, ".d"], outname, inname); display (sys.lti); # display sampling time if (tsam == -2) disp ("Static gain."); elseif (tsam == 0) disp ("Continuous-time model."); else disp ("Discrete-time model."); endif endfunction function __disp_mat__ (m, mname, rname, cname) MAX_LEN = 12; # max length of row name and column name [mrows, mcols] = size (m); row_name = strjust (strvcat (" ", rname), "left"); row_name = row_name(:, 1 : min (MAX_LEN, end)); row_name = horzcat (repmat (" ", mrows+1, 3), row_name); mat = cell (1, mcols); for k = 1 : mcols cname{k} = cname{k}(:, 1 : min (MAX_LEN, end)); acol = vertcat (cname(k), cellstr (deblank (num2str (m(:, k), 4)))); mat{k} = strjust (strvcat (acol{:}), "right"); endfor lcols = cellfun (@columns, mat); lcols_max = 2 + max (horzcat (lcols, 1)); for k = 1 : mcols mat{k} = horzcat (repmat (" ", mrows+1, lcols_max-lcols(k)), mat{k}); endfor tsize = terminal_size (); dispcols = max (1, floor ((tsize(2) - columns (row_name)) / lcols_max)); disprows = max (1, ceil (mcols / dispcols)); disp ([mname, " ="]); for k = 1 : disprows disp (horzcat (row_name, mat{1+(k-1)*dispcols : min (mcols, k*dispcols)})); disp (""); endfor endfunction control/inst/@ss/ss.m0000644000076500000240000001456412273411257014063 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} ss (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} ss (@var{d}) ## @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}) ## @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}) ## @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}, @var{d}, @dots{}) ## @deftypefnx {Function File} {@var{sys} =} ss (@var{a}, @var{b}, @var{c}, @var{d}, @var{tsam}, @dots{}) ## Create or convert to state-space model. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model to be converted to state-space. ## @item a ## State matrix (n-by-n). ## @item b ## Input matrix (n-by-m). ## @item c ## Output matrix (p-by-n). ## If @var{c} is empty @code{[]} or not specified, an identity matrix is assumed. ## @item d ## Feedthrough matrix (p-by-m). ## If @var{d} is empty @code{[]} or not specified, a zero matrix is assumed. ## @item tsam ## Sampling time in seconds. If @var{tsam} is not specified, a continuous-time model is assumed. ## @item @dots{} ## Optional pairs of properties and values. ## Type @command{set (ss)} for more information. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## State-space model. ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'a', 'b', 'c', 'd', 'e' ## State-space matrices. See 'Inputs' for details. ## ## @item 'stname' ## The name of the states in @var{sys}. ## Cell vector containing strings for each state. ## Default names are @code{@{'x1', 'x2', ...@}} ## ## @item 'scaled' ## Logical. If set to true, no automatic scaling is used, ## e.g. for frequency response plots. ## ## @item 'tsam' ## Sampling time. See 'Inputs' for details. ## ## @item 'inname' ## The name of the input channels in @var{sys}. ## Cell vector of length m containing strings. ## Default names are @code{@{'u1', 'u2', ...@}} ## ## @item 'outname' ## The name of the output channels in @var{sys}. ## Cell vector of length p containing strings. ## Default names are @code{@{'y1', 'y2', ...@}} ## ## @item 'ingroup' ## Struct with input group names as field names and ## vectors of input indices as field values. ## Default is an empty struct. ## ## @item 'outgroup' ## Struct with output group names as field names and ## vectors of output indices as field values. ## Default is an empty struct. ## ## @item 'name' ## String containing the name of the model. ## ## @item 'notes' ## String or cell of string containing comments. ## ## @item 'userdata' ## Any data type. ## @end table ## ## @strong{Example} ## @example ## @group ## octave:1> a = [1 2 3; 4 5 6; 7 8 9]; ## octave:2> b = [10; 11; 12]; ## octave:3> stname = @{"V", "A", "kJ"@}; ## octave:4> sys = ss (a, b, [], [], "stname", stname) ## @end group ## @end example ## ## @example ## @group ## sys.a = ## V A kJ ## V 1 2 3 ## A 4 5 6 ## kJ 7 8 9 ## @end group ## @end example ## ## @example ## @group ## sys.b = ## u1 ## V 10 ## A 11 ## kJ 12 ## @end group ## @end example ## ## @example ## @group ## sys.c = ## V A kJ ## y1 1 0 0 ## y2 0 1 0 ## y3 0 0 1 ## @end group ## @end example ## ## @example ## @group ## sys.d = ## u1 ## y1 0 ## y2 0 ## y3 0 ## ## Continuous-time model. ## octave:5> ## @end group ## @end example ## ## @seealso{tf, dss} ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.3 function sys = ss (a = [], b = [], c = [], d = [], varargin) ## model precedence: frd > ss > zpk > tf > double ## inferiorto ("frd"); superiorto ("zpk", "tf", "double"); argc = 0; # initialize argument count tsam = 0; # initialize sampling time if (nargin == 1) if (isa (a, "ss")) # already in ss form sys = ss (sssys) sys = a; return; elseif (isa (a, "lti")) # another lti object sys = ss (sys) [sys, alti] = __sys2ss__ (a); sys.lti = alti; # preserve lti properties return; elseif (is_real_matrix (a)) # static gain sys = ss (5), sys = ss (matrix) d = a; a = []; else print_usage (); endif elseif (nargin > 4) # default case sys = ss (a, b, c, d, "prop1", val1, ...) argc = numel (varargin); # number of additional arguments after d if (issample (varargin{1}, -10)) # sys = ss (a, b, c, d, tsam, "prop1, "val1", ...) tsam = varargin{1}; # sampling time, could be 0 as well argc--; # tsam is not a property-value pair if (argc > 0) # if there are any properties and values ... varargin = varargin(2:end); # remove tsam from property-value list endif endif endif # nothing to do for ss (), ss (a, b), ss (a, b, c), ss (a, b, c, d) [a, b, c, d, tsam] = __adjust_ss_data__ (a, b, c, d, tsam); [p, m, n] = __ss_dim__ (a, b, c, d); # determine number of outputs, inputs and states stname = repmat ({""}, n, 1); # cell with empty state names ssdata = struct ("a", a, "b", b, "c", c, "d", d, "e", [], "stname", {stname}, "scaled", false); # struct for ss-specific data ltisys = lti (p, m, tsam); # parent class for general lti data sys = class (ssdata, "ss", ltisys); # create ss object if (argc > 0) # if there are any properties and values, ... sys = set (sys, varargin{:}); # use the general set function endif endfunction control/inst/@tf/0000755000076500000240000000000012273411257013232 5ustar lukasstaffcontrol/inst/@tf/__c2d__.m0000644000076500000240000000544512273411257014664 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert the continuous TF model into its discrete-time equivalent. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function sys = __c2d__ (sys, tsam, method = "zoh", w0 = 0) if (strncmpi (method, "m", 1)) # "matched" ## TODO: move this code to @zpk/__c2d__.m once ZPK models are implemented if (! issiso (sys)) error ("tf: c2d: require SISO system for matched pole/zero method"); endif [z_c, p_c, k_c] = zpkdata (sys, "vector"); p_d = exp (p_c * tsam); z_d = exp (z_c * tsam); if (any (! isfinite (p_d)) || any (! isfinite (z_d))) error ("tf: c2d: discrete-time poles and zeros are not finite"); endif ## continuous-time zeros at infinity are mapped to -1 in discrete-time ## except for one. for non-proper transfer functions, no zeros at -1 are added. np = length (p_c); # number of poles nz = length (z_c); # number of finite zeros, np-nz number of infinite zeros z_d = vertcat (z_d, repmat (-1, np-nz-1, 1)); ## the discrete-time gain k_d is matched at a certain frequency (w_c, w_d) ## to continuous-time gain k_c. the dc gain is taken (w_c=0, w_d=1) unless ## there are continuous-time poles/zeros near 0. then w_c=1/tsam is taken. w_c = 0; # dc gain tol = sqrt (eps); # poles/zeros below tol are assumed to be zero while (any (abs ([p_c; z_c] - w_c) < tol)) w_c += 0.1 / tsam; endwhile w_d = exp (w_c * tsam); k_d = real (k_c * prod (w_c - z_c) / prod (w_c - p_c) * prod (w_d - p_d) / prod (w_d - z_d)); tmp = zpk (z_d, p_d, k_d, tsam); sys.num = tmp.num; sys.den = tmp.den; else [p, m] = size (sys); for i = 1 : p for j = 1 : m idx = substruct ("()", {i, j}); tmp = subsref (sys, idx); tmp = c2d (ss (tmp), tsam, method, w0); [num, den] = tfdata (tmp, "tfpoly"); sys.num(i, j) = num; sys.den(i, j) = den; endfor endfor endif sys.tfvar = "z"; endfunction control/inst/@tf/__ctranspose__.m0000644000076500000240000000252612273411257016372 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Conjugate transpose of TF models. ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function sys = __ctranspose__ (sys, ct) num = sys.num; den = sys.den; if (ct) # continuous-time num = cellfun (@conj_ct, num, "uniformoutput", false); den = cellfun (@conj_ct, den, "uniformoutput", false); else # discrete-time num = cellfun (@conj_dt, num, "uniformoutput", false); den = cellfun (@conj_dt, den, "uniformoutput", false); ## TODO: shall I make "den" a monic polynomial? endif sys.num = num.'; sys.den = den.'; endfunction control/inst/@tf/__d2c__.m0000644000076500000240000000432612273411257014661 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert the discrete TF model into its continuous-time equivalent. ## Author: Lukas Reichlin ## Created: September 2011 ## Version: 0.2 function sys = __d2c__ (sys, tsam, method = "zoh", w0 = 0) if (strncmpi (method, "m", 1)) # "matched" ## TODO: move this code to @zpk/__d2c__.m once ZPK models are implemented if (! issiso (sys)) error ("tf: d2c: require SISO system for matched pole/zero method"); endif [z_d, p_d, k_d] = zpkdata (sys, "vector"); if (any (abs (p_d) < eps) || any (abs (z_d) < eps)) error ("tf: d2c: discrete-time poles and zeros at 0 not supported because log(0) is -Inf"); endif z_d_orig = z_d; z_d(abs (z_d+1) < sqrt (eps)) = []; p_c = log (p_d) / tsam; z_c = log (z_d) / tsam; w_c = 0; w_d = 1; tol = sqrt (eps); while (any (abs ([p_d; z_d_orig] - w_d) < tol)) w_c += 0.1 / tsam; endwhile w_d = exp (w_c * tsam); k_c = real (k_d * prod (w_d - z_d_orig) / prod (w_d - p_d) * prod (w_c - p_c) / prod (w_c - z_c)); tmp = zpk (z_c, p_c, k_c); sys.num = tmp.num; sys.den = tmp.den; else [p, m] = size (sys); for i = 1 : p for j = 1 : m idx = substruct ("()", {i, j}); tmp = subsref (sys, idx); tmp = d2c (ss (tmp), method, w0); [num, den] = tfdata (tmp, "tfpoly"); sys.num(i, j) = num; sys.den(i, j) = den; endfor endfor endif sys.tfvar = "s"; endfunction control/inst/@tf/__freqresp__.m0000644000076500000240000000270312273411257016035 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Frequency response of TF models. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.5 function H = __freqresp__ (sys, w, cellflag = false) [num, den, tsam] = tfdata (sys, "vector"); if (isct (sys)) # continuous system s = i * w; else # discrete system s = exp (i * w * abs (tsam)); endif s = reshape (s, 1, 1, []); if (issiso (sys)) H = polyval (num, s) ./ polyval (den, s); else H = cellfun (@(x, y) polyval (x, s) ./ polyval (y, s), num, den, "uniformoutput", false); H = cell2mat (H); endif if (cellflag) [p, m] = size (sys); l = length (s); H = mat2cell (H, p, m, ones (1, l))(:); endif endfunction control/inst/@tf/__get__.m0000644000076500000240000000245712273411257014773 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Access property values of TF objects. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function val = __get__ (sys, prop) switch (prop) # {, } case "num" val = cellfun (@get, sys.num, "uniformoutput", false); case "den" val = cellfun (@get, sys.den, "uniformoutput", false); case {"tfvar", "variable", "var"} val = sys.tfvar; case "inv" val = sys.inv; otherwise error ("tf: get: invalid property name '%s'", prop); endswitch endfunction control/inst/@tf/__minreal__.m0000644000076500000240000000422012273411257015631 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Minimal realization of TF models. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function sys = __minreal__ (sys, tol) ## TODO: Once ZPK models are implemented, convert TF to ZPK, ## do cancellations over there and convert back to TF sqrt_eps = sqrt (eps); # treshold for zero [p, m] = size (sys); for ny = 1 : p for nu = 1 : m sisosys = __sys_prune__ (sys, ny, nu); [zer, gain] = zero (sisosys); pol = pole (sisosys); for k = length (zer) : -1 : 1 # reversed because of deleted zeros [~, idx] = min (abs (zer(k) - pol)); # find best match if (tol == "def") if (abs (zer(k)) < sqrt_eps) # catch case zer(k) = 0 t = 1000 * eps; else t = 1000 * abs (zer(k)) * sqrt_eps; endif else t = tol; endif if (abs (zer(k) - pol(idx)) < t) zer(k) = []; pol(idx) = []; endif endfor num = real (gain * poly (zer)); den = real (poly (pol)); num_idx = find (abs (num) < sqrt_eps); # suppress numerical noise den_idx = find (abs (den) < sqrt_eps); # in polynomial coefficients num(num_idx) = 0; den(den_idx) = 0; sys.num{ny, nu} = tfpoly (num); sys.den{ny, nu} = tfpoly (den); endfor endfor endfunction control/inst/@tf/__pole__.m0000644000076500000240000000211512273411257015142 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Poles of TF object. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function pol = __pole__ (sys) if (issiso (sys)) pol = roots (sys.den{1}); else warning ("tf: pole: converting to minimal state-space for pole computation of mimo tf"); pol = pole (ss (sys)); endif endfunction control/inst/@tf/__property_names__.m0000644000076500000240000000341312273411257017254 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{sys}) ## @deftypefnx {Function File} {[@var{props}, @var{vals}] =} __property_names__ (@var{sys}, @var{"specific"}) ## Return the list of properties as well as the assignable values for a ss object sys. ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function [props, vals] = __property_names__ (sys, flg) ## cell vector of tf-specific properties props = {"num"; "den"; "tfvar"; "inv"}; ## cell vector of tf-specific assignable values vals = {"p-by-m cell array of row vectors (m = number of inputs)"; "p-by-m cell array of row vectors (p = number of outputs)"; "string (usually s or z)"; "logical (true for negative powers of TF variable)"}; if (nargin == 1) [ltiprops, ltivals] = __property_names__ (sys.lti); props = [props; ltiprops]; vals = [vals; ltivals]; endif endfunction control/inst/@tf/__set__.m0000644000076500000240000000330512273411257015000 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Set or modify properties of TF objects. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function sys = __set__ (sys, prop, val) switch (prop) # {, } case "num" num = __vec2tfpoly__ (val); __tf_dim__ (num, sys.den); sys.num = num; case "den" den = __vec2tfpoly__ (val); __tf_dim__ (sys.num, den); sys.den = den; case {"tfvar", "variable", "var"} if (ischar (val)) sys.tfvar = val; else error ("tf: set: invalid transfer function variable"); endif case "inv" if (! isdt (sys)) error ("tf: set: property 'inv' requires discrete-time system"); elseif (! isscalar (val)) error ("tf: set: property 'inv' must be a scalar logical"); else sys.inv = logical (val); endif otherwise error ("tf: set: invalid property name '%s'", prop); endswitch endfunction control/inst/@tf/__sys2frd__.m0000644000076500000240000000220612273411257015600 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## TF to FRD conversion. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [retsys, retlti] = __sys2frd__ (sys, w = []) if (isempty (w)) # case sys = frd (sys) w = __frequency_vector__ (sys); endif H = __freqresp__ (sys, w); retsys = frd (H, w); # tsam is set below retlti = sys.lti; # preserve lti properties endfunction control/inst/@tf/__sys2ss__.m0000644000076500000240000001257212273411257015461 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## TF to SS conversion. ## Reference: ## Varga, A.: Computation of irreducible generalized state-space realizations. ## Kybernetika, 26:89-106, 1990 ## Special thanks to Vasile Sima and Andras Varga for their advice. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.5 function [retsys, retlti] = __sys2ss__ (sys) ## TODO: determine appropriate tolerance from number of inputs ## (since we multiply all denominators in a row), index, ... ## default tolerance from TB01UD is TOLDEF = N*N*EPS ## SECRET WISH: a routine which accepts individual denominators for ## each channel and which supports descriptor systems [p, m] = size (sys); [num, den] = tfdata (sys); len_num = cellfun (@length, num); len_den = cellfun (@length, den); ## check for properness ## tfpoly ensures that there are no leading zeros tmp = len_num > len_den; if (any (tmp(:))) # non-proper transfer function ## separation into strictly proper and polynomial part [numq, numr] = cellfun (@deconv, num, den, "uniformoutput", false); numq = cellfun (@__remove_leading_zeros__, numq, "uniformoutput", false); numr = cellfun (@__remove_leading_zeros__, numr, "uniformoutput", false); ## minimal state-space realization for the proper part [a1, b1, c1] = __proper_tf2ss__ (numr, den, p, m); e1 = eye (size (a1)); ## minimal realization for the polynomial part [e2, a2, b2, c2] = __polynomial_tf2ss__ (numq, p, m); ## assemble irreducible descriptor realization e = blkdiag (e1, e2); a = blkdiag (a1, a2); b = vertcat (b1, b2); c = horzcat (c1, c2); retsys = dss (a, b, c, [], e); else # proper transfer function [a, b, c, d] = __proper_tf2ss__ (num, den, p, m); retsys = ss (a, b, c, d); endif retlti = sys.lti; # preserve lti properties such as tsam endfunction ## transfer function to state-space conversion for proper models function [a, b, c, d] = __proper_tf2ss__ (num, den, p, m) ## new cells for the TF of same row denominators numc = cell (p, m); denc = cell (p, 1); ## set zero denominators to 1 for convolution zero_idx = cellfun (@(x) all (x == 0), den); den(zero_idx) = 1; ## multiply all denominators in a row and ## update each numerator accordingly ## except for single-input models and those ## with equal denominators in a row for i = 1 : p if (m == 1 || isequal (den{i,:})) denc(i) = den{i,1}; numc(i,:) = num(i,:); else denc(i) = __conv__ (den{i,:}); for j = 1 : m idx = setdiff (1:m, j); numc(i,j) = __conv__ (num{i,j}, den{i,idx}); endfor endif endfor ## set numerators to zero if their denominators are zero numc(zero_idx) = 0; len_numc = cellfun (@length, numc); len_denc = cellfun (@length, denc); ## check for properness ## tfpoly ensures that there are no leading zeros ## tmp = len_numc > repmat (len_denc, 1, m); ## if (any (tmp(:))) ## error ("tf: tf2ss: system must be proper"); ## endif ## create arrays and fill in the data ## in a way that Slicot TD04AD can use max_len_denc = max (len_denc(:)); ucoeff = zeros (p, m, max_len_denc); dcoeff = zeros (p, max_len_denc); index = len_denc-1; for i = 1 : p len = len_denc(i); dcoeff(i, 1:len) = denc{i}; for j = 1 : m ucoeff(i, j, len-len_numc(i,j)+1 : len) = numc{i,j}; endfor endfor tol = min (sqrt (eps), eps*prod (index)); [a, b, c, d] = __sl_td04ad__ (ucoeff, dcoeff, index, tol); endfunction ## realization of the polynomial part according to Andras' paper function [e2, a2, b2, c2] = __polynomial_tf2ss__ (numq, p, m) len_numq = cellfun (@length, numq); max_len_numq = max (len_numq(:)); numq = cellfun (@(x) prepad (x, max_len_numq, 0, 2), numq, "uniformoutput", false); f = @(y) cellfun (@(x) x(y), numq); s = 1 : max_len_numq; D = arrayfun (f, s, "uniformoutput", false); e2 = diag (ones (p*(max_len_numq-1), 1), -p); a2 = eye (p*max_len_numq); b2 = vertcat (D{:}); c2 = horzcat (zeros (p, p*(max_len_numq-1)), -eye (p)); ## remove uncontrollable part [a2, e2, b2, c2] = __sl_tg01jd__ (a2, e2, b2, c2, 0.0, true, 1, 2); endfunction ## convolution for more than two arguments function vec = __conv__ (vec, varargin) if (nargin == 1) return; else for k = 1 : nargin-1 vec = conv (vec, varargin{k}); endfor endif endfunction ## remove leading zeros from polynomial vector function p = __remove_leading_zeros__ (p) idx = find (p != 0); if (isempty (idx)) p = 0; else p = p(idx(1) : end); # p(idx) would remove all zeros endif endfunction control/inst/@tf/__sys_connect__.m0000644000076500000240000001052412273411257016535 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{retsys} =} __sys_connect__ (@var{sys}, @var{M}) ## This function is part of the Model Abstraction Layer. No argument checking. ## For internal use only. ## @example ## @group ## Problem: Solve the system equations of ## Y(s) = G(s) E(s) ## E(s) = U(s) + M Y(s) ## in order to build ## Y(s) = H(s) U(s) ## Solution: ## Y(s) = G(s) [I - M G(s)]^-1 U(s) ## = [I - G(s) M]^-1 G(s) U(s) ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function sys = __sys_connect__ (sys, M) [p, m] = size (sys); num = sys.num; den = sys.den; ## TODO: Implementation for MIMO models. There are three possibilities: ## 1. An _algebraic_ solution of the inversion problem in order ## to not introduce unwanted zero/pole pairs. Difficult. ## 2. A numeric solution of the inversion problem. Afterwards, ## elimination of _all_ zero/pole pairs by minreal. Bad. ## 3. Conversion to state-space, solving the problem there and ## converting back to transfer function. Easier, but obviously, ## this way needs MIMO __sys2ss__ and __sys2tf__ implementations ## as described in Thomas Kailath's classic "Linear Systems". ## Possibly this is the way to go, but it works for proper systems ## only unless descriptor state-space models are implemented. ## WARNING: The code below is a cheap hack to quickly enable SISO TF connections. ## TODO: Check for den = 0, e.g. in feedback (tf (1), tf (-1)) if (p == 2 && m == 2 && num{1,2} == 0 && num{2,1} == 0 ... && M(1,1) == 0 && M(2,2) == 0) ## mtimes, feedback sys.num(1,1) = num{1,1} * den{2,2}; sys.num(1,2) = M(1,2) * num{1,1} * num{2,2}; sys.num(2,1) = M(2,1) * num{1,1} * num{2,2}; sys.num(2,2) = num{2,2} * den{1,1}; sys.den(:) = den{1,1} * den{2,2} - M(1,2) * M(2,1) * num{1,1} * num{2,2}; elseif (p == 3 && m == 4 && num{1,3} == 0 && num{1,4} == 0 ... && num{2,1} == 0 && num{2,2} == 0 && num{2,4} == 0 ... && num{3,1} == 0 && num{3,2} == 0 && num{3,3} == 0 ... && M == [0, 1, 0; 0, 0, 1; 0, 0, 0; 0, 0, 0]) ## horzcat [sys1, sys2], plus, minus sys.num(:) = tfpoly (0); sys.den(:) = tfpoly (1); sys.num(1,1) = num{1,1}; sys.num(1,2) = num{1,2}; sys.num(1,3) = num{1,1} * num{2,3}; sys.num(1,4) = num{1,2} * num{3,4}; sys.num(2,3) = num{2,3}; sys.num(3,4) = num{3,4}; sys.den(1,3) = den{2,3}; sys.den(1,4) = den{3,4}; sys.den(2,3) = den{2,3}; sys.den(3,4) = den{3,4}; elseif (p == 3 && m == 3 && num{1,3} == 0 ... && num{2,1} == 0 && num{2,2} == 0 && num{2,3} == 1 ... && num{3,1} == 0 && num{3,2} == 0 && num{3,3} == 1 ... && M == [0, 1, 0; 0, 0, 1; 0, 0, 0]) ## plus, minus sys.num(1,3) = num{1,1} * den{1,2} + num{1,2} * den{1,1}; sys.den(1,3) = den{1,1} * den{1,2}; elseif (p == 4 && m == 3 && num{1,2} == 0 && num{1,3} == 0 ... && num{2,1} == 0 && num{2,3} == 0 ... && num{3,1} == 0 && num{3,2} == 0 && num{3,3} == 1 ... && num{4,1} == 0 && num{4,2} == 0 && num{4,3} == 1 ... && M == [0, 0, 1, 0; 0, 0, 0, 1; 0, 0, 0, 0]) ## vertcat [sys1; sys2] sys.num(1,3) = num{1,1}; sys.num(2,3) = num{2,2}; sys.den(1,3) = den{1,1}; sys.den(2,3) = den{2,2}; else ## MIMO case, convert to state-space and back. warning ("tf: converting to minimal state-space for MIMO TF interconnections"); sys = tf (__sys_connect__ (ss (sys), M)); endif endfunction control/inst/@tf/__sys_data__.m0000644000076500000240000000176712273411257016026 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Used by tfdata instead of multiple get calls. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function [num, den, tfvar] = __sys_data__ (sys) num = sys.num; den = sys.den; tfvar = sys.tfvar; endfunction control/inst/@tf/__sys_group__.m0000644000076500000240000000337612273411257016247 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Block diagonal concatenation of two TF models. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function retsys = __sys_group__ (sys1, sys2) if (! isa (sys1, "tf")) sys1 = tf (sys1); endif if (! isa (sys2, "tf")) sys2 = tf (sys2); endif retsys = tf (); retsys.lti = __lti_group__ (sys1.lti, sys2.lti); [p1, m1] = size (sys1); [p2, m2] = size (sys2); empty12 = tfpolyzeros (p1, m2); empty21 = tfpolyzeros (p2, m1); retsys.num = [sys1.num, empty12 ; empty21, sys2.num]; empty12 = tfpolyones (p1, m2); empty21 = tfpolyones (p2, m1); retsys.den = [sys1.den, empty12 ; empty21, sys2.den]; if (sys1.tfvar == sys2.tfvar) retsys.tfvar = sys1.tfvar; elseif (sys1.tfvar == "x") retsys.tfvar = sys2.tfvar; else retsys.tfvar = sys1.tfvar; endif if (sys1.inv || sys2.inv) retsys.inv = true; endif endfunction control/inst/@tf/__sys_inverse__.m0000644000076500000240000000330512273411257016556 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Inversion of TF models. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function sys = __sys_inverse__ (sys) nvec = size (sys); num = sys.num; den = sys.den; if (all (nvec == 1)) # SISO if (num{1,1} == 0) # catch case num = 0 sys.num(1,1) = tfpoly (0); sys.den(1,1) = tfpoly (1); else sys.num = den; sys.den = num; endif elseif (all (nvec == 2)) # 2x2 MIMO sys.num(1,1) = -den{1,1}*den{1,2}*den{2,1}*num{2,2}; sys.num(1,2) = den{1,1}*den{2,1}*den{2,2}*num{1,2}; sys.num(2,1) = den{1,1}*den{1,2}*den{2,2}*num{2,1}; sys.num(2,2) = -den{1,2}*den{2,1}*den{2,2}*num{1,1}; sys.den(:) = den{1,1}*den{2,2}*num{1,2}*num{2,1} - den{1,2}*den{2,1}*num{1,1}*num{2,2}; else ## I've calculated 3x3 systems with sage but the formula is quite long [num, den] = tfdata (inv (ss (sys)), "tfpoly"); sys.num = num; sys.den = den; endif endfunction control/inst/@tf/__sys_prune__.m0000644000076500000240000000225612273411257016240 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Submodel extraction and reordering for TF objects. ## This file is part of the Model Abstraction Layer. ## For internal use only. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function sys = __sys_prune__ (sys, out_idx, in_idx) [sys.lti, out_idx, in_idx] = __lti_prune__ (sys.lti, out_idx, in_idx); sys.num = sys.num(out_idx, in_idx); sys.den = sys.den(out_idx, in_idx); endfunction control/inst/@tf/__transpose__.m0000644000076500000240000000174712273411257016233 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Transpose of TF models. ## Author: Lukas Reichlin ## Created: February 2010 ## Version: 0.1 function sys = __transpose__ (sys) num = sys.num; den = sys.den; sys.num = num.'; sys.den = den.'; endfunction control/inst/@tf/__zero__.m0000644000076500000240000000232212273411257015162 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Transmission zeros of TF object. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function [zer, gain, info] = __zero__ (sys) if (issiso (sys)) num = get (sys.num{1}); den = get (sys.den{1}); zer = roots (num); gain = num(1) / den(1); info = []; else warning ("tf: zero: converting to minimal state-space for zero computation of mimo tf"); [zer, gain, info] = zero (ss (sys)); endif endfunction control/inst/@tf/display.m0000644000076500000240000000537412273411257015066 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Display routine for TF objects. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.4 function display (sys) sysname = inputname (1); [inname, outname, tsam] = __lti_data__ (sys.lti); [inname, m] = __labels__ (inname, "u"); [outname, p] = __labels__ (outname, "y"); disp (""); if (sys.inv && ! isct (sys)) [num, den] = filtdata (sys); else num = sys.num; den = sys.den; endif for nu = 1 : m disp (["Transfer function '", sysname, "' from input '", inname{nu}, "' to output ..."]); disp (""); for ny = 1 : p __disp_frac__ (num{ny, nu}, den{ny, nu}, sys.tfvar, outname{ny}); endfor endfor display (sys.lti); # display sampling time if (tsam == -2) disp ("Static gain."); elseif (tsam == 0) disp ("Continuous-time model."); else disp ("Discrete-time model."); endif endfunction function __disp_frac__ (num, den, tfvar, name) MAX_LEN = 12; # max length of output name tfp = isa (num, "tfpoly"); if (num == tfpoly (0)) str = [" ", name, ": 0"]; elseif ((tfp && den == 1) || (! tfp && isequal (den, 1))) ## elseif (den == tfpoly (1)) doesn't work because it ## would mistakingly accept non-tfpoly denominators like [0, 1] str = [" ", name, ": "]; numstr = tfpoly2str (num, tfvar); str = [str, numstr]; ##elseif (length (den) == 1) # de-comment for non-development use ## str = [" ", name, ": "]; ## num = num * (1/get (den)); ## numstr = tfpoly2str (num, tfvar); ## str = [str, numstr]; else numstr = tfpoly2str (num, tfvar); denstr = tfpoly2str (den, tfvar); fracstr = repmat ("-", 1, max (length (numstr), length (denstr))); str = strjust (strvcat (numstr, fracstr, denstr), "center"); namestr = name(:, 1 : min (MAX_LEN, end)); namestr = [namestr, ": "]; namestr = strjust (strvcat (" ", namestr, " "), "left"); namestr = horzcat (repmat (" ", 3, 1), namestr); str = [namestr, str]; endif disp (str); disp (""); endfunction control/inst/@tf/tf.m0000644000076500000240000002037012273411257014023 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{s} =} tf (@var{'s'}) ## @deftypefnx {Function File} {@var{z} =} tf (@var{'z'}, @var{tsam}) ## @deftypefnx {Function File} {@var{sys} =} tf (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} tf (@var{num}, @var{den}, @dots{}) ## @deftypefnx {Function File} {@var{sys} =} tf (@var{num}, @var{den}, @var{tsam}, @dots{}) ## Create or convert to transfer function model. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model to be converted to transfer function. ## @item num ## Numerator or cell of numerators. Each numerator must be a row vector ## containing the coefficients of the polynomial in descending powers of ## the transfer function variable. ## num@{i,j@} contains the numerator polynomial from input j to output i. ## In the SISO case, a single vector is accepted as well. ## @item den ## Denominator or cell of denominators. Each denominator must be a row vector ## containing the coefficients of the polynomial in descending powers of ## the transfer function variable. ## den@{i,j@} contains the denominator polynomial from input j to output i. ## In the SISO case, a single vector is accepted as well. ## @item tsam ## Sampling time in seconds. If @var{tsam} is not specified, a continuous-time ## model is assumed. ## @item @dots{} ## Optional pairs of properties and values. ## Type @command{set (tf)} for more information. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Transfer function model. ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'num' ## Numerator. See 'Inputs' for details. ## ## @item 'den' ## Denominator. See 'Inputs' for details. ## ## @item 'tfvar' ## String containing the transfer function variable. ## ## @item 'inv' ## Logical. True for negative powers of the transfer function variable. ## ## @item 'tsam' ## Sampling time. See 'Inputs' for details. ## ## @item 'inname' ## The name of the input channels in @var{sys}. ## Cell vector of length m containing strings. ## Default names are @code{@{'u1', 'u2', ...@}} ## ## @item 'outname' ## The name of the output channels in @var{sys}. ## Cell vector of length p containing strings. ## Default names are @code{@{'y1', 'y2', ...@}} ## ## @item 'ingroup' ## Struct with input group names as field names and ## vectors of input indices as field values. ## Default is an empty struct. ## ## @item 'outgroup' ## Struct with output group names as field names and ## vectors of output indices as field values. ## Default is an empty struct. ## ## @item 'name' ## String containing the name of the model. ## ## @item 'notes' ## String or cell of string containing comments. ## ## @item 'userdata' ## Any data type. ## @end table ## ## @strong{Example} ## @example ## @group ## octave:1> s = tf ('s'); ## octave:2> G = 1/(s+1) ## ## Transfer function 'G' from input 'u1' to output ... ## ## 1 ## y1: ----- ## s + 1 ## ## Continuous-time model. ## @end group ## @end example ## @example ## @group ## octave:3> z = tf ('z', 0.2); ## octave:4> H = 0.095/(z-0.9) ## ## Transfer function 'H' from input 'u1' to output ... ## ## 0.095 ## y1: ------- ## z - 0.9 ## ## Sampling time: 0.2 s ## Discrete-time model. ## @end group ## @end example ## @example ## @group ## octave:5> num = @{[1, 5, 7], [1]; [1, 7], [1, 5, 5]@}; ## octave:6> den = @{[1, 5, 6], [1, 2]; [1, 8, 6], [1, 3, 2]@}; ## octave:7> sys = tf (num, den) ## @end group ## @end example ## ## @example ## @group ## Transfer function 'sys' from input 'u1' to output ... ## ## s^2 + 5 s + 7 ## y1: ------------- ## s^2 + 5 s + 6 ## ## s + 7 ## y2: ------------- ## s^2 + 8 s + 6 ## @end group ## @end example ## ## @example ## @group ## Transfer function 'sys' from input 'u2' to output ... ## ## 1 ## y1: ----- ## s + 2 ## ## s^2 + 5 s + 5 ## y2: ------------- ## s^2 + 3 s + 2 ## ## Continuous-time model. ## octave:8> ## @end group ## @end example ## ## @seealso{filt, ss, dss} ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.3 function sys = tf (num = {}, den = {}, varargin) ## model precedence: frd > ss > zpk > tf > double ## inferiorto ("frd", "ss", "zpk"); # error if de-commented. bug in octave? superiorto ("double"); argc = 0; # initialize argument count switch (nargin) case 0 # sys = tf () tsam = -2; # undefined sampling time tfvar = "x"; # undefined transfer function variable case 1 if (isa (num, "tf")) # already in tf form sys = tf (tfsys) sys = num; return; elseif (isa (num, "lti")) # another lti object sys = tf (sys) [sys, numlti] = __sys2tf__ (num); sys.lti = numlti; # preserve lti properties return; elseif (is_real_matrix (num)) # static gain sys = tf (4), sys = tf (matrix) num = num2cell (num); num = __vec2tfpoly__ (num); [p, m] = size (num); den = tfpolyones (p, m); # denominators are all 1 tsam = -2; # undefined sampling time tfvar = "x"; # undefined transfer function variable elseif (ischar (num)) # s = tf ("s") tfvar = num; num = __vec2tfpoly__ ([1, 0]); den = __vec2tfpoly__ ([1]); tsam = 0; else print_usage (); endif case 2 if (ischar (num) && issample (den, -1)) # z = tf ("z", 0.3) tfvar = num; tsam = den; num = __vec2tfpoly__ ([1, 0]); den = __vec2tfpoly__ ([1]); else # sys = tf (num, den) num = __vec2tfpoly__ (num); den = __vec2tfpoly__ (den); tfvar = "s"; tsam = 0; endif otherwise # default case sys = tf (num, den, ...) num = __vec2tfpoly__ (num); den = __vec2tfpoly__ (den); argc = numel (varargin); # number of additional arguments after num and den if (issample (varargin{1}, -10)) # sys = tf (num, den, tsam, "prop1", val1, ...) tsam = varargin{1}; # sampling time, could be 0 as well argc--; # tsam is not a property-value pair if (tsam == 0) tfvar = "s"; else tfvar = "z"; endif if (argc > 0) # if there are any properties and values ... varargin = varargin(2:end); # remove tsam from property-value list endif else # sys = tf (num, den, "prop1", val1, ...) tsam = 0; # continuous-time tfvar = "s"; endif endswitch [p, m] = __tf_dim__ (num, den); # determine number of outputs and inputs tfdata = struct ("num", {num}, "den", {den}, "tfvar", tfvar, "inv", false); # struct for tf-specific data ltisys = lti (p, m, tsam); # parent class for general lti data sys = class (tfdata, "tf", ltisys); # create tf object if (argc > 0) # if there are any properties and values, ... sys = set (sys, varargin{:}); # use the general set function endif endfunction control/inst/@tfpoly/0000755000076500000240000000000012273411257014136 5ustar lukasstaffcontrol/inst/@tfpoly/__make_equally_long__.m0000644000076500000240000000222412273411257020600 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Make two polynomials equally long by adding leading zeros ## to the shorter one. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function [a, b] = __make_equally_long__ (a, b) la = length (a.poly); lb = length (b.poly); lmax = max (la, lb); a.poly = [zeros(1, lmax-la), a.poly]; b.poly = [zeros(1, lmax-lb), b.poly]; endfunction control/inst/@tfpoly/__remove_leading_zeros__.m0000644000076500000240000000223112273411257021310 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Remove leading zeros from a polynomial, except for polynomials ## which are of length 1. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function p = __remove_leading_zeros__ (p) idx = find (p.poly != 0); if (isempty (idx)) p.poly = 0; else p.poly = p.poly(idx(1) : end); # p.poly(idx) would remove all zeros endif endfunction control/inst/@tfpoly/conj_ct.m0000644000076500000240000000176212273411257015741 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Conjugate of continuous-time polynomial. Replace s by -s. ## For internal use only. ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function p = conj_ct (p) p.poly(2:2:end) = -p.poly(2:2:end); endfunction control/inst/@tfpoly/conj_dt.m0000644000076500000240000000174712273411257015745 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Conjugate of discrete-time polynomial. Replace z by 1/z. ## For internal use only. ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function p = conj_dt (p) p.poly = fliplr (p.poly); endfunction control/inst/@tfpoly/display.m0000644000076500000240000000167012273411257015765 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Display routine. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function display (p) tfpoly2str (p, "s") endfunction control/inst/@tfpoly/eq.m0000644000076500000240000000202712273411257014722 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Test for equal coefficients. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function bool = eq (a, b) a = tfpoly (a); b = tfpoly (b); bool = (length (a) == length (b)) && all (a.poly == b.poly); endfunction control/inst/@tfpoly/get.m0000644000076500000240000000171412273411257015076 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Return coefficients as a row vector. For internal use only. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function coeffs = get (p) coeffs = p.poly; endfunction control/inst/@tfpoly/length.m0000644000076500000240000000170312273411257015576 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Number of coefficients. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function l = length (p) l = length (p.poly); endfunction control/inst/@tfpoly/minus.m0000644000076500000240000000222112273411257015444 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Subtraction of two polynomials. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function a = minus (a, b) if (! isa (a, "tfpoly")) a = tfpoly (a); endif if (! isa (b, "tfpoly")) b = tfpoly (b); endif [a, b] = __make_equally_long__ (a, b); a.poly = a.poly - b.poly; a = __remove_leading_zeros__ (a); endfunction control/inst/@tfpoly/mpower.m0000644000076500000240000000242412273411257015627 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Power of a polynomial. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function p = mpower (a, b) if (! isa (b, "double") && ! is_real_scalar (b)) error ("tfpoly: mpower: power must be a natural number"); endif c = uint64 (b); if (c != b) error ("tfpoly: mpower: power must be a positive integer"); endif if (c == 0) p = tfpoly (1); return; endif p = a; for k = 1 : (c-1) p.poly = conv (p.poly, a.poly); endfor endfunction control/inst/@tfpoly/mtimes.m0000644000076500000240000000211412273411257015610 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Multiplication of two polynomials. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function a = mtimes (a, b) if (! isa (a, "tfpoly")) a = tfpoly (a); endif if (! isa (b, "tfpoly")) b = tfpoly (b); endif a.poly = conv (a.poly, b.poly); endfunction control/inst/@tfpoly/ne.m0000644000076500000240000000204412273411257014716 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Return true if polynomials are not equal. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function bool = ne (a, b) a = tfpoly (a); b = tfpoly (b); bool = (length (a) != length (b)) || any (a.poly != b.poly); endfunction control/inst/@tfpoly/numel.m0000644000076500000240000000170112273411257015433 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Number of coefficients. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function n = numel (p) n = numel (p.poly); endfunction control/inst/@tfpoly/plus.m0000644000076500000240000000221512273411257015277 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Addition of two polynomials. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function a = plus (a, b) if (! isa (a, "tfpoly")) a = tfpoly (a); endif if (! isa (b, "tfpoly")) b = tfpoly (b); endif [a, b] = __make_equally_long__ (a, b); a.poly = a.poly + b.poly; a = __remove_leading_zeros__ (a); endfunction control/inst/@tfpoly/roots.m0000644000076500000240000000166512273411257015472 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Compute the roots of a TFPOLY object. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function r = roots (p) r = roots (p.poly); endfunction control/inst/@tfpoly/subsref.m0000644000076500000240000000237712273411257015776 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Evaluate polynomial. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function b = subsref (a, s) if (isempty (s)) error ("tfpoly: missing index"); endif switch (s(1).type) case "()" idx = s(1).subs; if (numel (idx) == 1) b = polyval (a.poly, idx{1}); else error ("tfpoly: need exactly one index"); endif otherwise error ("tfpoly: invalid subscript type"); endswitch endfunction control/inst/@tfpoly/tfpoly.m0000644000076500000240000000257112273411257015636 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Class constructor. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function p = tfpoly (a) superiorto ("double"); switch (nargin) case 0 p = struct ("poly", []); p = class (p, "tfpoly"); case 1 if (isa (a, "tfpoly")) p = a; return; elseif (is_real_vector (a)) p.poly = reshape (a, 1, []); p = class (p, "tfpoly"); p = __remove_leading_zeros__ (p); else error ("tfpoly: argument must be a real vector"); endif otherwise print_usage (); endswitch endfunction control/inst/@tfpoly/tfpoly2str.m0000644000076500000240000000473012273411257016450 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{str} =} tfpoly2str (@var{p}) ## @deftypefnx {Function File} {@var{str} =} tfpoly2str (@var{p}, @var{tfvar}) ## Return the string of a polynomial with string @var{tfvar} as variable. ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function str = tfpoly2str (p, tfvar = "x") str = ""; lp = numel (p.poly); if (lp > 0) ## first element (highest order) a = p.poly(1); if (a < 0) cs = "-"; else cs = ""; endif if (lp == 1) str = [cs, num2str(abs (a), 4)]; else if (abs (a) == 1) str = [cs, __variable__(tfvar, lp-1)]; else str = [cs, __coefficient__(a), " ", __variable__(tfvar, lp-1)]; endif endif if (lp > 1) ## elements in the middle for k = 2 : lp-1 a = p.poly(k); if (a != 0) if (a < 0) cs = " - "; else cs = " + "; endif if (abs (a) == 1) str = [str, cs, __variable__(tfvar, lp-k)]; else str = [str, cs, __coefficient__(a), " ", __variable__(tfvar, lp-k)]; endif endif endfor ## last element (lowest order) a = p.poly(lp); if (a != 0) if (a < 0) cs = " - "; else cs = " + "; endif str = [str, cs, num2str(abs (a), 4)]; endif endif endif endfunction function str = __coefficient__ (a) b = abs (a); if (b == 1) str = ""; else str = num2str (b, 4); endif endfunction function str = __variable__ (tfvar, n) if (n == 1) str = tfvar; else str = [tfvar, "^", num2str(n)]; endif endfunction control/inst/@tfpoly/uminus.m0000644000076500000240000000170012273411257015632 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Unitary minus operator. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function a = uminus (a) a.poly = -a.poly; endfunction control/inst/@tfpoly/uplus.m0000644000076500000240000000167612273411257015476 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Unitary plus operator. For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function a = uplus (a) a.poly = +a.poly; endfunction control/inst/__adjust_frd_data__.m0000644000076500000240000000322612273411257016614 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Common code for adjusting FRD model data. ## Used by @frd/frd.m and @frd/__set__.m ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.1 function [H, w] = __adjust_frd_data__ (H, w); w = reshape (w, [], 1); lw = length (w); if (ndims (H) != 3 && ! isempty (H)) if (isscalar (H)) H = reshape (H, 1, 1, []); if (lw > 1) H = repmat (H, [1, 1, lw]); # needed for "frd1 + scalar2" or "scalar1 * frd2) endif elseif (isvector (H) && length (H) == lw) # SISO system (H is a vector) H = reshape (H, 1, 1, []); elseif (ismatrix (H)) H = reshape (H, rows (H), []); if (lw > 1) H = repmat (H, [1, 1, lw]); # needed for "frd1 + matrix2" or "matrix1 * frd2) endif else error ("frd: first argument H invalid"); endif elseif (isempty (H)) H = zeros (0, 0, 0); endif endfunction control/inst/__adjust_iddata__.m0000644000076500000240000000212212273411257016270 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.1 function [y, u] = __adjust_iddata__ (y, u) if (iscell (y)) y = reshape (y, [], 1); else y = {y}; endif if (isempty (u)) u = {}; # avoid [](nx0) and the like elseif (iscell (u)) u = reshape (u, [], 1); else u = {u}; endif endfunction control/inst/__adjust_iddata_tsam__.m0000644000076500000240000000302012273411257017312 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Check whether tsam is a e-by-1 cell array of valid sampling times. ## If not, it tries to convert tsam accordingly. ## Empty tsam are filled with default value -1. ## Author: Lukas Reichlin ## Created: February 2012 ## Version: 0.1 function tsam = __adjust_iddata_tsam__ (tsam, e) if (isempty (tsam)) tsam = num2cell (-ones (e, 1)); elseif (iscell (tsam)) tsam = reshape (tsam, [], 1); else tsam = {tsam}; endif tmp = cellfun (@issample, tsam, {-1}); if (any (! tmp)) error ("iddata: invalid sampling time"); endif nt = numel (tsam); if (nt == 1 && e > 1) tsam = repmat (tsam, e, 1); elseif (nt != e) error ("iddata: there are %d experiments, but only %d sampling times", ... e, nt); endif endfunction control/inst/__adjust_labels__.m0000644000076500000240000000245712273411257016317 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Check whether a cell contains the required number of strings. ## Used by set and __set__. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function name = __adjust_labels__ (name, req_len) if (iscell (name)) name = reshape (name, [], 1); else # catch the siso case, name = {name}; # e.g. sys = set (sys, "inname", "u_1") endif if (! iscellstr (name) || numel (name) != req_len) error ("lti: set: cell must contain %d strings", req_len); endif endfunction control/inst/__adjust_ss_data__.m0000644000076500000240000000312612273411257016465 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Common code for adjusting SS model data. ## Used by @ss/ss.m, others possibly follow. ## Author: Lukas Reichlin ## Created: October 2010 ## Version: 0.2 function [a, b, c, d, tsam] = __adjust_ss_data__ (a, b, c, d, tsam); if (isempty (a)) # static system a = []; # avoid [](nx0) or [](0xn) tsam = -2; endif if (isempty (d)) if (all (size (c) == 0)) # ss (a, b), ss (a, b, [], [], ...), but allow c(0xn) and d(0xm) c = eye (size (a)); d = zeros (rows (a), columns (b)); else # ss (a, b, c), ss (a, b, c, [], ...) d = zeros (rows (c), columns (b)); endif endif if (isempty (b) && isempty (c)) # sys = ss ([], [], [], d) b = zeros (0, columns (d)); c = zeros (rows(d), 0); tsam = -2; endif endfunction control/inst/__axis_limits__.m0000644000076500000240000000426712273411257016031 0ustar lukasstaff## Copyright (C) 1998, 2000, 2004, 2005, 2007 ## Auburn University. All rights reserved. ## ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {Function File} {} __axis_limits__ (@var{axdata}) ## Determine axis limits for 2-D data (column vectors); leaves a 10% ## margin around the plots. ## Inserts margins of +/- 0.1 if data is one-dimensional ## (or a single point). ## ## @strong{Input} ## @table @var ## @item axdata ## @var{n} by 2 matrix of data [@var{x}, @var{y}]. ## @end table ## ## @strong{Output} ## @table @var ## @item axvec ## Vector of axis limits appropriate for call to @command{axis} function. ## @end table ## @end deftypefn function axvec = __axis_limits__ (axdata) if (nargin < 1 || isempty (axdata)) axdata = 0; endif ## compute axis limits minv = min (axdata); maxv = max (axdata); delv = (maxv-minv)/2; # breadth of the plot midv = (minv + maxv)/2; # midpoint of the plot axmid = [midv(1), midv(1), midv(2), midv(2)]; axdel = [-0.1, 0.1, -0.1, 0.1]; # default plot width (if less than 2-d data) if (max (delv) == 0) if (midv(1) != 0) axdel(1:2) = [-0.1*midv(1), 0.1*midv(1)]; endif if (midv(2) != 0) axdel(3:4) = [-0.1*midv(2), 0.1*midv(2)]; endif else ## they're at least one-dimensional tolv = max(1e-8, 1e-8*abs(midv)); if (abs (delv(1)) >= tolv(1)) axdel(1:2) = 1.1*[-delv(1),delv(1)]; endif if (abs (delv(2)) >= tolv(2)) axdel(3:4) = 1.1*[-delv(2),delv(2)]; endif endif axvec = axmid + axdel; endfunction control/inst/__axis_margin__.m0000644000076500000240000000362412273411257016001 0ustar lukasstaff## Copyright (C) 1998, 2000, 2004, 2005, 2007 ## Auburn University. All rights reserved. ## ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {Function File} {} __axis_margin__ (@var{axdata}) ## Determine axis limits for 2-D data (column vectors); leaves a 10% ## margin around the plots. ## Inserts margins of +/- 0.1 if data is one-dimensional ## (or a single point). ## ## @strong{Input} ## @table @var ## @item axdata ## @var{n} by 2 matrix of data [@var{x}, @var{y}]. ## @end table ## ## @strong{Output} ## @table @var ## @item axvec ## Vector of axis limits appropriate for call to @command{axis} function. ## @end table ## @end deftypefn function axvec = __axis_margin__ (axdata) ## compute axis limits minv = axdata(1); maxv = axdata(2); delv = (maxv-minv)/2; # breadth of the plot midv = (minv + maxv)/2; # midpoint of the plot axmid = [midv, midv]; axdel = [-0.1, 0.1]; # default plot width (if less than 2-d data) if (delv == 0) if (midv != 0) axdel = [-0.1*midv, 0.1*midv]; endif else ## they're at least one-dimensional tolv = max(1e-8, 1e-8*abs(midv)); if (abs (delv) >= tolv) axdel = 1.1*[-delv,delv]; endif endif axvec = axmid + axdel; endfunction control/inst/__conred_check_feedback_sign__.m0000644000076500000240000000231712273411257020731 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check the feedback sign. ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function negfb = __conred_check_feedback_sign__ (fbsign, key = "feedback") if (! ischar (fbsign)) error ("conred: key '%s' requires string value", key); endif switch (fbsign) case "+" negfb = false; case "-" negfb = true; otherwise error ("conred: key '%s' has an invalid value", key); endswitch endfunction control/inst/__conred_sb16ad__.m0000644000076500000240000001351312273411257016110 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Kr}, @var{info}] =} __conred_sb16ad__ (@var{method}, @dots{}) ## Backend for btaconred and spaconred. ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function [Kr, info] = __conred_sb16ad__ (method, varargin) if (nargin < 3) print_usage (); endif if (method != "bta" && method != "spa") error ("modred: invalid method"); endif G = varargin{1}; K = varargin{2}; varargin = varargin(3:end); if (! isa (G, "lti")) error ("%sconred: first argument must be an LTI system", method); endif if (! isa (K, "lti")) error ("%sconred: second argument must be an LTI system", method); endif if (nargin > 3) # *conred (G, K, ...) if (is_real_scalar (varargin{1})) # *conred (G, K, nr) varargin = horzcat (varargin(2:end), {"order"}, varargin(1)); endif if (isstruct (varargin{1})) # *conred (G, K, opt, ...), *conred (G, K, nr, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif ## order placed at the end such that nr from *conred (G, K, nr, ...) ## and *conred (G, K, nr, opt, ...) overrides possible nr's from ## key/value-pairs and inside opt struct (later keys override former keys, ## nr > key/value > opt) endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("%sconred: keys and values must come in pairs", method); endif [a, b, c, d, tsam, scaled] = ssdata (G); [ac, bc, cc, dc, tsamc, scaledc] = ssdata (K); [p, m] = size (G); [pc, mc] = size (K); dt = isdt (G); if (p != mc || m != pc) error ("%sconred: dimensions of controller (%dx%d) and plant (%dx%d) don't match", ... method, pc, mc, p, c); endif ## default arguments alpha = __modred_default_alpha__ (dt); tol1 = 0.0; tol2 = 0.0; jobc = jobo = 0; bf = true; # balancing-free weight = 3; equil = scaled && scaledc; ordsel = 1; ncr = 0; negfb = false; # positive feedback controller ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case "weight" switch (lower (val(1))) case "n" # none weight = 0; case {"l", "o"} # left, output weight = 1; case {"r", "i"} # right, input weight = 2; case {"b", "p"} # both, performance weight = 3; otherwise error ("%sconred: '%s' is an invalid value for key weight", method, val); endswitch case {"order", "ncr", "nr"} [ncr, ordsel] = __modred_check_order__ (val, rows (ac)); case "tol1" tol1 = __modred_check_tol__ (val, "tol1"); case "tol2" tol2 = __modred_check_tol__ (val, "tol2"); case "alpha" alpha = __modred_check_alpha__ (val, dt); case "method" switch (tolower (val)) case "sr" bf = false; case "bfsr" bf = true; otherwise error ("modred: '%s' is an invalid approach", val); endswitch case {"jobc", "gram-ctrb"} jobc = __modred_check_gram__ (val, "gram-ctrb"); case {"jobo", "gram-obsv"} jobo = __modred_check_gram__ (val, "gram-obsv"); case {"equil", "equilibrate", "equilibration", "scale", "scaling"} scaled = __modred_check_equil__ (val); case "feedback" negfb = __conred_check_feedback_sign__ (val); otherwise warning ("%sconred: invalid property name '%s' ignored", method, key); endswitch endfor ## handle model reduction approach if (method == "bta" && ! bf) # 'B': use the square-root Balance & Truncate method jobmr = 0; elseif (method == "bta" && bf) # 'F': use the balancing-free square-root Balance & Truncate method jobmr = 1; elseif (method == "spa" && ! bf) # 'S': use the square-root Singular Perturbation Approximation method jobmr = 2; elseif (method == "spa" && bf) # 'P': use the balancing-free square-root Singular Perturbation Approximation method jobmr = 3; else error ("%smodred: invalid jobmr option"); # this should never happen endif ## handle negative feedback controllers if (negfb) [ac, bc, cc, dc] = ssdata (-K); endif ## perform model order reduction [acr, bcr, ccr, dcr, ncr, hsvc, ncs] = __sl_sb16ad__ (a, b, c, d, dt, equil, ncr, ordsel, alpha, jobmr, ... ac, bc, cc, dc, ... weight, jobc, jobo, tol1, tol2); ## assemble reduced order controller Kr = ss (acr, bcr, ccr, dcr, tsamc); ## handle negative feedback controllers if (negfb) Kr = -Kr; endif ## assemble info struct info = struct ("ncr", ncr, "ncs", ncs, "hsvc", hsvc); endfunction control/inst/__dss2ss__.m0000644000076500000240000000663012273411257014721 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert descriptor state-space system into regular state-space form. ## Author: Lukas Reichlin ## Created: September 2011 ## Version: 0.2 function [a, b, c, d, e] = __dss2ss__ (a, b, c, d, e) if (isempty (e)) return; elseif (rcond (e) < eps) # check for singularity ## check whether regular state-space representation is possible [~, ~, ~, ~, ranke, rnka22] = __sl_tg01fd__ (a, e, b, c, false, 0); if (ranke+rnka22 < rows (a)) error ("ss: dss2ss: this descriptor system cannot be converted to regular state-space form"); endif endif [a, b, c, d] = __sl_sb10jd__ (a, b, c, d, e); e = []; endfunction ## Test from SLICOT TG01FD %!shared a, b, c, e, ranke, rnka22, q, z, a_exp, b_exp, c_exp, e_exp, q_exp, z_exp %! %! e = [1, 2, 0, 0; 0, 1, 0, 1; 3, 9, 6, 3; 0, 0, 2, 0]; %! a = [-1, 0, 0, 3; 0, 0, 1, 2; 1, 1, 0, 4; 0, 0, 0, 0]; %! b = [1, 0; 0, 0; 0, 1; 1, 1]; %! c = [-1, 0, 1, 0; 0, 1, -1, 1]; %! %! [a, e, b, c, ranke, rnka22, q, z] = __sl_tg01fd__ (a, e, b, c, true, 0.0); %! %! e_exp = [10.1587 5.8230 1.3021 0.0000; %! 0.0000 -2.4684 -0.1896 0.0000; %! 0.0000 0.0000 1.0338 0.0000; %! 0.0000 0.0000 0.0000 0.0000]; %! %! a_exp = [ 2.0278 0.1078 3.9062 -2.1571; %! -0.0980 0.2544 1.6053 -0.1269; %! 0.2713 0.7760 -0.3692 -0.4853; %! 0.0690 -0.5669 -2.1974 0.3086]; %! %! b_exp = [-0.2157 -0.9705; %! 0.3015 0.9516; %! 0.7595 0.0991; %! 1.1339 0.3780]; %! %! c_exp = [ 0.3651 -1.0000 -0.4472 -0.8165; %! -1.0954 1.0000 -0.8944 0.0000]; %! %! q_exp = [-0.2157 -0.5088 0.6109 0.5669; %! -0.1078 -0.2544 -0.7760 0.5669; %! -0.9705 0.1413 -0.0495 -0.1890; %! 0.0000 0.8102 0.1486 0.5669]; %! %! z_exp = [-0.3651 0.0000 0.4472 0.8165; %! -0.9129 0.0000 0.0000 -0.4082; %! 0.0000 -1.0000 0.0000 0.0000; %! -0.1826 0.0000 -0.8944 0.4082]; %! %!assert (a, a_exp, 1e-4); %!assert (e, e_exp, 1e-4); %!assert (b, b_exp, 1e-4); %!assert (c, c_exp, 1e-4); %!assert (q, q_exp, 1e-4); %!assert (z, z_exp, 1e-4); %!assert (ranke, 3); %!assert (rnka22, 1); ## test error %!shared mms %! %! mm = tf([3, 5, 0], [4, 1]); %! mms = ss (mm); %!error (__dss2ss__ (mms.a, mms.b, mms.c, mms.d, mms.e)); ## Realizable descriptor system with singular E matrix %!test %! A = [1 0; 0 1]; %! B = [1; 0]; %! C = [1 0]; %! D = 0; %! E = [1 0; 0 0]; %! %! sys = dss (A, B, C, D, E); %! [Ao, Bo, Co, Do] = ssdata (sys); %! %! assert (Ao, 1, 1e-4); %! assert (Bo, 1, 1e-4); %! assert (Co, 1, 1e-4); %! assert (Do, 0, 1e-4); control/inst/__dss_bilin__.m0000644000076500000240000000460312273411257015444 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## 1. Discrete -> continuous ## _ ## E = alpha*E + A ## _ ## A = beta (A - alpha*E) ## _ ## B = sqrt(2*alpha*beta) * B ## _ -1 ## C = sqrt(2*alpha*beta) * C * (alpha*E + A) * E ## _ -1 ## D = D - C * (alpha*E + A) * B ## ## ## 2. Continuous -> discrete ## _ ## E = beta*E - A ## _ ## A = alpha (beta*E + A) ## _ ## B = sqrt(2*alpha*beta) * B ## _ -1 ## C = sqrt(2*alpha*beta) * C * (beta*E - A) * E ## _ -1 ## D = D + C * (beta*E - A) * B ## Special thanks to Andras Varga for the formulae. ## Author: Lukas Reichlin ## Created: October 2011 ## Version: 0.1 function [Ar, Br, Cr, Dr, Er] = __dss_bilin__ (A, B, C, D, E, beta, discrete) if (discrete) EpA = E + A; s2b = sqrt (2*beta); if (rcond (EpA) < eps) error ("d2c: E+A singular"); endif CiEpA = C / EpA; Er = EpA; Ar = beta * (A - E); Br = s2b * B; Cr = s2b * CiEpA * E; Dr = D - CiEpA * B; ## Er = E + A; ## Ar = beta * (A - E); ## Br = sqrt (2*beta) * B; ## Cr = sqrt (2*beta) * C / (E + A) * E; ## Dr = D - C / (E + A) * B; else bEmA = beta*E - A; s2b = sqrt (2*beta); if (rcond (bEmA) < eps) error ("c2d: beta*E-A singular"); endif CibEmA = C / bEmA; Er = bEmA; Ar = beta*E + A; Br = s2b * B; Cr = s2b * CibEmA * E; Dr = D + CibEmA * B; ## Er = beta*E - A; ## Ar = beta*E + A; ## Br = sqrt (2*beta) * B; ## Cr = sqrt (2*beta) * C / (beta*E - A) * E; ## Dr = D + C / (beta*E - A) * B; endif endfunction control/inst/__frd_dim__.m0000644000076500000240000000300112273411257015071 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Number of outputs and inputs of transfer function numerator and ## denominator. For internal use only. ## Author: Lukas Reichlin ## Created: February 2010 ## Version: 0.1 function [p, m, l] = __frd_dim__ (H, w) if (! isnumeric (H)) error ("frd: H must be a 3-dimensional numeric array"); endif lw = length (w); if (! isempty (w) && (! is_real_vector (w) || any (w < 0) ... || ! issorted (w) || w(1) > w(end) ... || length (unique (w)) != lw)) error ("frd: w must be a vector of positive real numbers in ascending order"); endif [p, m, l] = size (H); if (l != lw) error ("frd: H (%dx%dx%d) and w (%d) must have equal length", p, m, l, lw); endif endfunction control/inst/__frequency_response__.m0000644000076500000240000000552712273411257017423 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Return frequency response H and frequency vector w. ## If w is empty, it will be calculated by __frequency_vector__. ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.5 function [H, w] = __frequency_response__ (args, mimoflag = false, wbounds = "std", cellflag = false) isc = iscell (args); if (! isc) args = {args}; endif sys_idx = cellfun (@isa, args, {"lti"}); # look for LTI models w_idx = cellfun (@(x) is_real_vector (x) && length (x) > 1, args); # look for frequency vectors r_idx = cellfun (@iscell, args); # look for frequency ranges {wmin, wmax} sys_cell = args(sys_idx); # extract LTI models frd_idx = cellfun (@isa, sys_cell, {"frd"}); # look for FRD models ## check arguments if (! mimoflag && ! all (cellfun (@issiso, sys_cell))) error ("frequency_response: require SISO systems"); endif ## determine frequencies if (any (r_idx)) # if there are frequency ranges r = args(r_idx){end}; # take the last one if (numel (r) == 2 && issample (r{1}) && issample (r{2})) w = __frequency_vector__ (sys_cell, wbounds, r{1}, r{2}); else error ("frequency_response: invalid cell"); endif elseif (any (w_idx)) # are there any frequency vectors? w = args(w_idx){end}; w = repmat ({w}, 1, numel (sys_cell)); else # there are neither frequency ranges nor vectors w = __frequency_vector__ (sys_cell, wbounds); endif w_frd = w(frd_idx); # temporarily save frequency vectors of FRD models w(frd_idx) = {[]}; # freqresp returns all frequencies of FRD models for w=[] ## compute frequency response H for all LTI models H = cellfun (@__freqresp__, sys_cell, w, {cellflag}, "uniformoutput", false); ## restore frequency vectors of FRD models in w w(frd_idx) = w_frd; if (! isc) # for old non-multiplot functions H = H{1}; w = w{1}; endif endfunction control/inst/__frequency_vector__.m0000644000076500000240000001331412273411257017060 0ustar lukasstaff## Copyright (C) 1996, 2000, 2004, 2005, 2006, 2007 ## Auburn University. All rights reserved. ## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{w} =} __frequency_vector__ (@var{sys}) ## Get default range of frequencies based on cutoff frequencies of system ## poles and zeros. ## Frequency range is the interval ## @iftex ## @tex ## $ [ 10^{w_{min}}, 10^{w_{max}} ] $ ## @end tex ## @end iftex ## @ifnottex ## [10^@var{wmin}, 10^@var{wmax}] ## @end ifnottex ## ## Used by @command{__frequency_response__} ## @end deftypefn ## Adapted-By: Lukas Reichlin ## Date: October 2009 ## Version: 0.4 function w = __frequency_vector__ (sys_cell, wbounds = "std", wmin, wmax) isc = iscell (sys_cell); if (! isc) # __sys2frd__ methods pass LTI models not in cells sys_cell = {sys_cell}; endif idx = cellfun (@(x) isa (x, "lti"), sys_cell); sys_cell = sys_cell(idx); len = numel (sys_cell); [dec_min, dec_max, zp] = cellfun (@__frequency_range__, sys_cell, {wbounds}, "uniformoutput", false); if (strcmpi (wbounds, "std")) # plots with explicit frequencies if (nargin == 4) # w = {wmin, wmax} dec_min = log10 (wmin); dec_max = log10 (wmax); else dec_min = min (cell2mat (dec_min)); dec_max = max (cell2mat (dec_max)); endif zp = horzcat (zp{:}); ## include zeros and poles for nice peaks in plots idx = find (zp > 10^dec_min & zp < 10^dec_max); zp = zp(idx); w = logspace (dec_min, dec_max, 500); w = unique ([w, zp]); # unique also sorts frequency vector w = repmat ({w}, 1, len); # return cell of frequency vectors elseif (strcmpi (wbounds, "ext")) # plots with implicit frequencies if (nargin == 4) dec_min = repmat ({log10 (wmin)}, 1, len); dec_max = repmat ({log10 (wmax)}, 1, len); endif idx = cellfun (@(zp, dec_min, dec_max) find (zp > 10^dec_min & zp < 10^dec_max), ... zp, dec_min, dec_max, "uniformoutput", false); zp = cellfun (@(zp, idx) zp(idx), zp, idx, "uniformoutput", false); w = cellfun (@logspace, dec_min, dec_max, {500}, "uniformoutput", false); w = cellfun (@(w, zp) unique ([w, zp]), w, zp, "uniformoutput", false); ## unique also sorts frequency vector else error ("frequency_vector: invalid argument 'wbounds'"); endif if (! isc) # for __sys2frd__ methods w = w{1}; endif endfunction function [dec_min, dec_max, zp] = __frequency_range__ (sys, wbounds = "std") if (isa (sys, "frd")) w = get (sys, "w"); dec_min = log10 (w(1)); dec_max = log10 (w(end)); zp = []; return; endif zer = zero (sys); pol = pole (sys); tsam = abs (get (sys, "tsam")); # tsam could be -1 discrete = ! isct (sys); # static gains (tsam = -2) are assumed continuous ## make sure zer, pol are row vectors pol = reshape (pol, 1, []); zer = reshape (zer, 1, []); ## check for natural frequencies away from omega = 0 if (discrete) ## The 2nd conditions prevents log(0) in the next log command iiz = find (abs(zer-1) > norm(zer)*eps && abs(zer) > norm(zer)*eps); iip = find (abs(pol-1) > norm(pol)*eps && abs(pol) > norm(pol)*eps); ## avoid dividing empty matrices, it would work but looks nasty if (! isempty (iiz)) czer = log (zer(iiz))/tsam; else czer = []; endif if (! isempty (iip)) cpol = log (pol(iip))/tsam; else cpol = []; endif else ## continuous iip = find (abs(pol) > norm(pol)*eps); iiz = find (abs(zer) > norm(zer)*eps); if (! isempty (zer)) czer = zer(iiz); else czer = []; endif if (! isempty (pol)) cpol = pol(iip); else cpol = []; endif endif if (isempty (iip) && isempty (iiz)) ## no poles/zeros away from omega = 0; pick defaults dec_min = 0; # -1 dec_max = 2; # 3 else dec_min = floor (log10 (min (abs ([cpol, czer])))); dec_max = ceil (log10 (max (abs ([cpol, czer])))); endif ## expand to show the entirety of the "interesting" portion of the plot switch (wbounds) case "std" # standard if (dec_min == dec_max) dec_min -= 2; dec_max += 2; else dec_min--; dec_max++; endif case "ext" # extended (for nyquist) if (any (abs (pol) < sqrt (eps))) # look for integrators ## dec_min -= 0.5; dec_max += 2; else dec_min -= 2; dec_max += 2; endif otherwise error ("frequency_range: second argument invalid"); endswitch ## run discrete frequency all the way to pi if (discrete) dec_max = log10 (pi/tsam); endif ## include zeros and poles for nice peaks in plots zp = [abs(zer), abs(pol)]; endfunction control/inst/__iddata_dim__.m0000644000076500000240000000437312273411257015561 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## Author: Lukas Reichlin ## Created: October 2011 ## Version: 0.1 function [p, m, e] = __iddata_dim__ (y, u) e = numel (y); # number of experiments if (isempty (u)) # time series data, outputs only [p, m] = cellfun (@__experiment_dim__, y, "uniformoutput", false); elseif (e == numel (u)) # outputs and inputs present [p, m] = cellfun (@__experiment_dim__, y, u, "uniformoutput", false); else error ("iddata: require input and output data with matching number of experiments"); endif if (e > 1 && ! isequal (p{:})) error ("iddata: require identical number of output channels for all experiments"); endif if (e > 1 && ! isequal (m{:})) error ("iddata: require identical number of input channels for all experiments"); endif p = p{1}; m = m{1}; endfunction function [p, m] = __experiment_dim__ (y, u = []) if (! is_matrix (y, u)) error ("iddata: inputs and outputs must be real or complex matrices"); endif [ly, p] = size (y); [lu, m] = size (u); if (! isempty (u) && ly != lu) error ("iddata: matrices 'y' (%dx%d) and 'u' (%dx%d) must have the same number of samples (rows)", ... ly, p, lu, m); endif if (ly < p) warning ("iddata:transpose", "iddata: more outputs than samples - matrice 'y' should probably be transposed"); endif if (lu < m) warning ("iddata:transpose", "iddata: more inputs than samples - matrice 'u' should probably be transposed"); endif endfunction control/inst/__is_stable__.m0000644000076500000240000000235112273411257015441 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Determine whether all poles in a vector are stable. ## Author: Lukas Reichlin ## Created: December 2010 ## Version: 0.2 function bool = __is_stable__ (pol, ct = true, tol = 0) if (! is_real_scalar (tol) || tol < 0) error ("isstable: tolerance must be a real-valued, non-negative scalar"); endif if (ct) # continuous-time bool = all (real (pol) < -tol*(1 + abs (pol))); else # discrete-time bool = all (abs (pol) < 1 - tol); endif endfunction control/inst/__labels__.m0000644000076500000240000000240112273411257014732 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Return default labels if cell "name" contains only empty strings. ## If not, check whether individual strings of the cell "name" are ## empty and mark them with "?". Used by display routines. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function [name, n] = __labels__ (name, variable = "x") n = numel (name); if (n == 0 || isequal ("", name{:})) name = strseq (variable, 1:n); else idx = cellfun (@isempty, name); name(idx) = "?"; endif endfunction control/inst/__modred_ab09id__.m0000644000076500000240000001325212273411257016100 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Gr}, @var{info}] =} __modred_ab09id__ (@var{method}, @dots{}) ## Backend for btamodred and spamodred. ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function [Gr, info] = __modred_ab09id__ (method, varargin) if (nargin < 2) print_usage (); endif if (method != "bta" && method != "spa") error ("modred: invalid method"); endif G = varargin{1}; varargin = varargin(2:end); if (! isa (G, "lti")) error ("%smodred: first argument must be an LTI system", method); endif if (nargin > 2) # *modred (G, ...) if (is_real_scalar (varargin{1})) # *modred (G, nr) varargin = horzcat (varargin(2:end), {"order"}, varargin(1)); endif if (isstruct (varargin{1})) # *modred (G, opt, ...), *modred (G, nr, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif ## order placed at the end such that nr from *modred (G, nr, ...) ## and *modred (G, nr, opt, ...) overrides possible nr's from ## key/value-pairs and inside opt struct (later keys override former keys, ## nr > key/value > opt) endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("%smodred: keys and values must come in pairs", method); endif [a, b, c, d, tsam, scaled] = ssdata (G); [p, m] = size (G); dt = isdt (G); ## default arguments alpha = __modred_default_alpha__ (dt); av = bv = cv = dv = []; jobv = 0; aw = bw = cw = dw = []; jobw = 0; alphac = alphao = 0.0; tol1 = 0.0; tol2 = 0.0; jobc = jobo = 0; bf = true; # balancing-free weight = 1; equil = 0; ordsel = 1; nr = 0; ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case {"left", "output", "v"} [av, bv, cv, dv, jobv] = __modred_check_weight__ (val, dt, p, []); case {"right", "input", "w"} [aw, bw, cw, dw, jobw] = __modred_check_weight__ (val, dt, [], m); case {"order", "n", "nr"} [nr, ordsel] = __modred_check_order__ (val, rows (a)); case "tol1" tol1 = __modred_check_tol__ (val, "tol1"); case "tol2" tol2 = __modred_check_tol__ (val, "tol2"); case "alpha" alpha = __modred_check_alpha__ (val, dt); case "method" switch (tolower (val)) case "sr" bf = false; case "bfsr" bf = true; otherwise error ("modred: '%s' is an invalid approach", val); endswitch case {"jobc", "gram-ctrb"} jobc = __modred_check_gram__ (val, "gram-ctrb"); case {"jobo", "gram-obsv"} jobo = __modred_check_gram__ (val, "gram-obsv"); case {"alphac", "alpha-ctrb"} alphac = __modred_check_alpha_gram__ (val, "alpha-ctrb"); case {"alphao", "alpha-obsv"} alphao = __modred_check_alpha_gram__ (val, "alpha-obsv"); case {"equil", "equilibrate", "equilibration", "scale", "scaling"} scaled = __modred_check_equil__ (val); otherwise warning ("%smodred: invalid property name '%s' ignored", method, key); endswitch endfor ## handle type of frequency weighting if (jobv && jobw) weight = 3; # 'B': both left and right weightings V and W are used elseif (jobv) weight = 1; # 'L': only left weighting V is used (W = I) elseif (jobw) weight = 2; # 'R': only right weighting W is used (V = I) else weight = 0; # 'N': no weightings are used (V = I, W = I) endif ## handle model reduction approach if (method == "bta" && ! bf) # 'B': use the square-root Balance & Truncate method job = 0; elseif (method == "bta" && bf) # 'F': use the balancing-free square-root Balance & Truncate method job = 1; elseif (method == "spa" && ! bf) # 'S': use the square-root Singular Perturbation Approximation method job = 2; elseif (method == "spa" && bf) # 'P': use the balancing-free square-root Singular Perturbation Approximation method job = 3; else error ("modred: invalid job option"); # this should never happen endif ## perform model order reduction [ar, br, cr, dr, nr, hsv, ns] = __sl_ab09id__ (a, b, c, d, dt, equil, nr, ordsel, alpha, job, ... av, bv, cv, dv, ... aw, bw, cw, dw, ... weight, jobc, jobo, alphac, alphao, ... tol1, tol2); ## assemble reduced order model Gr = ss (ar, br, cr, dr, tsam); ## assemble info struct info = struct ("nr", nr, "ns", ns, "hsv", hsv); endfunction control/inst/__modred_check_alpha__.m0000644000076500000240000000241012273411257017244 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check alpha for model reduction commands ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function alpha = __modred_check_alpha__ (alpha, dt) if (! is_real_scalar (alpha)) error ("modred: argument alpha must be a real scalar"); endif if (dt) # discrete-time if (alpha < 0 || alpha > 1) error ("modred: require 0 <= ALPHA <= 1"); endif else # continuous-time if (alpha > 0) error ("modred: require ALPHA <= 0"); endif endif endfunction control/inst/__modred_check_alpha_gram__.m0000644000076500000240000000221212273411257020252 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check alpha for combination methods of grammians ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function alpha = __modred_check_alpha_gram__ (alpha, key) if (! is_real_scalar (alpha)) error ("modred: argument '%s' must be a real scalar", key); endif if (abs (alpha) > 1) error ("modred: require -1 <= %s <= 1", key); endif endfunction control/inst/__modred_check_equil__.m0000644000076500000240000000211412273411257017277 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check equilibration for model reduction commands ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function scaled = __modred_check_equil__ (equil) if (isscalar (equil)) scaled = ! logical (equil); else error ("modred: property 'equil' must be a logical value"); endif endfunction control/inst/__modred_check_gram__.m0000644000076500000240000000235512273411257017115 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check choice of frequency-weighted grammians. ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function job = __modred_check_gram__ (choice, key) if (! ischar (choice)) error ("modred: key '%s' requires string value", key); endif switch (tolower (choice (1))) case "s" # standard job = 0; case "e" # enhanced job = 1; otherwise error ("modred: key '%s' has an invalid value", key); endswitch endfunction control/inst/__modred_check_order__.m0000644000076500000240000000233312273411257017276 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check order for model reduction commands ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function [nr, ordsel] = __modred_check_order__ (nr, n) if (! issample (nr, 0) || nr != round (nr)) error ("modred: order of reduced model must be an integer >= 0"); endif if (nr > n) error ("modred: order of reduced model (%d) can't be larger than the original one (%d)", ... nr, n); endif ordsel = 0; endfunction control/inst/__modred_check_tol__.m0000644000076500000240000000205012273411257016755 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check tolerance for model reduction commands ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function tol = __modred_check_tol__ (tol, str = "") if (! is_real_scalar (tol)) error ("modred: argument %s must be a real scalar", str); endif endfunction control/inst/__modred_check_weight__.m0000644000076500000240000000265212273411257017456 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## check weightings for model reduction commands ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function [a, b, c, d, job] = __modred_check_weight__ (sys, dt, p = [], m = []) sys = ss (sys); # could be non-lti, therefore ssdata would fail if (dt != isdt (sys)) error ("modred: ct/dt"); # TODO: error message endif [pw, mw] = size (sys); if (! isempty (p) && mw != p) error ("modred: left weight requires %d inputs", p); endif if (! isempty (m) && pw != m) error ("modred: right weight requires %d outputs", m); endif [a, b, c, d] = ssdata (sys); job = 1; ## TODO: check system size endfunction control/inst/__modred_default_alpha__.m0000644000076500000240000000210612273411257017615 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## default alpha for model reduction commands ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function alpha = __modred_default_alpha__ (dt) if (dt) # discrete-time alpha = 1; # ALPHA <= 0 else # continuous-time alpha = 0; # 0 <= ALPHA <= 1 endif endfunction control/inst/__opt2cell__.m0000644000076500000240000000225612273411257015224 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert option struct to a cell with field names as keys and ## field values as values. ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function c = __opt2cell__ (opt) if (! isstruct (opt)) error ("opt2cell: argument must be a struct"); endif key = fieldnames (opt); val = struct2cell (opt); c = [key.'; val.'](:).'; # reshape to {key1, val1, key2, val2, ...} endfunction control/inst/__remove_trailing_zeros__.m0000644000076500000240000000213712273411257020106 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Remove trailing zeros from a polynomial, except for polynomials ## which are of length 1. For internal use only. ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function p = __remove_trailing_zeros__ (p) idx = find (p != 0); if (isempty (idx)) p = 0; else p = p(1 : idx(end)); endif endfunction control/inst/__slicot_identification__.m0000644000076500000240000001650312273411257020046 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{sys}, @var{x0}], @var{info} =} __slicot_identification__ (@var{method}, @var{dat}, @dots{}) ## Backend for moesp, moen4 and n4sid. ## @end deftypefn ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function [sys, x0, info] = __slicot_identification__ (method, nout, dat, varargin) ## determine identification method switch (method) case "moesp" meth = 0; case "n4sid" meth = 1; case "moen4" meth = 2; otherwise error ("ident: invalid method"); # should never happen endswitch if (! isa (dat, "iddata") || ! dat.timedomain) error ("%s: first argument must be a time-domain 'iddata' dataset", method); endif if (nargin > 3) # ident (dat, ...) if (is_real_scalar (varargin{1})) # ident (dat, n, ...) varargin = horzcat (varargin(2:end), {"order"}, varargin(1)); endif if (isstruct (varargin{1})) # ident (dat, opt, ...), ident (dat, n, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("%s: keys and values must come in pairs", method); endif [ns, p, m, e] = size (dat); # dataset dimensions tsam = dat.tsam; ## multi-experiment data requires equal sampling times if (e > 1 && ! isequal (tsam{:})) error ("%s: require equally sampled experiments", method); else tsam = tsam{1}; endif ## default arguments alg = 0; conct = 1; # no connection between experiments ctrl = 1; # don't confirm order n rcond = 0.0; tol = 0.0; # -1.0; s = []; n = []; conf = []; noise = "n"; ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case {"n", "order"} if (! issample (val, 0) || val != round (val)) error ("%s: 'n' must be a positive integer", method); endif n = val; case "s" if (! issample (val, 0) || val != round (val)) error ("%s: 's' must be a positive integer", method); endif s = val; case {"alg", "algorithm"} if (strncmpi (val, "c", 1)) alg = 0; # Cholesky algorithm applied to correlation matrix elseif (strncmpi (val, "f", 1)) alg = 1; # fast QR algorithm elseif (strncmpi (val, "q", 1)) alg = 2; # QR algorithm applied to block Hankel matrices else error ("%s: invalid algorithm", method); endif case "tol" if (! is_real_scalar (val)) error ("%s: tolerance 'tol' must be a real scalar", method); endif tol = val; case "rcond" if (! is_real_scalar (val)) error ("%s: 'rcond' must be a real scalar", method); endif rcond = val; case "confirm" conf = logical (val); case {"noiseinput", "noiseinputs", "noise", "input", "inputs"} noise = val; otherwise warning ("%s: invalid property name '%s' ignored", method, key); endswitch endfor ## handle s/nobr and n nsmp = sum (ns); # total number of samples nobr = fix ((nsmp+1)/(2*(m+p+1))); if (e > 1) nobr = min (nobr, fix (min (ns) / 2)); endif if (isempty (s) && isempty (n)) ctrl = 0; # confirm system order estimate n = 0; elseif (isempty (s)) s = min (2*n, n+10); # upper bound for n nobr = min (nobr, s); elseif (isempty (n)) nobr = __check_s__ (s, nobr, method); ctrl = 0; # confirm system order estimate n = 0; else # s & n non-empty nobr = __check_s__ (s, nobr, method); if (n >= nobr) error ("%s: n=%d, but require n < %d (s)", method, n, nobr); endif endif if (! isempty (conf)) ctrl = ! conf; endif if (nout == 0) ## compute singular values [sv, nrec] = __sl_ib01ad__ (dat.y, dat.u, nobr, n, meth, alg, conct, ctrl, rcond, tol); ## there is no 'logbar' function svl = log10 (sv); base = floor (min (svl)); clf bar (svl, "basevalue", base) xlim ([0, length(sv)+1]) yl = ylim; ylim ([base, yl(2)]) title ("Singular Values") ylabel ("Logarithm of Singular Values") xlabel (sprintf ("Estimated System Order with current Tolerance: %d", nrec)) grid on else ## perform system identification [a, b, c, d, q, ry, s, k, x0] = __sl_ident__ (dat.y, dat.u, nobr, n, meth, alg, conct, ctrl, rcond, tol); ## compute noise variance matrix factor L ## L L' = Ry, e = L v ## v becomes white noise with identity covariance matrix l = chol (ry, "lower"); ## assemble model [inname, outname] = get (dat, "inname", "outname"); if (strncmpi (noise, "e", 1)) # add error inputs e, not normalized sys = ss (a, [b, k], c, [d, eye(p)], tsam); in_u = __labels__ (inname, "u"); in_e = __labels__ (outname, "y"); in_e = cellfun (@(x) ["e@", x], in_e, "uniformoutput", false); inname = [in_u; in_e]; elseif (strncmpi (noise, "v", 1)) # add error inputs v, normalized sys = ss (a, [b, k*l], c, [d, l], tsam); in_u = __labels__ (inname, "u"); in_v = __labels__ (outname, "y"); in_v = cellfun (@(x) ["v@", x], in_v, "uniformoutput", false); inname = [in_u; in_v]; elseif (strncmpi (noise, "k", 1)) # Kalman predictor sys = ss ([a-k*c], [b-k*d, k], c, [d, zeros(p)], tsam); in_u = __labels__ (inname, "u"); in_y = __labels__ (outname, "y"); inname = [in_u; in_y]; else # no error inputs, default sys = ss (a, b, c, d, tsam); endif sys = set (sys, "inname", inname, "outname", outname); ## return x0 as vector for single-experiment data ## instead of a cell containing one vector if (numel (x0) == 1) x0 = x0{1}; endif ## assemble info struct ## Kalman gain matrix K ## state covariance matrix Q ## output covariance matrix Ry ## state-output cross-covariance matrix S ## noise variance matrix factor L info = struct ("K", k, "Q", q, "Ry", ry, "S", s, "L", l); endif endfunction function nobr = __check_s__ (s, nobr, method) if (s <= nobr) nobr = s; else error ("%s: require upper bound s <= %d, but the requested s is %d", method, nobr, s); endif endfunction control/inst/__ss_dim__.m0000644000076500000240000000422112273411257014750 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Number of outputs (p), inputs (m) and states (n) of state space matrices. ## For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.2 function [p, m, n] = __ss_dim__ (a, b, c, d, e = []) ## TODO: create oct-file? if (! is_real_matrix (a, b, c, d, e)) error ("ss: system matrices must be real"); endif [arows, acols] = size (a); [brows, bcols] = size (b); [crows, ccols] = size (c); [drows, dcols] = size (d); m = bcols; # = dcols n = arows; # = acols p = crows; # = drows if (arows != acols) error ("ss: system matrix a(%dx%d) is not square", arows, acols); endif if (brows != arows) error ("ss: system matrices a(%dx%d) and b(%dx%d) are incompatible", arows, acols, brows, bcols); endif if (ccols != acols) error ("ss: system matrices a(%dx%d) and c(%dx%d) are incompatible", arows, acols, crows, ccols); endif if (bcols != dcols) error ("ss: system matrices b(%dx%d) and d(%dx%d) are incompatible", brows, bcols, drows, dcols); endif if (crows != drows) error ("ss: system matrices c(%dx%d) and d(%dx%d) are incompatible", crows, ccols, drows, dcols); endif if (! isempty (e) && ! size_equal (e, a)) error ("ss: system matrices a(%dx%d) and e(%dx%d) are incompatible", arows, acols, rows (e), columns (e)); endif endfunction control/inst/__str2idx__.m0000644000076500000240000000254412273411257015077 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Get input/output indices from in/outgroup and in/outname. ## For internal use only. ## Author: Lukas Reichlin ## Created: October 2013 ## Version: 0.1 function idx = __str2idx__ (group, name, str, id) if (isfield (group, str)) idx = group.(str)(:); else tmp = strcmp (name, str)(:); switch (nnz (tmp)) case 1 idx = find (tmp); case 0 error ("lti: %sgroup or %sname '%s' not found", id, id, str); otherwise error ("lti: %sname '%s' is ambiguous", id, str); ## FIXME: error for structure arrays endswitch endif endfunction control/inst/__tf_dim__.m0000644000076500000240000000233112273411257014734 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Number of outputs and inputs of transfer function numerator and ## denominator. For internal use only. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.1 function [nrows, ncols] = __tf_dim__ (num, den) [nrows, ncols] = size (num); [drows, dcols] = size (den); if (nrows != drows || ncols != dcols) error ("tf: num(%dx%d) and den(%dx%d) must have equal dimensions", nrows, ncols, drows, dcols); endif endfunction control/inst/__time_response__.m0000644000076500000240000003315612273411257016357 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Common code for the time response functions step, impulse and initial. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.4.1 function [y, t, x] = __time_response__ (response, args, sysname, plotflag) sys_idx = find (cellfun (@isa, args, {"lti"})); # look for LTI models, 'find' needed for plot styles sys_cell = cellfun (@ss, args(sys_idx), "uniformoutput", false); # convert to state-space if (! size_equal (sys_cell{:})) error ("%s: models must have equal sizes", response); endif vec_idx = find (cellfun (@is_real_matrix, args)); # indices of vector arguments n_vec = length (vec_idx); # number of vector arguments n_sys = length (sys_cell); # number of LTI systems tfinal = []; dt = []; x0 = []; ## extract tfinal/t, dt, x0 from args if (strcmpi (response, "initial")) if (n_vec < 1) error ("initial: require initial state vector 'x0'"); else # initial state vector x0 specified arg = args{vec_idx(1)}; if (is_real_vector (arg)) x0 = arg; else error ("initial: initial state vector 'x0' must be a vector of real values"); endif if (n_vec > 1) # tfinal or time vector t specified arg = args{vec_idx(2)}; if (issample (arg)) tfinal = arg; elseif (isempty (arg)) ## tfinal = []; # nothing to do here elseif (is_real_vector (arg)) dt = abs (arg(2) - arg(1)); # assume that t is regularly spaced tfinal = arg(end); else warning ("initial: argument number %d ignored", vec_idx(2)); endif if (n_vec > 2) # sampling time dt specified arg = args{vec_idx(3)}; if (issample (arg)) dt = arg; else warning ("initial: argument number %d ignored", vec_idx(3)); endif if (n_vec > 3) warning ("initial: ignored"); endif endif endif endif else # step or impulse response if (n_vec > 0) # tfinal or time vector t specified arg = args{vec_idx(1)}; if (issample (arg)) tfinal = arg; elseif (isempty (arg)) ## tfinal = []; # nothing to do here elseif (is_real_vector (arg)) dt = abs (arg(2) - arg(1)); # assume that t is regularly spaced tfinal = arg(end); else warning ("%s: argument number %d ignored", response, vec_idx(1)); endif if (n_vec > 1) # sampling time dt specified arg = args{vec_idx(2)}; if (issample (arg)) dt = arg; else warning ("%s: argument number %d ignored", response, vec_idx(2)); endif if (n_vec > 2) warning ("%s: ignored", response); endif endif endif endif ## TODO: share common code between initial and step/impulse [tfinal, dt] = cellfun (@__sim_horizon__, sys_cell, {tfinal}, {dt}, "uniformoutput", false); tfinal = max ([tfinal{:}]); ct_idx = cellfun (@isct, sys_cell); sys_dt_cell = sys_cell; tmp = cellfun (@c2d, sys_cell(ct_idx), dt(ct_idx), {"zoh"}, "uniformoutput", false); sys_dt_cell(ct_idx) = tmp; ## time vector t = @cellfun (@(dt) reshape (0 : dt : tfinal, [], 1), dt, "uniformoutput", false); ## function [y, x_arr] = __initial_response__ (sys, sys_dt, t, x0) ## function [y, x_arr] = __step_response__ (sys_dt, t) ## function [y, x_arr] = __impulse_response__ (sys, sys_dt, t) ## function [y, x_arr] = __ramp_response__ (sys_dt, t) switch (response) case "initial" [y, x] = cellfun (@__initial_response__, sys_dt_cell, t, {x0}, "uniformoutput", false); case "step" [y, x] = cellfun (@__step_response__, sys_dt_cell, t, "uniformoutput", false); case "impulse" [y, x] = cellfun (@__impulse_response__, sys_cell, sys_dt_cell, t, "uniformoutput", false); case "ramp" [y, x] = cellfun (@__ramp_response__, sys_dt_cell, t, "uniformoutput", false); otherwise error ("time_response: invalid response type"); endswitch if (plotflag) # display plot [p, m] = size (sys_cell{1}); switch (response) case "initial" str = "Response to Initial Conditions"; cols = 1; ## yfinal = zeros (p, 1); case "step" str = "Step Response"; cols = m; ## yfinal = dcgain (sys_cell{1}); case "impulse" str = "Impulse Response"; cols = m; ## yfinal = zeros (p, m); case "ramp" str = "Ramp Response"; cols = m; otherwise error ("time_response: invalid response type"); endswitch style_idx = find (cellfun (@ischar, args)); outname = get (sys_cell{end}, "outname"); outname = __labels__ (outname, "y"); colororder = get (gca, "colororder"); rc = rows (colororder); for k = 1 : n_sys # for every system if (k == n_sys) lim = numel (args); else lim = sys_idx(k+1); endif style = args(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); if (isempty (style)) color = colororder(1+rem (k-1, rc), :); style = {"color", color}; endif if (ct_idx(k)) # continuous-time system for i = 1 : p # for every output for j = 1 : cols # for every input (except for initial where cols=1) if (p != 1 || cols != 1) subplot (p, cols, (i-1)*cols+j); endif plot (t{k}, y{k}(:, i, j), style{:}); hold on; grid on; if (k == n_sys) axis tight ylim (__axis_margin__ (ylim)) if (j == 1) ylabel (outname{i}); if (i == 1) title (str); endif endif endif endfor endfor else # discrete-time system for i = 1 : p # for every output for j = 1 : cols # for every input (except for initial where cols=1) if (p != 1 || cols != 1) subplot (p, cols, (i-1)*cols+j); endif stairs (t{k}, y{k}(:, i, j), style{:}); hold on; grid on; if (k == n_sys) axis tight; ylim (__axis_margin__ (ylim)) if (j == 1) ylabel (outname{i}); if (i == 1) title (str); endif endif endif endfor endfor endif endfor xlabel ("Time [s]"); if (p == 1 && m == 1) legend (sysname) endif hold off; endif endfunction function [y, x_arr] = __initial_response__ (sys_dt, t, x0) [F, G, C, D] = ssdata (sys_dt); # system must be proper n = rows (F); # number of states m = columns (G); # number of inputs p = rows (C); # number of outputs l_t = length (t); ## preallocate memory y = zeros (l_t, p); x_arr = zeros (l_t, n); ## initial conditions x = reshape (x0, [], 1); # make sure that x is a column vector if (n != length (x0) || ! is_real_vector (x0)) error ("initial: x0 must be a real vector with %d elements", n); endif ## simulation for k = 1 : l_t y(k, :) = C * x; x_arr(k, :) = x; x = F * x; endfor endfunction function [y, x_arr] = __step_response__ (sys_dt, t) [F, G, C, D] = ssdata (sys_dt); # system must be proper n = rows (F); # number of states m = columns (G); # number of inputs p = rows (C); # number of outputs l_t = length (t); ## preallocate memory y = zeros (l_t, p, m); x_arr = zeros (l_t, n, m); for j = 1 : m # for every input channel ## initial conditions x = zeros (n, 1); u = zeros (m, 1); u(j) = 1; ## simulation for k = 1 : l_t y(k, :, j) = C * x + D * u; x_arr(k, :, j) = x; x = F * x + G * u; endfor endfor endfunction function [y, x_arr] = __impulse_response__ (sys, sys_dt, t) [~, B] = ssdata (sys); [F, G, C, D, dt] = ssdata (sys_dt); # system must be proper dt = abs (dt); # use 1 second if tsam is unspecified (-1) discrete = ! isct (sys); n = rows (F); # number of states m = columns (G); # number of inputs p = rows (C); # number of outputs l_t = length (t); ## preallocate memory y = zeros (l_t, p, m); x_arr = zeros (l_t, n, m); for j = 1 : m # for every input channel ## initial conditions u = zeros (m, 1); u(j) = 1; if (discrete) x = zeros (n, 1); # zero by definition y(1, :, j) = D * u / dt; x_arr(1, :, j) = x; x = G * u / dt; else x = B * u; # B, not G! y(1, :, j) = C * x; x_arr(1, :, j) = x; x = F * x; endif ## simulation for k = 2 : l_t y (k, :, j) = C * x; x_arr(k, :, j) = x; x = F * x; endfor endfor if (discrete) y *= dt; x_arr *= dt; endif endfunction function [y, x_arr] = __ramp_response__ (sys_dt, t) [F, G, C, D] = ssdata (sys_dt); # system must be proper n = rows (F); # number of states m = columns (G); # number of inputs p = rows (C); # number of outputs l_t = length (t); ## preallocate memory y = zeros (l_t, p, m); x_arr = zeros (l_t, n, m); for j = 1 : m # for every input channel ## initial conditions x = zeros (n, 1); u = zeros (m, l_t); u(j, :) = t; ## simulation for k = 1 : l_t y(k, :, j) = C * x + D * u(:, k); x_arr(k, :, j) = x; x = F * x + G * u(:, k); endfor endfor endfunction function [tfinal, dt] = __sim_horizon__ (sys, tfinal, Ts) ## code based on __stepimp__.m of Kai P. Mueller and A. Scottedward Hodel TOL = 1.0e-10; # values below TOL are assumed to be zero N_MIN = 50; # min number of points N_MAX = 2000; # max number of points N_DEF = 1000; # default number of points T_DEF = 10; # default simulation time ev = pole (sys); n = length (ev); # number of states/poles continuous = isct (sys); discrete = ! continuous; if (discrete) dt = Ts = abs (get (sys, "tsam")); ## perform bilinear transformation on poles in z for k = 1 : n pol = ev(k); if (abs (pol + 1) < TOL) ev(k) = 0; else ev(k) = 2 / Ts * (pol - 1) / (pol + 1); endif endfor endif ## remove poles near zero from eigenvalue array ev nk = n; for k = 1 : n if (abs (real (ev(k))) < TOL) ev(k) = 0; nk -= 1; endif endfor if (nk == 0) if (isempty (tfinal)) tfinal = T_DEF; endif if (continuous) dt = tfinal / N_DEF; endif else ev = ev(find (ev)); ev_max = max (abs (ev)); if (continuous) dt = 0.2 * pi / ev_max; endif if (isempty (tfinal)) ev_min = min (abs (real (ev))); tfinal = 5.0 / ev_min; ## round up yy = 10^(ceil (log10 (tfinal)) - 1); tfinal = yy * ceil (tfinal / yy); endif if (continuous) N = tfinal / dt; if (N < N_MIN) dt = tfinal / N_MIN; endif if (N > N_MAX) dt = tfinal / N_MAX; endif endif endif if (continuous && ! isempty (Ts)) # catch case cont. system with dt specified dt = Ts; endif endfunction control/inst/__tito_dim__.m0000644000076500000240000000307412273411257015307 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## Extract nmeas and ncon from plant P which has been partitioned by mktito. ## Author: Lukas Reichlin ## Created: January 2014 ## Version: 0.1 function [nmeas, ncon] = __tito_dim__ (P, name) [p, m] = size (P); outgroup = P.outgroup; ingroup = P.ingroup; if (! isfield (outgroup, "V")) error ("%s: missing outgroup 'V'", name); endif if (! isfield (ingroup, "U")) error ("%s: missing ingroup 'U'", name); endif nmeas = numel (outgroup.V); ncon = numel (ingroup.U); ## check whether indices of V and U are in ascending order ## and at the end of the outputs/inputs if (! isequal (outgroup.V(:), (p-nmeas+1:p)(:))) error ("%s: outgroup 'V' invalid", name); endif if (! isequal (ingroup.U(:), (m-ncon+1:m)(:))) error ("%s: ingroup 'U' invalid", name); endif endfunction control/inst/__vec2tfpoly__.m0000644000076500000240000000210312273411257015564 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Convert a (cell of) row vector(s) to a cell of tfpoly objects. ## Used by tf and __set__. ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function ndr = __vec2tfpoly__ (nd) if (! iscell (nd)) nd = {nd}; endif ndr = cellfun (@tfpoly, nd, "uniformoutput", false); endfunction control/inst/Anderson.m0000644000076500000240000000506012273411257014451 0ustar lukasstaff%% -*- texinfo -*- %% Frequency-weighted coprime factorization controller reduction. % =============================================================================== % Coprime Factorization Controller Reduction Lukas Reichlin December 2011 % =============================================================================== % Reference: Anderson, B.D.O.: Controller Reduction: Concepts and Approaches % IEEE Transactions of Automatic Control, Vol. 34, No. 8, August 1989 % =============================================================================== % Tabula Rasa clear all, close all, clc % Plant A = [ -0.161 -6.004 -0.58215 -9.9835 -0.40727 -3.982 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 ]; B = [ 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ]; C = [ 0.0 0.0 6.4432e-3 2.3196e-3 7.1252e-2 1.0002 0.10455 0.99551 ]; G = ss (A, B, C); % LQG Design H = [ 0.0 0.0 0.0 0.0 0.55 11.0 1.32 18.0 ]; q1 = 1e-6; q2 = 100; % [100, 1000, 2000] Q = q1 * H.' * H; R = 1; W = q2 * B * B.'; V = 1; F = lqr (G, Q, R) L = lqe (G, W, V) % Coprime Factorization using Balanced Truncation Approximation Kr = arrayfun (@(k) cfconred (G, F, L, k), 8:-1:2, 'uniformoutput', false); % 'method', 'bfsr-bta' T = cellfun (@(Kr) feedback (G*Kr), Kr, 'uniformoutput', false); figure (1) step (T{:}, 200) % Coprime Factorization using Singular Perturbation Approximation Kr = arrayfun (@(k) cfconred (G, F, L, k, 'method', 'bfsr-spa'), 8:-1:2, 'uniformoutput', false); T = cellfun (@(Kr) feedback (G*Kr), Kr, 'uniformoutput', false); figure (2) step (T{:}, 200) % Frequency-Weighted Coprime Factorization using BTA Kr = arrayfun (@(k) fwcfconred (G, F, L, k), 8:-1:2, 'uniformoutput', false); T = cellfun (@(Kr) feedback (G*Kr), Kr, 'uniformoutput', false); figure (3) step (T{:}, 300) control/inst/arx.m0000644000076500000240000002551612273411257013502 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{n}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{n}, @var{opt}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{opt}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}] =} arx (@var{dat}, @var{'na'}, @var{na}, @var{'nb'}, @var{nb}) ## Estimate ARX model using QR factorization. ## @iftex ## @tex ## $$ A(q) \\, y(t) = B(q) \\, u(t) \\, + \\, e(t) $$ ## @end tex ## @end iftex ## @ifnottex ## ## @example ## A(q) y(t) = B(q) u(t) + e(t) ## @end example ## ## @end ifnottex ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata identification dataset containing the measurements, i.e. time-domain signals. ## @item n ## The desired order of the resulting model @var{sys}. ## @item @dots{} ## Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## ## @strong{Outputs} ## @table @var ## @item sys ## Discrete-time transfer function model. ## If the second output argument @var{x0} is returned, ## @var{sys} becomes a state-space model. ## @item x0 ## Initial state vector. If @var{dat} is a multi-experiment dataset, ## @var{x0} becomes a cell vector containing an initial state vector ## for each experiment. ## @end table ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'na' ## Order of the polynomial A(q) and number of poles. ## ## @item 'nb' ## Order of the polynomial B(q)+1 and number of zeros+1. ## @var{nb} <= @var{na}. ## ## @item 'nk' ## Input-output delay specified as number of sampling instants. ## Scalar positive integer. This corresponds to a call to command ## @command{nkshift}, followed by padding the B polynomial with ## @var{nk} leading zeros. ## @end table ## ## ## @strong{Algorithm}@* ## Uses the formulae given in [1] on pages 318-319, ## 'Solving for the LS Estimate by QR Factorization'. ## For the initial conditions, SLICOT IB01CD is used by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @strong{References}@* ## [1] Ljung, L. (1999) ## @cite{System Identification: Theory for the User: Second Edition}. ## Prentice Hall, New Jersey, USA. ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function [sys, varargout] = arx (dat, varargin) ## TODO: delays if (nargin < 2) print_usage (); endif if (! isa (dat, "iddata") || ! dat.timedomain) error ("arx: first argument must be a time-domain iddata dataset"); endif ## p: outputs, m: inputs, ex: experiments [~, p, m, ex] = size (dat); # dataset dimensions if (is_real_scalar (varargin{1})) # arx (dat, n, ...) varargin = horzcat (varargin(2:end), {"na"}, varargin(1), {"nb"}, varargin(1)); endif if (isstruct (varargin{1})) # arx (dat, opt, ...), arx (dat, n, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("arx: keys and values must come in pairs"); endif ## default arguments na = []; nb = []; nk = 0; ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case "na" na = __check_n__ (val, "na"); case "nb" nb = __check_n__ (val, "nb"); case "nk" nk = __check_n__ (val, "nk"); if (! issample (val, 0)) error ("arx: channel-wise 'nk' matrices not supported yet"); endif otherwise warning ("arx: invalid property name '%s' ignored", key); endswitch endfor if (any (nk(:) != 0)) dat = nkshift (dat, nk); endif ## extract data Y = dat.y; U = dat.u; tsam = dat.tsam; ## multi-experiment data requires equal sampling times if (ex > 1 && ! isequal (tsam{:})) error ("arx: require equally sampled experiments"); else tsam = tsam{1}; endif if (is_real_scalar (na, nb)) na = repmat (na, p, 1); # na(p-by-1) nb = repmat (nb, p, m); # nb(p-by-m) elseif (! (is_real_vector (na) && is_real_matrix (nb) ... && rows (na) == p && rows (nb) == p && columns (nb) == m)) error ("arx: require na(%dx1) instead of (%dx%d) and nb(%dx%d) instead of (%dx%d)", ... p, rows (na), columns (na), p, m, rows (nb), columns (nb)); endif max_nb = max (nb, [], 2); # one maximum for each row/output, max_nb(p-by-1) n = max (na, max_nb); # n(p-by-1) ## create empty cells for numerator and denominator polynomials num = cell (p, m+p); den = cell (p, m+p); ## MIMO (p-by-m) models are identified as p MISO (1-by-m) models ## For multi-experiment data, minimize the trace of the error for i = 1 : p # for every output Phi = cell (ex, 1); # one regression matrix per experiment for e = 1 : ex # for every experiment ## avoid warning: toeplitz: column wins anti-diagonal conflict ## therefore set first row element equal to y(1) PhiY = toeplitz (Y{e}(1:end-1, i), [Y{e}(1, i); zeros(na(i)-1, 1)]); ## create MISO Phi for every experiment PhiU = arrayfun (@(x) toeplitz (U{e}(1:end-1, x), [U{e}(1, x); zeros(nb(i,x)-1, 1)]), 1:m, "uniformoutput", false); Phi{e} = (horzcat (-PhiY, PhiU{:}))(n(i):end, :); endfor ## compute parameter vector Theta Theta = __theta__ (Phi, Y, i, n); ## extract polynomial matrices A and B from Theta ## A is a scalar polynomial for output i, i=1:p ## B is polynomial row vector (1-by-m) for output i A = [1; Theta(1:na(i))]; # a0 = 1, a1 = Theta(1), an = Theta(n) ThetaB = Theta(na(i)+1:end); # all polynomials from B are in one column vector B = mat2cell (ThetaB, nb(i,:)); # now separate the polynomials, one for each input B = reshape (B, 1, []); # make B a row cell (1-by-m) B = cellfun (@(B) [zeros(1+nk, 1); B], B, "uniformoutput", false); # b0 = 0 (leading zero required by filt) ## add error inputs Be = repmat ({0}, 1, p); # there are as many error inputs as system outputs (p) Be(i) = [zeros(1,nk), 1]; # inputs m+1:m+p are zero, except m+i which is one num(i, :) = [B, Be]; # numerator polynomials for output i, individual for each input den(i, :) = repmat ({A}, 1, m+p); # in a row (output i), all inputs have the same denominator polynomial endfor ## A(q) y(t) = B(q) u(t) + e(t) ## there is only one A per row ## B(z) and A(z) are a Matrix Fraction Description (MFD) ## y = A^-1(q) B(q) u(t) + A^-1(q) e(t) ## since A(q) is a diagonal polynomial matrix, its inverse is trivial: ## the corresponding transfer function has common row denominators. sys = filt (num, den, tsam); # filt creates a transfer function in z^-1 ## compute initial state vector x0 if requested ## this makes only sense for state-space models, therefore convert TF to SS if (nargout > 1) sys = prescale (ss (sys(:,1:m))); x0 = __sl_ib01cd__ (Y, U, sys.a, sys.b, sys.c, sys.d, 0.0); ## return x0 as vector for single-experiment data ## instead of a cell containing one vector if (numel (x0) == 1) x0 = x0{1}; endif varargout{1} = x0; endif endfunction function Theta = __theta__ (Phi, Y, i, n) if (numel (Phi) == 1) # single-experiment dataset ## use "square-root algorithm" A = horzcat (Phi{1}, Y{1}(n(i)+1:end, i)); # [Phi, Y] R0 = triu (qr (A, 0)); # 0 for economy-size R (without zero rows) R1 = R0(1:end-1, 1:end-1); # R1 is triangular - can we exploit this in R1\R2? R2 = R0(1:end-1, end); Theta = __ls_svd__ (R1, R2); # R1 \ R2 ## Theta = Phi \ Y(n+1:end, :); # naive formula ## Theta = __ls_svd__ (Phi{1}, Y{1}(n(i)+1:end, i)); else # multi-experiment dataset ## TODO: find more sophisticated formula than ## Theta = (Phi1' Phi1 + Phi2' Phi2 + ...) \ (Phi1' Y1 + Phi2' Y2 + ...) ## covariance matrix C = (Phi1' Phi + Phi2' Phi2 + ...) tmp = cellfun (@(Phi) Phi.' * Phi, Phi, "uniformoutput", false); ## rc = cellfun (@rcond, tmp); # also test C? QR or SVD? C = plus (tmp{:}); ## PhiTY = (Phi1' Y1 + Phi2' Y2 + ...) tmp = cellfun (@(Phi, Y) Phi.' * Y(n(i)+1:end, i), Phi, Y, "uniformoutput", false); PhiTY = plus (tmp{:}); ## pseudoinverse Theta = C \ Phi'Y Theta = __ls_svd__ (C, PhiTY); endif endfunction function x = __ls_svd__ (A, b) ## solve the problem Ax=b ## x = A\b would also work, ## but this way we have better control and warnings ## solve linear least squares problem by pseudoinverse ## the pseudoinverse is computed by singular value decomposition ## M = U S V* ---> M+ = V S+ U* ## Th = Ph \ Y = Ph+ Y ## Th = V S+ U* Y, S+ = 1 ./ diag (S) [U, S, V] = svd (A, 0); # 0 for "economy size" decomposition S = diag (S); # extract main diagonal r = sum (S > eps*S(1)); if (r < length (S)) warning ("arx: rank-deficient coefficient matrix"); warning ("sampling time too small"); warning ("persistence of excitation"); endif V = V(:, 1:r); S = S(1:r); U = U(:, 1:r); x = V * (S .\ (U' * b)); # U' is the conjugate transpose endfunction function val = __check_n__ (val, str = "n") if (! is_real_matrix (val) || fix (val) != val) error ("arx: argument '%s' must be a positive integer", str); endif endfunction control/inst/augw.m0000644000076500000240000001235112273411257013644 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{P} =} augw (@var{G}, @var{W1}, @var{W2}, @var{W3}) ## Extend plant for stacked S/KS/T problem. Subsequently, the robust control problem ## can be solved by h2syn or hinfsyn. ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model of plant. ## @item W1 ## @acronym{LTI} model of performance weight. Bounds the largest singular values of sensitivity @var{S}. ## Model must be empty @code{[]}, SISO or of appropriate size. ## @item W2 ## @acronym{LTI} model to penalize large control inputs. Bounds the largest singular values of @var{KS}. ## Model must be empty @code{[]}, SISO or of appropriate size. ## @item W3 ## @acronym{LTI} model of robustness and noise sensitivity weight. Bounds the largest singular values of ## complementary sensitivity @var{T}. Model must be empty @code{[]}, SISO or of appropriate size. ## @end table ## ## All inputs must be proper/realizable. ## Scalars, vectors and matrices are possible instead of @acronym{LTI} models. ## ## @strong{Outputs} ## @table @var ## @item P ## State-space model of augmented plant. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## ## | W1 | -W1*G | z1 = W1 r - W1 G u ## | 0 | W2 | z2 = W2 u ## P = | 0 | W3*G | z3 = W3 G u ## |----+-------| ## | I | -G | e = r - G u ## @end group ## @end example ## @example ## @group ## +------+ z1 ## +---------------------------------------->| W1 |-----> ## | +------+ ## | +------+ z2 ## | +---------------------->| W2 |-----> ## | | +------+ ## r + e | +--------+ u | +--------+ y +------+ z3 ## ----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |-----> ## ^ - +--------+ +--------+ | +------+ ## | | ## +----------------------------------------+ ## @end group ## @end example ## @example ## @group ## +--------+ ## | |-----> z1 (p1x1) z1 = W1 e ## r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u ## | |-----> z3 (p3x1) z3 = W3 y ## u (mx1) ----->| |-----> e (px1) e = r - y ## +--------+ ## @end group ## @end example ## @example ## @group ## +--------+ ## r ----->| |-----> z ## | P(s) | ## u +---->| |-----+ e ## | +--------+ | ## | | ## | +--------+ | ## +-----| K(s) |<----+ ## +--------+ ## @end group ## @end example ## ## @strong{References}@* ## [1] Skogestad, S. and Postlethwaite I. (2005) ## @cite{Multivariable Feedback Control: Analysis and Design: ## Second Edition}. Wiley. ## ## @seealso{h2syn, hinfsyn, mixsyn} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2009 ## Version: 0.3 function P = augw (G, W1 = [], W2 = [], W3 = []) if (nargin == 0 || nargin > 4) print_usage (); endif G = ss (G); [p, m] = size (G); [W1, p1, m1] = __adjust_weighting__ (W1, p); [W2, p2, m2] = __adjust_weighting__ (W2, m); [W3, p3, m3] = __adjust_weighting__ (W3, p); ## Pr = [1; 0; 0; 1]; ## Pu = [-1; 0; 1; -1]*G + [0; 1; 0; 0]; Pr = ss ([eye(m1,p) ; zeros(m2,p); zeros(m3,p); eye(p,p) ]); Pu1 = ss ([-eye(m1,p) ; zeros(m2,p); eye(m3,p) ; -eye(p,p) ]); Pu2 = ss ([zeros(m1,m); eye(m2,m) ; zeros(m3,m); zeros(p,m) ]); Pu = Pu1 * G + Pu2; P = append (W1, W2, W3, eye (p, p)) * [Pr, Pu]; P = mktito (P, p, m); endfunction function [W, p, m] = __adjust_weighting__ (W, s) W = ss (W); [p, m] = size (W); if (m == 0 || m == s) # model is empty or has s inputs return; elseif (m == 1) # model is SISO or SIMO tmp = W; for k = 2 : s W = append (W, tmp); # stack single-input model s times endfor [p, m] = size (W); # weighting function now of correct size else # model is MIMO or MISO error ("augw: %s must have 1 or %d inputs", inputname (1), s); endif endfunction control/inst/BMWengine.m0000644000076500000240000000672112273411257014520 0ustar lukasstaff## -*- texinfo -*- ## @deftypefn{Function File} {@var{sys} =} BMWengine () ## @deftypefnx{Function File} {@var{sys} =} BMWengine (@var{"scaled"}) ## @deftypefnx{Function File} {@var{sys} =} BMWengine (@var{"unscaled"}) ## Model of the BMW 4-cylinder engine at ETH Zurich's control laboratory. ## @example ## @group ## OPERATING POINT ## Drosselklappenstellung alpha_DK = 10.3 Grad ## Saugrohrdruck p_s = 0.48 bar ## Motordrehzahl n = 860 U/min ## Lambda-Messwert lambda = 1.000 ## Relativer Wandfilminhalt nu = 1 ## @end group ## @end example ## @example ## @group ## INPUTS ## U_1 Sollsignal Drosselklappenstellung [Grad] ## U_2 Relative Einspritzmenge [-] ## U_3 Zuendzeitpunkt [Grad KW] ## M_L Lastdrehmoment [Nm] ## @end group ## @end example ## @example ## @group ## STATES ## X_1 Drosselklappenstellung [Grad] ## X_2 Saugrohrdruck [bar] ## X_3 Motordrehzahl [U/min] ## X_4 Messwert Lamba-Sonde [-] ## X_5 Relativer Wandfilminhalt [-] ## @end group ## @end example ## @example ## @group ## OUTPUTS ## Y_1 Motordrehzahl [U/min] ## Y_2 Messwert Lambda-Sonde [-] ## @end group ## @end example ## @example ## @group ## SCALING ## U_1N, X_1N 1 Grad ## U_2N, X_4N, X_5N, Y_2N 0.05 ## U_3N 1.6 Grad KW ## X_2N 0.05 bar ## X_3N, Y_1N 200 U/min ## @end group ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.1.1 ## TODO: translate German terminology function sys = BMWengine (flg = "scaled") if (nargin > 1) print_usage (); endif switch (tolower (flg)) case "unscaled" ## Linearisiertes Modell, nicht skaliert Apu = [ -40.0000 0 0 0 0 0.1683 -2.9471 -0.0016 0 0 26.6088 920.3932 -0.1756 0 259.1700 -0.5852 14.1941 0.0061 -5.7000 -5.7000 0.6600 -1.1732 -0.0052 0 -15.0000 ]; Bpu = [ 40.0000 0 0 0 0 0 0 181.4190 1.5646 0 -3.9900 0 0 4.5000 0 ]; Bdpu = [ 0 0 -15.9000 0 0 ]; Cpu = [ 0 0 1 0 0 0 0 0 1 0 ]; sys = ss (Apu, [Bpu, Bdpu], Cpu); case "scaled" ## Skaliertes Zustandsraummodell Ap = [ -40.0000 0 0 0 0 3.3659 -2.9471 -6.5157 0 0 0.1330 0.2301 -0.1756 0 0.0648 -11.7043 14.1941 24.3930 -5.7000 -5.7000 13.2003 -1.1732 -20.9844 0 -15.0000 ]; Bp = [ 40.0000 0 0 0 0 0 0 0.0454 0.0125 0 -3.9900 0 0 4.5000 0 ]; Bdp = [ 0 0 -1.5900 0 0 ]; Cp = [ 0 0 1 0 0 0 0 0 1 0 ]; sys = ss (Ap, [Bp, Bdp], Cp, [], "scaled", true); otherwise print_usage (); endswitch endfunction control/inst/bode.m0000644000076500000240000000761412273411257013620 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} bode (@var{sys}) ## @deftypefnx {Function File} {} bode (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx {Function File} {} bode (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) ## @deftypefnx {Function File} {} bode (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} bode (@var{sys}) ## @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} bode (@var{sys}, @var{w}) ## Bode diagram of frequency response. If no output arguments are given, ## the response is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. Must be a single-input and single-output (SISO) system. ## @item w ## Optional vector of frequency values. If @var{w} is not specified, ## it is calculated by the zeros and poles of the system. ## Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, ## where @var{wmin} and @var{wmax} denote minimum and maximum frequencies ## in rad/s. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item mag ## Vector of magnitude. Has length of frequency vector @var{w}. ## @item pha ## Vector of phase. Has length of frequency vector @var{w}. ## @item w ## Vector of frequency values used. ## @end table ## ## @seealso{nichols, nyquist, sigma} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.5 function [mag_r, pha_r, w_r] = bode (varargin) if (nargin == 0) print_usage (); endif [H, w] = __frequency_response__ (varargin, false, "std", false); H = cellfun (@reshape, H, {[]}, {1}, "uniformoutput", false); mag = cellfun (@abs, H, "uniformoutput", false); pha = cellfun (@(H) unwrap (arg (H)) * 180 / pi, H, "uniformoutput", false); if (! nargout) mag_db = cellfun (@(mag) 20 * log10 (mag), mag, "uniformoutput", false); tmp = cellfun (@isa, varargin, {"lti"}); sys_idx = find (tmp); tmp = cellfun (@ischar, varargin); style_idx = find (tmp); len = numel (H); mag_args = {}; pha_args = {}; legend_args = {}; for k = 1:len if (k == len) lim = nargin; else lim = sys_idx(k+1); endif style = varargin(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); mag_args = cat (2, mag_args, w(k), mag_db(k), style); pha_args = cat (2, pha_args, w(k), pha(k), style); try legend_args = cat (2, legend_args, inputname(sys_idx(k))); # watch out for bode (lticell{:}) end_try_catch endfor subplot (2, 1, 1) semilogx (mag_args{:}) axis ("tight") ylim (__axis_margin__ (ylim)) grid ("on") title ("Bode Diagram") ylabel ("Magnitude [dB]") subplot (2, 1, 2) semilogx (pha_args{:}) axis ("tight") ylim (__axis_margin__ (ylim)) grid ("on") xlabel ("Frequency [rad/s]") ylabel ("Phase [deg]") legend (legend_args) else mag_r = mag{1}; pha_r = pha{1}; w_r = w{1}; endif endfunction control/inst/bodemag.m0000644000076500000240000000676712273411257014315 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} bodemag (@var{sys}) ## @deftypefnx {Function File} {} bodemag (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx {Function File} {} bodemag (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) ## @deftypefnx {Function File} {} bodemag (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx {Function File} {[@var{mag}, @var{w}] =} bodemag (@var{sys}) ## @deftypefnx {Function File} {[@var{mag}, @var{w}] =} bodemag (@var{sys}, @var{w}) ## Bode magnitude diagram of frequency response. If no output arguments are given, ## the response is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. Must be a single-input and single-output (SISO) system. ## @item w ## Optional vector of frequency values. If @var{w} is not specified, ## it is calculated by the zeros and poles of the system. ## Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, ## where @var{wmin} and @var{wmax} denote minimum and maximum frequencies ## in rad/s. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item mag ## Vector of magnitude. Has length of frequency vector @var{w}. ## @item w ## Vector of frequency values used. ## @end table ## ## @seealso{bode, nichols, nyquist, sigma} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.5 function [mag_r, w_r] = bodemag (varargin) if (nargin == 0) print_usage (); endif [H, w] = __frequency_response__ (varargin, false, "std"); H = cellfun (@reshape, H, {[]}, {1}, "uniformoutput", false); mag = cellfun (@abs, H, "uniformoutput", false); if (! nargout) mag_db = cellfun (@(mag) 20 * log10 (mag), mag, "uniformoutput", false); tmp = cellfun (@isa, varargin, {"lti"}); sys_idx = find (tmp); tmp = cellfun (@ischar, varargin); style_idx = find (tmp); len = numel (H); mag_args = {}; legend_args = {}; for k = 1:len if (k == len) lim = nargin; else lim = sys_idx(k+1); endif style = varargin(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); mag_args = cat (2, mag_args, w(k), mag_db(k), style); try legend_args = cat (2, legend_args, inputname(sys_idx(k))); # watch out for bodemag (lticell{:}) end_try_catch endfor semilogx (mag_args{:}) axis ("tight") ylim (__axis_margin__ (ylim)) grid ("on") title ("Bode Magnitude Diagram") xlabel ("Frequency [rad/s]") ylabel ("Magnitude [dB]") legend (legend_args) else mag_r = mag{1}; w_r = w{1}; endif endfunction control/inst/Boeing707.m0000644000076500000240000000433312273411257014343 0ustar lukasstaff## Copyright (C) 1997, 2000, 2004, 2005, 2006, 2007 Kai P. Mueller ## ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} Boeing707 () ## Creates a linearized state-space model of a Boeing 707-321 aircraft ## at @var{v}=80 m/s ## @iftex ## @tex ## ($M = 0.26$, $G_{a0} = -3^{\\circ}$, ${\\alpha}_0 = 4^{\\circ}$, ${\\kappa}= 50^{\\circ}$). ## @end tex ## @end iftex ## @ifnottex ## (@var{M} = 0.26, @var{Ga0} = -3 deg, @var{alpha0} = 4 deg, @var{kappa} = 50 deg). ## @end ifnottex ## ## System inputs: (1) thrust and (2) elevator angle. ## ## System outputs: (1) airspeed and (2) pitch angle. ## ## @strong{Reference}: R. Brockhaus: @cite{Flugregelung} (Flight ## Control), Springer, 1994. ## @end deftypefn ## Author: Kai P. Mueller ## Created: September 28, 1997 function outsys = Boeing707 () if (nargin != 0) print_usage (); endif a = [-0.46E-01, 0.10681415316, 0.0, -0.17121680433; -0.1675901504661613, -0.515, 1.0, 0.6420630320636088E-02; 0.1543104215347786, -0.547945, -0.906, -0.1521689385990753E-02; 0.0, 0.0, 1.0, 0.0]; b = [0.1602300107479095, 0.2111848453E-02; 0.8196877780963616E-02, -0.3025E-01; 0.9173594317692437E-01, -0.75283075; 0.0, 0.0]; c = [1.0, 0.0, 0.0, 0.0; 0.0, 0.0, 0.0, 1.0]; d = zeros (2, 2); inam = {"thrust"; "rudder"}; onam = {"speed"; "pitch"}; ## snam = {"x1"; "x2"; "x3"; "x4"}; outsys = ss (a, b, c, d, "inname", inam, "outname", onam); endfunction control/inst/bstmodred.m0000644000076500000240000003037712273411257014674 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @var{opt}, @dots{}) ## ## Model order reduction by Balanced Stochastic Truncation (BST) method. ## The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order ## @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} ## approximates the one from original system @var{G}. ## ## BST is a relative error method which tries to minimize ## @iftex ## @tex ## $$ || G^{-1} (G-G_r) ||_{\\infty} = \\min $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 ## ||G (G-Gr)|| = min ## inf ## @end example ## @end ifnottex ## ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model to be reduced. ## @item nr ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Gr ## Reduced order state-space model. ## @item info ## Struct containing additional information. ## @table @var ## @item info.n ## The order of the original system @var{G}. ## @item info.ns ## The order of the @var{alpha}-stable subsystem of the original system @var{G}. ## @item info.hsv ## The Hankel singular values of the phase system corresponding ## to the @var{alpha}-stable part of the original system @var{G}. ## The @var{ns} Hankel singular values are ordered decreasingly. ## @item info.nu ## The order of the @var{alpha}-unstable subsystem of both the original ## system @var{G} and the reduced-order system @var{Gr}. ## @item info.nr ## The order of the obtained reduced order system @var{Gr}. ## @end table ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'nr' ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is the sum of NU and the number of ## Hankel singular values greater than @code{MAX(TOL1,NS*EPS)}; ## @var{nr} can be further reduced to ensure that ## @code{HSV(NR-NU) > HSV(NR+1-NU)}. ## ## @item 'method' ## Approximation method for the H-infinity norm. ## Valid values corresponding to this key are: ## @table @var ## @item 'sr-bta', 'b' ## Use the square-root Balance & Truncate method. ## @item 'bfsr-bta', 'f' ## Use the balancing-free square-root Balance & Truncate method. Default method. ## @item 'sr-spa', 's' ## Use the square-root Singular Perturbation Approximation method. ## @item 'bfsr-spa', 'p' ## Use the balancing-free square-root Singular Perturbation Approximation method. ## @end table ## ## @item 'alpha' ## Specifies the ALPHA-stability boundary for the eigenvalues ## of the state dynamics matrix @var{G.A}. For a continuous-time ## system, ALPHA <= 0 is the boundary value for ## the real parts of eigenvalues, while for a discrete-time ## system, 0 <= ALPHA <= 1 represents the ## boundary value for the moduli of eigenvalues. ## The ALPHA-stability domain does not include the boundary. ## Default value is 0 for continuous-time systems and ## 1 for discrete-time systems. ## ## @item 'beta' ## Use @code{[G, beta*I]} as new system @var{G} to combine ## absolute and relative error methods. ## BETA > 0 specifies the absolute/relative error weighting ## parameter. A large positive value of BETA favours the ## minimization of the absolute approximation error, while a ## small value of BETA is appropriate for the minimization ## of the relative error. ## BETA = 0 means a pure relative error method and can be ## used only if rank(G.D) = rows(G.D) which means that ## the feedthrough matrice must not be rank-deficient. ## Default value is 0. ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of reduced system. ## For model reduction, the recommended value of @var{tol1} lies ## in the interval [0.00001, 0.001]. @var{tol1} < 1. ## If @var{tol1} <= 0 on entry, the used default value is ## @var{tol1} = NS*EPS, where NS is the number of ## ALPHA-stable eigenvalues of A and EPS is the machine ## precision. ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the phase system (see METHOD) corresponding ## to the ALPHA-stable part of the given system. ## The recommended value is TOL2 = NS*EPS. TOL2 <= TOL1 < 1. ## This value is used by default if @var{'tol2'} is not specified ## or if TOL2 <= 0 on entry. ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on system @var{G} prior to order reduction. ## Default value is true if @code{G.scaled == false} and ## false if @code{G.scaled == true}. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## ## BST is often suitable to perform model reduction in order to obtain ## low order design models for controller synthesis. ## ## Approximation Properties: ## @itemize @bullet ## @item ## Guaranteed stability of reduced models ## @item ## Approximates simultaneously gain and phase ## @item ## Preserves non-minimum phase zeros ## @item ## Guaranteed a priori error bound ## @iftex ## @tex ## $$ || G^{-1} (G-G_r) ||_{\\infty} \\leq 2 \\sum_{j=r+1}^{n} {1+\\sigma_j \\over 1-\\sigma_j} - 1 $$ ## @end tex ## @end iftex ## @end itemize ## ## @strong{Algorithm}@* ## Uses SLICOT AB09HD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2011 ## Version: 0.1 function [Gr, info] = bstmodred (G, varargin) if (nargin == 0) print_usage (); endif if (! isa (G, "lti")) error ("bstmodred: first argument must be an LTI system"); endif if (nargin > 1) # bstmodred (G, ...) if (is_real_scalar (varargin{1})) # bstmodred (G, nr) varargin = horzcat (varargin(2:end), {"order"}, varargin(1)); endif if (isstruct (varargin{1})) # bstmodred (G, opt, ...), bstmodred (G, nr, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif ## order placed at the end such that nr from bstmodred (G, nr, ...) ## and bstmodred (G, nr, opt, ...) overrides possible nr's from ## key/value-pairs and inside opt struct (later keys override former keys, ## nr > key/value > opt) endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("bstmodred: keys and values must come in pairs"); endif [a, b, c, d, tsam, scaled] = ssdata (G); dt = isdt (G); ## default arguments alpha = __modred_default_alpha__ (dt); beta = 0; tol1 = 0; tol2 = 0; ordsel = 1; nr = 0; job = 1; ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case {"order", "nr"} [nr, ordsel] = __modred_check_order__ (val, rows (a)); case "tol1" tol1 = __modred_check_tol__ (val, "tol1"); case "tol2" tol2 = __modred_check_tol__ (val, "tol2"); case "alpha" alpha = __modred_check_alpha__ (val, dt); case "beta" if (! issample (val, 0)) error ("bstmodred: argument %s must be BETA >= 0", varargin{k}); endif beta = val; case "method" # approximation method switch (tolower (val)) case {"sr-bta", "b"} # 'B': use the square-root Balance & Truncate method job = 0; case {"bfsr-bta", "f"} # 'F': use the balancing-free square-root Balance & Truncate method job = 1; case {"sr-spa", "s"} # 'S': use the square-root Singular Perturbation Approximation method job = 2; case {"bfsr-spa", "p"} # 'P': use the balancing-free square-root Singular Perturbation Approximation method job = 3; otherwise error ("bstmodred: '%s' is an invalid approximation method", val); endswitch case {"equil", "equilibrate", "equilibration", "scale", "scaling"} scaled = __modred_check_equil__ (val); otherwise warning ("bstmodred: invalid property name '%s' ignored", key); endswitch endfor ## perform model order reduction [ar, br, cr, dr, nr, hsv, ns] = __sl_ab09hd__ (a, b, c, d, dt, scaled, job, nr, ordsel, alpha, beta, ... tol1, tol2); ## assemble reduced order model Gr = ss (ar, br, cr, dr, tsam); ## assemble info struct n = rows (a); nu = n - ns; info = struct ("n", n, "ns", ns, "hsv", hsv, "nu", nu, "nr", nr); endfunction %!shared Mo, Me, Info, HSVe %! A = [ -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 %! -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 %! 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 %! 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 %! 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 %! 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 ]; %! %! B = [ 0.0000 0.0000 %! 12.500 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 12.500 %! 0.0000 0.0000 ]; %! %! C = [ 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 ]; %! %! D = [ 0.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 ]; %! %! G = ss (A, B, C, D, "scaled", true); %! %! [Gr, Info] = bstmodred (G, "beta", 1.0, "tol1", 0.1, "tol2", 0.0); %! [Ao, Bo, Co, Do] = ssdata (Gr); %! %! Ae = [ 1.2729 0.0000 6.5947 0.0000 -3.4229 %! 0.0000 0.8169 0.0000 2.4821 0.0000 %! -2.9889 0.0000 -2.9028 0.0000 -0.3692 %! 0.0000 -3.3921 0.0000 -3.1126 0.0000 %! -1.4767 0.0000 -2.0339 0.0000 -0.6107 ]; %! %! Be = [ 0.1331 -0.1331 %! -0.0862 -0.0862 %! -2.6777 2.6777 %! -3.5767 -3.5767 %! -2.3033 2.3033 ]; %! %! Ce = [ -0.6907 -0.6882 0.0779 0.0958 -0.0038 %! 0.0676 0.0000 0.6532 0.0000 -0.7522 %! 0.6907 -0.6882 -0.0779 0.0958 0.0038 ]; %! %! De = [ 0.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 ]; %! %! HSVe = [ 0.8803 0.8506 0.8038 0.4494 0.3973 0.0214 0.0209 ].'; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); %!assert (Info.hsv, HSVe, 1e-4); control/inst/btaconred.m0000644000076500000240000002110312273411257014635 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @var{opt}, @dots{}) ## ## Controller reduction by frequency-weighted Balanced Truncation Approximation (BTA). ## Given a plant @var{G} and a stabilizing controller @var{K}, determine a reduced ## order controller @var{Kr} such that the closed-loop system is stable and closed-loop ## performance is retained. ## ## The algorithm tries to minimize the frequency-weighted error ## @iftex ## @tex ## $$ || V \\ (K - K_r) \\ W ||_{\\infty} = \\min $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ||V (K-Kr) W|| = min ## inf ## @end example ## @end ifnottex ## where @var{V} and @var{W} denote output and input weightings. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model of the plant. ## It has m inputs, p outputs and n states. ## @item K ## @acronym{LTI} model of the controller. ## It has p inputs, m outputs and nc states. ## @item ncr ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Kr ## State-space model of reduced order controller. ## @item info ## Struct containing additional information. ## @table @var ## @item info.ncr ## The order of the obtained reduced order controller @var{Kr}. ## @item info.ncs ## The order of the alpha-stable part of original controller @var{K}. ## @item info.hsvc ## The Hankel singular values of the alpha-stable part of @var{K}. ## The @var{ncs} Hankel singular values are ordered decreasingly. ## @end table ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'ncr' ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically such that states with ## Hankel singular values @var{info.hsvc} > @var{tol1} are retained. ## ## @item 'method' ## Order reduction approach to be used as follows: ## @table @var ## @item 'sr', 'b' ## Use the square-root Balance & Truncate method. ## @item 'bfsr', 'f' ## Use the balancing-free square-root Balance & Truncate method. Default method. ## @end table ## ## @item 'weight' ## Specifies the type of frequency-weighting as follows: ## @table @var ## @item 'none' ## No weightings are used (V = I, W = I). ## ## @item 'left', 'output' ## Use stability enforcing left (output) weighting ## @iftex ## @tex ## $$ V = (I - G K)^{-1} G, \\qquad W = I $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 ## V = (I-G*K) *G , W = I ## @end example ## @end ifnottex ## ## @item 'right', 'input' ## Use stability enforcing right (input) weighting ## @iftex ## @tex ## $$ V = I, \\qquad W = (I - G K)^{-1} G $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 ## V = I , W = (I-G*K) *G ## @end example ## @end ifnottex ## ## @item 'both', 'performance' ## Use stability and performance enforcing weightings ## @iftex ## @tex ## $$ V = (I - G K)^{-1} G, \\qquad W = (I - G K)^{-1} $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 -1 ## V = (I-G*K) *G , W = (I-G*K) ## @end example ## @end ifnottex ## Default value. ## @end table ## ## @item 'feedback' ## Specifies whether @var{K} is a positive or negative feedback controller: ## @table @var ## @item '+' ## Use positive feedback controller. Default value. ## @item '-' ## Use negative feedback controller. ## @end table ## ## @item 'alpha' ## Specifies the ALPHA-stability boundary for the eigenvalues ## of the state dynamics matrix @var{K.A}. For a continuous-time ## controller, ALPHA <= 0 is the boundary value for ## the real parts of eigenvalues, while for a discrete-time ## controller, 0 <= ALPHA <= 1 represents the ## boundary value for the moduli of eigenvalues. ## The ALPHA-stability domain does not include the boundary. ## Default value is 0 for continuous-time controllers and ## 1 for discrete-time controllers. ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced controller. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsvc(1), where c lies in the interval [0.00001, 0.001]. ## Default value is info.ncs*eps*info.hsvc(1). ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the ALPHA-stable part of the given ## controller. TOL2 <= TOL1. ## If not specified, ncs*eps*info.hsvc(1) is chosen. ## ## @item 'gram-ctrb' ## Specifies the choice of frequency-weighted controllability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to standard Enns' method [1]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified Enns' method of [2]. ## @end table ## ## @item 'gram-obsv' ## Specifies the choice of frequency-weighted observability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to standard Enns' method [1]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified Enns' method of [2]. ## @end table ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on @var{G} and @var{K} prior to order reduction. ## Default value is false if both @code{G.scaled == true, K.scaled == true} ## and true otherwise. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT SB16AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function [Kr, info] = btaconred (varargin) [Kr, info] = __conred_sb16ad__ ("bta", varargin{:}); endfunction %!shared Mo, Me, Info, HSVCe %! A = [ -1. 0. 4. %! 0. 2. 0. %! 0. 0. -3. ]; %! %! B = [ 1. %! 1. %! 1. ]; %! %! C = [ 1. 1. 1. ]; %! %! D = [ 0. ]; %! %! G = ss (A, B, C, D, "scaled", true); %! %! AC = [ -26.4000, 6.4023, 4.3868; %! 32.0000, 0, 0; %! 0, 8.0000, 0 ]; %! %! BC = [ -16 %! 0 %! 0 ]; %! %! CC = [ 9.2994 1.1624 0.1090 ]; %! %! DC = [ 0 ]; %! %! K = ss (AC, BC, CC, DC, "scaled", true); %! %! [Kr, Info] = btaconred (G, K, 2, "weight", "input", "feedback", "+"); %! [Ao, Bo, Co, Do] = ssdata (Kr); %! %! Ae = [ 9.1900 0.0000 %! 0.0000 -34.5297 ]; %! %! Be = [ -11.9593 %! 86.3137 ]; %! %! Ce = [ 2.8955 -1.3566 ]; %! %! De = [ 0.0000 ]; %! %! HSVCe = [ 3.8253 0.2005 ].'; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); %!assert (Info.hsvc, HSVCe, 1e-4); control/inst/btamodred.m0000644000076500000240000002373112273411257014646 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @var{opt}, @dots{}) ## ## Model order reduction by frequency weighted Balanced Truncation Approximation (BTA) method. ## The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order ## @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} ## approximates the one from original system @var{G}. ## ## BTA is an absolute error method which tries to minimize ## @iftex ## @tex ## $$ || G - G_r ||_{\\infty} = \\min $$ ## $$ || V \\ (G - G_r) \\ W ||_{\\infty} = \\min $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ||G-Gr|| = min ## inf ## ## ||V (G-Gr) W|| = min ## inf ## @end example ## @end ifnottex ## where @var{V} and @var{W} denote output and input weightings. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model to be reduced. ## @item nr ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Gr ## Reduced order state-space model. ## @item info ## Struct containing additional information. ## @table @var ## @item info.n ## The order of the original system @var{G}. ## @item info.ns ## The order of the @var{alpha}-stable subsystem of the original system @var{G}. ## @item info.hsv ## The Hankel singular values of the @var{alpha}-stable part of ## the original system @var{G}, ordered decreasingly. ## @item info.nu ## The order of the @var{alpha}-unstable subsystem of both the original ## system @var{G} and the reduced-order system @var{Gr}. ## @item info.nr ## The order of the obtained reduced order system @var{Gr}. ## @end table ## @end table ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'nr' ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically such that states with ## Hankel singular values @var{info.hsv} > @var{tol1} are retained. ## ## @item 'left', 'output' ## @acronym{LTI} model of the left/output frequency weighting @var{V}. ## Default value is an identity matrix. ## ## @item 'right', 'input' ## @acronym{LTI} model of the right/input frequency weighting @var{W}. ## Default value is an identity matrix. ## ## @item 'method' ## Approximation method for the L-infinity norm to be used as follows: ## @table @var ## @item 'sr', 'b' ## Use the square-root Balance & Truncate method. ## @item 'bfsr', 'f' ## Use the balancing-free square-root Balance & Truncate method. Default method. ## @end table ## ## @item 'alpha' ## Specifies the ALPHA-stability boundary for the eigenvalues ## of the state dynamics matrix @var{G.A}. For a continuous-time ## system, ALPHA <= 0 is the boundary value for ## the real parts of eigenvalues, while for a discrete-time ## system, 0 <= ALPHA <= 1 represents the ## boundary value for the moduli of eigenvalues. ## The ALPHA-stability domain does not include the boundary. ## Default value is 0 for continuous-time systems and ## 1 for discrete-time systems. ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced model. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. ## Default value is info.ns*eps*info.hsv(1). ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the ALPHA-stable part of the given ## model. TOL2 <= TOL1. ## If not specified, ns*eps*info.hsv(1) is chosen. ## ## @item 'gram-ctrb' ## Specifies the choice of frequency-weighted controllability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to a combination method [4] ## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified combination method of [4]. ## @end table ## ## @item 'gram-obsv' ## Specifies the choice of frequency-weighted observability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to a combination method [4] ## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified combination method of [4]. ## @end table ## ## @item 'alpha-ctrb' ## Combination method parameter for defining the ## frequency-weighted controllability Grammian. ## abs(alphac) <= 1. ## If alphac = 0, the choice of ## Grammian corresponds to the method of Enns [1], while if ## alphac = 1, the choice of Grammian corresponds ## to the method of Lin and Chiu [2,3]. ## Default value is 0. ## ## @item 'alpha-obsv' ## Combination method parameter for defining the ## frequency-weighted observability Grammian. ## abs(alphao) <= 1. ## If alphao = 0, the choice of ## Grammian corresponds to the method of Enns [1], while if ## alphao = 1, the choice of Grammian corresponds ## to the method of Lin and Chiu [2,3]. ## Default value is 0. ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on system @var{G} prior to order reduction. ## This is done by state transformations. ## Default value is true if @code{G.scaled == false} and ## false if @code{G.scaled == true}. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## ## Approximation Properties: ## @itemize @bullet ## @item ## Guaranteed stability of reduced models ## @item ## Lower guaranteed error bound ## @item ## Guaranteed a priori error bound ## @iftex ## @tex ## $$ \\sigma_{r+1} \\leq || (G-G_r) ||_{\\infty} \\leq 2 \\sum_{j=r+1}^{n} \\sigma_j $$ ## @end tex ## @end iftex ## @end itemize ## ## ## @strong{References}@* ## [1] Enns, D. ## @cite{Model reduction with balanced realizations: An error bound ## and a frequency weighted generalization}. ## Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984. ## ## [2] Lin, C.-A. and Chiu, T.-Y. ## @cite{Model reduction via frequency-weighted balanced realization}. ## Control Theory and Advanced Technology, vol. 8, ## pp. 341-351, 1992. ## ## [3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. ## @cite{New results on frequency weighted balanced reduction ## technique}. ## Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995. ## ## [4] Varga, A. and Anderson, B.D.O. ## @cite{Square-root balancing-free methods for the frequency-weighted ## balancing related model reduction}. ## (report in preparation) ## ## ## @strong{Algorithm}@* ## Uses SLICOT AB09ID by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function [Gr, info] = btamodred (varargin) [Gr, info] = __modred_ab09id__ ("bta", varargin{:}); endfunction %!shared Mo, Me, Info, HSVe %! A = [ -26.4000, 6.4023, 4.3868; %! 32.0000, 0, 0; %! 0, 8.0000, 0 ]; %! %! B = [ 16 %! 0 %! 0 ]; %! %! C = [ 9.2994 1.1624 0.1090 ]; %! %! D = [ 0 ]; %! %! G = ss (A, B, C, D); % "scaled", false %! %! AV = [ -1.0000, 0, 4.0000, -9.2994, -1.1624, -0.1090; %! 0, 2.0000, 0, -9.2994, -1.1624, -0.1090; %! 0, 0, -3.0000, -9.2994, -1.1624, -0.1090; %! 16.0000, 16.0000, 16.0000, -26.4000, 6.4023, 4.3868; %! 0, 0, 0, 32.0000, 0, 0; %! 0, 0, 0, 0, 8.0000, 0 ]; %! %! BV = [ 1 %! 1 %! 1 %! 0 %! 0 %! 0 ]; %! %! CV = [ 1 1 1 0 0 0 ]; %! %! DV = [ 0 ]; %! %! V = ss (AV, BV, CV, DV); %! %! [Gr, Info] = btamodred (G, 2, "left", V); %! [Ao, Bo, Co, Do] = ssdata (Gr); %! %! Ae = [ 9.1900 0.0000 %! 0.0000 -34.5297 ]; %! %! Be = [ 11.9593 %! 16.9329 ]; %! %! Ce = [ 2.8955 6.9152 ]; %! %! De = [ 0.0000 ]; %! %! HSVe = [ 3.8253 0.2005 ].'; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); %!assert (Info.hsv, HSVe, 1e-4); control/inst/care.m0000644000076500000240000001501012273411257013606 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) ## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} care (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) ## Solve continuous-time algebraic Riccati equation (ARE). ## ## @strong{Inputs} ## @table @var ## @item a ## Real matrix (n-by-n). ## @item b ## Real matrix (n-by-m). ## @item q ## Real matrix (n-by-n). ## @item r ## Real matrix (m-by-m). ## @item s ## Optional real matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. ## @item e ## Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. ## @end table ## ## @strong{Outputs} ## @table @var ## @item x ## Unique stabilizing solution of the continuous-time Riccati equation (n-by-n). ## @item l ## Closed-loop poles (n-by-1). ## @item g ## Corresponding gain matrix (m-by-n). ## @end table ## ## @strong{Equations} ## @example ## @group ## -1 ## A'X + XA - XB R B'X + Q = 0 ## ## -1 ## A'X + XA - (XB + S) R (B'X + S') + Q = 0 ## ## -1 ## G = R B'X ## ## -1 ## G = R (B'X + S') ## ## L = eig (A - B*G) ## @end group ## @end example ## @example ## @group ## -1 ## A'XE + E'XA - E'XB R B'XE + Q = 0 ## ## -1 ## A'XE + E'XA - (E'XB + S) R (B'XE + S') + Q = 0 ## ## -1 ## G = R B'XE ## ## -1 ## G = R (B'XE + S) ## ## L = eig (A - B*G, E) ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB02OD and SG02AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{dare, lqr, dlqr, kalman} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.5.1 function [x, l, g] = care (a, b, q, r, s = [], e = []) ## TODO: extract feedback matrix g from SB02OD (and SG02AD) if (nargin < 4 || nargin > 6) print_usage (); endif if (! is_real_square_matrix (a, q, r)) ## error ("care: a, q, r must be real and square"); error ("care: %s, %s, %s must be real and square", ... inputname (1), inputname (3), inputname (4)); endif if (! is_real_matrix (b) || rows (a) != rows (b)) ## error ("care: a and b must have the same number of rows"); error ("care: %s and %s must have the same number of rows", ... inputname (1), inputname (2)); endif if (columns (r) != columns (b)) ## error ("care: b and r must have the same number of columns"); error ("care: %s and %s must have the same number of columns", ... inputname (2), inputname (4)); endif if (! is_real_matrix (s) && ! size_equal (s, b)) ## error ("care: s(%dx%d) must be real and identically dimensioned with b(%dx%d)", ## rows (s), columns (s), rows (b), columns (b)); error ("care: %s(%dx%d) must be real and identically dimensioned with %s(%dx%d)", ... inputname (5), rows (s), columns (s), inputname (2), rows (b), columns (b)); endif if (! isempty (e) && (! is_real_square_matrix (e) || ! size_equal (e, a))) ## error ("care: a and e must have the same number of rows"); error ("care: %s and %s must have the same number of rows", ... inputname (1), inputname (6)); endif ## check stabilizability if (! isstabilizable (a, b, e, [], 0)) ## error ("care: (a, b) not stabilizable"); error ("care: (%s, %s) not stabilizable", ... inputname (1), inputname (2)); endif ## check positive semi-definiteness if (isempty (s)) t = zeros (size (b)); else t = s; endif m = [q, t; t.', r]; if (isdefinite (m) < 0) ## error ("care: require [q, s; s.', r] >= 0"); error ("care: require [%s, %s; %s.', %s] >= 0", ... inputname (3), inputname (5), inputname (5), inputname (4)); endif ## solve the riccati equation if (isempty (e)) if (isempty (s)) [x, l] = __sl_sb02od__ (a, b, q, r, b, false, false); g = r \ (b.'*x); # gain matrix else [x, l] = __sl_sb02od__ (a, b, q, r, s, false, true); g = r \ (b.'*x + s.'); # gain matrix endif else if (isempty (s)) [x, l] = __sl_sg02ad__ (a, e, b, q, r, b, false, false); g = r \ (b.'*x*e); # gain matrix else [x, l] = __sl_sg02ad__ (a, e, b, q, r, s, false, true); g = r \ (b.'*x*e + s.'); # gain matrix endif endif endfunction %!shared x, l, g, xe, le, ge %! a = [-3 2 %! 1 1]; %! %! b = [ 0 %! 1]; %! %! c = [ 1 -1]; %! %! r = 3; %! %! [x, l, g] = care (a, b, c.'*c, r); %! %! xe = [ 0.5895 1.8216 %! 1.8216 8.8188]; %! %! le = [-3.5026 %! -1.4370]; %! %! ge = [ 0.6072 2.9396]; %! %!assert (x, xe, 1e-4); %!assert (l, le, 1e-4); %!assert (g, ge, 1e-4); %!shared x, l, g, xe, le, ge %! a = [ 0.0 1.0 %! 0.0 0.0]; %! %! b = [ 0.0 %! 1.0]; %! %! c = [ 1.0 0.0 %! 0.0 1.0 %! 0.0 0.0]; %! %! d = [ 0.0 %! 0.0 %! 1.0]; %! %! [x, l, g] = care (a, b, c.'*c, d.'*d); %! %! xe = [ 1.7321 1.0000 %! 1.0000 1.7321]; %! %! le = [-0.8660 + 0.5000i %! -0.8660 - 0.5000i]; %! %! ge = [ 1.0000 1.7321]; %! %!assert (x, xe, 1e-4); %!assert (l, le, 1e-4); %!assert (g, ge, 1e-4); %!shared x, xe %! a = [ 0.0 1.0 %! 0.0 0.0 ]; %! %! e = [ 1.0 0.0 %! 0.0 1.0 ]; %! %! b = [ 0.0 %! 1.0 ]; %! %! c = [ 1.0 0.0 %! 0.0 1.0 %! 0.0 0.0 ]; %! %! d = [ 0.0 %! 0.0 %! 1.0 ]; %! %! x = care (a, b, c.'*c, d.'*d, [], e); %! %! xe = [ 1.7321 1.0000 %! 1.0000 1.7321 ]; %! %!assert (x, xe, 1e-4); control/inst/cfconred.m0000644000076500000240000002667312273411257014500 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @var{opt}, @dots{}) ## ## Reduction of state-feedback-observer based controller by coprime factorization (CF). ## Given a plant @var{G}, state feedback gain @var{F} and full observer gain @var{L}, ## determine a reduced order controller @var{Kr}. ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model of the open-loop plant (A,B,C,D). ## It has m inputs, p outputs and n states. ## @item F ## Stabilizing state feedback matrix (m-by-n). ## @item L ## Stabilizing observer gain matrix (n-by-p). ## @item ncr ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Kr ## State-space model of reduced order controller. ## @item info ## Struct containing additional information. ## @table @var ## @item info.hsv ## The Hankel singular values of the extended system?!?. ## The @var{n} Hankel singular values are ordered decreasingly. ## @item info.ncr ## The order of the obtained reduced order controller @var{Kr}. ## @end table ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'ncr' ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically such that states with ## Hankel singular values @var{info.hsv} > @var{tol1} are retained. ## ## @item 'method' ## Order reduction approach to be used as follows: ## @table @var ## @item 'sr-bta', 'b' ## Use the square-root Balance & Truncate method. ## @item 'bfsr-bta', 'f' ## Use the balancing-free square-root Balance & Truncate method. Default method. ## @item 'sr-spa', 's' ## Use the square-root Singular Perturbation Approximation method. ## @item 'bfsr-spa', 'p' ## Use the balancing-free square-root Singular Perturbation Approximation method. ## @end table ## ## @item 'cf' ## Specifies whether left or right coprime factorization is ## to be used as follows: ## @table @var ## @item 'left', 'l' ## Use left coprime factorization. Default method. ## @item 'right', 'r' ## Use right coprime factorization. ## @end table ## ## @item 'feedback' ## Specifies whether @var{F} and @var{L} are fed back positively or negatively: ## @table @var ## @item '+' ## A+BK and A+LC are both Hurwitz matrices. ## @item '-' ## A-BK and A-LC are both Hurwitz matrices. Default value. ## @end table ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced system. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. ## Default value is n*eps*info.hsv(1). ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the coprime factorization controller. ## TOL2 <= TOL1. ## If not specified, n*eps*info.hsv(1) is chosen. ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on system @var{G} prior to order reduction. ## Default value is true if @code{G.scaled == false} and ## false if @code{G.scaled == true}. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT SB16BD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function [Kr, info] = cfconred (G, F, L, varargin) if (nargin < 3) print_usage (); endif if (! isa (G, "lti")) error ("cfconred: first argument must be an LTI system"); endif if (! is_real_matrix (F)) error ("cfconred: second argument must be a real matrix"); endif if (! is_real_matrix (L)) error ("cfconred: third argument must be a real matrix"); endif if (nargin > 3) # cfconred (G, F, L, ...) if (is_real_scalar (varargin{1})) # cfconred (G, F, L, nr) varargin = horzcat (varargin(2:end), {"order"}, varargin(1)); endif if (isstruct (varargin{1})) # cfconred (G, F, L, opt, ...), cfconred (G, F, L, nr, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif ## order placed at the end such that nr from cfconred (G, F, L, nr, ...) ## and cfconred (G, F, L, nr, opt, ...) overrides possible nr's from ## key/value-pairs and inside opt struct (later keys override former keys, ## nr > key/value > opt) endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("cfconred: keys and values must come in pairs"); endif [a, b, c, d, tsam, scaled] = ssdata (G); [p, m] = size (G); n = rows (a); [mf, nf] = size (F); [nl, pl] = size (L); dt = isdt (G); jobd = any (d(:)); if (mf != m || nf != n) error ("cfconred: dimensions of state-feedback matrix (%dx%d) and plant (%dx%d, %d states) don't match", ... mf, nf, p, m, n); endif if (nl != n || pl != p) error ("cfconred: dimensions of observer matrix (%dx%d) and plant (%dx%d, %d states) don't match", ... nl, pl, p, m, n); endif ## default arguments tol1 = 0.0; tol2 = 0.0; jobcf = 0; jobmr = 2; # balancing-free BTA equil = scaled; # equil: 0 means "S", 1 means "N" ordsel = 1; ncr = 0; negfb = true; # A-BK, A-LC Hurwitz ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case {"order", "ncr", "nr"} [ncr, ordsel] = __modred_check_order__ (val, n); case "tol1" tol1 = __modred_check_tol__ (val, "tol1"); case "tol2" tol2 = __modred_check_tol__ (val, "tol2"); case "cf" switch (lower (val(1))) case "l" jobcf = 0; case "r" jobcf = 1; otherwise error ("cfconred: '%s' is an invalid coprime factorization", val); endswitch case "method" # approximation method switch (tolower (val)) case {"sr-bta", "b"} # 'B': use the square-root Balance & Truncate method jobmr = 0; case {"bfsr-bta", "f"} # 'F': use the balancing-free square-root Balance & Truncate method jobmr = 1; case {"sr-spa", "s"} # 'S': use the square-root Singular Perturbation Approximation method jobmr = 2; case {"bfsr-spa", "p"} # 'P': use the balancing-free square-root Singular Perturbation Approximation method jobmr = 3; otherwise error ("cfconred: '%s' is an invalid approach", val); endswitch case {"equil", "equilibrate", "equilibration", "scale", "scaling"} equil = __modred_check_equil__ (val); case "feedback" negfb = __conred_check_feedback_sign__ (val); otherwise warning ("cfconred: invalid property name '%s' ignored", key); endswitch endfor ## A - B*F --> A + B*F ; A - L*C --> A + L*C if (negfb) F = -F; L = -L; endif ## perform model order reduction [acr, bcr, ccr, dcr, ncr, hsv] = __sl_sb16bd__ (a, b, c, d, dt, equil, ncr, ordsel, jobd, jobmr, ... F, L, jobcf, tol1, tol2); ## assemble reduced order controller Kr = ss (acr, bcr, ccr, dcr, tsam); ## assemble info struct info = struct ("ncr", ncr, "hsv", hsv); endfunction %!shared Mo, Me, Info, HSVe %! A = [ 0 1.0000 0 0 0 0 0 0 %! 0 0 0 0 0 0 0 0 %! 0 0 -0.0150 0.7650 0 0 0 0 %! 0 0 -0.7650 -0.0150 0 0 0 0 %! 0 0 0 0 -0.0280 1.4100 0 0 %! 0 0 0 0 -1.4100 -0.0280 0 0 %! 0 0 0 0 0 0 -0.0400 1.850 %! 0 0 0 0 0 0 -1.8500 -0.040 ]; %! %! B = [ 0.0260 %! -0.2510 %! 0.0330 %! -0.8860 %! -4.0170 %! 0.1450 %! 3.6040 %! 0.2800 ]; %! %! C = [ -.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026 ]; %! %! D = [ 0.0 ]; %! %! G = ss (A, B, C, D); % "scaled", false %! %! F = [ 4.4721e-002 6.6105e-001 4.6986e-003 3.6014e-001 1.0325e-001 -3.7541e-002 -4.2685e-002 3.2873e-002 ]; %! %! L = [ 4.1089e-001 %! 8.6846e-002 %! 3.8523e-004 %! -3.6194e-003 %! -8.8037e-003 %! 8.4205e-003 %! 1.2349e-003 %! 4.2632e-003 ]; %! %! [Kr, Info] = cfconred (G, F, L, 4, "method", "bfsr-bta", "cf", "left", "feedback", "+"); %! [Ao, Bo, Co, Do] = ssdata (Kr); %! %! Ae = [ 0.5946 -0.7336 0.1914 -0.3368 %! 0.5960 -0.0184 -0.1088 0.0207 %! 1.2253 0.2043 0.1009 -1.4948 %! -0.0330 -0.0243 1.3440 0.0035 ]; %! %! Be = [ 0.0015 %! -0.0202 %! 0.0159 %! -0.0544 ]; %! %! Ce = [ 0.3534 0.0274 0.0337 -0.0320 ]; %! %! De = [ 0.0000 ]; %! %! HSVe = [ 4.9078 4.8745 3.8455 3.7811 1.2289 1.1785 0.5176 0.1148 ].'; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); %!assert (Info.hsv, HSVe, 1e-4); control/inst/covar.m0000644000076500000240000000434512273411257014017 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{p}, @var{q}] =} covar (@var{sys}, @var{w}) ## Return the steady-state covariance. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item w ## Intensity of Gaussian white noise inputs which drive @var{sys}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item p ## Output covariance. ## @item q ## State covariance. ## @end table ## ## @seealso{lyap, dlyap} ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.1 function [p, q] = covar (sys, w) if (nargin != 2) print_usage (); endif if (! isa (sys, "lti")) error ("covar: first argument must be an LTI model"); endif if (! isstable (sys)) error ("covar: system must be stable"); endif [a, b, c, d] = ssdata (sys); if (isct (sys)) if (any (d(:))) error ("covar: system is not strictly proper"); endif q = lyap (a, b*w*b.'); p = c*q*c.'; else q = dlyap (a, b*w*b.'); p = c*q*c.' + d*w*d.'; endif endfunction ## continuous-time %!shared p, q, p_exp, q_exp %! sys = ss (-1, 1, 1, 0); %! [p, q] = covar (sys, 5); %! p_exp = 2.5000; %! q_exp = 2.5000; %!assert (p, p_exp, 1e-4); %!assert (q, q_exp, 1e-4); ## discrete-time %!shared p, q, p_exp, q_exp %! sys = ss ([-0.2, -0.5; 1, 0], [2; 0], [1, 0.5], [0], 0.1); %! [p, q] = covar (sys, 5); %! p_exp = 30.3167; %! q_exp = [27.1493, -3.6199; -3.6199, 27.1493]; %!assert (p, p_exp, 1e-4); %!assert (q, q_exp, 1e-4); control/inst/ctrb.m0000644000076500000240000000427112273411257013635 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## Copyright (C) 2009 Luca Favatella ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{co} =} ctrb (@var{sys}) ## @deftypefnx {Function File} {@var{co} =} ctrb (@var{a}, @var{b}) ## Return controllability matrix. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item a ## State matrix (n-by-n). ## @item b ## Input matrix (n-by-m). ## @end table ## ## @strong{Outputs} ## @table @var ## @item co ## Controllability matrix. ## @end table ## ## @strong{Equation} ## @iftex ## @tex ## $$ C_o = [ B \\ \\ AB \\ \\ A^2B \\ \\ldots \\ A^{n-1}B ] $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## 2 n-1 ## Co = [ B AB A B ... A B ] ## @end example ## @end ifnottex ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function co = ctrb (a, b) if (nargin == 1) # ctrb (sys) if (! isa (a, "lti")) error ("ctrb: argument must be an lti system"); endif [a, b] = ssdata (a); elseif (nargin == 2) # ctrb (a, b) if (! is_real_square_matrix (a) || ! is_real_matrix (b) || rows (a) != rows (b)) error ("ctrb: invalid arguments (a, b)"); endif else print_usage (); endif n = rows (a); # number of states k = 0:n-1; # exponents for a tmp = arrayfun (@(x) a^x*b, k, "uniformoutput", false); co = horzcat (tmp{:}); endfunction %!assert (ctrb ([1, 0; 0, -0.5], [8; 8]), [8, 8; 8, -4]); control/inst/ctrbf.m0000644000076500000240000000713712273411257014007 0ustar lukasstaff## Copyright (C) 2010 Benjamin Fernandez ## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys}) ## @deftypefnx{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys}, @var{tol}) ## @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C}) ## @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C}, @var{TOL}) ## If Co=ctrb(A,B) has rank r <= n = SIZE(A,1), then there is a ## similarity transformation Tc such that Tc = [t1 t2] where t1 ## is the controllable subspace and t2 is orthogonal to t1 ## ## @example ## @group ## Abar = Tc \\ A * Tc , Bbar = Tc \\ B , Cbar = C * Tc ## @end group ## @end example ## ## and the transformed system has the form ## ## @example ## @group ## | Ac A12| | Bc | ## Abar = |----------|, Bbar = | ---|, Cbar = [Cc | Cnc]. ## | 0 Anc| | 0 | ## @end group ## @end example ## ## where (Ac,Bc) is controllable, and Cc(sI-Ac)^(-1)Bc = C(sI-A)^(-1)B. ## and the system is stabilizable if Anc has no eigenvalues in ## the right half plane. The last output K is a vector of length n ## containing the number of controllable states. ## @end deftypefn ## Author: Benjamin Fernandez ## Created: 2010-04-30 ## Version: 0.1 function [ac, bc, cc, z, ncont] = ctrbf (a, b = [], c, tol = []) if (nargin < 1 || nargin > 4) print_usage (); endif islti = isa (a, "lti"); if (islti) if (nargin > 2) print_usage (); endif sys = a; tol = b; [a, b, c] = ssdata (sys); else if (nargin < 3) print_usage (); endif sys = ss (a, b, c); [a, b, c] = ssdata (sys); endif if (isempty (tol)) tol = 0; # default tolerance elseif (! is_real_scalar (tol)) error ("ctrbf: tol must be a real scalar"); endif [ac, bc, cc, z, ncont] = __sl_tb01ud__ (a, b, c, tol); if (islti) ac = set (sys, "a", ac, "b", bc, "c", cc, "scaled", false); bc = z; cc = ncont; endif endfunction %!shared Ao, Bo, Co, Zo, Ae, Be, Ce, Ze, NCONT %! A = [ -1.0 0.0 0.0 %! -2.0 -2.0 -2.0 %! -1.0 0.0 -3.0 ]; %! %! B = [ 1.0 0.0 0.0 %! 0.0 2.0 1.0 ].'; %! %! C = [ 0.0 2.0 1.0 %! 1.0 0.0 0.0 ]; %! %! [Ao, Bo, Co, Zo, NCONT] = ctrbf (A, B, C); %! %! Ae = [ -3.0000 2.2361 %! 0.0000 -1.0000 ]; %! %! Be = [ 0.0000 -2.2361 %! 1.0000 0.0000 ]; %! %! Ce = [ -2.2361 0.0000 %! 0.0000 1.0000 ]; %! %! Ze = [ 0.0000 1.0000 0.0000 %! -0.8944 0.0000 -0.4472 %! -0.4472 0.0000 0.8944 ]; %! %!assert (Ao(1:NCONT, 1:NCONT), Ae, 1e-4); %!assert (Bo(1:NCONT, :), Be, 1e-4); %!assert (Co(:, 1:NCONT), Ce, 1e-4); %!assert (Zo, Ze, 1e-4); control/inst/dare.m0000644000076500000240000001373212273411257013620 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) ## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) ## Solve discrete-time algebraic Riccati equation (ARE). ## ## @strong{Inputs} ## @table @var ## @item a ## Real matrix (n-by-n). ## @item b ## Real matrix (n-by-m). ## @item q ## Real matrix (n-by-n). ## @item r ## Real matrix (m-by-m). ## @item s ## Optional real matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. ## @item e ## Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. ## @end table ## ## @strong{Outputs} ## @table @var ## @item x ## Unique stabilizing solution of the discrete-time Riccati equation (n-by-n). ## @item l ## Closed-loop poles (n-by-1). ## @item g ## Corresponding gain matrix (m-by-n). ## @end table ## ## @strong{Equations} ## @example ## @group ## -1 ## A'XA - X - A'XB (B'XB + R) B'XA + Q = 0 ## ## -1 ## A'XA - X - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0 ## ## -1 ## G = (B'XB + R) B'XA ## ## -1 ## G = (B'XB + R) (B'XA + S') ## ## L = eig (A - B*G) ## @end group ## @end example ## @example ## @group ## -1 ## A'XA - E'XE - A'XB (B'XB + R) B'XA + Q = 0 ## ## -1 ## A'XA - E'XE - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0 ## ## -1 ## G = (B'XB + R) B'XA ## ## -1 ## G = (B'XB + R) (B'XA + S') ## ## L = eig (A - B*G, E) ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB02OD and SG02AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{care, lqr, dlqr, kalman} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.5.1 function [x, l, g] = dare (a, b, q, r, s = [], e = []) ## TODO: extract feedback matrix g from SB02OD (and SG02AD) if (nargin < 4 || nargin > 6) print_usage (); endif if (! is_real_square_matrix (a, q, r)) ## error ("dare: a, q, r must be real and square"); error ("dare: %s, %s, %s must be real and square", ... inputname (1), inputname (3), inputname (4)); endif if (! is_real_matrix (b) || rows (a) != rows (b)) ## error ("dare: a and b must have the same number of rows"); error ("dare: %s and %s must have the same number of rows", ... inputname (1), inputname (2)); endif if (columns (r) != columns (b)) ## error ("dare: b and r must have the same number of columns"); error ("dare: %s and %s must have the same number of columns", ... inputname (2), inputname (4)); endif if (! is_real_matrix (s) && ! size_equal (s, b)) ## error ("dare: s(%dx%d) must be real and identically dimensioned with b(%dx%d)", ## rows (s), columns (s), rows (b), columns (b)); error ("dare: %s(%dx%d) must be real and identically dimensioned with %s(%dx%d)", ... inputname (5), rows (s), columns (s), inputname (2), rows (b), columns (b)); endif if (! isempty (e) && (! is_real_square_matrix (e) || ! size_equal (e, a))) ## error ("dare: a and e must have the same number of rows"); error ("dare: %s and %s must have the same number of rows", ... inputname (1), inputname (6)); endif ## check stabilizability if (! isstabilizable (a, b, e, [], 1)) ## error ("dare: (a, b) not stabilizable"); error ("dare: (%s, %s) not stabilizable", ... inputname (1), inputname (2)); endif ## check positive semi-definiteness if (isempty (s)) t = zeros (size (b)); else t = s; endif m = [q, t; t.', r]; if (isdefinite (m) < 0) ## error ("dare: require [q, s; s.', r] >= 0"); error ("dare: require [%s, %s; %s.', %s] >= 0", ... inputname (3), inputname (5), inputname (5), inputname (4)); endif ## solve the riccati equation if (isempty (e)) if (isempty (s)) [x, l] = __sl_sb02od__ (a, b, q, r, b, true, false); g = (r + b.'*x*b) \ (b.'*x*a); # gain matrix else [x, l] = __sl_sb02od__ (a, b, q, r, s, true, true); g = (r + b.'*x*b) \ (b.'*x*a + s.'); # gain matrix endif else if (isempty (s)) [x, l] = __sl_sg02ad__ (a, e, b, q, r, b, true, false); g = (r + b.'*x*b) \ (b.'*x*a); # gain matrix else [x, l] = __sl_sg02ad__ (a, e, b, q, r, s, true, true); g = (r + b.'*x*b) \ (b.'*x*a + s.'); # gain matrix endif endif endfunction %!shared x, l, g, xe, le, ge %! a = [ 0.4 1.7 %! 0.9 3.8]; %! %! b = [ 0.8 %! 2.1]; %! %! c = [ 1 -1]; %! %! r = 3; %! %! [x, l, g] = dare (a, b, c.'*c, r); %! %! xe = [ 1.5354 1.2623 %! 1.2623 10.5596]; %! %! le = [-0.0022 %! 0.2454]; %! %! ge = [ 0.4092 1.7283]; %! %!assert (x, xe, 1e-4); %!assert (sort (l), sort (le), 1e-4); %!assert (g, ge, 1e-4); ## TODO: add more tests (nonempty s and/or e) control/inst/db2mag.m0000644000076500000240000000261312273411257014035 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{mag} =} db2mag (@var{db}) ## Convert Decibels (dB) to Magnitude. ## ## @strong{Inputs} ## @table @var ## @item db ## Decibel (dB) value(s). Both real-valued scalars and matrices are accepted. ## @end table ## ## @strong{Outputs} ## @table @var ## @item mag ## Magnitude value(s). ## @end table ## ## @seealso{mag2db} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2012 ## Version: 0.1 function mag = db2mag (db) if (nargin != 1 || ! is_real_matrix (db)) print_usage (); endif mag = 10.^(db./20); endfunction %!assert (db2mag (40), 100); %!assert (db2mag (-20), 0.1); control/inst/dlqe.m0000644000076500000240000000733212273411257013631 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## Copyright (C) 2012 Megan Zagrobelny ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{m}, @var{p}, @var{z}, @var{e}] =} dlqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}, @var{s}) ## Kalman filter for discrete-time systems. ## ## @example ## @group ## x[k] = Ax[k] + Bu[k] + Gw[k] (State equation) ## y[k] = Cx[k] + Du[k] + v[k] (Measurement Equation) ## E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S ## @end group ## @end example ## ## @strong{Inputs} ## @table @var ## @item a ## State transition matrix of discrete-time system (n-by-n). ## @item g ## Process noise matrix of discrete-time system (n-by-g). ## If @var{g} is empty @code{[]}, an identity matrix is assumed. ## @item c ## Measurement matrix of discrete-time system (p-by-n). ## @item q ## Process noise covariance matrix (g-by-g). ## @item r ## Measurement noise covariance matrix (p-by-p). ## @item s ## Optional cross term covariance matrix (g-by-p), s = cov(w,v). ## If @var{s} is empty @code{[]} or not specified, a zero matrix is assumed. ## @end table ## ## @strong{Outputs} ## @table @var ## @item m ## Kalman filter gain matrix (n-by-p). ## @item p ## Unique stabilizing solution of the discrete-time Riccati equation (n-by-n). ## Symmetric matrix. ## @item z ## Error covariance (n-by-n), cov(x(k|k)-x) ## @item e ## Closed-loop poles (n-by-1). ## @end table ## ## @strong{Equations} ## @example ## @group ## x[k|k] = x[k|k-1] + M(y[k] - Cx[k|k-1] - Du[k]) ## ## x[k+1|k] = Ax[k|k] + Bu[k] for S=0 ## ## x[k+1|k] = Ax[k|k] + Bu[k] + G*S*(C*P*C' + R)^-1*(y[k] - C*x[k|k-1]) for non-zero S ## ## ## E = eig(A - A*M*C) for S=0 ## ## E = eig(A - A*M*C - G*S*(C*P*C' + Rv)^-1*C) for non-zero S ## ## @end group ## @end example ## @seealso{dare, care, dlqr, lqr, lqe} ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function [m, p, z, e] = dlqe (a, g, c, q, r, s = []) if (nargin < 5 || nargin > 6) print_usage (); endif if (isempty (g)) [p, e] = dare (a.', c.', q, r, s); # dlqe (a, [], c, q, r, s), g=I elseif (columns (g) != rows (q) || ! issquare (q)) error ("dlqe: matrices g(%dx%d) and q(%dx%d) have incompatible dimensions", ... rows (g), columns (g), rows (q), columns (q)); elseif (isempty (s)) [p, e] = dare (a.', c.', g*q*g.', r); elseif (columns (g) != rows (s)) error ("dlqe: matrices g(%dx%d) and s(%dx%d) have incompatible dimensions", ... rows (g), columns (g), rows (s), columns (s)); else [p, e] = dare (a.', c.', g*q*g.', r, g*s); endif m = p*c.' / (c*p*c.' + r); z = p - m*c*p; z = (z + z.') / 2; endfunction control/inst/dlqr.m0000644000076500000240000000606512273411257013650 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{sys}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{sys}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} dlqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) ## Linear-quadratic regulator for discrete-time systems. ## ## @strong{Inputs} ## @table @var ## @item sys ## Continuous or discrete-time @acronym{LTI} model (p-by-m, n states). ## @item a ## State transition matrix of discrete-time system (n-by-n). ## @item b ## Input matrix of discrete-time system (n-by-m). ## @item q ## State weighting matrix (n-by-n). ## @item r ## Input weighting matrix (m-by-m). ## @item s ## Optional cross term matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. ## @item e ## Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. ## @end table ## ## @strong{Outputs} ## @table @var ## @item g ## State feedback matrix (m-by-n). ## @item x ## Unique stabilizing solution of the discrete-time Riccati equation (n-by-n). ## @item l ## Closed-loop poles (n-by-1). ## @end table ## ## @strong{Equations} ## @example ## @group ## x[k+1] = A x[k] + B u[k], x[0] = x0 ## ## inf ## J(x0) = SUM (x' Q x + u' R u + 2 x' S u) ## k=0 ## ## L = eig (A - B*G) ## @end group ## @end example ## @seealso{dare, care, lqr} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.2 function [g, x, l] = dlqr (a, b, q, r = [], s = [], e = []) if (nargin < 3 || nargin > 6) print_usage (); endif if (isa (a, "lti")) s = r; r = q; q = b; [a, b, c, d, e, tsam] = dssdata (a, []); elseif (nargin < 4) print_usage (); else tsam = 1; # any value > 0 could be used here endif if (issample (tsam, -1)) [x, l, g] = dare (a, b, q, r, s, e); else [x, l, g] = care (a, b, q, r, s, e); endif endfunction control/inst/dlyap.m0000644000076500000240000001155712273411257014021 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{x} =} dlyap (@var{a}, @var{b}) ## @deftypefnx{Function File} {@var{x} =} dlyap (@var{a}, @var{b}, @var{c}) ## @deftypefnx{Function File} {@var{x} =} dlyap (@var{a}, @var{b}, @var{[]}, @var{e}) ## Solve discrete-time Lyapunov or Sylvester equations. ## ## @strong{Equations} ## @example ## @group ## AXA' - X + B = 0 (Lyapunov Equation) ## ## AXB' - X + C = 0 (Sylvester Equation) ## ## AXA' - EXE' + B = 0 (Generalized Lyapunov Equation) ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB03MD, SB04QD and SG03AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{dlyapchol, lyap, lyapchol} ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.2.1 function [x, scale] = dlyap (a, b, c, e) scale = 1; switch (nargin) case 2 # Lyapunov equation if (! is_real_square_matrix (a, b)) ## error ("dlyap: a, b must be real and square"); error ("dlyap: %s, %s must be real and square", ... inputname (1), inputname (2)); endif if (rows (a) != rows (b)) ## error ("dlyap: a, b must have the same number of rows"); error ("dlyap: %s, %s must have the same number of rows", ... inputname (1), inputname (2)); endif [x, scale] = __sl_sb03md__ (a, -b, true); # AXA' - X = -B ## x /= scale; # 0 < scale <= 1 case 3 # Sylvester equation if (! is_real_square_matrix (a, b)) ## error ("dlyap: a, b must be real and square"); error ("dlyap: %s, %s must be real and square", ... inputname (1), inputname (2)); endif if (! is_real_matrix (c) || rows (c) != rows (a) || columns (c) != columns (b)) ## error ("dlyap: c must be a real (%dx%d) matrix", rows (a), columns (b)); error ("dlyap: %s must be a real (%dx%d) matrix", ... rows (a), columns (b), inputname (3)); endif x = __sl_sb04qd__ (-a, b, c); # AXB' - X = -C case 4 # generalized Lyapunov equation if (! isempty (c)) print_usage (); endif if (! is_real_square_matrix (a, b, e)) ## error ("dlyap: a, b, e must be real and square"); error ("dlyap: %s, %s, %s must be real and square", ... inputname (1), inputname (2), inputname (4)); endif if (rows (b) != rows (a) || rows (e) != rows (a)) ## error ("dlyap: a, b, e must have the same number of rows"); error ("dlyap: %s, %s, %s must have the same number of rows", ... inputname (1), inputname (2), inputname (4)); endif if (! issymmetric (b)) ## error ("dlyap: b must be symmetric"); error ("dlyap: %s must be symmetric", ... inputname (2)); endif [x, scale] = __sl_sg03ad__ (a, e, -b, true); # AXA' - EXE' = -B ## x /= scale; # 0 < scale <= 1 otherwise print_usage (); endswitch if (scale < 1) warning ("dlyap: solution scaled by %g to prevent overflow", scale); endif endfunction ## Lyapunov %!shared X, X_exp %! A = [3.0 1.0 1.0 %! 1.0 3.0 0.0 %! 0.0 0.0 3.0]; %! %! B = [25.0 24.0 15.0 %! 24.0 32.0 8.0 %! 15.0 8.0 40.0]; %! %! X = dlyap (A.', -B); %! %! X_exp = [2.0000 1.0000 1.0000 %! 1.0000 3.0000 0.0000 %! 1.0000 0.0000 4.0000]; %! %!assert (X, X_exp, 1e-4); ## Sylvester %!shared X, X_exp %! A = [1.0 2.0 3.0 %! 6.0 7.0 8.0 %! 9.0 2.0 3.0]; %! %! B = [7.0 2.0 3.0 %! 2.0 1.0 2.0 %! 3.0 4.0 1.0]; %! %! C = [271.0 135.0 147.0 %! 923.0 494.0 482.0 %! 578.0 383.0 287.0]; %! %! X = dlyap (-A, B, C); %! %! X_exp = [2.0000 3.0000 6.0000 %! 4.0000 7.0000 1.0000 %! 5.0000 3.0000 2.0000]; %! %!assert (X, X_exp, 1e-4); ## Generalized Lyapunov ## TODO: add a test control/inst/dlyapchol.m0000644000076500000240000000635112273411257014663 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{u} =} dlyapchol (@var{a}, @var{b}) ## @deftypefnx{Function File} {@var{u} =} dlyapchol (@var{a}, @var{b}, @var{e}) ## Compute Cholesky factor of discrete-time Lyapunov equations. ## ## @strong{Equations} ## @example ## @group ## A U' U A' - U' U + B B' = 0 (Lyapunov Equation) ## ## A U' U A' - E U' U E' + B B' = 0 (Generalized Lyapunov Equation) ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB03OD and SG03BD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{dlyap, lyap, lyapchol} ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.2.1 function [u, scale] = dlyapchol (a, b, e) switch (nargin) case 2 if (! is_real_square_matrix (a)) ## error ("dlyapchol: a must be real and square"); error ("dlyapchol: %s must be real and square", ... inputname (1)); endif if (! is_real_matrix (b)) ## error ("dlyapchol: b must be real") error ("dlyapchol: %s must be real", ... inputname (2)) endif if (rows (a) != rows (b)) ## error ("dlyapchol: a and b must have the same number of rows"); error ("dlyapchol: %s and %s must have the same number of rows", ... inputname (1), inputname (2)); endif [u, scale] = __sl_sb03od__ (a.', b.', true); ## NOTE: TRANS = 'T' not suitable because we need U' U, not U U' case 3 if (! is_real_square_matrix (a, e)) ## error ("dlyapchol: a, e must be real and square"); error ("dlyapchol: %s, %s must be real and square", ... inputname (1), inputname (3)); endif if (! is_real_matrix (b)) ## error ("dlyapchol: b must be real"); error ("dlyapchol: %s must be real", ... inputname (2)); endif if (rows (b) != rows (a) || rows (e) != rows (a)) ## error ("dlyapchol: a, b, e must have the same number of rows"); error ("dlyapchol: %s, %s, %s must have the same number of rows", ... inputname (1), inputname (2), inputname (3)); endif [u, scale] = __sl_sg03bd__ (a.', e.', b.', true); ## NOTE: TRANS = 'T' not suitable because we need U' U, not U U' otherwise print_usage (); endswitch if (scale < 1) warning ("dlyapchol: solution scaled by %g to prevent overflow", scale); endif endfunction ## TODO: add tests control/inst/dss.m0000644000076500000240000000757512273411257013506 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} dss (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} dss (@var{d}) ## @deftypefnx {Function File} {@var{sys} =} dss (@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @dots{}) ## @deftypefnx {Function File} {@var{sys} =} dss (@var{a}, @var{b}, @var{c}, @var{d}, @var{e}, @var{tsam}, @dots{}) ## Create or convert to descriptor state-space model. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model to be converted to state-space. ## @item a ## State matrix (n-by-n). ## @item b ## Input matrix (n-by-m). ## @item c ## Output matrix (p-by-n). ## @item d ## Feedthrough matrix (p-by-m). ## @item e ## Descriptor matrix (n-by-n). ## @item tsam ## Sampling time in seconds. If @var{tsam} is not specified, ## a continuous-time model is assumed. ## @item @dots{} ## Optional pairs of properties and values. ## Type @command{set (dss)} for more information. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Descriptor state-space model. ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'a', 'b', 'c', 'd', 'e' ## State-space matrices. See 'Inputs' for details. ## ## @item 'stname' ## The name of the states in @var{sys}. ## Cell vector containing strings for each state. ## Default names are @code{@{'x1', 'x2', ...@}} ## ## @item 'scaled' ## Logical. If set to true, no automatic scaling is used, ## e.g. for frequency response plots. ## ## @item 'tsam' ## Sampling time. See 'Inputs' for details. ## ## @item 'inname' ## The name of the input channels in @var{sys}. ## Cell vector of length m containing strings. ## Default names are @code{@{'u1', 'u2', ...@}} ## ## @item 'outname' ## The name of the output channels in @var{sys}. ## Cell vector of length p containing strings. ## Default names are @code{@{'y1', 'y2', ...@}} ## ## @item 'ingroup' ## Struct with input group names as field names and ## vectors of input indices as field values. ## Default is an empty struct. ## ## @item 'outgroup' ## Struct with output group names as field names and ## vectors of output indices as field values. ## Default is an empty struct. ## ## @item 'name' ## String containing the name of the model. ## ## @item 'notes' ## String or cell of string containing comments. ## ## @item 'userdata' ## Any data type. ## @end table ## ## @strong{Equations} ## @example ## @group ## . ## E x = A x + B u ## y = C x + D u ## @end group ## @end example ## ## @seealso{ss, tf} ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2010 ## Version: 0.1 function sys = dss (varargin) switch (nargin) case {0, 1} # static gain (dss (5)) or empty (useful for "set (dss)") sys = ss (varargin{:}); case {2, 3, 4} print_usage (); otherwise # general case sys = ss (varargin{[1:4, 6:end]}, "e", varargin{5}); endswitch endfunction ## NOTE: The author prefers "dss (e, a, b, c, d)" since we write ## . ## E x = A x + B u, y = C x + D u ## ## but this would break compatibility to a widespread ## commercial implementation of the octave language. ## There's no way to tell e and d apart if n = m = p. control/inst/estim.m0000644000076500000240000000516212273411257014024 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{est} =} estim (@var{sys}, @var{l}) ## @deftypefnx {Function File} {@var{est} =} estim (@var{sys}, @var{l}, @var{sensors}, @var{known}) ## Return state estimator for a given estimator gain. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item l ## State feedback matrix. ## @item sensors ## Indices of measured output signals y from @var{sys}. If omitted, all outputs are measured. ## @item known ## Indices of known input signals u (deterministic) to @var{sys}. All other inputs to @var{sys} ## are assumed stochastic. If argument @var{known} is omitted, no inputs u are known. ## @end table ## ## @strong{Outputs} ## @table @var ## @item est ## State-space model of estimator. ## @end table ## @seealso{kalman, place} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.2 function est = estim (sys, l, sensors = [], known = []) if (nargin < 2 || nargin > 4) print_usage (); endif if (! isa (sys, "lti")) error ("estim: first argument must be an LTI system"); endif [a, b, c, d, e, tsam] = dssdata (sys, []); if (isempty (sensors)) sensors = 1 : rows (c); endif m = length (known); n = rows (a); p = length (sensors); b = b(:, known); c = c(sensors, :); d = d(sensors, known); f = a - l*c; g = [b - l*d, l]; h = [c; eye(n)]; j = [d, zeros(p, p); zeros(n, m), zeros(n, p)]; ## k = e; est = dss (f, g, h, j, e, tsam); ## TODO: inname, stname, outname endfunction %!shared m, m_exp %! sys = ss (-2, 1, 1, 3); %! est = estim (sys, 5); %! [a, b, c, d] = ssdata (est); %! m = [a, b; c, d]; %! m_exp = [-7, 5; 1, 0; 1, 0]; %!assert (m, m_exp, 1e-4); %!shared m, m_exp %! sys = ss (-1, 2, 3, 4); %! est = estim (sys, 5); %! [a, b, c, d] = ssdata (est); %! m = [a, b; c, d]; %! m_exp = [-16, 5; 3, 0; 1, 0]; %!assert (m, m_exp, 1e-4); control/inst/filt.m0000644000076500000240000001323012273411257013634 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} filt (@var{num}, @var{den}, @dots{}) ## @deftypefnx {Function File} {@var{sys} =} filt (@var{num}, @var{den}, @var{tsam}, @dots{}) ## Create discrete-time transfer function model from data in DSP format. ## ## @strong{Inputs} ## @table @var ## @item num ## Numerator or cell of numerators. Each numerator must be a row vector ## containing the coefficients of the polynomial in ascending powers of z^-1. ## num@{i,j@} contains the numerator polynomial from input j to output i. ## In the SISO case, a single vector is accepted as well. ## @item den ## Denominator or cell of denominators. Each denominator must be a row vector ## containing the coefficients of the polynomial in ascending powers of z^-1. ## den@{i,j@} contains the denominator polynomial from input j to output i. ## In the SISO case, a single vector is accepted as well. ## @item tsam ## Sampling time in seconds. If @var{tsam} is not specified, ## default value -1 (unspecified) is taken. ## @item @dots{} ## Optional pairs of properties and values. ## Type @command{set (filt)} for more information. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Discrete-time transfer function model. ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'num' ## Numerator. See 'Inputs' for details. ## ## @item 'den' ## Denominator. See 'Inputs' for details. ## ## @item 'tfvar' ## String containing the transfer function variable. ## ## @item 'inv' ## Logical. True for negative powers of the transfer function variable. ## ## @item 'tsam' ## Sampling time. See 'Inputs' for details. ## ## @item 'inname' ## The name of the input channels in @var{sys}. ## Cell vector of length m containing strings. ## Default names are @code{@{'u1', 'u2', ...@}} ## ## @item 'outname' ## The name of the output channels in @var{sys}. ## Cell vector of length p containing strings. ## Default names are @code{@{'y1', 'y2', ...@}} ## ## @item 'ingroup' ## Struct with input group names as field names and ## vectors of input indices as field values. ## Default is an empty struct. ## ## @item 'outgroup' ## Struct with output group names as field names and ## vectors of output indices as field values. ## Default is an empty struct. ## ## @item 'name' ## String containing the name of the model. ## ## @item 'notes' ## String or cell of string containing comments. ## ## @item 'userdata' ## Any data type. ## @end table ## ## @strong{Example} ## @example ## @group ## 3 z^-1 ## H(z^-1) = ------------------- ## 1 + 4 z^-1 + 2 z^-2 ## ## octave:1> H = filt ([0, 3], [1, 4, 2]) ## ## Transfer function 'H' from input 'u1' to output ... ## ## 3 z^-1 ## y1: ------------------- ## 1 + 4 z^-1 + 2 z^-2 ## ## Sampling time: unspecified ## Discrete-time model. ## @end group ## @end example ## ## @seealso{tf} ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function sys = filt (num = {}, den = {}, tsam = -1, varargin) switch (nargin) case 0 # filt () sys = tf (); ## sys.inv = true; return; case 1 # filt (sys), filt (matrix) if (isa (num, "lti") || is_real_matrix (num)) sys = tf (num); ## sys.inv = true; # would be a problem for continuous-time LTI models return; else print_usage (); endif otherwise # filt (num, den, ...) if (! iscell (num)) num = {num}; endif if (! iscell (den)) den = {den}; endif ## convert from z^-1 to z ## expand each channel by z^x, where x is the largest exponent of z^-1 (z^-x) ## remove trailing zeros ## such that polynomials are as short as possible num = cellfun (@__remove_trailing_zeros__, num, "uniformoutput", false); den = cellfun (@__remove_trailing_zeros__, den, "uniformoutput", false); ## make numerator and denominator polynomials equally long ## by adding trailing zeros lnum = cellfun (@length, num, "uniformoutput", false); lden = cellfun (@length, den, "uniformoutput", false); lmax = cellfun (@max, lnum, lden, "uniformoutput", false); num = cellfun (@postpad, num, lmax, "uniformoutput", false); den = cellfun (@postpad, den, lmax, "uniformoutput", false); ## use standard tf constructor ## sys is stored in standard z form, not z^-1 ## so we can mix it with regular transfer function models ## property "inv", true displays sys in z^-1 form sys = tf (num, den, tsam, "inv", true, varargin{:}); endswitch endfunction %!shared num, den, n1, d1, n2, d2, n2e, d2e %! num = [0, 3]; %! den = [1, 4, 2]; %! sys = filt (num, den); %! [n1, d1] = filtdata (sys, "vector"); %! [n2, d2] = tfdata (sys, "vector"); %! n2e = [3, 0]; %! d2e = [1, 4, 2]; %!assert (n1, num, 1e-4); %!assert (d1, den, 1e-4); %!assert (n2, n2e, 1e-4); %!assert (d2, d2e, 1e-4); control/inst/fitfrd.m0000644000076500000240000000567512273411257014172 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{sys}, @var{n}] =} fitfrd (@var{dat}, @var{n}) ## @deftypefnx{Function File} {[@var{sys}, @var{n}] =} fitfrd (@var{dat}, @var{n}, @var{flag}) ## Fit frequency response data with a state-space system. ## If requested, the returned system is stable and minimum-phase. ## ## @strong{Inputs} ## @table @var ## @item dat ## @acronym{LTI} model containing frequency response data of a SISO system. ## @item n ## The desired order of the system to be fitted. @code{n <= length(dat.w)}. ## @item flag ## The flag controls whether the returned system is stable and minimum-phase. ## @table @var ## @item 0 ## The system zeros and poles are not constrained. Default value. ## @item 1 ## The system zeros and poles will have negative real parts in the ## continuous-time case, or moduli less than 1 in the discrete-time case. ## @end table ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## State-space model of order @var{n}, fitted to frequency response data @var{dat}. ## @item n ## The order of the obtained system. The value of @var{n} ## could only be modified if inputs @code{n > 0} and @code{flag = 1}. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT SB10YD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2011 ## Version: 0.1 function [sys, n] = fitfrd (dat, n, flag = 0) if (nargin == 0 || nargin > 3) print_usage (); endif if (! isa (dat, "frd")) dat = frd (dat); endif if (! issiso (dat)) error ("fitfrd: require SISO system"); endif if (! issample (n, 0) || n != round (n)) error ("fitfrd: second argument must be an integer >= 0"); endif [H, w, tsam] = frdata (dat, "vector"); dt = isdt (dat); if (n > length (w)) error ("fitfrd: require n <= length (dat.w)"); endif [a, b, c, d, n] = __sl_sb10yd__ (real (H), imag (H), w, n, dt, logical (flag)); sys = ss (a, b, c, d, tsam); endfunction %!shared Yo, Ye %! SYS = ss (-1, 1, 1, 0); %! T = 0:0.1:50; %! Ye = step (SYS, T); %! W = logspace (-2, 2, 100); %! FR = frd (SYS, W); %! N = 1; %! SYSID = fitfrd (FR, N, 1); %! Yo = step (SYSID, T); %!assert (Yo, Ye, 1e-2); control/inst/fwcfconred.m0000644000076500000240000002354512273411257015030 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} fwcfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @var{opt}, @dots{}) ## ## Reduction of state-feedback-observer based controller by frequency-weighted coprime factorization (FW CF). ## Given a plant @var{G}, state feedback gain @var{F} and full observer gain @var{L}, ## determine a reduced order controller @var{Kr} by using stability enforcing frequency weights. ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model of the open-loop plant (A,B,C,D). ## It has m inputs, p outputs and n states. ## @item F ## Stabilizing state feedback matrix (m-by-n). ## @item L ## Stabilizing observer gain matrix (n-by-p). ## @item ncr ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Kr ## State-space model of reduced order controller. ## @item info ## Struct containing additional information. ## @table @var ## @item info.hsv ## The Hankel singular values of the extended system?!?. ## The @var{n} Hankel singular values are ordered decreasingly. ## @item info.ncr ## The order of the obtained reduced order controller @var{Kr}. ## @end table ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'ncr' ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically such that states with ## Hankel singular values @var{info.hsv} > @var{tol1} are retained. ## ## @item 'method' ## Order reduction approach to be used as follows: ## @table @var ## @item 'sr', 'b' ## Use the square-root Balance & Truncate method. ## @item 'bfsr', 'f' ## Use the balancing-free square-root Balance & Truncate method. Default method. ## @end table ## ## @item 'cf' ## Specifies whether left or right coprime factorization is ## to be used as follows: ## @table @var ## @item 'left', 'l' ## Use left coprime factorization. ## @item 'right', 'r' ## Use right coprime factorization. Default method. ## @end table ## ## @item 'feedback' ## Specifies whether @var{F} and @var{L} are fed back positively or negatively: ## @table @var ## @item '+' ## A+BK and A+LC are both Hurwitz matrices. ## @item '-' ## A-BK and A-LC are both Hurwitz matrices. Default value. ## @end table ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced system. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. ## Default value is n*eps*info.hsv(1). ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT SB16CD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function [Kr, info] = fwcfconred (G, F, L, varargin) if (nargin < 3) print_usage (); endif if (! isa (G, "lti")) error ("fwcfconred: first argument must be an LTI system"); endif if (! is_real_matrix (F)) error ("fwcfconred: second argument must be a real matrix"); endif if (! is_real_matrix (L)) error ("fwcfconred: third argument must be a real matrix"); endif if (nargin > 3) # fwcfconred (G, F, L, ...) if (is_real_scalar (varargin{1})) # fwcfconred (G, F, L, nr) varargin = horzcat (varargin(2:end), {"order"}, varargin(1)); endif if (isstruct (varargin{1})) # fwcfconred (G, F, L, opt, ...), fwcfconred (G, F, L, nr, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif ## order placed at the end such that nr from fwcfconred (G, F, L, nr, ...) ## and fwcfconred (G, F, L, nr, opt, ...) overrides possible nr's from ## key/value-pairs and inside opt struct (later keys override former keys, ## nr > key/value > opt) endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("fwcfconred: keys and values must come in pairs"); endif [a, b, c, d, tsam] = ssdata (G); [p, m] = size (G); n = rows (a); [mf, nf] = size (F); [nl, pl] = size (L); dt = isdt (G); jobd = any (d(:)); if (mf != m || nf != n) error ("fwcfconred: dimensions of state-feedback matrix (%dx%d) and plant (%dx%d, %d states) don't match", ... mf, nf, p, m, n); endif if (nl != n || pl != p) error ("fwcfconred: dimensions of observer matrix (%dx%d) and plant (%dx%d, %d states) don't match", ... nl, pl, p, m, n); endif ## default arguments tol1 = 0.0; jobcf = 1; jobmr = 1; # balancing-free BTA ordsel = 1; ncr = 0; negfb = true; # A-BK, A-LC Hurwitz ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case {"order", "ncr", "nr"} [ncr, ordsel] = __modred_check_order__ (val, n); case {"tol1", "tol"} tol1 = __modred_check_tol__ (val, "tol1"); case "cf" switch (lower (val(1))) case "l" jobcf = 0; case "r" jobcf = 1; otherwise error ("cfconred: '%s' is an invalid coprime factorization", val); endswitch case "method" # approximation method switch (tolower (val)) case {"sr-bta", "b", "sr"} # 'B': use the square-root Balance & Truncate method jobmr = 0; case {"bfsr-bta", "f", "bfsr"} # 'F': use the balancing-free square-root Balance & Truncate method jobmr = 1; otherwise error ("cfconred: '%s' is an invalid approach", val); endswitch case "feedback" negfb = __conred_check_feedback_sign__ (val); otherwise warning ("fwcfconred: invalid property name '%s' ignored", key); endswitch endfor ## A - B*F --> A + B*F ; A - L*C --> A + L*C if (negfb) F = -F; L = -L; endif ## perform model order reduction [acr, bcr, ccr, ncr, hsv] = __sl_sb16cd__ (a, b, c, d, dt, ncr, ordsel, jobd, jobmr, ... F, L, jobcf, tol1); ## assemble reduced order controller Kr = ss (acr, bcr, ccr, [], tsam); ## assemble info struct info = struct ("ncr", ncr, "hsv", hsv); endfunction %!shared Mo, Me, Info, HSVe %! A = [ 0 1.0000 0 0 0 0 0 0 %! 0 0 0 0 0 0 0 0 %! 0 0 -0.0150 0.7650 0 0 0 0 %! 0 0 -0.7650 -0.0150 0 0 0 0 %! 0 0 0 0 -0.0280 1.4100 0 0 %! 0 0 0 0 -1.4100 -0.0280 0 0 %! 0 0 0 0 0 0 -0.0400 1.850 %! 0 0 0 0 0 0 -1.8500 -0.040 ]; %! %! B = [ 0.0260 %! -0.2510 %! 0.0330 %! -0.8860 %! -4.0170 %! 0.1450 %! 3.6040 %! 0.2800 ]; %! %! C = [ -.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026 ]; %! %! D = [ 0.0 ]; %! %! G = ss (A, B, C, D); % "scaled", false %! %! F = [ 4.472135954999638e-002 6.610515358414598e-001 4.698598960657579e-003 3.601363251422058e-001 1.032530880771415e-001 -3.754055214487997e-002 -4.268536964759344e-002 3.287284547842979e-002 ]; %! %! L = [ 4.108939884667451e-001 %! 8.684600000000012e-002 %! 3.852317308197148e-004 %! -3.619366874815911e-003 %! -8.803722876359955e-003 %! 8.420521094001852e-003 %! 1.234944428038507e-003 %! 4.263205617645322e-003 ]; %! %! [Kr, Info] = fwcfconred (G, F, L, 2, "method", "bfsr", "cf", "right", "feedback", "+"); %! [Ao, Bo, Co, Do] = ssdata (Kr); %! %! Ae = [ -0.4334 0.4884 %! -0.1950 -0.1093 ]; %! %! Be = [ -0.4231 %! -0.1785 ]; %! %! Ce = [ -0.0326 -0.2307 ]; %! %! De = [ 0.0000 ]; %! %! HSVe = [ 3.3073 0.7274 0.1124 0.0784 0.0242 0.0182 0.0101 0.0094 ].'; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); %!assert (Info.hsv, HSVe, 1e-4); control/inst/gensig.m0000644000076500000240000000537412273411257014164 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}) ## @deftypefnx{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}, @var{tfinal}) ## @deftypefnx{Function File} {[@var{u}, @var{t}] =} gensig (@var{sigtype}, @var{tau}, @var{tfinal}, @var{tsam}) ## Generate periodic signal. Useful in combination with lsim. ## ## @strong{Inputs} ## @table @var ## @item sigtype = "sin" ## Sine wave. ## @item sigtype = "cos" ## Cosine wave. ## @item sigtype = "square" ## Square wave. ## @item sigtype = "pulse" ## Periodic pulse. ## @item tau ## Duration of one period in seconds. ## @item tfinal ## Optional duration of the signal in seconds. Default duration is 5 periods. ## @item tsam ## Optional sampling time in seconds. Default spacing is tau/64. ## @end table ## ## @strong{Outputs} ## @table @var ## @item u ## Vector of signal values. ## @item t ## Time vector of the signal. ## @end table ## ## @seealso{lsim} ## @end deftypefn ## Author: Lukas Reichlin ## Created: August 2009 ## Version: 0.4 function [u, t] = gensig (sigtype, tau, tfinal, tsam) if (nargin < 2 || nargin > 4) print_usage (); endif if (! ischar (sigtype)) error ("gensig: first argument must be a string"); endif if (! issample (tau)) error ("gensig: second argument is not a valid period"); endif if (nargin < 3) tfinal = 5 * tau; elseif (! issample (tfinal)) error ("gensig: third argument is not a valid final time"); endif if (nargin < 4) tsam = tau / 64; elseif (! issample (tsam)) error ("gensig: fourth argument is not a valid sampling time"); endif t = reshape (0 : tsam : tfinal, [], 1); switch (lower (sigtype(1:2))) case "si" u = sin (2*pi/tau * t); case "co" u = cos (2*pi/tau * t); case "sq" u = double (rem (t, tau) >= tau/2); case "pu" u = double (rem (t, tau) < (1 - 1000*eps) * tsam); otherwise error ("gensig: '%s' is an invalid signal type", sigtype); endswitch endfunction control/inst/gram.m0000644000076500000240000000754612273411257013641 0ustar lukasstaff## Copyright (C) 1996, 2000, 2003, 2004, 2005, 2007 ## Auburn University. All rights reserved. ## ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{W} =} gram (@var{sys}, @var{mode}) ## @deftypefnx {Function File} {@var{Wc} =} gram (@var{a}, @var{b}) ## @code{gram (@var{sys}, "c")} returns the controllability gramian of ## the (continuous- or discrete-time) system @var{sys}. ## @code{gram (@var{sys}, "o")} returns the observability gramian of the ## (continuous- or discrete-time) system @var{sys}. ## @code{gram (@var{a}, @var{b})} returns the controllability gramian ## @var{Wc} of the continuous-time system @math{dx/dt = a x + b u}; ## i.e., @var{Wc} satisfies @math{a Wc + m Wc' + b b' = 0}. ## ## @end deftypefn ## Author: A. S. Hodel ## Adapted-By: Lukas Reichlin ## Date: October 2009 ## Version: 0.2 function W = gram (argin1, argin2) if (nargin != 2) print_usage (); endif if (ischar (argin2)) # the function was called as "gram (sys, mode)" sys = argin1; if (! isa (sys, "lti")) error ("gram: first argument must be an LTI model"); endif [a, b, c] = ssdata (sys); if (strncmpi (argin2, "o", 1)) a = a.'; b = c.'; elseif (! strncmpi (argin2, "c", 1)) print_usage (); endif else # the function was called as "gram (a, b)" a = argin1; b = argin2; ## assume that a and b are matrices of continuous-time system ## values of b, c and d are irrelevant for system stability sys = ss (a, b, zeros (1, columns (a)), zeros (1, columns (b))); endif if (! isstable (sys)) error ("gram: system matrix a must be stable"); endif if (isct (sys)) W = lyap (a, b*b.'); # let lyap do the error checking about dimensions else # discrete-time system W = dlyap (a, b*b.'); # let dlyap do the error checking about dimensions endif endfunction %!test %! a = [-1 0 0; 1/2 -1 0; 1/2 0 -1]; %! b = [1 0; 0 -1; 0 1]; %! c = [0 0 1; 1 1 0]; ## it doesn't matter what the value of c is %! Wc = gram (ss (a, b, c), "c"); %! assert (a * Wc + Wc * a.' + b * b.', zeros (size (a))) %!test %! a = [-1 0 0; 1/2 -1 0; 1/2 0 -1]; %! b = [1 0; 0 -1; 0 1]; ## it doesn't matter what the value of b is %! c = [0 0 1; 1 1 0]; %! Wo = gram (ss (a, b, c), "o"); %! assert (a.' * Wo + Wo * a + c.' * c, zeros (size (a))) %!test %! a = [-1 0 0; 1/2 -1 0; 1/2 0 -1]; %! b = [1 0; 0 -1; 0 1]; %! Wc = gram (a, b); %! assert (a * Wc + Wc * a.' + b * b.', zeros (size (a))) %!test %! a = [-1 0 0; 1/2 1 0; 1/2 0 -1] / 2; %! b = [1 0; 0 -1; 0 1]; %! c = [0 0 1; 1 1 0]; ## it doesn't matter what the value of c is %! d = zeros (rows (c), columns (b)); ## it doesn't matter what the value of d is %! Ts = 0.1; ## Ts != 0 %! Wc = gram (ss (a, b, c, d, Ts), "c"); %! assert (a * Wc * a.' - Wc + b * b.', zeros (size (a)), 1e-12) %!test %! a = [-1 0 0; 1/2 1 0; 1/2 0 -1] / 2; %! b = [1 0; 0 -1; 0 1]; ## it doesn't matter what the value of b is %! c = [0 0 1; 1 1 0]; %! d = zeros (rows (c), columns (b)); ## it doesn't matter what the value of d is %! Ts = 0.1; ## Ts != 0 %! Wo = gram (ss (a, b, c, d, Ts), "o"); %! assert (a.' * Wo * a - Wo + c.' * c, zeros (size (a)), 1e-12) control/inst/h2syn.m0000644000076500000240000002032212273411257013741 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} h2syn (@var{P}, @var{nmeas}, @var{ncon}) ## @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} h2syn (@var{P}) ## H-2 control synthesis for @acronym{LTI} plant. ## ## @strong{Inputs} ## @table @var ## @item P ## Generalized plant. Must be a proper/realizable @acronym{LTI} model. ## If @var{P} is constructed with @command{mktito} or @command{augw}, ## arguments @var{nmeas} and @var{ncon} can be omitted. ## @item nmeas ## Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the ## inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used ## to calculate the H-2 norm. ## @item ncon ## Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the ## outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited ## by a harmonic test signal. ## @end table ## ## @strong{Outputs} ## @table @var ## @item K ## State-space model of the H-2 optimal controller. ## @item N ## State-space model of the lower LFT of @var{P} and @var{K}. ## @item info ## Structure containing additional information. ## @item info.gamma ## H-2 norm of @var{N}. ## @item info.rcond ## Vector @var{rcond} contains estimates of the reciprocal condition ## numbers of the matrices which are to be inverted and ## estimates of the reciprocal condition numbers of the ## Riccati equations which have to be solved during the ## computation of the controller @var{K}. For details, ## see the description of the corresponding SLICOT routine. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## ## gamma = min||N(K)|| N = lft (P, K) ## K 2 ## ## +--------+ ## w ----->| |-----> z ## | P(s) | ## u +---->| |-----+ v ## | +--------+ | ## | | ## | +--------+ | ## +-----| K(s) |<----+ ## +--------+ ## ## +--------+ ## w ----->| N(s) |-----> z ## +--------+ ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB10HD and SB10ED by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{augw, lqr, dlqr, kalman} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2009 ## Version: 0.3 function [K, varargout] = h2syn (P, nmeas, ncon) ## check input arguments if (nargin != 1 && nargin != 3) print_usage (); endif if (! isa (P, "lti")) error ("h2syn: first argument must be an LTI system"); endif if (nargin == 1) [nmeas, ncon] = __tito_dim__ (P, "h2syn"); endif if (! is_real_scalar (nmeas)) error ("h2syn: second argument 'nmeas' invalid"); endif if (! is_real_scalar (ncon)) error ("h2syn: third argument 'ncon' invalid"); endif [a, b, c, d, tsam] = ssdata (P); ## check assumptions A1 - A3 m = columns (b); p = rows (c); m1 = m - ncon; p1 = p - nmeas; d11 = d(1:p1, 1:m1); if (isct (P) && any (d11(:))) warning ("h2syn: setting matrice D11 to zero"); d(1:p1, 1:m1) = 0; endif if (! isstabilizable (P(:, m1+1:m))) error ("h2syn: (A, B2) must be stabilizable"); endif if (! isdetectable (P(p1+1:p, :))) error ("h2syn: (C2, A) must be detectable"); endif ## H-2 synthesis if (isct (P)) # continuous plant [ak, bk, ck, dk, rcond] = __sl_sb10hd__ (a, b, c, d, ncon, nmeas); else # discrete plant [ak, bk, ck, dk, rcond] = __sl_sb10ed__ (a, b, c, d, ncon, nmeas); endif ## controller K = ss (ak, bk, ck, dk, tsam); if (nargout > 1) N = lft (P, K); varargout{1} = N; if (nargout > 2) gamma = norm (N, 2); varargout{2} = gamma; if (nargout > 3) varargout{3} = struct ("gamma", gamma, "rcond", rcond); endif endif endif endfunction ## continuous-time case %!shared M, M_exp %! A = [-1.0 0.0 4.0 5.0 -3.0 -2.0 %! -2.0 4.0 -7.0 -2.0 0.0 3.0 %! -6.0 9.0 -5.0 0.0 2.0 -1.0 %! -8.0 4.0 7.0 -1.0 -3.0 0.0 %! 2.0 5.0 8.0 -9.0 1.0 -4.0 %! 3.0 -5.0 8.0 0.0 2.0 -6.0]; %! %! B = [-3.0 -4.0 -2.0 1.0 0.0 %! 2.0 0.0 1.0 -5.0 2.0 %! -5.0 -7.0 0.0 7.0 -2.0 %! 4.0 -6.0 1.0 1.0 -2.0 %! -3.0 9.0 -8.0 0.0 5.0 %! 1.0 -2.0 3.0 -6.0 -2.0]; %! %! C = [ 1.0 -1.0 2.0 -4.0 0.0 -3.0 %! -3.0 0.0 5.0 -1.0 1.0 1.0 %! -7.0 5.0 0.0 -8.0 2.0 -2.0 %! 9.0 -3.0 4.0 0.0 3.0 7.0 %! 0.0 1.0 -2.0 1.0 -6.0 -2.0]; %! %! D = [ 0.0 0.0 0.0 -4.0 -1.0 %! 0.0 0.0 0.0 1.0 0.0 %! 0.0 0.0 0.0 0.0 1.0 %! 3.0 1.0 0.0 1.0 -3.0 %! -2.0 0.0 1.0 7.0 1.0]; %! %! P = ss (A, B, C, D); %! K = h2syn (P, 2, 2); %! M = [K.A, K.B; K.C, K.D]; %! %! KA = [ 88.0015 -145.7298 -46.2424 82.2168 -45.2996 -31.1407 %! 25.7489 -31.4642 -12.4198 9.4625 -3.5182 2.7056 %! 54.3008 -102.4013 -41.4968 50.8412 -20.1286 -26.7191 %! 108.1006 -198.0785 -45.4333 70.3962 -25.8591 -37.2741 %! -115.8900 226.1843 47.2549 -47.8435 -12.5004 34.7474 %! 59.0362 -101.8471 -20.1052 36.7834 -16.1063 -26.4309]; %! %! KB = [ 3.7345 3.4758 %! -0.3020 0.6530 %! 3.4735 4.0499 %! 4.3198 7.2755 %! -3.9424 -10.5942 %! 2.1784 2.5048]; %! %! KC = [ -2.3346 3.2556 0.7150 -0.9724 0.6962 0.4074 %! 7.6899 -8.4558 -2.9642 7.0365 -4.2844 0.1390]; %! %! KD = [ 0.0000 0.0000 %! 0.0000 0.0000]; %! %! M_exp = [KA, KB; KC, KD]; %! %!assert (M, M_exp, 1e-4); ## discrete-time case %!shared M, M_exp %! A = [-0.7 0.0 0.3 0.0 -0.5 -0.1 %! -0.6 0.2 -0.4 -0.3 0.0 0.0 %! -0.5 0.7 -0.1 0.0 0.0 -0.8 %! -0.7 0.0 0.0 -0.5 -1.0 0.0 %! 0.0 0.3 0.6 -0.9 0.1 -0.4 %! 0.5 -0.8 0.0 0.0 0.2 -0.9]; %! %! B = [-1.0 -2.0 -2.0 1.0 0.0 %! 1.0 0.0 1.0 -2.0 1.0 %! -3.0 -4.0 0.0 2.0 -2.0 %! 1.0 -2.0 1.0 0.0 -1.0 %! 0.0 1.0 -2.0 0.0 3.0 %! 1.0 0.0 3.0 -1.0 -2.0]; %! %! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0 %! -3.0 0.0 1.0 -1.0 1.0 0.0 %! 0.0 2.0 0.0 -4.0 0.0 -2.0 %! 1.0 -3.0 0.0 0.0 3.0 1.0 %! 0.0 1.0 -2.0 1.0 0.0 -2.0]; %! %! D = [ 1.0 -1.0 -2.0 0.0 0.0 %! 0.0 1.0 0.0 1.0 0.0 %! 2.0 -1.0 -3.0 0.0 1.0 %! 0.0 1.0 0.0 1.0 -1.0 %! 0.0 0.0 1.0 2.0 1.0]; %! %! P = ss (A, B, C, D, 1); # value of sampling time doesn't matter %! K = h2syn (P, 2, 2); %! M = [K.A, K.B; K.C, K.D]; %! %! KA = [-0.0551 -2.1891 -0.6607 -0.2532 0.6674 -1.0044 %! -1.0379 2.3804 0.5031 0.3960 -0.6605 1.2673 %! -0.0876 -2.1320 -0.4701 -1.1461 1.2927 -1.5116 %! -0.1358 -2.1237 -0.9560 -0.7144 0.6673 -0.7957 %! 0.4900 0.0895 0.2634 -0.2354 0.1623 -0.2663 %! 0.1672 -0.4163 0.2871 -0.1983 0.4944 -0.6967]; %! %! KB = [-0.5985 -0.5464 %! 0.5285 0.6087 %! -0.7600 -0.4472 %! -0.7288 -0.6090 %! 0.0532 0.0658 %! -0.0663 0.0059]; %! %! KC = [ 0.2500 -1.0200 -0.3371 -0.2733 0.2747 -0.4444 %! 0.0654 0.2095 0.0632 0.2089 -0.1895 0.1834]; %! %! KD = [-0.2181 -0.2070 %! 0.1094 0.1159]; %! %! M_exp = [KA, KB; KC, KD]; %! %!assert (M, M_exp, 1e-4); control/inst/hinfsyn.m0000644000076500000240000003712112273411257014361 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}) ## @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}, @dots{}) ## @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @dots{}) ## @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{opt}, @dots{}) ## H-infinity control synthesis for @acronym{LTI} plant. ## ## @strong{Inputs} ## @table @var ## @item P ## Generalized plant. Must be a proper/realizable @acronym{LTI} model. ## If @var{P} is constructed with @command{mktito} or @command{augw}, ## arguments @var{nmeas} and @var{ncon} can be omitted. ## @item nmeas ## Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the ## inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used ## to calculate the H-infinity norm. ## @item ncon ## Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the ## outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited ## by a harmonic test signal. ## @item @dots{} ## Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by function @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item K ## State-space model of the H-infinity (sub-)optimal controller. ## @item N ## State-space model of the lower LFT of @var{P} and @var{K}. ## @item info ## Structure containing additional information. ## @item info.gamma ## L-infinity norm of @var{N}. ## @item info.rcond ## Vector @var{rcond} contains estimates of the reciprocal condition ## numbers of the matrices which are to be inverted and ## estimates of the reciprocal condition numbers of the ## Riccati equations which have to be solved during the ## computation of the controller @var{K}. For details, ## see the description of the corresponding SLICOT routine. ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'method' ## String specifying the desired kind of controller: ## @table @var ## @item 'optimal', 'opt', 'o' ## Compute optimal controller using gamma iteration. ## Default selection for compatibility reasons. ## @item 'suboptimal', 'sub', 's' ## Compute (sub-)optimal controller. For stability reasons, ## suboptimal controllers are to be preferred over optimal ones. ## @end table ## @item 'gmax' ## The maximum value of the H-infinity norm of @var{N}. ## It is assumed that @var{gmax} is sufficiently large ## so that the controller is admissible. Default value is 1e15. ## @item 'gmin' ## Initial lower bound for gamma iteration. Default value is 0. ## @var{gmin} is only meaningful for optimal discrete-time controllers. ## @item 'tolgam' ## Tolerance used for controlling the accuracy of @var{gamma} ## and its distance to the estimated minimal possible ## value of @var{gamma}. Default value is 0.01. ## If @var{tolgam} = 0, then a default value equal to @code{sqrt(eps)} ## is used, where @var{eps} is the relative machine precision. ## For suboptimal controllers, @var{tolgam} is ignored. ## @item 'actol' ## Upper bound for the poles of the closed-loop system @var{N} ## used for determining if it is stable. ## @var{actol} >= 0 for stable systems. ## For suboptimal controllers, @var{actol} is ignored. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## ## gamma = min||N(K)|| N = lft (P, K) ## K inf ## ## +--------+ ## w ----->| |-----> z ## | P(s) | ## u +---->| |-----+ v ## | +--------+ | ## | | ## | +--------+ | ## +-----| K(s) |<----+ ## +--------+ ## ## +--------+ ## w ----->| N(s) |-----> z ## +--------+ ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB10FD, SB10DD and SB10AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{augw, mixsyn} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2009 ## Version: 0.3 function [K, varargout] = hinfsyn (P, varargin) ## check input arguments if (nargin == 0) print_usage (); endif if (! isa (P, "lti")) error ("hinfsyn: first argument must be an LTI system"); endif if (nargin == 1 || (nargin > 1 && ! is_real_scalar (varargin{1}))) # hinfsyn (P, ...) [nmeas, ncon] = __tito_dim__ (P, "hinfsyn"); elseif (nargin >= 3) # hinfsyn (P, nmeas, ncon, ...) nmeas = varargin{1}; ncon = varargin{2}; varargin = varargin(3:end); else print_usage (); endif if (! is_real_scalar (nmeas)) error ("hinfsyn: second argument 'nmeas' invalid"); endif if (! is_real_scalar (ncon)) error ("hinfsyn: third argument 'ncon' invalid"); endif if (numel (varargin) > 0 && isstruct (varargin{1})) # hinfsyn (P, nmeas, ncon, opt, ...), hinfsyn (P, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("hinfsyn: keys and values must come in pairs"); endif ## default arguments gmax = 1e15; gmin = 0; tolgam = 0.01; actol = eps; # tolerance for stability margin method = "opt"; ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case "gmax" if (! is_real_scalar (val) || val < 0) error ("hinfsyn: 'gmax' must be a real-valued, non-negative scalar"); endif gmax = val; case "gmin" if (! is_real_scalar (val) || val < 0) error ("hinfsyn: 'gmin' must be a real-valued, non-negative scalar"); endif gmin = val; case "tolgam" if (! is_real_scalar (val) || val < 0) error ("hinfsyn: 'tolgam' must be a real-valued, non-negative scalar"); endif tolgam = val; case "actol" if (! is_real_scalar (val) || val < 0) error ("hinfsyn: 'actol' must be a real-valued, non-negative scalar"); endif actol = val; case "method" ## NOTE: I called this "method" because of the dark side, ## maybe something like "type" would make more sense ... if (strncmpi (val, "s", 1)) method = "sub"; # sub-optimal elseif (strncmpi (val, "o", 1) || strncmpi (val, "ric", 1)) method = "opt"; # optimal else error ("hinfsyn: invalid method '%s'", val); endif otherwise warning ("hinfsyn: invalid property name '%s' ignored", key); endswitch endfor [a, b, c, d, tsam] = ssdata (P); ## check assumption A1 m = columns (b); p = rows (c); m1 = m - ncon; p1 = p - nmeas; if (! isstabilizable (P(:, m1+1:m))) error ("hinfsyn: (A, B2) must be stabilizable"); endif if (! isdetectable (P(p1+1:p, :))) error ("hinfsyn: (C2, A) must be detectable"); endif ## H-infinity synthesis switch (method) case "sub" # sub-optimal controller if (isct (P)) # continuous-time plant [ak, bk, ck, dk, rcond] = __sl_sb10fd__ (a, b, c, d, ncon, nmeas, gmax); else # discrete-time plant [ak, bk, ck, dk, rcond] = __sl_sb10dd__ (a, b, c, d, ncon, nmeas, gmax); endif case "opt" # optimal controller if (isct (P)) # continuous-time plant [ak, bk, ck, dk, ~, ~, ~, ~, ~, rcond] = __sl_sb10ad__ (a, b, c, d, ncon, nmeas, gmax, tolgam, -actol); else # discrete-time plant ## NOTE: check whether it is an alternative to compute the bilinear transformation ## of P, use __sl_sb10ad__ for a continuous-time controller and then ## discretize the controller. ## estimate gamma Pt = d2c (P, "tustin"); [at, bt, ct, dt] = ssdata (Pt); [~, ~, ~, ~, ~, ~, ~, ~, gamma] = __sl_sb10ad__ (at, bt, ct, dt, ncon, nmeas, gmax, tolgam, -actol); ## gamma iteration - bisection method using __sl_sb10dd__ gmax = 1.2*gamma; while (gmax > eps && (gmax - gmin)/gmax > tolgam) gmid = (gmax + gmin)/2; try [ak, bk, ck, dk, rcond] = __sl_sb10dd__ (a, b, c, d, ncon, nmeas, gmid); ## check for stability K = ss (ak, bk, ck, dk, tsam); N = lft (P, K); if (isstable (N, actol)) gmax = norm (N, inf); else gmin = gmid; endif catch # cannot find solution gmin = gmid; end_try_catch endwhile endif otherwise error ("hinfsyn: this should never happen"); endswitch ## controller K = ss (ak, bk, ck, dk, tsam); if (nargout > 1) N = lft (P, K); varargout{1} = N; if (nargout > 2) gamma = norm (N, inf); varargout{2} = gamma; if (nargout > 3) varargout{3} = struct ("gamma", gamma, "rcond", rcond); endif endif endif endfunction ## sub-optimal controller, continuous-time case %!shared M, M_exp %! A = [-1.0 0.0 4.0 5.0 -3.0 -2.0 %! -2.0 4.0 -7.0 -2.0 0.0 3.0 %! -6.0 9.0 -5.0 0.0 2.0 -1.0 %! -8.0 4.0 7.0 -1.0 -3.0 0.0 %! 2.0 5.0 8.0 -9.0 1.0 -4.0 %! 3.0 -5.0 8.0 0.0 2.0 -6.0]; %! %! B = [-3.0 -4.0 -2.0 1.0 0.0 %! 2.0 0.0 1.0 -5.0 2.0 %! -5.0 -7.0 0.0 7.0 -2.0 %! 4.0 -6.0 1.0 1.0 -2.0 %! -3.0 9.0 -8.0 0.0 5.0 %! 1.0 -2.0 3.0 -6.0 -2.0]; %! %! C = [ 1.0 -1.0 2.0 -4.0 0.0 -3.0 %! -3.0 0.0 5.0 -1.0 1.0 1.0 %! -7.0 5.0 0.0 -8.0 2.0 -2.0 %! 9.0 -3.0 4.0 0.0 3.0 7.0 %! 0.0 1.0 -2.0 1.0 -6.0 -2.0]; %! %! D = [ 1.0 -2.0 -3.0 0.0 0.0 %! 0.0 4.0 0.0 1.0 0.0 %! 5.0 -3.0 -4.0 0.0 1.0 %! 0.0 1.0 0.0 1.0 -3.0 %! 0.0 0.0 1.0 7.0 1.0]; %! %! P = ss (A, B, C, D); %! K = hinfsyn (P, 2, 2, "method", "sub", "gmax", 15); %! M = [K.A, K.B; K.C, K.D]; %! %! KA = [ -2.8043 14.7367 4.6658 8.1596 0.0848 2.5290 %! 4.6609 3.2756 -3.5754 -2.8941 0.2393 8.2920 %! -15.3127 23.5592 -7.1229 2.7599 5.9775 -2.0285 %! -22.0691 16.4758 12.5523 -16.3602 4.4300 -3.3168 %! 30.6789 -3.9026 -1.3868 26.2357 -8.8267 10.4860 %! -5.7429 0.0577 10.8216 -11.2275 1.5074 -10.7244]; %! %! KB = [ -0.1581 -0.0793 %! -0.9237 -0.5718 %! 0.7984 0.6627 %! 0.1145 0.1496 %! -0.6743 -0.2376 %! 0.0196 -0.7598]; %! %! KC = [ -0.2480 -0.1713 -0.0880 0.1534 0.5016 -0.0730 %! 2.8810 -0.3658 1.3007 0.3945 1.2244 2.5690]; %! %! KD = [ 0.0554 0.1334 %! -0.3195 0.0333]; %! %! M_exp = [KA, KB; KC, KD]; %! %!assert (M, M_exp, 1e-4); ## sub-optimal controller, discrete-time case %!shared M, M_exp %! A = [-0.7 0.0 0.3 0.0 -0.5 -0.1 %! -0.6 0.2 -0.4 -0.3 0.0 0.0 %! -0.5 0.7 -0.1 0.0 0.0 -0.8 %! -0.7 0.0 0.0 -0.5 -1.0 0.0 %! 0.0 0.3 0.6 -0.9 0.1 -0.4 %! 0.5 -0.8 0.0 0.0 0.2 -0.9]; %! %! B = [-1.0 -2.0 -2.0 1.0 0.0 %! 1.0 0.0 1.0 -2.0 1.0 %! -3.0 -4.0 0.0 2.0 -2.0 %! 1.0 -2.0 1.0 0.0 -1.0 %! 0.0 1.0 -2.0 0.0 3.0 %! 1.0 0.0 3.0 -1.0 -2.0]; %! %! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0 %! -3.0 0.0 1.0 -1.0 1.0 0.0 %! 0.0 2.0 0.0 -4.0 0.0 -2.0 %! 1.0 -3.0 0.0 0.0 3.0 1.0 %! 0.0 1.0 -2.0 1.0 0.0 -2.0]; %! %! D = [ 1.0 -1.0 -2.0 0.0 0.0 %! 0.0 1.0 0.0 1.0 0.0 %! 2.0 -1.0 -3.0 0.0 1.0 %! 0.0 1.0 0.0 1.0 -1.0 %! 0.0 0.0 1.0 2.0 1.0]; %! %! P = ss (A, B, C, D, 1); # value of sampling time doesn't matter %! K = hinfsyn (P, 2, 2, "method", "sub", "gmax", 111.294); %! M = [K.A, K.B; K.C, K.D]; %! %! KA = [-18.0030 52.0376 26.0831 -0.4271 -40.9022 18.0857 %! 18.8203 -57.6244 -29.0938 0.5870 45.3309 -19.8644 %! -26.5994 77.9693 39.0368 -1.4020 -60.1129 26.6910 %! -21.4163 62.1719 30.7507 -0.9201 -48.6221 21.8351 %! -0.8911 4.2787 2.3286 -0.2424 -3.0376 1.2169 %! -5.3286 16.1955 8.4824 -0.2489 -12.2348 5.1590]; %! %! KB = [ 16.9788 14.1648 %! -18.9215 -15.6726 %! 25.2046 21.2848 %! 20.1122 16.8322 %! 1.4104 1.2040 %! 5.3181 4.5149]; %! %! KC = [ -9.1941 27.5165 13.7364 -0.3639 -21.5983 9.6025 %! 3.6490 -10.6194 -5.2772 0.2432 8.1108 -3.6293]; %! %! KD = [ 9.0317 7.5348 %! -3.4006 -2.8219]; %! %! M_exp = [KA, KB; KC, KD]; %! %!assert (M, M_exp, 1e-4); ## optimal controller, discrete-time case??? -- test for bisection method %!shared M, M_exp, GAM_exp, GAM %! A = [-0.7 0.0 0.3 0.0 -0.5 -0.1 %! -0.6 0.2 -0.4 -0.3 0.0 0.0 %! -0.5 0.7 -0.1 0.0 0.0 -0.8 %! -0.7 0.0 0.0 -0.5 -1.0 0.0 %! 0.0 0.3 0.6 -0.9 0.1 -0.4 %! 0.5 -0.8 0.0 0.0 0.2 -0.9]; %! %! B = [-1.0 -2.0 -2.0 1.0 0.0 %! 1.0 0.0 1.0 -2.0 1.0 %! -3.0 -4.0 0.0 2.0 -2.0 %! 1.0 -2.0 1.0 0.0 -1.0 %! 0.0 1.0 -2.0 0.0 3.0 %! 1.0 0.0 3.0 -1.0 -2.0]; %! %! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0 %! -3.0 0.0 1.0 -1.0 1.0 0.0 %! 0.0 2.0 0.0 -4.0 0.0 -2.0 %! 1.0 -3.0 0.0 0.0 3.0 1.0 %! 0.0 1.0 -2.0 1.0 0.0 -2.0]; %! %! D = [ 1.0 -1.0 -2.0 0.0 0.0 %! 0.0 1.0 0.0 1.0 0.0 %! 2.0 -1.0 -3.0 0.0 1.0 %! 0.0 1.0 0.0 1.0 -1.0 %! 0.0 0.0 1.0 2.0 1.0]; %! %! P = ss (A, B, C, D, 1); %! [K, ~, GAM] = hinfsyn (P, 2, 2, "gmax", 1000, "tolgam", 1e-4); %! M = [K.A, K.B; K.C, K.D]; %! %! KA = [-18.0030 52.0376 26.0831 -0.4271 -40.9022 18.0857 %! 18.8203 -57.6244 -29.0938 0.5870 45.3309 -19.8644 %! -26.5994 77.9693 39.0368 -1.4020 -60.1129 26.6910 %! -21.4163 62.1719 30.7507 -0.9201 -48.6221 21.8351 %! -0.8911 4.2787 2.3286 -0.2424 -3.0376 1.2169 %! -5.3286 16.1955 8.4824 -0.2489 -12.2348 5.1590]; %! %! KB = [ 16.9788 14.1648 %! -18.9215 -15.6726 %! 25.2046 21.2848 %! 20.1122 16.8322 %! 1.4104 1.2040 %! 5.3181 4.5149]; %! %! KC = [ -9.1941 27.5165 13.7364 -0.3639 -21.5983 9.6025 %! 3.6490 -10.6194 -5.2772 0.2432 8.1108 -3.6293]; %! %! KD = [ 9.0317 7.5348 %! -3.4006 -2.8219]; %! %! M_exp = [KA, KB; KC, KD]; %! GAM_exp = 111.294; %! %!assert (M, M_exp, 1e-1); %!assert (GAM, GAM_exp, 1e-3); control/inst/hnamodred.m0000644000076500000240000003401112273411257014637 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @var{opt}, @dots{}) ## ## Model order reduction by frequency weighted optimal Hankel-norm (HNA) method. ## The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order ## @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} ## approximates the one from original system @var{G}. ## ## HNA is an absolute error method which tries to minimize ## @iftex ## @tex ## $$ || G - G_r ||_H = \\min $$ ## $$ || V \\ (G - G_r) \\ W ||_H = \\min $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ||G-Gr|| = min ## H ## ## ||V (G-Gr) W|| = min ## H ## @end example ## @end ifnottex ## where @var{V} and @var{W} denote output and input weightings. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model to be reduced. ## @item nr ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically according ## to the description of key @var{"order"}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Gr ## Reduced order state-space model. ## @item info ## Struct containing additional information. ## @table @var ## @item info.n ## The order of the original system @var{G}. ## @item info.ns ## The order of the @var{alpha}-stable subsystem of the original system @var{G}. ## @item info.hsv ## The Hankel singular values corresponding to the projection @code{op(V)*G1*op(W)}, ## where G1 denotes the @var{alpha}-stable part of the original system @var{G}. ## The @var{ns} Hankel singular values are ordered decreasingly. ## @item info.nu ## The order of the @var{alpha}-unstable subsystem of both the original ## system @var{G} and the reduced-order system @var{Gr}. ## @item info.nr ## The order of the obtained reduced order system @var{Gr}. ## @end table ## @end table ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'nr' ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is the sum of @var{info.nu} and the number of ## Hankel singular values greater than @code{max(tol1, ns*eps*info.hsv(1)}; ## ## @item 'method' ## Specifies the computational approach to be used. ## Valid values corresponding to this key are: ## @table @var ## @item 'descriptor' ## Use the inverse free descriptor system approach. ## @item 'standard' ## Use the inversion based standard approach. ## @item 'auto' ## Switch automatically to the inverse free ## descriptor approach in case of badly conditioned ## feedthrough matrices in V or W. Default method. ## @end table ## ## ## @item 'left', 'v' ## @acronym{LTI} model of the left/output frequency weighting. ## The weighting must be antistable. ## @iftex ## @math{|| V \\ (G-G_r) \\dots ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || V (G-Gr) . || = min ## H ## @end example ## @end ifnottex ## ## @item 'right', 'w' ## @acronym{LTI} model of the right/input frequency weighting. ## The weighting must be antistable. ## @iftex ## @math{|| \\dots (G-G_r) \\ W ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || . (G-Gr) W || = min ## H ## @end example ## @end ifnottex ## ## ## @item 'left-inv', 'inv-v' ## @acronym{LTI} model of the left/output frequency weighting. ## The weighting must have only antistable zeros. ## @iftex ## @math{|| inv(V) \\ (G-G_r) \\dots ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || inv(V) (G-Gr) . || = min ## H ## @end example ## @end ifnottex ## ## @item 'right-inv', 'inv-w' ## @acronym{LTI} model of the right/input frequency weighting. ## The weighting must have only antistable zeros. ## @iftex ## @math{|| \\dots (G-G_r) \\ inv(W) ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || . (G-Gr) inv(W) || = min ## H ## @end example ## @end ifnottex ## ## ## @item 'left-conj', 'conj-v' ## @acronym{LTI} model of the left/output frequency weighting. ## The weighting must be stable. ## @iftex ## @math{|| conj(V) \\ (G-G_r) \\dots ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || V (G-Gr) . || = min ## H ## @end example ## @end ifnottex ## ## @item 'right-conj', 'conj-w' ## @acronym{LTI} model of the right/input frequency weighting. ## The weighting must be stable. ## @iftex ## @math{|| \\dots (G-G_r) \\ conj(W) ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || . (G-Gr) W || = min ## H ## @end example ## @end ifnottex ## ## ## @item 'left-conj-inv', 'conj-inv-v' ## @acronym{LTI} model of the left/output frequency weighting. ## The weighting must be minimum-phase. ## @iftex ## @math{|| conj(inv(V)) \\ (G-G_r) \\dots ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || V (G-Gr) . || = min ## H ## @end example ## @end ifnottex ## ## @item 'right-conj-inv', 'conj-inv-w' ## @acronym{LTI} model of the right/input frequency weighting. ## The weighting must be minimum-phase. ## @iftex ## @math{|| \\dots (G-G_r) \\ conj(inv(W)) ||_H = \\min} ## @end iftex ## @ifnottex ## @example ## || . (G-Gr) W || = min ## H ## @end example ## @end ifnottex ## ## ## @item 'alpha' ## Specifies the ALPHA-stability boundary for the eigenvalues ## of the state dynamics matrix @var{G.A}. For a continuous-time ## system, ALPHA <= 0 is the boundary value for ## the real parts of eigenvalues, while for a discrete-time ## system, 0 <= ALPHA <= 1 represents the ## boundary value for the moduli of eigenvalues. ## The ALPHA-stability domain does not include the boundary. ## Default value is 0 for continuous-time systems and ## 1 for discrete-time systems. ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced model. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. ## @var{tol1} < 1. ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the ALPHA-stable part of the given ## model. @var{tol2} <= @var{tol1} < 1. ## If not specified, ns*eps*info.hsv(1) is chosen. ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on system @var{G} prior to order reduction. ## Default value is true if @code{G.scaled == false} and ## false if @code{G.scaled == true}. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## ## Approximation Properties: ## @itemize @bullet ## @item ## Guaranteed stability of reduced models ## @item ## Lower guaranteed error bound ## @item ## Guaranteed a priori error bound ## @iftex ## @tex ## $$ \\sigma_{r+1} \\leq || (G-G_r) ||_{\\infty} \\leq 2 \\sum_{j=r+1}^{n} \\sigma_j $$ ## @end tex ## @end iftex ## @end itemize ## ## @strong{Algorithm}@* ## Uses SLICOT AB09JD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2011 ## Version: 0.1 function [Gr, info] = hnamodred (G, varargin) if (nargin == 0) print_usage (); endif if (! isa (G, "lti")) error ("hnamodred: first argument must be an LTI system"); endif if (nargin > 1) # hnamodred (G, ...) if (is_real_scalar (varargin{1})) # hnamodred (G, nr) varargin = horzcat (varargin(2:end), {"order"}, varargin(1)); endif if (isstruct (varargin{1})) # hnamodred (G, opt, ...), hnamodred (G, nr, opt, ...) varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end)); endif ## order placed at the end such that nr from hnamodred (G, nr, ...) ## and hnamodred (G, nr, opt, ...) overrides possible nr's from ## key/value-pairs and inside opt struct (later keys override former keys, ## nr > key/value > opt) endif nkv = numel (varargin); # number of keys and values if (rem (nkv, 2)) error ("hnamodred: keys and values must come in pairs"); endif [a, b, c, d, tsam, scaled] = ssdata (G); [p, m] = size (G); dt = isdt (G); ## default arguments alpha = __modred_default_alpha__ (dt); av = bv = cv = dv = []; jobv = 0; aw = bw = cw = dw = []; jobw = 0; jobinv = 2; tol1 = 0; tol2 = 0; ordsel = 1; nr = 0; ## handle keys and values for k = 1 : 2 : nkv key = lower (varargin{k}); val = varargin{k+1}; switch (key) case {"left", "v", "wo"} [av, bv, cv, dv, jobv] = __modred_check_weight__ (val, dt, p, p); ## TODO: correct error messages for non-square weights case {"right", "w", "wi"} [aw, bw, cw, dw, jobw] = __modred_check_weight__ (val, dt, m, m); case {"left-inv", "inv-v"} [av, bv, cv, dv] = __modred_check_weight__ (val, dt, p, p); jobv = 2; case {"right-inv", "inv-w"} [aw, bw, cw, dw] = __modred_check_weight__ (val, dt, m, m); jobv = 2 case {"left-conj", "conj-v"} [av, bv, cv, dv] = __modred_check_weight__ (val, dt, p, p); jobv = 3; case {"right-conj", "conj-w"} [aw, bw, cw, dw] = __modred_check_weight__ (val, dt, m, m); jobv = 3 case {"left-conj-inv", "conj-inv-v"} [av, bv, cv, dv] = __modred_check_weight__ (val, dt, p, p); jobv = 4; case {"right-conj-inv", "conj-inv-w"} [aw, bw, cw, dw] = __modred_check_weight__ (val, dt, m, m); jobv = 4 case {"order", "nr"} [nr, ordsel] = __modred_check_order__ (val, rows (a)); case "tol1" tol1 = __modred_check_tol__ (val, "tol1"); case "tol2" tol2 = __modred_check_tol__ (val, "tol2"); case "alpha" alpha = __modred_check_alpha__ (val, dt); case "method" switch (tolower (val(1))) case {"d", "n"} # "descriptor" jobinv = 0; case {"s", "i"} # "standard" jobinv = 1; case "a" # {"auto", "automatic"} jobinv = 2; otherwise error ("hnamodred: invalid computational approach"); endswitch case {"equil", "equilibrate", "equilibration", "scale", "scaling"} scaled = __modred_check_equil__ (val); otherwise warning ("hnamodred: invalid property name '%s' ignored", key); endswitch endfor ## perform model order reduction [ar, br, cr, dr, nr, hsv, ns] = __sl_ab09jd__ (a, b, c, d, dt, scaled, nr, ordsel, alpha, ... jobv, av, bv, cv, dv, ... jobw, aw, bw, cw, dw, ... jobinv, tol1, tol2); ## assemble reduced order model Gr = ss (ar, br, cr, dr, tsam); ## assemble info struct n = rows (a); nu = n - ns; info = struct ("n", n, "ns", ns, "hsv", hsv, "nu", nu, "nr", nr); endfunction %!shared Mo, Me, Info, HSVe %! A = [ -3.8637 -7.4641 -9.1416 -7.4641 -3.8637 -1.0000 %! 1.0000, 0 0 0 0 0 %! 0 1.0000 0 0 0 0 %! 0 0 1.0000 0 0 0 %! 0 0 0 1.0000 0 0 %! 0 0 0 0 1.0000 0 ]; %! %! B = [ 1 %! 0 %! 0 %! 0 %! 0 %! 0 ]; %! %! C = [ 0 0 0 0 0 1 ]; %! %! D = [ 0 ]; %! %! G = ss (A, B, C, D); # "scaled", false %! %! AV = [ 0.2000 -1.0000 %! 1.0000 0 ]; %! %! BV = [ 1 %! 0 ]; %! %! CV = [ -1.8000 0 ]; %! %! DV = [ 1 ]; %! %! V = ss (AV, BV, CV, DV); %! %! [Gr, Info] = hnamodred (G, "left", V, "tol1", 1e-1, "tol2", 1e-14); %! [Ao, Bo, Co, Do] = ssdata (Gr); %! %! Ae = [ -0.2391 0.3072 1.1630 1.1967 %! -2.9709 -0.2391 2.6270 3.1027 %! 0.0000 0.0000 -0.5137 -1.2842 %! 0.0000 0.0000 0.1519 -0.5137 ]; %! %! Be = [ -1.0497 %! -3.7052 %! 0.8223 %! 0.7435 ]; %! %! Ce = [ -0.4466 0.0143 -0.4780 -0.2013 ]; %! %! De = [ 0.0219 ]; %! %! HSVe = [ 2.6790 2.1589 0.8424 0.1929 0.0219 0.0011 ].'; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); %!assert (Info.hsv, HSVe, 1e-4); control/inst/hsvd.m0000644000076500000240000000655512273411257013656 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{hsv} =} hsvd (@var{sys}) ## @deftypefnx{Function File} {@var{hsv} =} hsvd (@var{sys}, @var{"offset"}, @var{offset}) ## @deftypefnx{Function File} {@var{hsv} =} hsvd (@var{sys}, @var{"alpha"}, @var{alpha}) ## Hankel singular values of the stable part of an @acronym{LTI} model. If no output arguments are ## given, the Hankel singular values are displayed in a plot. ## ## @strong{Algorithm}@* ## Uses SLICOT AB13AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.4 function hsv_r = hsvd (sys, prop = "offset", val = 1e-8) if (nargin != 1 && nargin != 3) print_usage (); endif if (! isa (sys, "lti")) error ("hsvd: first argument must be an LTI system"); endif if (! is_real_scalar (val)) error ("hsvd: third argument must be a real scalar"); endif [a, b, c, ~, ~, scaled] = ssdata (sys); discrete = ! isct (sys); switch (tolower (prop(1))) case "o" # offset if (discrete) alpha = 1 - val; else alpha = - val; endif case "a" # alpha alpha = val; otherwise error ("hsvd: second argument invalid"); endswitch [hsv, ns] = __sl_ab13ad__ (a, b, c, discrete, alpha, scaled); if (nargout) hsv_r = hsv; else bar ((1:ns) + (rows (a) - ns), hsv); title (["Hankel Singular Values of Stable Part of ", inputname(1)]); xlabel ("State"); ylabel ("State Energy"); grid ("on"); endif endfunction %!shared hsv, hsv_exp %! a = [ -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 %! -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 %! 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 %! 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 %! 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 %! 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300]; %! %! b = [ 0.0000 0.0000 %! 12.5000 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 12.500 %! 0.0000 0.0000]; %! %! c = [ 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000]; %! %! sys = ss (a, b, c, [], "scaled", true); %! hsv = hsvd (sys, "alpha", 0.0); %! %! hsv_exp = [2.5139; 2.0846; 1.9178; 0.7666; 0.5473; 0.0253; 0.0246]; %! %!assert (hsv, hsv_exp, 1e-4); control/inst/impulse.m0000644000076500000240000000663612273411257014370 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {} impulse (@var{sys}) ## @deftypefnx{Function File} {} impulse (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx{Function File} {} impulse (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx{Function File} {} impulse (@var{sys1}, @dots{}, @var{t}) ## @deftypefnx{Function File} {} impulse (@var{sys1}, @dots{}, @var{tfinal}) ## @deftypefnx{Function File} {} impulse (@var{sys1}, @dots{}, @var{tfinal}, @var{dt}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{t}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{tfinal}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} impulse (@var{sys}, @var{tfinal}, @var{dt}) ## Impulse response of @acronym{LTI} system. ## If no output arguments are given, the response is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item t ## Time vector. Should be evenly spaced. If not specified, it is calculated by ## the poles of the system to reflect adequately the response transients. ## @item tfinal ## Optional simulation horizon. If not specified, it is calculated by ## the poles of the system to reflect adequately the response transients. ## @item dt ## Optional sampling time. Be sure to choose it small enough to capture transient ## phenomena. If not specified, it is calculated by the poles of the system. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item y ## Output response array. Has as many rows as time samples (length of t) ## and as many columns as outputs. ## @item t ## Time row vector. ## @item x ## State trajectories array. Has @code{length (t)} rows and as many columns as states. ## @end table ## ## @seealso{initial, lsim, step} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function [y_r, t_r, x_r] = impulse (varargin) if (nargin == 0) print_usage (); endif if (nargout) sysname = {}; else sys_idx = find (cellfun (@isa, varargin, {"lti"})); len = length (sys_idx); sysname = cell (len, 1); for k = 1 : len try sysname{k} = inputname(sys_idx(k)); catch sysname{k} = ""; end_try_catch endfor endif [y, t, x] = __time_response__ ("impulse", varargin, sysname, ! nargout); if (nargout) y_r = y{1}; t_r = t{1}; x_r = x{1}; endif endfunction control/inst/initial.m0000644000076500000240000001246612273411257014341 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {} initial (@var{sys}, @var{x0}) ## @deftypefnx{Function File} {} initial (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{x0}) ## @deftypefnx{Function File} {} initial (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}, @var{x0}) ## @deftypefnx{Function File} {} initial (@var{sys1}, @dots{}, @var{x0}, @var{t}) ## @deftypefnx{Function File} {} initial (@var{sys1}, @dots{}, @var{x0}, @var{tfinal}) ## @deftypefnx{Function File} {} initial (@var{sys1}, @dots{}, @var{x0}, @var{tfinal}, @var{dt}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{t}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{tfinal}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} initial (@var{sys}, @var{x0}, @var{tfinal}, @var{dt}) ## Initial condition response of state-space model. ## If no output arguments are given, the response is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## State-space model. ## @item x0 ## Vector of initial conditions for each state. ## @item t ## Optional time vector. Should be evenly spaced. If not specified, it is calculated ## by the poles of the system to reflect adequately the response transients. ## @item tfinal ## Optional simulation horizon. If not specified, it is calculated by ## the poles of the system to reflect adequately the response transients. ## @item dt ## Optional sampling time. Be sure to choose it small enough to capture transient ## phenomena. If not specified, it is calculated by the poles of the system. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item y ## Output response array. Has as many rows as time samples (length of t) ## and as many columns as outputs. ## @item t ## Time row vector. ## @item x ## State trajectories array. Has @code{length (t)} rows and as many columns as states. ## @end table ## ## @strong{Example} ## @example ## @group ## . ## Continuous Time: x = A x , y = C x , x(0) = x0 ## ## Discrete Time: x[k+1] = A x[k] , y[k] = C x[k] , x[0] = x0 ## @end group ## @end example ## ## @seealso{impulse, lsim, step} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function [y_r, t_r, x_r] = initial (varargin) if (nargin < 2) print_usage (); endif if (nargout) sysname = {}; else sys_idx = find (cellfun (@isa, varargin, {"lti"})); len = length (sys_idx); sysname = cell (len, 1); for k = 1 : len try sysname{k} = inputname(sys_idx(k)); catch sysname{k} = ""; end_try_catch endfor endif [y, t, x] = __time_response__ ("initial", varargin, sysname, ! nargout); if (nargout) y_r = y{1}; t_r = t{1}; x_r = x{1}; endif endfunction %!shared initial_c, initial_c_exp, initial_d, initial_d_exp %! %! A = [ -2.8 2.0 -1.8 %! -2.4 -2.0 0.8 %! 1.1 1.7 -1.0 ]; %! %! B = [ -0.8 0.5 0 %! 0 0.7 2.3 %! -0.3 -0.1 0.5 ]; %! %! C = [ -0.1 0 -0.3 %! 0.9 0.5 1.2 %! 0.1 -0.1 1.9 ]; %! %! D = [ -0.5 0 0 %! 0.1 0 0.3 %! -0.8 0 0 ]; %! %! x_0 = [1, 2, 3]; %! %! sysc = ss (A, B, C, D); %! %! [yc, tc, xc] = initial (sysc, x_0, 0.2, 0.1); %! initial_c = [yc, tc, xc]; %! %! sysd = c2d (sysc, 2); %! %! [yd, td, xd] = initial (sysd, x_0, 4); %! initial_d = [yd, td, xd]; %! %! ## expected values computed by the "dark side" %! %! yc_exp = [ -1.0000 5.5000 5.6000 %! -0.9872 5.0898 5.7671 %! -0.9536 4.6931 5.7598 ]; %! %! tc_exp = [ 0.0000 %! 0.1000 %! 0.2000 ]; %! %! xc_exp = [ 1.0000 2.0000 3.0000 %! 0.5937 1.6879 3.0929 %! 0.2390 1.5187 3.0988 ]; %! %! initial_c_exp = [yc_exp, tc_exp, xc_exp]; %! %! yd_exp = [ -1.0000 5.5000 5.6000 %! -0.6550 3.1673 4.2228 %! -0.5421 2.6186 3.4968 ]; %! %! td_exp = [ 0 %! 2 %! 4 ]; %! %! xd_exp = [ 1.0000 2.0000 3.0000 %! -0.4247 1.5194 2.3249 %! -0.3538 1.2540 1.9250 ]; %! %! initial_d_exp = [yd_exp, td_exp, xd_exp]; %! %!assert (initial_c, initial_c_exp, 1e-4) %!assert (initial_d, initial_d_exp, 1e-4) control/inst/isctrb.m0000644000076500000240000000722612273411257014174 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{sys}) ## @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{sys}, @var{tol}) ## @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}) ## @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{e}) ## @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{[]}, @var{tol}) ## @deftypefnx {Function File} {[@var{bool}, @var{ncon}] =} isctrb (@var{a}, @var{b}, @var{e}, @var{tol}) ## Logical check for system controllability. ## For numerical reasons, @code{isctrb (sys)} ## should be used instead of @code{rank (ctrb (sys))}. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. Descriptor state-space models are possible. ## If @var{sys} is not a state-space model, it is converted to ## a minimal state-space realization, so beware of pole-zero ## cancellations which may lead to wrong results! ## @item a ## State matrix (n-by-n). ## @item b ## Input matrix (n-by-m). ## @item e ## Descriptor matrix (n-by-n). ## If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. ## @item tol ## Optional roundoff parameter. Default value is 0. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool = 0 ## System is not controllable. ## @item bool = 1 ## System is controllable. ## @item ncon ## Number of controllable states. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT AB01OD and TG01HD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{isobsv} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.5 function [bool, ncont] = isctrb (a, b = [], e = [], tol = []) if (nargin < 1 || nargin > 4) print_usage (); elseif (isa (a, "lti")) # isctrb (sys), isctrb (sys, tol) if (nargin > 2) print_usage (); endif if (! isa (a, "ss")) warning ("isctrb: converting to minimal state-space realization"); endif tol = b; [a, b, c, d, e] = dssdata (a, []); elseif (nargin < 2) # isctrb (a, b), isctrb (a, b, tol) print_usage (); elseif (! is_real_square_matrix (a) || ! is_real_matrix (b) || rows (a) != rows (b)) error ("isctrb: a(%dx%d), b(%dx%d) not conformal", rows (a), columns (a), rows (b), columns (b)); elseif (! isempty (e) && (! is_real_square_matrix (e) || ! size_equal (e, a))) error ("isctrb: a(%dx%d), e(%dx%d) not conformal", rows (a), columns (a), rows (e), columns (e)); endif if (isempty (tol)) tol = 0; # default tolerance elseif (! is_real_scalar (tol)) error ("isctrb: tol must be a real scalar"); endif if (isempty (e)) [~, ~, ~, ncont] = __sl_ab01od__ (a, b, tol); else [~, ~, ~, ~, ~, ~, ncont] = __sl_tg01hd__ (a, e, b, zeros (1, columns (a)), tol); endif bool = (ncont == rows (a)); endfunction control/inst/isdetectable.m0000644000076500000240000000660412273411257015335 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} isdetectable (@var{sys}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{sys}, @var{tol}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{tol}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{tol}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{[]}, @var{dflg}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{[]}, @var{dflg}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{[]}, @var{tol}, @var{dflg}) ## @deftypefnx {Function File} {@var{bool} =} isdetectable (@var{a}, @var{c}, @var{e}, @var{tol}, @var{dflg}) ## Logical test for system detectability. ## All unstable modes must be observable or all unobservable states must be stable. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. ## @item a ## State transition matrix. ## @item c ## Measurement matrix. ## @item e ## Descriptor matrix. ## If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. ## @item tol ## Optional tolerance for stability. Default value is 0. ## @item dflg = 0 ## Matrices (@var{a}, @var{c}) are part of a continuous-time system. Default Value. ## @item dflg = 1 ## Matrices (@var{a}, @var{c}) are part of a discrete-time system. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool = 0 ## System is not detectable. ## @item bool = 1 ## System is detectable. ## @end table ## ## ## @strong{Algorithm}@* ## Uses SLICOT AB01OD and TG01HD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## See @command{isstabilizable} for description of computational method. ## @seealso{isstabilizable, isstable, isctrb, isobsv} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.3 function bool = isdetectable (a, c = [], e = [], tol = [], dflg = 0) if (nargin == 0) print_usage (); elseif (isa (a, "lti")) # isdetectable (sys), isdetectable (sys, tol) if (nargin > 2) print_usage (); endif bool = isstabilizable (a.', c); # transpose is overloaded elseif (nargin < 2 || nargin > 5) print_usage (); else # isdetectable (a, c, ...) bool = isstabilizable (a.', c.', e.', tol, dflg); # arguments checked inside endif endfunction control/inst/isobsv.m0000644000076500000240000000535412273411257014213 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{sys}) ## @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{sys}, @var{tol}) ## @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}) ## @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{e}) ## @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{[]}, @var{tol}) ## @deftypefnx {Function File} {[@var{bool}, @var{nobs}] =} isobsv (@var{a}, @var{c}, @var{e}, @var{tol}) ## Logical check for system observability. ## For numerical reasons, @code{isobsv (sys)} ## should be used instead of @code{rank (obsv (sys))}. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. Descriptor state-space models are possible. ## @item a ## State matrix (n-by-n). ## @item c ## Measurement matrix (p-by-n). ## @item e ## Descriptor matrix (n-by-n). ## If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. ## @item tol ## Optional roundoff parameter. Default value is 0. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool = 0 ## System is not observable. ## @item bool = 1 ## System is observable. ## @item nobs ## Number of observable states. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT AB01OD and TG01HD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{isctrb} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.4 function [bool, nobs] = isobsv (a, c = [], e = [], tol = []) if (nargin == 0) print_usage (); elseif (isa (a, "lti")) # isobsv (sys), isobsv (sys, tol) if (nargin > 2) print_usage (); endif [bool, nobs] = isctrb (a.', c); # transpose is overloaded elseif (nargin < 2 || nargin > 4) print_usage (); else # isobsv (a, c), isobsv (a, c, e), ... [bool, nobs] = isctrb (a.', c.', e.', tol); endif endfunction control/inst/issample.m0000644000076500000240000000661712273411257014526 0ustar lukasstaff## Copyright (C) 1996, 2000, 2002, 2003, 2004, 2005, 2007 ## Auburn University. All rights reserved. ## ## This program is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} issample (@var{ts}) ## @deftypefnx {Function File} {@var{bool} =} issample (@var{ts}, @var{flg}) ## Return true if @var{ts} is a valid sampling time. ## ## @strong{Inputs} ## @table @var ## @item ts ## Alleged sampling time to be tested. ## @item flg = 1 ## Accept real scalars @var{ts} > 0. Default Value. ## @item flg = 0 ## Accept real scalars @var{ts} >= 0. ## @item flg = -1 ## Accept real scalars @var{ts} > 0 and @var{ts} == -1. ## @item flg = -10 ## Accept real scalars @var{ts} >= 0 and @var{ts} == -1. ## @item flg = -2 ## Accept real scalars @var{ts} >= 0, @var{ts} == -1 and @var{ts} == -2. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool ## True if conditions are met and false otherwise. ## @end table ## ## @end deftypefn ## Author: A. S. Hodel ## Created: July 1995 ## Adapted-By: Lukas Reichlin ## Date: September 2009 ## Version: 0.3 function bool = issample (tsam, flg = 1) if (nargin < 1 || nargin > 2) print_usage (); endif switch (flg) case 1 # discrete bool = is_real_scalar (tsam) && (tsam > 0); case 0 # continuous or discrete bool = is_real_scalar (tsam) && (tsam >= 0); case -1 # discrete, tsam unspecified bool = is_real_scalar (tsam) && (tsam > 0 || tsam == -1); case -10 # continuous or discrete, tsam unspecified bool = is_real_scalar (tsam) && (tsam >= 0 || tsam == -1); case -2 # accept static gains bool = is_real_scalar (tsam) && (tsam >= 0 || tsam == -1 || tsam == -2); otherwise print_usage (); endswitch endfunction ## flg == 1 %!assert (issample (1)) %!assert (issample (pi)) %!assert (issample (0), false) %!assert (issample (-1), false) %!assert (issample (-1, 1), false) %!assert (issample ("a"), false) %!assert (issample (eye (2)), false) %!assert (issample (2+2i), false) ## flg == 0 %!assert (issample (1, 0)) %!assert (issample (0, 0)) %!assert (issample (-1, 0), false) %!assert (issample (pi, 0)) %!assert (issample ("b", 0), false) %!assert (issample (rand (3,2), 0), false) %!assert (issample (2+2i, 0), false) %!assert (issample (0+2i, 0), false) ## flg == -1 %!assert (issample (-1, -1)) %!assert (issample (0, -1), false) %!assert (issample (1, -1)) %!assert (issample (pi, -1)) %!assert (issample (-pi, -1), false) %!assert (issample ("b", -1), false) %!assert (issample (rand (3,2), -1), false) %!assert (issample (-2+2i, -1), false) ## errors %!error (issample (-1, "ab")) %!error (issample ()) %!error (issample (-1, -1, -1)) %!error (issample (1, pi)) %!error (issample (5, rand (2,3))) %!error (issample (0, 1+2i)) control/inst/isstabilizable.m0000644000076500000240000001225512273411257015705 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{bool} =} isstabilizable (@var{sys}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{sys}, @var{tol}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{tol}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{tol}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{[]}, @var{dflg}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{[]}, @var{dflg}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{[]}, @var{tol}, @var{dflg}) ## @deftypefnx {Function File} {@var{bool} =} isstabilizable (@var{a}, @var{b}, @var{e}, @var{tol}, @var{dflg}) ## Logical check for system stabilizability. ## All unstable modes must be controllable or all uncontrollable states must be stable. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. If @var{sys} is not a state-space system, it is converted to ## a minimal state-space realization, so beware of pole-zero cancellations ## which may lead to wrong results! ## @item a ## State transition matrix. ## @item b ## Input matrix. ## @item e ## Descriptor matrix. ## If @var{e} is empty @code{[]} or not specified, an identity matrix is assumed. ## @item tol ## Optional tolerance for stability. Default value is 0. ## @item dflg = 0 ## Matrices (@var{a}, @var{b}) are part of a continuous-time system. Default Value. ## @item dflg = 1 ## Matrices (@var{a}, @var{b}) are part of a discrete-time system. ## @end table ## ## @strong{Outputs} ## @table @var ## @item bool = 0 ## System is not stabilizable. ## @item bool = 1 ## System is stabilizable. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT AB01OD and TG01HD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @example ## @group ## * Calculate staircase form (SLICOT AB01OD) ## * Extract unobservable part of state transition matrix ## * Calculate eigenvalues of unobservable part ## * Check whether ## real (ev) < -tol*(1 + abs (ev)) continuous-time ## abs (ev) < 1 - tol discrete-time ## @end group ## @end example ## @seealso{isdetectable, isstable, isctrb, isobsv} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.5 function bool = isstabilizable (a, b = [], e = [], tol = [], dflg = 0) if (nargin < 1 || nargin > 5) print_usage (); elseif (isa (a, "lti")) # isstabilizable (sys), isstabilizable (sys, tol) if (nargin > 2) print_usage (); endif if (! isa (a, "ss")) warning ("isstabilizable: converting to minimal state-space realization"); endif tol = b; dflg = ! isct (a); [a, b, c, d, e] = dssdata (a, []); elseif (nargin == 1) # isstabilizable (a, b, ...) print_usage (); elseif (! is_real_square_matrix (a) || rows (a) != rows (b)) error ("isstabilizable: a must be square and conformal to b"); elseif (! isempty (e) && (! is_real_square_matrix (e) || ! size_equal (a, e))) error ("isstabilizable: e must be square and conformal to a"); endif if (isempty (tol)) tol = 0; # default tolerance elseif (! is_real_scalar (tol)) error ("isstabilizable: tol must be a real scalar"); endif if (isempty (e)) ## controllability staircase form [ac, ~, ~, ncont] = __sl_ab01od__ (a, b, tol); ## extract uncontrollable part of staircase form uncont_idx = ncont+1 : rows (a); auncont = ac(uncont_idx, uncont_idx); ## calculate poles of uncontrollable part pol = eig (auncont); else ## controllability staircase form - output matrix c has no influence [ac, ec, ~, ~, ~, ~, ncont] = __sl_tg01hd__ (a, e, b, zeros (1, columns (a)), tol); ## extract uncontrollable part of staircase form uncont_idx = ncont+1 : rows (a); auncont = ac(uncont_idx, uncont_idx); euncont = ec(uncont_idx, uncont_idx); ## calculate poles of uncontrollable part pol = eig (auncont, euncont); ## remove infinite poles tolinf = norm ([auncont, euncont], 2); idx = find (abs (pol) < tolinf/eps); pol = pol(idx); endif ## check whether uncontrollable poles are stable bool = __is_stable__ (pol, ! dflg, tol); endfunction control/inst/kalman.m0000644000076500000240000000740412273411257014147 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{[]}, @var{sensors}, @var{known}) ## @deftypefnx {Function File} {[@var{est}, @var{g}, @var{x}] =} kalman (@var{sys}, @var{q}, @var{r}, @var{s}, @var{sensors}, @var{known}) ## Design Kalman estimator for @acronym{LTI} systems. ## ## @strong{Inputs} ## @table @var ## @item sys ## Nominal plant model. ## @item q ## Covariance of white process noise. ## @item r ## Covariance of white measurement noise. ## @item s ## Optional cross term covariance. Default value is 0. ## @item sensors ## Indices of measured output signals y from @var{sys}. If omitted, all outputs are measured. ## @item known ## Indices of known input signals u (deterministic) to @var{sys}. All other inputs to @var{sys} ## are assumed stochastic. If argument @var{known} is omitted, no inputs u are known. ## @end table ## ## @strong{Outputs} ## @table @var ## @item est ## State-space model of the Kalman estimator. ## @item g ## Estimator gain. ## @item x ## Solution of the Riccati equation. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## u +-------+ ^ ## +---------------------------->| |-------> y ## | +-------+ + y | est | ^ ## u ----+--->| |----->(+)------>| |-------> x ## | sys | ^ + +-------+ ## w -------->| | | ## +-------+ | v ## ## Q = cov (w, w') R = cov (v, v') S = cov (w, v') ## @end group ## @end example ## ## @seealso{care, dare, estim, lqr} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.3 function [est, k, x] = kalman (sys, q, r, s = [], sensors = [], deterministic = []) ## TODO: type "current" for discrete-time systems if (nargin < 3 || nargin > 6 || ! isa (sys, "lti")) print_usage (); endif [a, b, c, d, e] = dssdata (sys, []); if (isempty (sensors)) sensors = 1 : rows (c); endif stochastic = setdiff (1 : columns (b), deterministic); c = c(sensors, :); g = b(:, stochastic); h = d(sensors, stochastic); if (isempty (s)) rbar = r + h*q*h.'; sbar = g * q*h.'; else rbar = r + h*q*h.'+ h*s + s.'*h.'; sbar = g * (q*h.' + s); endif if (isct (sys)) [x, l, k] = care (a.', c.', g*q*g.', rbar, sbar, e.'); else [x, l, k] = dare (a.', c.', g*q*g.', rbar, sbar, e.'); endif k = k.'; est = estim (sys, k, sensors, deterministic); endfunction %!shared m, m_exp, g, g_exp, x, x_exp %! sys = ss (-2, 1, 1, 3); %! [est, g, x] = kalman (sys, 1, 1, 1); %! [a, b, c, d] = ssdata (est); %! m = [a, b; c, d]; %! m_exp = [-2.25, 0.25; 1, 0; 1, 0]; %! g_exp = 0.25; %! x_exp = 0; %!assert (m, m_exp, 1e-2); %!assert (g, g_exp, 1e-2); %!assert (x, x_exp, 1e-2); control/inst/lqe.m0000644000076500000240000001000312273411257013452 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## Copyright (C) 2012 Megan Zagrobelny ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{sys}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{sys}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{g}, @var{c}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{l}, @var{p}, @var{e}] =} lqe (@var{a}, @var{[]}, @var{c}, @var{q}, @var{r}, @var{s}) ## Kalman filter for continuous-time systems. ## ## @example ## @group ## . ## x = Ax + Bu + Gw (State equation) ## y = Cx + Du + v (Measurement Equation) ## E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S ## @end group ## @end example ## ## @strong{Inputs} ## @table @var ## @item sys ## Continuous or discrete-time @acronym{LTI} model (p-by-m, n states). ## @item a ## State matrix of continuous-time system (n-by-n). ## @item g ## Process noise matrix of continuous-time system (n-by-g). ## If @var{g} is empty @code{[]}, an identity matrix is assumed. ## @item c ## Measurement matrix of continuous-time system (p-by-n). ## @item q ## Process noise covariance matrix (g-by-g). ## @item r ## Measurement noise covariance matrix (p-by-p). ## @item s ## Optional cross term covariance matrix (g-by-p), s = cov(w,v). ## If @var{s} is empty @code{[]} or not specified, a zero matrix is assumed. ## @end table ## ## @strong{Outputs} ## @table @var ## @item l ## Kalman filter gain matrix (n-by-p). ## @item p ## Unique stabilizing solution of the continuous-time Riccati equation (n-by-n). ## Symmetric matrix. If @var{sys} is a discrete-time model, the solution of the ## corresponding discrete-time Riccati equation is returned. ## @item e ## Closed-loop poles (n-by-1). ## @end table ## ## @strong{Equations} ## @example ## @group ## . ## x = Ax + Bu + L(y - Cx -Du) ## ## E = eig(A - L*C) ## ## @end group ## @end example ## @seealso{dare, care, dlqr, lqr, dlqe} ## @end deftypefn ## Author: Lukas Reichlin ## Created: April 2012 ## Version: 0.1 function [l, p, e] = lqe (a, g, c, q = [], r = [], s = []) if (nargin < 3 || nargin > 6) print_usage (); endif if (isa (a, "lti")) [l, p, e] = lqr (a.', g, c, q); # lqe (sys, q, r, s), g=I, works like lqr (sys.', q, r, s).' elseif (isempty (g)) [l, p, e] = lqr (a.', c.', q, r, s); # lqe (a, [], c, q, r, s), g=I, works like lqr (a.', c.', q, r, s).' elseif (columns (g) != rows (q) || ! issquare (q)) error ("lqe: matrices g(%dx%d) and q(%dx%d) have incompatible dimensions", ... rows (g), columns (g), rows (q), columns (q)); elseif (isempty (s)) [l, p, e] = lqr (a.', c.', g*q*g.', r); elseif (columns (g) != rows (s)) error ("lqe: matrices g(%dx%d) and s(%dx%d) have incompatible dimensions", ... rows (g), columns (g), rows (s), columns (s)); else [l, p, e] = lqr (a.', c.', g*q*g.', r, g*s); endif l = l.'; ## NOTE: for discrete-time sys, the solution L' from DARE ## is different to L from DLQE (a, s) endfunction control/inst/lqr.m0000644000076500000240000000575112273411257013505 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{sys}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{sys}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e}) ## @deftypefnx {Function File} {[@var{g}, @var{x}, @var{l}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e}) ## Linear-quadratic regulator. ## ## @strong{Inputs} ## @table @var ## @item sys ## Continuous or discrete-time @acronym{LTI} model (p-by-m, n states). ## @item a ## State matrix of continuous-time system (n-by-n). ## @item b ## Input matrix of continuous-time system (n-by-m). ## @item q ## State weighting matrix (n-by-n). ## @item r ## Input weighting matrix (m-by-m). ## @item s ## Optional cross term matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed. ## @item e ## Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed. ## @end table ## ## @strong{Outputs} ## @table @var ## @item g ## State feedback matrix (m-by-n). ## @item x ## Unique stabilizing solution of the continuous-time Riccati equation (n-by-n). ## @item l ## Closed-loop poles (n-by-1). ## @end table ## ## @strong{Equations} ## @example ## @group ## . ## x = A x + B u, x(0) = x0 ## ## inf ## J(x0) = INT (x' Q x + u' R u + 2 x' S u) dt ## 0 ## ## L = eig (A - B*G) ## @end group ## @end example ## @seealso{care, dare, dlqr} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.2 function [g, x, l] = lqr (a, b, q, r = [], s = [], e = []) if (nargin < 3 || nargin > 6) print_usage (); endif if (isa (a, "lti")) s = r; r = q; q = b; [a, b, c, d, e, tsam] = dssdata (a, []); elseif (nargin < 4) print_usage (); else tsam = 0; endif if (issample (tsam, -1)) [x, l, g] = dare (a, b, q, r, s, e); else [x, l, g] = care (a, b, q, r, s, e); endif endfunction control/inst/lsim.m0000644000076500000240000002310112273411257013640 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {} lsim (@var{sys}, @var{u}) ## @deftypefnx{Function File} {} lsim (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{u}) ## @deftypefnx{Function File} {} lsim (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}, @var{u}) ## @deftypefnx{Function File} {} lsim (@var{sys1}, @dots{}, @var{u}, @var{t}) ## @deftypefnx{Function File} {} lsim (@var{sys1}, @dots{}, @var{u}, @var{t}, @var{x0}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{x0}) ## Simulate @acronym{LTI} model response to arbitrary inputs. If no output arguments are given, ## the system response is plotted on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. System must be proper, i.e. it must not have more zeros than poles. ## @item u ## Vector or array of input signal. Needs @code{length(t)} rows and as many columns ## as there are inputs. If @var{sys} is a single-input system, row vectors @var{u} ## of length @code{length(t)} are accepted as well. ## @item t ## Time vector. Should be evenly spaced. If @var{sys} is a continuous-time system ## and @var{t} is a real scalar, @var{sys} is discretized with sampling time ## @code{tsam = t/(rows(u)-1)}. If @var{sys} is a discrete-time system and @var{t} ## is not specified, vector @var{t} is assumed to be @code{0 : tsam : tsam*(rows(u)-1)}. ## @item x0 ## Vector of initial conditions for each state. If not specified, a zero vector is assumed. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item y ## Output response array. Has as many rows as time samples (length of t) ## and as many columns as outputs. ## @item t ## Time row vector. It is always evenly spaced. ## @item x ## State trajectories array. Has @code{length (t)} rows and as many columns as states. ## @end table ## ## @seealso{impulse, initial, step} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.4 function [y_r, t_r, x_r] = lsim (varargin) ## TODO: individual initial state vectors 'x0' for each system ## there would be conflicts with other arguments, ## maybe a cell {x0_1, x0_2, ..., x0_N} would be a solution? if (nargin < 2) print_usage (); endif sys_idx = find (cellfun (@isa, varargin, {"lti"})); # look for LTI models, 'find' needed for plot styles sys_cell = cellfun (@ss, varargin(sys_idx), "uniformoutput", false); # convert to state-space if (! size_equal (sys_cell{:})) error ("lsim: models must have equal sizes"); endif mat_idx = find (cellfun (@is_real_matrix, varargin)); # indices of matrix arguments n_mat = length (mat_idx); # number of vector arguments n_sys = length (sys_cell); # number of LTI systems t = []; x0 = []; if (n_mat < 1) error ("lsim: require input signal 'u'"); else arg = varargin{mat_idx(1)}; if (is_real_vector (arg)) u = reshape (arg, [], 1); # allow row vectors for single-input systems elseif (is_real_matrix (arg)); u = arg; else error ("lsim: input signal 'u' must be an array of real numbers"); endif if (n_mat > 1) # time vector t arg = varargin{mat_idx(2)}; if (is_real_vector (arg) || isempty (arg)) t = arg; else error ("lsim: time vector 't' must be real-valued or empty"); endif if (n_mat > 2) # initial state vector x0 arg = varargin{mat_idx(3)}; if (is_real_vector (arg)) x0 = arg; else error ("lsim: initial state vector 'x0' must be a real-valued vector"); endif if (n_mat > 3) warning ("lsim: ignored"); endif endif endif endif ## function [y, t, x_arr] = __linear_simulation__ (sys, u, t, x0) [y, t, x] = cellfun (@__linear_simulation__, sys_cell, {u}, {t}, {x0}, "uniformoutput", false); if (nargout == 0) # plot information [p, m] = size (sys_cell{1}); style_idx = find (cellfun (@ischar, varargin)); ct_idx = cellfun (@isct, sys_cell); str = "Linear Simulation Results"; outname = get (sys_cell{end}, "outname"); outname = __labels__ (outname, "y"); colororder = get (gca, "colororder"); rc = rows (colororder); sysname = cell (n_sys, 1); for k = 1 : n_sys # for every system if (k == n_sys) lim = nargin; else lim = sys_idx(k+1); endif style = varargin(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); if (isempty (style)) color = colororder(1+rem (k-1, rc), :); style = {"color", color}; endif try sysname{k} = inputname(sys_idx(k)); catch sysname{k} = ""; end_try_catch if (ct_idx(k)) # continuous-time system for i = 1 : p # for every output if (p != 1) subplot (p, 1, i); endif plot (t{k}, y{k}(:, i), style{:}); hold on; grid on; if (k == n_sys) axis tight ylim (__axis_margin__ (ylim)) ylabel (outname{i}); if (i == 1) title (str); endif endif endfor else # discrete-time system for i = 1 : p # for every output if (p != 1) subplot (p, 1, i); endif stairs (t{k}, y{k}(:, i), style{:}); hold on; grid on; if (k == n_sys) axis tight; ylim (__axis_margin__ (ylim)) ylabel (outname{i}); if (i == 1) title (str); endif endif endfor endif endfor xlabel ("Time [s]"); if (p == 1 && m == 1) legend (sysname) endif hold off; else # return values y_r = y{1}; t_r = t{1}; x_r = x{1}; endif endfunction function [y, t, x_arr] = __linear_simulation__ (sys, u, t, x0) method = "zoh"; [urows, ucols] = size (u); if (isct (sys)) # continuous-time system if (isempty (t)) # lsim (sys, u, [], ...) error ("lsim: invalid time vector"); elseif (length (t) == 1) # lsim (sys, u, tfinal, ...) dt = t / (urows - 1); tinitial = 0; tfinal = t; else # lsim (sys, u, t, ...) dt = t(2) - t(1); # assume that t is regularly spaced tinitial = t(1); tfinal = t(end); endif sys = c2d (sys, dt, method); # convert to discrete-time model else # discrete-time system dt = abs (get (sys, "tsam")); # use 1 second as default if tsam is unspecified (-1) if (isempty (t)) # lsim (sys, u) tinitial = 0; tfinal = dt * (urows - 1); elseif (length (t) == 1) # lsim (sys, u, tfinal) tinitial = 0; tfinal = t; else # lsim (sys, u, t, ...) warning ("lsim: spacing of time vector has no effect on sampling time of discrete-time system"); tinitial = t(1); tfinal = t(end); endif endif [A, B, C, D] = ssdata (sys); [p, m] = size (D); # number of outputs and inputs n = rows (A); # number of states ## time vector t = reshape (tinitial : dt : tfinal, [], 1); len_t = length (t); if (urows != len_t) error ("lsim: input vector u must have %d rows", len_t); endif if (ucols != m) error ("lsim: input vector u must have %d columns", m); endif ## preallocate memory y = zeros (len_t, p); x_arr = zeros (len_t, n); ## initial conditions if (isempty (x0)) x0 = zeros (n, 1); elseif (n != length (x0) || ! is_real_vector (x0)) error ("lsim: x0 must be a vector with %d elements", n); endif x = reshape (x0, [], 1); # make sure that x is a column vector ## simulation for k = 1 : len_t y(k, :) = C * x + D * u(k, :).'; x_arr(k, :) = x; x = A * x + B * u(k, :).'; endfor endfunction ## TODO: add test cases control/inst/ltimodels.m0000644000076500000240000003240612273411257014700 0ustar lukasstaff## Copyright (C) 2009 Luca Favatella ## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} test ltimodels ## @deftypefnx {Function File} ltimodels ## @deftypefnx {Function File} ltimodels (@var{systype}) ## Test suite and help for @acronym{LTI} models. ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.2 function ltimodels (systype = "general") %if (nargin > 1) print_usage (); %endif ## TODO: write documentation if (! ischar (systype)) error ("ltimodels: argument must be a string"); endif systype = lower (systype); switch (systype) case "ss" str = {"State Space (SS) Models"... "-----------------------"... ""}; case "tf" str = {"Transfer Function (TF) Models"... "-----------------------------"... ""}; otherwise # general str = {"Linear Time Invariant (LTI) Models"... "----------------------------------"... ""}; endswitch disp (""); disp (char (str)); endfunction ## ============================================================================== ## LTI Tests ## ============================================================================== ## isct, isdt %!shared ltisys %! ltisys = tf (12); %!assert (ltisys.ts, -2); %!assert (isct (ltisys)); %!assert (isdt (ltisys)); %!shared ltisys %! ltisys = ss (17); %!assert (ltisys.ts, -2); %!assert (isct (ltisys)); %!assert (isdt (ltisys)); %!shared ltisys %! ltisys = tf (1, [1 1]); %!assert (ltisys.ts, 0); %!assert (isct (ltisys)); %!assert (! isdt (ltisys)); %!shared ltisys, ts %! ts = 0.1; %! ltisys = ss (-1, 1, 1, 0, ts); %!assert (ltisys.ts, ts); %!assert (! isct (ltisys)); %!assert (isdt (ltisys)); ## ============================================================================== ## TF Tests ## ============================================================================== ## ============================================================================== ## SS Tests ## ============================================================================== ## staircase (SLICOT AB01OD) %!shared Ac, Bc, Ace, Bce %! A = [ 17.0 24.0 1.0 8.0 15.0 %! 23.0 5.0 7.0 14.0 16.0 %! 4.0 6.0 13.0 20.0 22.0 %! 10.0 12.0 19.0 21.0 3.0 %! 11.0 18.0 25.0 2.0 9.0 ]; %! %! B = [ -1.0 -4.0 %! 4.0 9.0 %! -9.0 -16.0 %! 16.0 25.0 %! -25.0 -36.0 ]; %! %! tol = 0; %! %! A = A.'; # There's a little mistake in the example %! # program of routine AB01OD in SLICOT 5.0 %! %! [Ac, Bc, U, ncont] = __sl_ab01od__ (A, B, tol); %! %! Ace = [ 12.8848 3.2345 11.8211 3.3758 -0.8982 %! 4.4741 -12.5544 5.3509 5.9403 1.4360 %! 14.4576 7.6855 23.1452 26.3872 -29.9557 %! 0.0000 1.4805 27.4668 22.6564 -0.0072 %! 0.0000 0.0000 -30.4822 0.6745 18.8680 ]; %! %! Bce = [ 31.1199 47.6865 %! 3.2480 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 %! 0.0000 0.0000 ]; %! %!assert (Ac, Ace, 1e-4); %!assert (Bc, Bce, 1e-4); ## controllability staircase form of descriptor state-space models (SLICOT TG01HD) %!shared ac, ec, bc, cc, q, z, ncont, ac_e, ec_e, bc_e, cc_e, q_e, z_e, ncont_e %! %! a = [ 2 0 2 0 -1 3 1 %! 0 1 0 0 1 0 0 %! 0 0 0 1 0 0 1 %! 0 0 2 0 -1 3 1 %! 0 0 0 1 0 0 1 %! 0 1 0 0 1 0 0 %! 0 0 0 1 0 0 1 ]; %! %! e = [ 0 0 1 0 0 0 0 %! 0 0 0 0 0 1 0 %! 0 0 0 0 0 0 1 %! 0 0 0 0 0 0 1 %! 0 0 0 1 0 0 0 %! 0 0 1 0 -1 0 0 %! 1 3 0 2 0 0 0 ]; %! %! b = [ 2 1 0 %! 0 0 0 %! 0 0 0 %! 0 0 0 %! 0 0 0 %! 0 0 0 %! 1 2 3 ]; %! %! c = [ 1 0 0 1 0 0 1 %! 0 -1 1 0 -1 1 0 ]; %! %! tol = 0; %! %! [ac, ec, bc, cc, q, z, ncont] = __sl_tg01hd__ (a, e, b, c, tol); %! %! ncont_e = 3; %! %! ac_e = [ 0.0000 0.0000 0.0000 0.0000 -1.2627 0.4334 0.4666 %! 0.0000 2.0000 0.0000 -3.7417 -0.8520 0.2924 -0.4342 %! 0.0000 0.0000 1.7862 0.3780 -0.2651 -0.7723 0.0000 %! 0.0000 0.0000 0.0000 3.7417 0.8520 -0.2924 0.4342 %! 0.0000 0.0000 0.0000 0.0000 -1.5540 0.5334 0.5742 %! 0.0000 0.0000 0.0000 0.0000 -0.6533 0.2242 0.2414 %! 0.0000 0.0000 0.0000 0.0000 -0.5892 0.2022 0.2177 ]; %! %! ec_e = [ -1.8325 1.0000 2.3752 0.0000 -0.8214 0.2819 1.8016 %! 0.4887 0.0000 0.3770 -0.5345 0.1874 0.5461 0.0000 %! -0.1728 0.0000 -0.1333 -1.1339 0.1325 0.3861 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.8520 -0.2924 0.4342 %! 0.0000 0.0000 0.0000 0.0000 -1.0260 -0.1496 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 1.1937 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 ]; %! %! bc_e = [ 1.0000 2.0000 3.0000 %! 2.0000 1.0000 0.0000 %! 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 ]; %! %! cc_e = [ 0.0000 1.0000 0.0000 0.0000 -1.2627 0.4334 0.4666 %! 0.3665 0.0000 -0.9803 -1.6036 0.1874 0.5461 0.0000 ]; %! %! q_e = [ 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.7071 0.0000 0.2740 -0.6519 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.8304 0.3491 -0.4342 %! 0.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.4003 0.1683 0.9008 %! 0.0000 0.0000 0.7071 0.0000 -0.2740 0.6519 0.0000 %! 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ]; %! %! z_e = [ 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 %! -0.6108 0.0000 0.7917 0.0000 0.0000 0.0000 0.0000 %! 0.4887 0.0000 0.3770 -0.5345 0.1874 0.5461 0.0000 %! 0.0000 0.0000 0.0000 0.0000 -0.4107 0.1410 0.9008 %! 0.6108 0.0000 0.4713 0.2673 -0.1874 -0.5461 0.0000 %! -0.1222 0.0000 -0.0943 -0.8018 -0.1874 -0.5461 0.0000 %! 0.0000 0.0000 0.0000 0.0000 -0.8520 0.2924 -0.4342 ]; %! %!assert (ac, ac_e, 1e-4); %!assert (ec, ec_e, 1e-4); %!assert (bc, bc_e, 1e-4); %!assert (cc, cc_e, 1e-4); %!assert (q, q_e, 1e-4); %!assert (z, z_e, 1e-4); %!assert (ncont, ncont_e); ## observability staircase form of descriptor state-space models (SLICOT TG01ID) %!shared ao, eo, bo, co, q, z, nobsv, ao_e, eo_e, bo_e, co_e, q_e, z_e, nobsv_e %! %! a = [ 2 0 0 0 0 0 0 %! 0 1 0 0 0 1 0 %! 2 0 0 2 0 0 0 %! 0 0 1 0 1 0 1 %! -1 1 0 -1 0 1 0 %! 3 0 0 3 0 0 0 %! 1 0 1 1 1 0 1 ]; %! %! e = [ 0 0 0 0 0 0 1 %! 0 0 0 0 0 0 3 %! 1 0 0 0 0 1 0 %! 0 0 0 0 1 0 2 %! 0 0 0 0 0 -1 0 %! 0 1 0 0 0 0 0 %! 0 0 1 1 0 0 0 ]; %! %! b = [ 1 0 %! 0 -1 %! 0 1 %! 1 0 %! 0 -1 %! 0 1 %! 1 0 ]; %! %! c = [ 2 0 0 0 0 0 1 %! 1 0 0 0 0 0 2 %! 0 0 0 0 0 0 3 ]; %! %! tol = 0; %! %! [ao, eo, bo, co, q, z, nobsv] = __sl_tg01id__ (a, e, b, c, tol); %! %! nobsv_e = 3; %! %! ao_e = [ 0.2177 0.2414 0.5742 0.4342 0.0000 -0.4342 0.4666 %! 0.2022 0.2242 0.5334 -0.2924 -0.7723 0.2924 0.4334 %! -0.5892 -0.6533 -1.5540 0.8520 -0.2651 -0.8520 -1.2627 %! 0.0000 0.0000 0.0000 3.7417 0.3780 -3.7417 0.0000 %! 0.0000 0.0000 0.0000 0.0000 1.7862 0.0000 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 2.0000 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ]; %! %! eo_e = [ 1.0000 0.0000 0.0000 0.4342 0.0000 0.0000 1.8016 %! 0.0000 1.1937 -0.1496 -0.2924 0.3861 0.5461 0.2819 %! 0.0000 0.0000 -1.0260 0.8520 0.1325 0.1874 -0.8214 %! 0.0000 0.0000 0.0000 0.0000 -1.1339 -0.5345 0.0000 %! 0.0000 0.0000 0.0000 0.0000 -0.1333 0.3770 2.3752 %! 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 %! 0.0000 0.0000 0.0000 0.0000 -0.1728 0.4887 -1.8325 ]; %! %! bo_e = [ 0.4666 0.0000 %! 0.4334 0.5461 %! -1.2627 0.1874 %! 0.0000 -1.6036 %! 0.0000 -0.9803 %! 1.0000 0.0000 %! 0.0000 0.3665 ]; %! %! co_e = [ 0.0000 0.0000 0.0000 0.0000 0.0000 2.0000 1.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 2.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.0000 ]; %! %! q_e = [ 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.7917 0.0000 -0.6108 %! 0.0000 0.5461 0.1874 -0.5345 0.3770 0.0000 0.4887 %! 0.9008 0.1410 -0.4107 0.0000 0.0000 0.0000 0.0000 %! 0.0000 -0.5461 -0.1874 0.2673 0.4713 0.0000 0.6108 %! 0.0000 -0.5461 -0.1874 -0.8018 -0.0943 0.0000 -0.1222 %! -0.4342 0.2924 -0.8520 0.0000 0.0000 0.0000 0.0000 ]; %! %! z_e = [ 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 %! 0.0000 -0.6519 0.2740 0.0000 0.7071 0.0000 0.0000 %! -0.4342 0.3491 0.8304 0.0000 0.0000 0.0000 0.0000 %! 0.0000 0.0000 0.0000 -1.0000 0.0000 0.0000 0.0000 %! 0.9008 0.1683 0.4003 0.0000 0.0000 0.0000 0.0000 %! 0.0000 0.6519 -0.2740 0.0000 0.7071 0.0000 0.0000 %! 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 ]; %! %!assert (ao, ao_e, 1e-4); %!assert (eo, eo_e, 1e-4); %!assert (bo, bo_e, 1e-4); %!assert (co, co_e, 1e-4); %!assert (q, q_e, 1e-4); %!assert (z, z_e, 1e-4); %!assert (nobsv, nobsv_e); ## ss2tf conversion by Slicot TB04BD ## Test provided by Slicot %!shared NUM, DEN, NUMe, DENe %! A = [ -1.0 0.0 0.0 %! 0.0 -2.0 0.0 %! 0.0 0.0 -3.0 ]; %! %! B = [ 0.0 1.0 -1.0 %! 1.0 1.0 0.0 ].'; %! %! C = [ 0.0 1.0 1.0 %! 1.0 1.0 1.0 ]; %! %! D = [ 1.0 0.0 %! 0.0 1.0 ]; %! %! [NUM, DEN] = tfdata (ss (A, B, C, D)); %! %! NUMe = {[1, 5, 7], [1]; [1], [1, 5, 5]}; %! %! DENe = {[1, 5, 6], [1, 2]; [1, 5, 6], [1, 3, 2]}; %! %!assert (NUM, NUMe, 1e-4); %!assert (DEN, DENe, 1e-4); ## ss2tf conversion by Slicot TB04BD ## Trivial test %!shared NUM, DEN, NUMe, DENe %! A = [ 0 ]; %! %! B = [ 1 ]; %! %! C = [ 1 ]; %! %! D = [ 0 ]; %! %! [NUM, DEN] = tfdata (ss (A, B, C, D)); %! %! NUMe = {[1]}; %! %! DENe = {[1, 0]}; %! %!assert (NUM, NUMe, 1e-4); %!assert (DEN, DENe, 1e-4); ## transfer function to state-space conversion ## test from SLICOT TD04AD %!shared Mo, Me %! INDEX = [ 3 3 ]; %! %! DCOEFF = [ 1.0 6.0 11.0 6.0 %! 1.0 6.0 11.0 6.0 ]; %! %! UCOEFF = zeros (2, 2, 4); %! %! u11 = [ 1.0 6.0 12.0 7.0 ]; %! u12 = [ 0.0 1.0 4.0 3.0 ]; %! u21 = [ 0.0 0.0 1.0 1.0 ]; %! u22 = [ 1.0 8.0 20.0 15.0 ]; %! %! UCOEFF(1,1,:) = u11; %! UCOEFF(1,2,:) = u12; %! UCOEFF(2,1,:) = u21; %! UCOEFF(2,2,:) = u22; %! %! [Ao, Bo, Co, Do] = __sl_td04ad__ (UCOEFF, DCOEFF, INDEX, 0); %! %! Ae = [ 0.5000 -0.8028 0.9387 %! 4.4047 -2.3380 2.5076 %! -5.5541 1.6872 -4.1620 ]; %! %! Be = [ -0.2000 -1.2500 %! 0.0000 -0.6097 %! 0.0000 2.2217 ]; %! %! Ce = [ 0.0000 -0.8679 0.2119 %! 0.0000 0.0000 0.9002 ]; %! %! De = [ 1.0000 0.0000 %! 0.0000 1.0000 ]; %! %! Mo = [Ao, Bo; Co, Do]; %! Me = [Ae, Be; Ce, De]; %! %!assert (Mo, Me, 1e-4); control/inst/lyap.m0000644000076500000240000001202112273411257013640 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{x} =} lyap (@var{a}, @var{b}) ## @deftypefnx{Function File} {@var{x} =} lyap (@var{a}, @var{b}, @var{c}) ## @deftypefnx{Function File} {@var{x} =} lyap (@var{a}, @var{b}, @var{[]}, @var{e}) ## Solve continuous-time Lyapunov or Sylvester equations. ## ## @strong{Equations} ## @example ## @group ## AX + XA' + B = 0 (Lyapunov Equation) ## ## AX + XB + C = 0 (Sylvester Equation) ## ## AXE' + EXA' + B = 0 (Generalized Lyapunov Equation) ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB03MD, SB04MD and SG03AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{lyapchol, dlyap, dlyapchol} ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.2.1 function [x, scale] = lyap (a, b, c, e) scale = 1; switch (nargin) case 2 # Lyapunov equation if (! is_real_square_matrix (a, b)) ## error ("lyap: a, b must be real and square"); error ("lyap: %s, %s must be real and square", ... inputname (1), inputname (2)); endif if (rows (a) != rows (b)) ## error ("lyap: a, b must have the same number of rows"); error ("lyap: %s, %s must have the same number of rows", ... inputname (1), inputname (2)); endif [x, scale] = __sl_sb03md__ (a, -b, false); # AX + XA' = -B ## x /= scale; # 0 < scale <= 1 case 3 # Sylvester equation if (! is_real_square_matrix (a, b)) ## error ("lyap: a, b must be real and square"); error ("lyap: %s, %s must be real and square", ... inputname (1), inputname (2)); endif if (! is_real_matrix (c) || rows (c) != rows (a) || columns (c) != columns (b)) ## error ("lyap: c must be a real (%dx%d) matrix", rows (a), columns (b)); error ("lyap: %s must be a real (%dx%d) matrix", ... rows (a), columns (b), inputname (3)); endif x = __sl_sb04md__ (a, b, -c); # AX + XB = -C case 4 # generalized Lyapunov equation if (! isempty (c)) print_usage (); endif if (! is_real_square_matrix (a, b, e)) ## error ("lyap: a, b, e must be real and square"); error ("lyap: %s, %s, %s must be real and square", ... inputname (1), inputname (2), inputname (4)); endif if (rows (b) != rows (a) || rows (e) != rows (a)) ## error ("lyap: a, b, e must have the same number of rows"); error ("lyap: %s, %s, %s must have the same number of rows", ... inputname (1), inputname (2), inputname (4)); endif if (! issymmetric (b)) ## error ("lyap: b must be symmetric"); error ("lyap: %s must be symmetric", ... inputname (2)); endif [x, scale] = __sl_sg03ad__ (a, e, -b, false); # AXE' + EXA' = -B ## x /= scale; # 0 < scale <= 1 otherwise print_usage (); endswitch if (scale < 1) warning ("lyap: solution scaled by %g to prevent overflow", scale); endif endfunction ## Lyapunov %!shared X, X_exp %! A = [1, 2; -3, -4]; %! Q = [3, 1; 1, 1]; %! X = lyap (A, Q); %! X_exp = [ 6.1667, -3.8333; %! -3.8333, 3.0000]; %!assert (X, X_exp, 1e-4); ## Sylvester %!shared X, X_exp %! A = [2.0 1.0 3.0 %! 0.0 2.0 1.0 %! 6.0 1.0 2.0]; %! %! B = [2.0 1.0 %! 1.0 6.0]; %! %! C = [2.0 1.0 %! 1.0 4.0 %! 0.0 5.0]; %! %! X = lyap (A, B, -C); %! %! X_exp = [-2.7685 0.5498 %! -1.0531 0.6865 %! 4.5257 -0.4389]; %! %!assert (X, X_exp, 1e-4); ## Generalized Lyapunov %!shared X, X_exp %! A = [ 3.0 1.0 1.0 %! 1.0 3.0 0.0 %! 1.0 0.0 2.0]; %! %! E = [ 1.0 3.0 0.0 %! 3.0 2.0 1.0 %! 1.0 0.0 1.0]; %! %! B = [-64.0 -73.0 -28.0 %! -73.0 -70.0 -25.0 %! -28.0 -25.0 -18.0]; %! %! X = lyap (A.', -B, [], E.'); %! %! X_exp = [-2.0000 -1.0000 0.0000 %! -1.0000 -3.0000 -1.0000 %! 0.0000 -1.0000 -3.0000]; %! %!assert (X, X_exp, 1e-4); control/inst/lyapchol.m0000644000076500000240000001072112273411257014513 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{u} =} lyapchol (@var{a}, @var{b}) ## @deftypefnx{Function File} {@var{u} =} lyapchol (@var{a}, @var{b}, @var{e}) ## Compute Cholesky factor of continuous-time Lyapunov equations. ## ## @strong{Equations} ## @example ## @group ## A U' U + U' U A' + B B' = 0 (Lyapunov Equation) ## ## A U' U E' + E U' U A' + B B' = 0 (Generalized Lyapunov Equation) ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB03OD and SG03BD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @seealso{lyap, dlyap, dlyapchol} ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.2.1 function [u, scale] = lyapchol (a, b, e) switch (nargin) case 2 if (! is_real_square_matrix (a)) ## error ("lyapchol: a must be real and square"); error ("lyapchol: %s must be real and square", ... inputname (1)); endif if (! is_real_matrix (b)) ## error ("lyapchol: b must be real") error ("lyapchol: %s must be real", ... inputname (2)) endif if (rows (a) != rows (b)) ## error ("lyapchol: a and b must have the same number of rows"); error ("lyapchol: %s and %s must have the same number of rows", ... inputname (1), inputname (2)); endif [u, scale] = __sl_sb03od__ (a.', b.', false); ## NOTE: TRANS = 'T' not suitable because we need U' U, not U U' case 3 if (! is_real_square_matrix (a, e)) ## error ("lyapchol: a, e must be real and square"); error ("lyapchol: %s, %s must be real and square", ... inputname (1), inputname (3)); endif if (! is_real_matrix (b)) ## error ("lyapchol: b must be real"); error ("lyapchol: %s must be real", ... inputname (2)); endif if (rows (b) != rows (a) || rows (e) != rows (a)) ## error ("lyapchol: a, b, e must have the same number of rows"); error ("lyapchol: %s, %s, %s must have the same number of rows", ... inputname (1), inputname (2), inputname (3)); endif [u, scale] = __sl_sg03bd__ (a.', e.', b.', false); ## NOTE: TRANS = 'T' not suitable because we need U' U, not U U' otherwise print_usage (); endswitch if (scale < 1) warning ("lyapchol: solution scaled by %g to prevent overflow", scale); endif endfunction %!shared U, U_exp, X, X_exp %! %! A = [ -1.0 37.0 -12.0 -12.0 %! -1.0 -10.0 0.0 4.0 %! 2.0 -4.0 7.0 -6.0 %! 2.0 2.0 7.0 -9.0 ].'; %! %! B = [ 1.0 2.5 1.0 3.5 %! 0.0 1.0 0.0 1.0 %! -1.0 -2.5 -1.0 -1.5 %! 1.0 2.5 4.0 -5.5 %! -1.0 -2.5 -4.0 3.5 ].'; %! %! U = lyapchol (A, B); %! %! X = U.' * U; # use lyap at home! %! %! U_exp = [ 1.0000 0.0000 0.0000 0.0000 %! 3.0000 1.0000 0.0000 0.0000 %! 2.0000 -1.0000 1.0000 0.0000 %! -1.0000 1.0000 -2.0000 1.0000 ].'; %! %! X_exp = [ 1.0000 3.0000 2.0000 -1.0000 %! 3.0000 10.0000 5.0000 -2.0000 %! 2.0000 5.0000 6.0000 -5.0000 %! -1.0000 -2.0000 -5.0000 7.0000 ]; %! %!assert (U, U_exp, 1e-4); %!assert (X, X_exp, 1e-4); %!shared U, U_exp, X, X_exp %! %! A = [ -1.0 3.0 -4.0 %! 0.0 5.0 -2.0 %! -4.0 4.0 1.0 ].'; %! %! E = [ 2.0 1.0 3.0 %! 2.0 0.0 1.0 %! 4.0 5.0 1.0 ].'; %! %! B = [ 2.0 -1.0 7.0 ].'; %! %! U = lyapchol (A, B, E); %! %! U_exp = [ 1.6003 -0.4418 -0.1523 %! 0.0000 0.6795 -0.2499 %! 0.0000 0.0000 0.2041 ]; %! %!assert (U, U_exp, 1e-4); control/inst/Madievski.m0000644000076500000240000001252112273411257014614 0ustar lukasstaff%% -*- texinfo -*- %% Demonstration of frequency-weighted controller reduction. %% The system considered in this example has been studied by Madievski and %% Anderson [1] and comprises four spinning disks. The disks are connected by a %% flexible rod, a motor applies torque to the third disk, and the angular %% displacement of the first disk is the variable of interest. The state-space %% model of eighth order is non-minimumphase and unstable. %% The continuous-time LQG controller used in [1] is open-loop stable and of %% eighth order like the plant. This eighth-order controller shall be reduced by %% frequency-weighted singular perturbation approximation (SPA). %% The major aim of this reduction is the preservation of the closed-loop %% transfer function. This means that the error in approximation of the %% controller @var{K} by the reduced-order controller @var{Kr} is minimized by %% @iftex %% @tex %% $$ \\underset{K_r}{\\min} \\ || W \\ (K - K_r) \\ V ||_{\\infty} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% min ||W (K-Kr) V|| %% Kr inf %% @end example %% @end ifnottex %% where weights @var{W} and @var{V} are dictated by the requirement to preserve %% (as far as possible) the closed-loop transfer function. In minimizing the %% error, they cause the approximation process for @var{K} to be more accurate at %% certain frequencies. Suggested by [1] is the use of the following stability %% and performance enforcing weights: %% @iftex %% @tex %% $$ W = (I - G K)^{-1} G, \\qquad V = (I - G K)^{-1} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% -1 -1 %% W = (I - G K) G, V = (I - G K) %% @end example %% @end ifnottex %% This example script reduces the eighth-order controller to orders four and two %% by the function call %% @code{Kr = spaconred (G, K, nr, 'feedback', '-')} %% where argument @var{nr} denotes the desired order (4 or 2). The key-value %% pair @code{'feedback', '-'} allows the reduction of negative feedback %% controllers while the default setting expects positive feedback controllers. %% The frequency responses of the original and reduced-order controllers are %% depicted in figure 1, the step responses of the closed loop in figure 2. %% There is no visible difference between the step responses of the closed-loop %% systems with original (blue) and fourth order (green) controllers. %% The second order controller (red) causes ripples in the step response, but %% otherwise the behavior of the system is unaltered. This leads to the %% conclusion that function @command{spaconred} is well suited to reduce the %% order of controllers considerably, while stability and performance are %% retained. %% @*@strong{Reference}@* %% [1] Madievski, A.G. and Anderson, B.D.O. %% @cite{Sampled-Data Controller Reduction Procedure}, %% IEEE Transactions of Automatic Control, %% Vol. 40, No. 11, November 1995 % =============================================================================== % Frequency Weighted Controller Reduction Lukas Reichlin December 2011 % =============================================================================== % Tabula Rasa clear all, close all, clc % Plant Ap1 = [ 0.0 1.0 0.0 0.0 ]; Ap2 = [ -0.015 0.765 -0.765 -0.015 ]; Ap3 = [ -0.028 1.410 -1.410 -0.028 ]; Ap4 = [ -0.04 1.85 -1.85 -0.04 ]; Ap = blkdiag (Ap1, Ap2, Ap3, Ap4); Bp = [ 0.026 -0.251 0.033 -0.886 -4.017 0.145 3.604 0.280 ]; Cp = [ -0.996 -0.105 0.261 0.009 -0.001 -0.043 0.002 -0.026 ]; Dp = [ 0.0 ]; P = ss (Ap, Bp, Cp, Dp); % Controller Ac = [ -0.4077 0.9741 0.1073 0.0131 0.0023 -0.0186 -0.0003 -0.0098 -0.0977 -0.1750 0.0215 -0.0896 -0.0260 0.0057 0.0109 -0.0105 0.0011 0.0218 -0.0148 0.7769 0.0034 -0.0013 -0.0014 0.0011 -0.0361 -0.5853 -0.7701 -0.3341 -0.0915 0.0334 0.0378 -0.0290 -0.1716 -2.6546 -0.0210 -1.4467 -0.4428 1.5611 0.1715 -0.1318 -0.0020 0.0950 0.0029 0.0523 -1.3950 -0.0338 -0.0062 0.0045 0.1607 2.3824 0.0170 1.2979 0.3721 -0.1353 -0.1938 1.9685 -0.0006 0.1837 0.0048 0.1010 0.0289 -0.0111 -1.8619 -0.0311 ]; Bc = [ -0.4105 -0.0868 -0.0004 0.0036 0.0081 -0.0085 -0.0004 -0.0132 ]; Cc = [ -0.0447 -0.6611 -0.0047 -0.3601 -0.1033 0.0375 0.0427 -0.0329 ]; Dc = [ 0.0 ]; K = ss (Ac, Bc, Cc, Dc); % Controller Reduction Kr4 = spaconred (P, K, 4, 'feedback', '-') Kr2 = spaconred (P, K, 2, 'feedback', '-') % Open Loop L = P * K; Lr4 = P * Kr4; Lr2 = P * Kr2; % Closed Loop T = feedback (L); Tr4 = feedback (Lr4); Tr2 = feedback (Lr2); % Frequency Range w = {1e-2, 1e1}; % Bode Plot of Controller figure (1) bode (K, Kr4, Kr2, w) legend ('K (8 states)', 'Kr (4 states)', 'Kr (2 states)', 'Location', 'SouthWest') % Step Response of Closed Loop figure (2) step (T, Tr4, Tr2, 100) legend ('K (8 states)', 'Kr (4 states)', 'Kr (2 states)', 'Location', 'SouthEast') control/inst/mag2db.m0000644000076500000240000000264512273411257014042 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{db} =} mag2db (@var{mag}) ## Convert Magnitude to Decibels (dB). ## ## @strong{Inputs} ## @table @var ## @item mag ## Magnitude value(s). Both real-valued scalars and matrices are accepted. ## @end table ## ## @strong{Outputs} ## @table @var ## @item db ## Decibel (dB) value(s). ## @end table ## ## @seealso{db2mag} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2012 ## Version: 0.1 function db = mag2db (mag) if (nargin != 1 || ! is_real_matrix (mag)) print_usage (); endif db = 20 .* log10 (mag); db(mag < 0) = NaN; endfunction %!assert (mag2db (100), 40); %!assert (mag2db (0.1), -20); control/inst/margin.m0000644000076500000240000003065512273411257014165 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{gamma}, @var{phi}, @var{w_gamma}, @var{w_phi}] =} margin (@var{sys}) ## @deftypefnx{Function File} {[@var{gamma}, @var{phi}, @var{w_gamma}, @var{w_phi}] =} margin (@var{sys}, @var{tol}) ## Gain and phase margin of a system. ## If no output arguments are given, both gain and phase margin are plotted on a bode diagram. ## Otherwise, the margins and their corresponding frequencies are computed and returned. ## A more robust criterion to assess the stability of a feedback system is the sensitivity Ms ## computed by command @command{sensitivity}. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. Must be a single-input and single-output (SISO) system. ## @item tol ## Imaginary parts below @var{tol} are assumed to be zero. ## If not specified, default value @code{sqrt (eps)} is taken. ## @end table ## ## @strong{Outputs} ## @table @var ## @item gamma ## Gain margin (as gain, not dBs). ## @item phi ## Phase margin (in degrees). ## @item w_gamma ## Frequency for the gain margin (in rad/s). ## @item w_phi ## Frequency for the phase margin (in rad/s). ## @end table ## ## @strong{Algorithm}@* ## Uses command @command{roots} to calculate the frequencies ## @var{w_gamma}, @var{w_phi} from special polynomials created ## from the transfer function of @var{sys} as listed below ## in section @guillemetleft{}Equations@guillemetright{}. ## ## @strong{Equations} ## @example ## @group ## CONTINUOUS-TIME SYSTEMS ## Gain Margin ## _ _ ## L(jw) = L(jw) BTW: L(jw) = L(-jw) = conj (L(jw)) ## ## num(jw) num(-jw) ## ------- = -------- ## den(jw) den(-jw) ## ## num(jw) den(-jw) = num(-jw) den(jw) ## ## imag (num(jw) den(-jw)) = 0 ## imag (num(-jw) den(jw)) = 0 ## @end group ## @end example ## @example ## @group ## Phase Margin ## |num(jw)| ## |L(jw)| = |-------| = 1 ## |den(jw)| ## _ 2 2 ## z z = Re z + Im z ## ## num(jw) num(-jw) ## ------- * -------- = 1 ## den(jw) den(-jw) ## ## num(jw) num(-jw) - den(jw) den(-jw) = 0 ## ## real (num(jw) num(-jw) - den(jw) den(-jw)) = 0 ## @end group ## @end example ## @example ## @group ## DISCRETE-TIME SYSTEMS ## Gain Margin ## jwT log z ## L(z) = L(1/z) BTW: z = e --> w = ----- ## j T ## num(z) num(1/z) ## ------ = -------- ## den(z) den(1/z) ## ## num(z) den(1/z) - num(1/z) den(z) = 0 ## @end group ## @end example ## @example ## @group ## Phase Margin ## |num(z)| ## |L(z)| = |------| = 1 ## |den(z)| ## @end group ## @end example ## @example ## @group ## L(z) L(1/z) = 1 ## ## num(z) num(1/z) ## ------ * -------- = 1 ## den(z) den(1/z) ## ## num(z) num(1/z) - den(z) den(1/z) = 0 ## @end group ## @end example ## @example ## @group ## PS: How to get L(1/z) ## 4 3 2 ## p(z) = a z + b z + c z + d z + e ## ## -4 -3 -2 -1 ## p(1/z) = a z + b z + c z + d z + e ## ## -4 2 3 4 ## = z ( a + b z + c z + d z + e z ) ## ## 4 3 2 4 ## = ( e z + d z + c z + b z + a ) / ( z ) ## @end group ## @end example ## ## @seealso{sensitivity, roots} ## @end deftypefn ## Author: Lukas Reichlin ## Created: July 2009 ## Version: 0.9.1 function [gamma_r, phi_r, w_gamma_r, w_phi_r] = margin (sys, tol = sqrt (eps)) ## TODO: multiplot feature: margin (sys1, "b", sys2, "r", ...) ## check whether arguments are OK if (nargin < 1 || nargin > 2) print_usage (); endif if (! isa (sys, "lti")) error ("margin: argument sys must be an LTI system"); endif if (! issiso (sys)) error ("margin: argument sys must be a SISO system"); endif ## get transfer function [num, den, tsam] = tfdata (sys, "vector"); continuous = isct (sys); tsam = abs (tsam); # use 1 second as default if tsam == -1 if (continuous) # CONTINUOUS-TIME SYSTEM ## create polynomials s -> jw l_num = length (num); l_den = length (den); num_jw = num .* i.^(l_num-1 : -1 : 0); den_jw = den .* i.^(l_den-1 : -1 : 0); ## GAIN MARGIN ## create gm polynomial gm_poly = imag (conv (num_jw, conj (den_jw))); ## find frequencies w w = roots (gm_poly); ## filter results [gamma, w_gamma] = gm_filter (w, num, den, tsam, tol, continuous); ## PHASE MARGIN ## create pm polynomials poly_1 = conv (num_jw, conj (num_jw)); poly_2 = conv (den_jw, conj (den_jw)); ## make polynomials equally long for subtraction [poly_1, poly_2] = poly_equalizer (poly_1, poly_2); ## subtract polynomials pm_poly = real (poly_1 - poly_2); ## find frequencies w w = roots (pm_poly); ## filter results [phi, w_phi] = pm_filter (w, num, den, tsam, tol, continuous); else # DISCRETE-TIME SYSTEM ## create polynomials z -> 1/z l_num = length (num); l_den = length (den); num_rev = fliplr (num); den_rev = fliplr (den); num_div = zeros (1, l_num); den_div = zeros (1, l_den); num_div(1) = 1; den_div(1) = 1; num_inv = conv (num_rev, den_div); den_inv = conv (den_rev, num_div); ## GAIN MARGIN ## create gm polynomial poly_1 = conv (num, den_inv); poly_2 = conv (num_inv, den); ## make polynomials equally long for subtraction [poly_1, poly_2] = poly_equalizer (poly_1, poly_2); ## subtract polynomials gm_poly = poly_1 - poly_2; ## find frequencies z z = roots (gm_poly); ## filter results idx = find (abs (abs (z) - 1) < tol); # find z with magnitude 1 if (length (idx) > 0) # if z with magnitude 1 exist z_gm = z(idx); w = log (z_gm) / (i*tsam); # get frequencies w from z [gamma, w_gamma] = gm_filter (w, num, den, tsam, tol, continuous); else # there are no z with magnitude 1 gamma = Inf; w_gamma = NaN; endif ## PHASE MARGIN ## create pm polynomials poly_1 = conv (num, num_inv); poly_2 = conv (den, den_inv); ## make polynomials equally long for subtraction [poly_1, poly_2] = poly_equalizer (poly_1, poly_2); ## subtract polynomials pm_poly = poly_1 - poly_2; ## find frequencies z z = roots (pm_poly); ## filter results idx = find (abs (abs (z) - 1) < tol); # find z with magnitude 1 if (length (idx) > 0) # if z with magnitude 1 exist z_gm = z(idx); w = log (z_gm) / (i*tsam); # get frequencies w from z [phi, w_phi] = pm_filter (w, num, den, tsam, tol, continuous); else # there are no z with magnitude 1 phi = 180; w_phi = NaN; endif endif if (nargout == 0) # show bode diagram [H, w] = __frequency_response__ (sys, false, "std"); H = reshape (H, [], 1); mag_db = 20 * log10 (abs (H)); pha = unwrap (arg (H)) * 180 / pi; gamma_db = 20 * log10 (gamma); wv = [min(w), max(w)]; ax_vec_mag = __axis_limits__ ([w(:), mag_db(:)]); ax_vec_mag(1:2) = wv; ax_vec_pha = __axis_limits__ ([w(:), pha(:)]); ax_vec_pha(1:2) = wv; wgm = [w_gamma, w_gamma]; mgmh = [-gamma_db, ax_vec_mag(3)]; mgm = [0, -gamma_db]; pgm = [ax_vec_pha(4), -180]; wpm = [w_phi, w_phi]; mpm = [0, ax_vec_mag(3)]; ppmh = [ax_vec_pha(4), phi - 180]; ppm = [phi - 180, -180]; title_str = sprintf ("GM = %g dB (at %g rad/s), PM = %g deg (at %g rad/s)", gamma_db, w_gamma, phi, w_phi); if (continuous) xl_str = "Frequency [rad/s]"; else xl_str = sprintf ("Frequency [rad/s] w_N = %g", pi/tsam); endif subplot (2, 1, 1) semilogx (w, mag_db, "b", wv, [0, 0], "-.k", wgm, mgmh, "-.k", wgm, mgm, "r", wpm, mpm, "-.k") axis (ax_vec_mag) grid ("on") title (title_str) ylabel ("Magnitude [dB]") subplot (2, 1, 2) semilogx (w, pha, "b", wv, [-180, -180], "-.k", wgm, pgm, "-.k", wpm, ppmh, "-.k", wpm, ppm, "r") axis (ax_vec_pha) grid ("on") xlabel (xl_str) ylabel ("Phase [deg]") else # return values gamma_r = gamma; phi_r = phi; w_gamma_r = w_gamma; w_phi_r = w_phi; endif endfunction function [poly_eq_1, poly_eq_2] = poly_equalizer (poly_1, poly_2) l_p1 = length (poly_1); l_p2 = length (poly_2); l_max = max (l_p1, l_p2); lead_zer_1 = zeros (1, l_max - l_p1); lead_zer_2 = zeros (1, l_max - l_p2); poly_eq_1 = horzcat (lead_zer_1, poly_1); poly_eq_2 = horzcat (lead_zer_2, poly_2); endfunction function [gamma, w_gamma] = gm_filter (w, num, den, tsam, tol, continuous) idx = find ((abs (imag (w)) < tol) & (real (w) > 0)); # find frequencies in R+ if (length (idx) > 0) # if frequencies in R+ exist w_gm = real (w(idx)); if (continuous) s = i * w_gm; else s = exp (i * w_gm * tsam); endif f_resp = polyval (num, s) ./ polyval (den, s); gm = (abs (f_resp)).^-1; ## find crossings between 0 and -1 idx = find ((real (f_resp) < 0) & (real (f_resp) >= -1)); if (length (idx) > 0) # if crossings between 0 and -1 exist gm = gm(idx); w_gm = w_gm(idx); [gamma, idx] = min (gm); w_gamma = w_gm(idx); else # there are no crossings between 0 and -1 idx = find (real (f_resp) < -1); # find crossings between -1 and -Inf if (length (idx) > 0) # if crossings between -1 and -Inf exist gm = gm(idx); w_gm = w_gm(idx); [gamma, idx] = max (gm); w_gamma = w_gm(idx); else gamma = Inf; w_gamma = NaN; endif endif else # there are no frequencies in R+ gamma = Inf; w_gamma = NaN; endif endfunction function [phi, w_phi] = pm_filter (w, num, den, tsam, tol, continuous) idx = find ((abs (imag (w)) < tol) & (real (w) > 0)); # find frequencies in R+ if (length (idx) > 0) # if frequencies in R+ exist w_pm = real (w(idx)); if (continuous) s = i * w_pm; else s = exp (i * w_pm * tsam); endif f_resp = polyval (num, s) ./ polyval (den, s); pm = 180 + arg (f_resp) ./ pi .* 180; [phi, idx] = min (pm); w_phi = w_pm(idx); else # there are no frequencies in R+ phi = 180; w_phi = NaN; endif endfunction %!shared margin_c, margin_c_exp, margin_d, margin_d_exp %! sysc = tf ([24], [1, 6, 11, 6]); %! [gamma_c, phi_c, w_gamma_c, w_phi_c] = margin (sysc); %! sysd = c2d (sysc, 0.3); %! [gamma_d, phi_d, w_gamma_d, w_phi_d] = margin (sysd); %! %! margin_c = [gamma_c, phi_c, w_gamma_c, w_phi_c]; %! margin_d = [gamma_d, phi_d, w_gamma_d, w_phi_d]; %! %! ## results from this implementation and the "dark side" diverge %! ## from the third digit after the decimal point on %! %! gamma_c_exp = 2.50; %! phi_c_exp = 35.43; %! w_gamma_c_exp = 3.32; %! w_phi_c_exp = 2.06; %! %! gamma_d_exp = 1.41; %! phi_d_exp = 18.60; %! w_gamma_d_exp = 2.48; %! w_phi_d_exp = 2.04; %! %! margin_c_exp = [gamma_c_exp, phi_c_exp, w_gamma_c_exp, w_phi_c_exp]; %! margin_d_exp = [gamma_d_exp, phi_d_exp, w_gamma_d_exp, w_phi_d_exp]; %! %!assert (margin_c, margin_c_exp, 1e-2); %!assert (margin_d, margin_d_exp, 1e-2); control/inst/MDSSystem.m0000644000076500000240000001310412273411257014526 0ustar lukasstaff%% -*- texinfo -*- %% Robust control of a mass-damper-spring system. %% Type @code{which MDSSystem} to locate, %% @code{edit MDSSystem} to open and simply %% @code{MDSSystem} to run the example file. % =============================================================================== % Robust Control of a Mass-Damper-Spring System Lukas Reichlin August 2011 % =============================================================================== % Reference: Gu, D.W., Petkov, P.Hr. and Konstantinov, M.M. % Robust Control Design with Matlab, Springer 2005 % =============================================================================== % Tabula Rasa clear all, close all, clc % =============================================================================== % System Model % =============================================================================== % +---------------+ % | d_m 0 0 | % +-----| 0 d_c 0 |<----+ % u_m | | 0 0 d_k | | y_m % u_c | +---------------+ | y_c % u_k | | y_k % | +---------------+ | % +---->| |-----+ % | G_nom | % u ----->| |-----> y % +---------------+ % Nominal Values m_nom = 3; % mass c_nom = 1; % damping coefficient k_nom = 2; % spring stiffness % Perturbations p_m = 0.4; % 40% uncertainty in the mass p_c = 0.2; % 20% uncertainty in the damping coefficient p_k = 0.3; % 30% uncertainty in the spring stiffness % State-Space Representation A = [ 0, 1 -k_nom/m_nom, -c_nom/m_nom ]; B1 = [ 0, 0, 0 -p_m, -p_c/m_nom, -p_k/m_nom ]; B2 = [ 0 1/m_nom ]; C1 = [ -k_nom/m_nom, -c_nom/m_nom 0, c_nom k_nom, 0 ]; C2 = [ 1, 0 ]; D11 = [ -p_m, -p_c/m_nom, -p_k/m_nom 0, 0, 0 0, 0, 0 ]; D12 = [ 1/m_nom 0 0 ]; D21 = [ 0, 0, 0 ]; D22 = [ 0 ]; inname = {'u_m', 'u_c', 'u_k', 'u'}; % input names outname = {'y_m', 'y_c', 'y_k', 'y'}; % output names G_nom = ss (A, [B1, B2], [C1; C2], [D11, D12; D21, D22], ... 'inputname', inname, 'outputname', outname); G = G_nom('y', 'u'); % extract output y and input u % =============================================================================== % Frequency Analysis of Uncertain System % =============================================================================== % Uncertainties: -1 <= delta_m, delta_c, delta_k <= 1 [delta_m, delta_c, delta_k] = ndgrid ([-1, 0, 1], [-1, 0, 1], [-1, 0, 1]); % Bode Plots of Perturbed Plants w = logspace (-1, 1, 100); % frequency vector Delta = arrayfun (@(m, c, k) diag ([m, c, k]), delta_m(:), delta_c(:), delta_k(:), 'uniformoutput', false); G_per = cellfun (@lft, Delta, {G_nom}, 'uniformoutput', false); figure (1) bode (G_per{:}, w) legend off % =============================================================================== % Mixed Sensitivity H-infinity Controller Design (S over KS Method) % =============================================================================== % +-------+ % +--------------------->| W_p |----------> e_p % | +-------+ % | +-------+ % | +---->| W_u |----------> e_u % | | +-------+ % | | +---------+ % | | ->| |-> % r + e | +-------+ u | | G_nom | % ----->(+)---+-->| K |----+--->| |----+----> y % ^ - +-------+ +---------+ | % | | % +-----------------------------------------+ % Weighting Functions s = tf ('s'); % transfer function variable W_p = 0.95 * (s^2 + 1.8*s + 10) / (s^2 + 8.0*s + 0.01); % performance weighting W_u = 10^-2; % control weighting % Synthesis K_mix = mixsyn (G, W_p, W_u); % mixed-sensitivity H-infinity synthesis % Interconnections L_mix = G * K_mix; % open loop T_mix = feedback (L_mix); % closed loop % =============================================================================== % H-infinity Loop-Shaping Design (Normalized Coprime Factor Perturbations) % =============================================================================== % Settings W1 = 8 * (2*s + 1) / (0.9*s); % precompensator W2 = 1; % postcompensator factor = 1.1; % suboptimal controller % Synthesis K_ncf = ncfsyn (G, W1, W2, factor); % positive feedback controller % Interconnections K_ncf = -K_ncf; % negative feedback controller L_ncf = G * K_ncf; % open loop T_ncf = feedback (L_ncf); % closed loop % =============================================================================== % Plot Results % =============================================================================== % Bode Plot figure (2) bode (K_mix, K_ncf) % Step Response figure (3) step (T_mix, T_ncf, 10) % step response for 10 seconds % =============================================================================== control/inst/mixsyn.m0000644000076500000240000002421212273411257014227 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} mixsyn (@var{G}, @var{W1}, @var{W2}, @var{W3}, @dots{}) ## Solve stacked S/KS/T H-infinity problem. ## Mixed-sensitivity is the name given to transfer function shaping problems in which ## the sensitivity function ## @iftex ## @tex ## $ S = (I + G K)^{-1} $ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 ## S = (I + G K) ## @end example ## @end ifnottex ## is shaped along with one or more other closed-loop transfer functions such as @var{K S} ## or the complementary sensitivity function ## @iftex ## @tex ## $ T = I - S = (I + G K)^{-1} G K $ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 ## T = I - S = (I + G K) ## @end example ## @end ifnottex ## in a typical one degree-of-freedom configuration, where @var{G} denotes the plant and ## @var{K} the (sub-)optimal controller to be found. The shaping of multivariable ## transfer functions is based on the idea that a satisfactory definition of gain ## (range of gain) for a matrix transfer function is given by the singular values ## @iftex ## @tex ## $\\sigma$ ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## of the transfer function. Hence the classical loop-shaping ideas of feedback design ## can be generalized to multivariable systems. In addition to the requirement that ## @var{K} stabilizes @var{G}, the closed-loop objectives are as follows [1]: ## @enumerate ## @item For @emph{disturbance rejection} make ## @iftex ## @tex ## $\\overline{\\sigma}(S)$ ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## small. ## @item For @emph{noise attenuation} make ## @iftex ## @tex ## $\\overline{\\sigma}(T)$ ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## small. ## @item For @emph{reference tracking} make ## @iftex ## @tex ## $\\overline{\\sigma}(T) \\approx \\underline{\\sigma}(T) \\approx 1$. ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## @item For @emph{input usage (control energy) reduction} make ## @iftex ## @tex ## $\\overline{\\sigma}(K S)$ ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## small. ## @item For @emph{robust stability} in the presence of an additive perturbation ## @iftex ## @tex ## $G_p = G + \\Delta$, ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## make ## @iftex ## @tex ## $\\overline{\\sigma}(K S)$ ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## small. ## @item For @emph{robust stability} in the presence of a multiplicative output perturbation ## @iftex ## @tex ## $G_p = (I + \\Delta) G$, ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## make ## @iftex ## @tex ## $\\overline{\\sigma}(T)$ ## @end tex ## @end iftex ## @ifnottex ## ## @end ifnottex ## small. ## @end enumerate ## In order to find a robust controller for the so-called stacked ## @iftex ## @tex ## $S/KS/T \\ H_{\\infty}$ ## @end tex ## @end iftex ## @ifnottex ## S/KS/T H-infinity ## @end ifnottex ## problem, the user function @command{mixsyn} minimizes the following criterion ## @iftex ## @tex ## $$ \\underset{K}{\\min} || N(K) ||_{\\infty}, \\quad N = | W_1 S; \\ W_2 K S; \\ W_3 T |$$ ## @end tex ## @end iftex ## @ifnottex ## @example ## | W1 S | ## min || N(K) || N = | W2 K S | ## K oo | W3 T | ## @end example ## @end ifnottex ## @code{[K, N] = mixsyn (G, W1, W2, W3)}. ## The user-defined weighting functions @var{W1}, @var{W2} and @var{W3} bound the largest ## singular values of the closed-loop transfer functions @var{S} (for performance), ## @var{K S} (to penalize large inputs) and @var{T} (for robustness and to avoid ## sensitivity to noise), respectively [1]. ## A few points are to be considered when choosing the weights. ## The weigths @var{Wi} must all be proper and stable. Therefore if one wishes, ## for example, to minimize @var{S} at low frequencies by a weighting @var{W1} including ## integral action, ## @iftex ## @tex ## ${1 \\over s}$ ## @end tex ## @end iftex ## @ifnottex ## @example ## 1 ## - ## s ## @end example ## @end ifnottex ## needs to be approximated by ## @iftex ## @tex ## ${1 \\over s + \\epsilon}$, where $\\epsilon \\ll 1$. ## @end tex ## @end iftex ## @ifnottex ## @example ## 1 ## ----- where e << 1. ## s + e ## @end example ## @end ifnottex ## Similarly one might be interested in weighting @var{K S} with a non-proper weight ## @var{W2} to ensure that @var{K} is small outside the system bandwidth. ## The trick here is to replace a non-proper term such as ## @iftex ## @tex ## $ 1 + \\tau_1 s $ by $ {1 + \\tau_1 s \\over 1 + \\tau_2 s} $, where ## @end tex ## @end iftex ## @ifnottex ## @example ## 1 + T1 s ## 1 + T1 s by --------, where T2 << T1. ## 1 + T2 s ## @end example ## @end ifnottex ## @iftex ## @tex ## $\\tau_2 \\ll \\tau_1$ ## @end tex ## @end iftex ## [1, 2]. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model of plant. ## @item W1 ## @acronym{LTI} model of performance weight. Bounds the largest singular values of sensitivity @var{S}. ## Model must be empty @code{[]}, SISO or of appropriate size. ## @item W2 ## @acronym{LTI} model to penalize large control inputs. Bounds the largest singular values of @var{KS}. ## Model must be empty @code{[]}, SISO or of appropriate size. ## @item W3 ## @acronym{LTI} model of robustness and noise sensitivity weight. Bounds the largest singular values of ## complementary sensitivity @var{T}. Model must be empty @code{[]}, SISO or of appropriate size. ## @item @dots{} ## Optional arguments of @command{hinfsyn}. Type @command{help hinfsyn} for more information. ## @end table ## ## All inputs must be proper/realizable. ## Scalars, vectors and matrices are possible instead of @acronym{LTI} models. ## ## @strong{Outputs} ## @table @var ## @item K ## State-space model of the H-infinity (sub-)optimal controller. ## @item N ## State-space model of the lower LFT of @var{P} and @var{K}. ## @item info ## Structure containing additional information. ## @item info.gamma ## L-infinity norm of @var{N}. ## @item info.rcond ## Vector @var{rcond} contains estimates of the reciprocal condition ## numbers of the matrices which are to be inverted and ## estimates of the reciprocal condition numbers of the ## Riccati equations which have to be solved during the ## computation of the controller @var{K}. For details, ## see the description of the corresponding SLICOT routine. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## ## | W1 S | ## gamma = min||N(K)|| N = | W2 K S | = lft (P, K) ## K inf | W3 T | ## @end group ## @end example ## @example ## @group ## +------+ z1 ## +---------------------------------------->| W1 |-----> ## | +------+ ## | +------+ z2 ## | +---------------------->| W2 |-----> ## | | +------+ ## r + e | +--------+ u | +--------+ y +------+ z3 ## ----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |-----> ## ^ - +--------+ +--------+ | +------+ ## | | ## +----------------------------------------+ ## @end group ## @end example ## @example ## @group ## +--------+ ## | |-----> z1 (p1x1) z1 = W1 e ## r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u ## | |-----> z3 (p3x1) z3 = W3 y ## u (mx1) ----->| |-----> e (px1) e = r - y ## +--------+ ## @end group ## @end example ## @example ## @group ## +--------+ ## r ----->| |-----> z ## | P(s) | ## u +---->| |-----+ e ## | +--------+ | ## | | ## | +--------+ | ## +-----| K(s) |<----+ ## +--------+ ## @end group ## @end example ## @example ## @group ## +--------+ ## r ----->| N(s) |-----> z ## +--------+ ## @end group ## @end example ## @example ## @group ## Extended Plant: P = augw (G, W1, W2, W3) ## Controller: K = mixsyn (G, W1, W2, W3) ## Entire System: N = lft (P, K) ## Open Loop: L = G * K ## Closed Loop: T = feedback (L) ## @end group ## @end example ## ## @strong{Algorithm}@* ## Relies on functions @command{augw} and @command{hinfsyn}, ## which use SLICOT SB10FD, SB10DD and SB10AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @strong{References}@* ## [1] Skogestad, S. and Postlethwaite I. (2005) ## @cite{Multivariable Feedback Control: Analysis and Design: ## Second Edition}. Wiley, Chichester, England.@* ## [2] Meinsma, G. (1995) ## @cite{Unstable and nonproper weights in H-infinity control} ## Automatica, Vol. 31, No. 11, pp. 1655-1658 ## ## @seealso{hinfsyn, augw} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2009 ## Version: 0.2 function [K, N, gamma, info] = mixsyn (G, W1 = [], W2 = [], W3 = [], varargin) if (nargin == 0) print_usage (); endif [p, m] = size (G); P = augw (G, W1, W2, W3); [K, N, gamma, info] = hinfsyn (P, p, m, varargin{:}); endfunction control/inst/mktito.m0000644000076500000240000000704312273411257014212 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{P} =} mktito (@var{P}, @var{nmeas}, @var{ncon}) ## Partition @acronym{LTI} plant @var{P} for robust controller synthesis. ## If a plant is partitioned this way, one can omit the inputs @var{nmeas} ## and @var{ncon} when calling the functions @command{hinfsyn} and @command{h2syn}. ## ## @strong{Inputs} ## @table @var ## @item P ## Generalized plant. ## @item nmeas ## Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the ## inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used ## to calculate the H-2/H-infinity norm. ## @item ncon ## Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the ## outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited ## by a harmonic test signal. ## @end table ## ## @strong{Outputs} ## @table @var ## @item P ## Partitioned plant. The input/output groups and names are overwritten with designations ## according to [1]. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## ## min||N(K)|| N = lft (P, K) ## K norm ## ## +--------+ ## w ----->| |-----> z ## | P(s) | ## u +---->| |-----+ v ## | +--------+ | ## | | ## | +--------+ | ## +-----| K(s) |<----+ ## +--------+ ## ## +--------+ ## w ----->| N(s) |-----> z ## +--------+ ## @end group ## @end example ## ## @strong{Reference}@* ## [1] Skogestad, S. and Postlethwaite, I. (2005) ## @cite{Multivariable Feedback Control: Analysis and Design: ## Second Edition}. Wiley, Chichester, England. ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2013 ## Version: 0.1 function P = mktito (P, nmeas, ncon) if (nargin != 3) print_usage (); endif if (! isa (P, "lti")) error ("mktito: first argument must be an LTI model"); endif [p, m] = size (P); if (! is_index (nmeas, p)) error ("mktito: second argument 'nmeas' invalid"); endif if (! is_index (ncon, m)) error ("mktito: third argument 'ncon' invalid"); endif outgroup = struct ("Z", 1:p-nmeas, "V", p-nmeas+1:p); outname = vertcat (strseq ("z", 1:p-nmeas), strseq ("v", 1:nmeas)); ingroup = struct ("W", 1:m-ncon, "U", m-ncon+1:m); inname = vertcat (strseq ("w", 1:m-ncon), strseq ("u", 1:ncon)); P = set (P, "outgroup", outgroup, "ingroup", ingroup, ... "outname", outname, "inname", inname); endfunction function bool = is_index (idx, s) ## (idx < s) and not (idx <= s) because we need at least one Z or W bool = is_real_scalar (idx) && fix (idx) == idx && idx > 0 && idx < s; endfunction control/inst/moen4.m0000644000076500000240000044615012273411257013733 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @var{n}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @var{opt}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moen4 (@var{dat}, @var{n}, @var{opt}, @dots{}) ## Estimate state-space model using combined subspace method: ## @acronym{MOESP} algorithm for finding the matrices A and C, ## and @acronym{N4SID} algorithm for finding the matrices B and D. ## If no output arguments are given, the singular values are ## plotted on the screen in order to estimate the system order. ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata set containing the measurements, i.e. time-domain signals. ## @item n ## The desired order of the resulting state-space system @var{sys}. ## If not specified, @var{n} is chosen automatically according ## to the singular values and tolerances. ## @item @dots{} ## Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## ## @strong{Outputs} ## @table @var ## @item sys ## Discrete-time state-space model. ## @item x0 ## Initial state vector. If @var{dat} is a multi-experiment dataset, ## @var{x0} becomes a cell vector containing an initial state vector ## for each experiment. ## @item info ## Struct containing additional information. ## @table @var ## @item info.K ## Kalman gain matrix. ## @item info.Q ## State covariance matrix. ## @item info.Ry ## Output covariance matrix. ## @item info.S ## State-output cross-covariance matrix. ## @item info.L ## Noise variance matrix factor. LL'=Ry. ## @end table ## @end table ## ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'n' ## The desired order of the resulting state-space system @var{sys}. ## @var{s} > @var{n} > 0. ## ## @item 's' ## The number of block rows @var{s} in the input and output ## block Hankel matrices to be processed. @var{s} > 0. ## In the MOESP theory, @var{s} should be larger than @var{n}, ## the estimated dimension of state vector. ## ## @item 'alg', 'algorithm' ## Specifies the algorithm for computing the triangular ## factor R, as follows: ## @table @var ## @item 'C' ## Cholesky algorithm applied to the correlation ## matrix of the input-output data. Default method. ## @item 'F' ## Fast QR algorithm. ## @item 'Q' ## QR algorithm applied to the concatenated block ## Hankel matrices. ## @end table ## ## @item 'tol' ## Absolute tolerance used for determining an estimate of ## the system order. If @var{tol} >= 0, the estimate is ## indicated by the index of the last singular value greater ## than or equal to @var{tol}. (Singular values less than @var{tol} ## are considered as zero.) When @var{tol} = 0, an internally ## computed default value, @var{tol} = @var{s}*@var{eps}*SV(1), is used, ## where SV(1) is the maximal singular value, and @var{eps} is ## the relative machine precision. ## When @var{tol} < 0, the estimate is indicated by the ## index of the singular value that has the largest ## logarithmic gap to its successor. Default value is 0. ## ## @item 'rcond' ## The tolerance to be used for estimating the rank of ## matrices. If the user sets @var{rcond} > 0, the given value ## of @var{rcond} is used as a lower bound for the reciprocal ## condition number; an m-by-n matrix whose estimated ## condition number is less than 1/@var{rcond} is considered to ## be of full rank. If the user sets @var{rcond} <= 0, then an ## implicitly computed, default tolerance, defined by ## @var{rcond} = m*n*@var{eps}, is used instead, where @var{eps} is the ## relative machine precision. Default value is 0. ## ## @item 'confirm' ## Specifies whether or not the user's confirmation of the ## system order estimate is desired, as follows: ## @table @var ## @item true ## User's confirmation. ## @item false ## No confirmation. Default value. ## @end table ## ## @item 'noiseinput' ## The desired type of noise input channels. ## @table @var ## @item 'n' ## No error inputs. Default value. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k $$ ## $$ y_k = C x_k + D u_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] ## y[k] = C x[k] + D u[k] ## @end example ## @end ifnottex ## ## @item 'e' ## Return @var{sys} as a (p-by-m+p) state-space model with ## both measured input channels u and noise channels e ## with covariance matrix @var{Ry}. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k + K e_k $$ ## $$ y_k = C x_k + D u_k + e_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] + K e[k] ## y[k] = C x[k] + D u[k] + e[k] ## @end example ## @end ifnottex ## ## @item 'v' ## Return @var{sys} as a (p-by-m+p) state-space model with ## both measured input channels u and white noise channels v ## with identity covariance matrix. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k + K L v_k $$ ## $$ y_k = C x_k + D u_k + L v_k $$ ## $$ e = L v, \\ L L^T = R_y $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] + K L v[k] ## y[k] = C x[k] + D u[k] + L v[k] ## e = L v, L L' = Ry ## @end example ## @end ifnottex ## ## @item 'k' ## Return @var{sys} as a Kalman predictor for simulation. ## @iftex ## @tex ## $$ \\widehat{x}_{k+1} = A \\widehat{x}_k + B u_k + K (y_k - \\widehat{y}_k) $$ ## $$ \\widehat{y}_k = C \\widehat{x}_k + D u_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ^ ^ ^ ## x[k+1] = A x[k] + B u[k] + K(y[k] - y[k]) ## ^ ^ ## y[k] = C x[k] + D u[k] ## @end example ## @end ifnottex ## ## @iftex ## @tex ## $$ \\widehat{x}_{k+1} = (A-KC) \\widehat{x}_k + (B-KD) u_k + K y_k $$ ## $$ \\widehat{y}_k = C \\widehat{x}_k + D u_k + 0 y_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ^ ^ ## x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k] ## ^ ^ ## y[k] = C x[k] + D u[k] + 0 y[k] ## @end example ## @end ifnottex ## @end table ## @end table ## ## ## @strong{Algorithm}@* ## Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function [sys, x0, info] = moen4 (varargin) if (nargin == 0) print_usage (); endif if (nargout == 0) __slicot_identification__ ("moen4", nargout, varargin{:}); else [sys, x0, info] = __slicot_identification__ ("moen4", nargout, varargin{:}); endif endfunction %!shared SYS, X0, INFO, Ae, Be, Ce, De, Ke, Qe, Rye, Se, X0e %! %! Y = [ 4.7661 5.5451 5.8503 5.3766 4.8833 5.4865 3.5378 5.3155 6.0530 4.3729 %! 4.7637 5.1886 5.9236 5.6818 4.8858 5.1495 3.5549 5.5329 6.0799 4.7417 %! 4.8394 4.8833 5.9212 5.8235 4.8931 4.8442 3.4938 5.4450 6.1287 5.0884 %! 5.0030 4.6000 5.9773 5.9529 4.7148 4.5414 3.4474 5.3961 6.0799 5.1861 %! 5.0176 4.2704 5.7405 6.0628 4.4511 4.2679 3.4401 5.2740 6.1678 5.0372 %! 5.0567 4.0384 5.3888 6.0897 4.2337 4.0604 3.4083 5.0274 6.1947 4.7856 %! 5.1544 3.8381 5.0005 6.0750 4.0433 3.9602 3.4108 4.7441 6.2362 4.5634 %! 5.3619 3.7112 4.8491 6.0262 3.8650 3.7893 3.4523 4.6684 6.0530 4.5341 %! 5.4254 3.5915 4.9444 5.9944 3.7576 3.6428 3.6818 4.6513 5.6525 4.7050 %! 5.5695 3.5353 5.1739 6.0775 3.6696 3.5256 4.0604 4.5146 5.2740 4.7417 %! 5.6818 3.4865 5.3693 5.8577 3.5939 3.4987 4.4413 4.2679 4.8589 4.6489 %! 5.7429 3.4767 5.4474 5.7014 3.5475 3.4547 4.8540 4.2606 4.5341 4.4315 %! 5.8039 3.4254 5.6037 5.7307 3.5060 3.4083 5.1544 4.2630 4.4560 4.2386 %! 5.9187 3.3815 5.7307 5.7844 3.4547 3.3790 5.4254 4.1898 4.6196 4.0652 %! 5.8210 3.3693 5.8503 5.8235 3.3986 3.3766 5.5964 4.2777 4.8662 3.9431 %! 5.4474 3.3644 5.9798 5.8943 3.3619 3.3619 5.5866 4.6000 5.1177 3.8113 %! 5.0616 3.3473 5.9920 5.7624 3.3400 3.3595 5.3546 4.9322 5.1666 3.6916 %! 4.6293 3.3815 6.0848 5.4157 3.3742 3.3693 5.0274 5.2838 5.0567 3.6525 %! 4.2679 3.4206 5.9407 4.9615 3.5207 3.3986 4.8638 5.5280 5.0030 3.8259 %! 4.0115 3.4132 5.8039 4.5952 3.7136 3.5793 4.7612 5.7405 5.0982 4.2240 %! 3.8503 3.4523 5.7917 4.3314 3.7576 3.9480 4.5707 5.8748 5.3253 4.4242 %! 3.7112 3.6355 5.6037 4.2972 3.7795 4.4120 4.3681 5.9554 5.5671 4.4291 %! 3.5695 4.0384 5.2643 4.5829 3.6965 4.5854 4.3974 5.9920 5.4670 4.3192 %! 3.5182 4.3754 4.9468 4.8613 3.7771 4.5146 4.5732 5.8455 5.2521 4.1385 %! 3.6525 4.7270 4.6196 5.1739 3.8870 4.3436 4.8418 5.5280 4.9468 3.9651 %! 3.8186 5.0567 4.5146 5.1666 3.9041 4.1556 5.2032 5.0616 4.8809 3.8870 %! 3.8626 5.2985 4.4340 4.9199 3.8503 3.9847 5.4523 4.7344 4.9810 3.8015 %! 4.0115 5.5329 4.2850 4.6074 3.9651 4.0433 5.6525 4.5341 5.2252 3.7014 %! 4.3534 5.4670 4.1214 4.3705 4.2826 4.3070 5.8552 4.5341 5.4596 3.6403 %! 4.7050 5.1959 3.9456 4.1825 4.5219 4.4218 5.9065 4.6977 5.7234 3.7673 %! 5.0836 4.8858 3.9847 4.0384 4.7148 4.3534 5.9529 4.7441 5.7917 4.1507 %! 5.3449 4.7637 4.2191 4.1458 4.9712 4.2240 5.8284 4.6196 5.9065 4.6489 %! 5.2740 4.8760 4.5463 4.4315 5.2203 4.0530 5.7917 4.6440 5.9920 4.9908 %! 5.1275 5.0420 4.8735 4.5561 5.5329 3.9407 5.7991 4.8320 5.8357 5.0884 %! 4.7612 5.2838 5.1544 4.4804 5.6525 3.8381 5.8137 5.1324 5.5280 5.0225 %! 4.4511 5.4914 5.3888 4.3754 5.7820 3.7307 5.8772 5.4108 5.1422 4.7832 %! 4.2215 5.5964 5.6135 4.3705 5.9554 3.6525 5.9554 5.6257 4.7759 4.6855 %! 4.0457 5.6721 5.8357 4.5585 6.0359 3.6110 5.7820 5.6037 4.4902 4.6660 %! 3.8748 5.7722 5.8845 4.8589 6.1190 3.5646 5.5182 5.3155 4.2362 4.7075 %! 3.7307 5.8308 5.9554 4.8955 6.1336 3.4963 5.1275 4.9615 4.0237 4.9126 %! 3.6623 5.9334 5.7624 4.7417 6.1532 3.4621 4.7637 4.6196 3.8870 5.1959 %! 3.5768 5.8992 5.4596 4.7441 6.1922 3.4547 4.4926 4.3583 3.7527 5.4157 %! 3.5427 5.9358 5.0616 4.8760 6.1434 3.4254 4.2337 4.1556 3.6818 5.6232 %! 3.4792 5.8943 4.7075 5.1055 6.1678 3.3790 4.0115 4.0335 3.8064 5.7405 %! 3.4547 5.9187 4.4584 5.2398 5.9920 3.4328 3.8552 3.8870 4.1458 5.8992 %! 3.3595 5.9944 4.2679 5.5182 5.6525 3.6232 3.6916 3.7722 4.6000 5.9285 %! 3.2985 5.9578 4.0530 5.6525 5.4596 3.9749 3.6355 3.6403 5.0030 6.0506 %! 3.2252 6.0311 3.9431 5.7234 5.4376 4.3803 3.8186 3.5329 5.3033 6.1532 %! 3.2008 6.0628 3.8259 5.8552 5.3400 4.7148 4.1556 3.4352 5.5524 5.9651 %! 3.2252 6.0408 3.9676 5.9627 5.0982 5.0738 4.5903 3.4279 5.6159 5.5866 %! 3.2276 6.0970 4.2801 5.9847 4.7856 5.3693 4.9883 3.4230 5.5231 5.3815 %! 3.2740 6.1239 4.4804 5.9847 4.4926 5.6037 5.0762 3.3986 5.6110 5.3717 %! 3.4572 6.1629 4.4926 6.0555 4.2362 5.7453 4.9077 3.6037 5.7136 5.4865 %! 3.8674 6.0408 4.3900 6.0628 4.0677 5.6525 4.6489 4.0237 5.8455 5.5671 %! 4.3217 5.8455 4.1971 6.0555 3.9334 5.4010 4.3778 4.4511 5.8992 5.8210 %! 4.4926 5.7722 4.1116 6.0701 3.8235 5.0152 4.2166 4.7930 5.9944 5.9138 %! 4.4315 5.7991 3.9822 5.7844 3.7307 4.7099 4.2875 4.9029 6.0921 5.9944 %! 4.2435 5.9236 3.8674 5.4401 3.6110 4.4169 4.5903 4.7808 6.0921 6.0115 %! 4.0506 5.9285 3.7673 5.0567 3.5646 4.2362 4.8467 4.5903 6.1434 5.9993 %! 3.8577 6.0018 3.8723 4.9419 3.5500 4.2362 5.1397 4.3363 6.1532 6.0188 %! 3.7307 6.0018 4.2362 5.0103 3.5573 4.2484 5.3888 4.1458 6.2337 5.8210 %! 3.7917 6.0604 4.6635 5.1348 3.5134 4.2215 5.6892 4.2166 6.1873 5.7282 %! 3.9212 5.8821 4.9712 5.3131 3.5158 4.2972 5.8845 4.4340 6.0140 5.7405 %! 3.9554 5.5109 5.0665 5.4792 3.6941 4.5903 6.0433 4.7148 5.8357 5.7649 %! 3.8479 5.3229 4.9029 5.6232 4.0726 4.8931 6.1703 5.0982 5.7746 5.8821 %! 3.7258 5.3717 4.6757 5.5622 4.4804 5.1348 6.2118 5.3595 5.6867 5.9260 %! 3.6110 5.4547 4.3925 5.3302 4.7050 5.4279 6.2508 5.5695 5.5378 5.7502 %! 3.7160 5.4376 4.0994 5.0103 4.6123 5.3790 6.2093 5.7722 5.3278 5.4157 %! 4.0921 5.1593 4.1141 4.6660 4.3851 5.3644 6.0140 5.9212 5.0543 4.9956 %! 4.4804 4.9029 4.3265 4.4145 4.2020 5.4523 5.7014 6.0555 4.7002 4.8613 %! 4.8149 4.5878 4.6440 4.2020 4.0262 5.5671 5.4694 5.9627 4.3949 4.9029 %! 5.0543 4.5024 4.9712 4.0482 3.9041 5.6721 5.4792 5.6428 4.1800 5.1031 %! 5.3033 4.5952 5.1593 4.0799 3.7746 5.7698 5.5573 5.4352 4.0433 5.3644 %! 5.4865 4.8247 5.3888 4.1898 3.6916 5.8308 5.7282 5.3888 3.8772 5.5964 %! 5.6721 5.0640 5.5768 4.1312 3.8455 5.9236 5.8821 5.5378 3.7527 5.7527 %! 5.7795 5.2716 5.6525 4.0042 4.2020 5.9651 5.9847 5.6818 3.7282 5.8455 %! 5.7991 5.4670 5.8039 3.9163 4.5854 6.0579 5.9016 5.7014 3.8699 5.9285 %! 5.6648 5.6159 5.9138 3.9602 4.9029 6.0506 5.5817 5.6159 4.2069 6.0066 %! 5.2911 5.5280 5.8870 4.1996 5.2569 6.0726 5.3717 5.6672 4.3558 5.8406 %! 4.8809 5.2545 5.7991 4.6245 5.5109 6.1116 5.4181 5.7405 4.4267 5.5182 %! 4.5585 4.8833 5.7307 4.8833 5.6403 6.0701 5.5109 5.8039 4.4535 5.1739 %! 4.1849 4.5170 5.7624 5.1373 5.8430 5.8967 5.6672 5.8821 4.5219 4.7392 %! 3.8894 4.1971 5.8137 5.3790 5.9749 5.7551 5.7917 5.9505 4.3925 4.4584 %! 3.7087 4.0018 5.8210 5.6232 5.9358 5.7185 5.6989 6.0726 4.1556 4.4267 %! 3.6232 3.8064 5.9285 5.7624 5.8210 5.8210 5.4840 6.1483 3.9651 4.6025 %! 3.5695 3.9041 6.0140 5.8333 5.5280 6.0018 5.1544 6.1165 3.8772 4.8223 %! 3.7185 3.9236 5.7649 5.6867 5.1715 6.0018 4.9810 6.1776 3.9700 5.1837 %! 4.0335 3.8699 5.4132 5.3668 4.8101 5.9016 5.0616 6.2020 4.2582 5.4303 %! 4.4120 3.8064 5.0982 5.2252 4.4535 5.5573 5.1959 6.2069 4.4218 5.6525 %! 4.6293 3.7209 4.6782 5.2398 4.3803 5.1739 5.3595 5.9920 4.3363 5.8210 %! 4.5585 3.8186 4.3729 5.3546 4.5659 4.8003 5.6159 5.5646 4.2997 5.7063 %! 4.3949 4.1409 4.3925 5.5085 4.8052 4.4315 5.7624 5.1788 4.3925 5.3693 %! 4.1800 4.5292 4.5903 5.5964 5.1251 4.1947 5.8577 4.9981 4.6757 5.0274 %! 4.1971 4.8052 4.9199 5.7527 5.3546 4.0066 5.9480 5.0518 4.7612 4.7050 %! 4.4315 5.0860 5.0176 5.8748 5.5891 3.8503 5.8357 5.2325 4.6587 4.4145 %! 4.7148 5.3400 4.8589 5.9065 5.7649 3.7478 5.7063 5.4840 4.4902 4.1458 %! 4.9615 5.5329 4.6757 5.8943 5.9236 3.6428 5.4987 5.6867 4.3070 3.9651 %! 5.3009 5.5768 4.6196 5.7429 5.9407 3.5915 5.1886 5.8992 4.1263 4.0335 %! 5.5671 5.6672 4.8345 5.4474 5.8577 3.5695 5.1177 5.8699 3.9724 4.3729 %! 5.6818 5.7917 5.0909 5.0250 5.6941 3.5280 5.1910 5.9773 4.0775 4.6831 ](:); %! %! %! %! U = [ 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 3.4100 %! 3.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 %! 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 %! 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 %! 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 6.4100 3.4100 6.4100 %! 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 6.4100 %! 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 %! 6.4100 3.4100 6.4100 3.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 %! 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 3.4100 %! 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 3.4100 %! 3.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 6.4100 3.4100 %! 3.4100 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 3.4100 %! 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 3.4100 %! 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 3.4100 %! 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 3.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 6.4100 6.4100 %! 3.4100 3.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 %! 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 %! 3.4100 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 3.4100 3.4100 %! 6.4100 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 %! 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 6.4100 3.4100 6.4100 3.4100 %! 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 6.4100 3.4100 %! 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 3.4100 %! 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 %! 6.4100 3.4100 3.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 %! 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 3.4100 3.4100 3.4100 3.4100 %! 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 %! 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 %! 3.4100 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 %! 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 %! 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 6.4100 %! 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 6.4100 %! 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 %! 6.4100 6.4100 3.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 3.4100 %! 6.4100 6.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 3.4100 6.4100 6.4100 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 %! 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 %! 3.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 %! 6.4100 3.4100 3.4100 3.4100 3.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 3.4100 3.4100 %! 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 %! 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 6.4100 %! 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 %! 3.4100 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 3.4100 %! 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 %! 3.4100 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 6.4100 %! 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 6.4100 %! 6.4100 3.4100 3.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 6.4100 6.4100 3.4100 6.4100 %! 3.4100 3.4100 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 %! 3.4100 6.4100 3.4100 6.4100 6.4100 3.4100 6.4100 3.4100 3.4100 3.4100 %! 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 3.4100 %! 3.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 6.4100 3.4100 %! 6.4100 6.4100 6.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 %! 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 %! 6.4100 6.4100 3.4100 6.4100 6.4100 3.4100 3.4100 6.4100 3.4100 3.4100 %! 6.4100 6.4100 6.4100 3.4100 6.4100 3.4100 3.4100 6.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 3.4100 3.4100 3.4100 6.4100 6.4100 3.4100 6.4100 %! 6.4100 6.4100 6.4100 3.4100 6.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 6.4100 3.4100 3.4100 6.4100 6.4100 6.4100 6.4100 %! 3.4100 6.4100 6.4100 3.4100 3.4100 3.4100 3.4100 6.4100 6.4100 6.4100 ](:); %! %! %! DAT = iddata (Y, U); %! %! [SYS, X0, INFO] = moen4 (DAT, "s", 15, "rcond", 0.0, "tol", -1.0, "confirm", false); %! %! Ae = [ 0.8924 0.3887 0.1285 0.1716 %! -0.0837 0.6186 -0.6273 -0.4582 %! 0.0052 0.1307 0.6685 -0.6755 %! 0.0055 0.0734 -0.2148 0.4788 ]; %! %! Ce = [ -0.4442 0.6663 0.3961 0.4102 ]; %! %! Be = [ -0.2142 %! -0.1968 %! 0.0525 %! 0.0361 ]; %! %! De = [ -0.0041 ]; %! %! Ke = [ -1.9513 %! -0.1867 %! 0.6348 %! -0.3486 ]; %! %! Qe = [ 0.0052 0.0005 -0.0017 0.0009 %! 0.0005 0.0000 -0.0002 0.0001 %! -0.0017 -0.0002 0.0006 -0.0003 %! 0.0009 0.0001 -0.0003 0.0002 ]; %! %! Rye = [ 0.0012 ]; %! %! Se = [ -0.0025 %! -0.0002 %! 0.0008 %! -0.0005 ]; %! %! X0e = [ -11.496422 %! -0.718576 %! -0.014211 %! 0.500073 ]; # X0e is not from SLICOT %! %! ## The SLICOT test for IB01CD uses COMUSE=C, not COMUSE=U. %! ## This means that they don't use the matrices B and D %! ## computed by IB01BD. They use only A and C from IB01BD, %! ## while B and D are from SLICOT routine IB01CD. %! ## Therefore they get slightly different matrices B and D %! ## and finally a different initial state vector X0. %! %!assert (SYS.A, Ae, 1e-4); %!assert (SYS.B, Be, 1e-4); %!assert (SYS.C, Ce, 1e-4); %!assert (SYS.D, De, 1e-4); %!assert (INFO.K, Ke, 1e-4); %!assert (INFO.Q, Qe, 1e-4); %!assert (INFO.Ry, Rye, 1e-4); %!assert (INFO.S, Se, 1e-4); %!assert (X0, X0e, 1e-4); ## [96-003] Data of a 120 MW power plant (Pont-sur-Sambre, France) %!shared SYS, Ae, Be, Ce, De %! U = [ -811 -592 421 -680 -681 %! -812 -619 477 -685 -651 %! -817 -565 538 -678 -677 %! -695 -725 536 -674 -702 %! -697 -571 531 -676 -685 %! -697 -618 533 -681 -721 %! -702 -579 549 -677 -699 %! -703 -487 575 -677 -694 %! -705 -449 561 -679 -678 %! -705 -431 563 -680 -692 %! -707 -502 561 -679 -686 %! -707 -583 530 -676 -751 %! -710 -458 540 -677 -700 %! -713 -469 543 -679 -731 %! -715 -506 549 -684 -635 %! -713 -590 532 -681 -715 %! -714 -582 528 -676 -696 %! -713 -575 538 -679 -690 %! -716 -382 557 -682 -701 %! -716 -476 558 -679 -690 %! -718 -425 565 -678 -686 %! -719 -409 562 -679 -615 %! -719 -508 523 -677 -737 %! -721 -569 523 -679 -722 %! -723 -434 542 -681 -735 %! -723 -395 544 -676 -704 %! -723 -428 542 -677 -729 %! -722 -402 537 -677 -706 %! -725 -380 534 -681 -696 %! -726 -324 549 -676 -701 %! -726 -211 578 -675 -614 %! -727 -113 569 -677 -738 %! -727 -208 554 -676 -737 %! -727 -320 523 -684 -697 %! -727 944 605 -680 -587 %! -729 396 654 -681 -708 %! -729 754 637 -682 -685 %! -727 419 522 -677 -715 %! -729 378 494 -668 -703 %! -728 363 493 -669 -682 %! -729 390 496 -665 -713 %! -729 106 483 -664 -711 %! -729 32 495 -661 -718 %! -729 517 585 -661 -641 %! -729 455 625 -659 -703 %! -730 521 649 -687 -669 %! -730 540 627 -689 -705 %! -731 514 605 -694 -682 %! -585 525 558 -685 -611 %! -586 534 520 -680 -668 %! -586 539 531 -681 -679 %! -585 519 507 -682 -663 %! -588 513 505 -667 -668 %! -587 509 539 -680 -616 %! -587 512 535 -668 -628 %! -588 514 557 -667 -648 %! -588 553 563 -676 -613 %! -589 519 559 -684 -638 %! -589 521 563 -682 -652 %! -588 518 547 -678 -597 %! -589 552 549 -688 -630 %! -589 520 535 -685 -623 %! -589 547 542 -678 -619 %! -589 549 531 -684 -524 %! -588 544 522 -1540 -580 %! -588 564 555 -1538 -584 %! -588 684 545 -1541 -564 %! -590 558 546 -1541 -609 %! -589 552 537 -1550 -601 %! -591 532 526 -1548 -580 %! -590 544 524 -1542 -565 %! -591 559 535 -1538 -604 %! -592 555 542 -1548 -629 %! -591 577 532 -1549 -587 %! -593 581 530 -1543 -585 %! -592 562 540 -1548 -583 %! -591 568 546 -1536 -587 %! -593 550 557 -1533 -569 %! -592 550 537 -1518 -527 %! -593 568 551 -1533 -582 %! -590 528 540 -1529 -492 %! -590 542 532 -1525 -585 %! -590 556 535 -1522 -606 %! -591 637 535 -1516 -571 %! -591 608 539 -1512 -582 %! -591 545 527 -1510 -577 %! -591 603 534 -1507 -548 %! -592 567 521 -1507 -539 %! -594 560 530 -1503 -583 %! -422 549 534 -1487 -542 %! 5 619 550 -1488 -609 %! 5 572 541 -1487 -545 %! 185 564 529 -1488 -528 %! 185 571 531 -1497 -512 %! 187 590 545 -1489 -472 %! 186 658 544 -1485 -501 %! 185 639 553 -1486 -512 %! 187 604 532 -1486 -490 %! 186 584 535 -1486 -505 %! 187 572 541 -1488 -479 %! 186 570 531 -1486 -477 %! 187 579 539 -1488 -482 %! 187 694 537 -1487 -512 %! 187 727 546 -1494 -474 %! 186 838 545 -1493 -470 %! 186 703 527 -1492 -510 %! 185 634 502 -1490 -486 %! 185 659 502 -1490 -446 %! 185 569 483 -1488 -472 %! 184 552 494 -1487 -468 %! 183 541 492 -1487 -411 %! 184 544 519 -1486 -471 %! 184 577 538 -1487 -420 %! 183 536 546 -1490 -455 %! 183 586 567 -1487 -421 %! 183 554 566 -1485 -449 %! 183 564 558 -1486 -455 %! 182 558 546 -1486 -426 %! 183 659 543 -1486 -398 %! 182 545 531 -1487 -425 %! 182 542 513 -1487 -297 %! 181 549 502 -1486 -407 %! 180 558 963 -1487 -391 %! 182 546 1105 -1487 -388 %! 179 540 1166 -1487 -387 %! 181 519 1221 -1487 -390 %! 179 510 1254 -1486 -382 %! 176 528 1289 -1487 -379 %! 178 510 1310 -1486 -389 %! 178 504 1313 -1484 -357 %! 178 507 1328 -1484 -383 %! 178 519 1333 -1489 -426 %! 183 509 1317 -1483 -377 %! 177 585 1331 -1483 -352 %! 176 765 1334 -1484 -375 %! 196 689 1328 -1484 -381 %! 178 570 1306 -1486 -398 %! 176 997 1338 -1483 -360 %! 176 839 1318 -1487 -366 %! 176 879 1324 -90 -394 %! 175 763 1302 -92 -436 %! 175 739 1307 -92 -430 %! 176 791 1336 -86 -446 %! 175 774 1301 -91 -405 %! 174 782 1304 -91 -414 %! 189 722 1335 -92 -360 %! 173 871 1341 -91 -417 %! 173 825 1337 -86 -383 %! 193 836 1332 -86 -449 %! 174 832 1325 -87 -432 %! 175 834 1326 -88 -383 %! 176 899 1332 -86 -341 %! 195 827 1296 -84 -371 %! 174 897 1311 -86 -366 %! 195 848 1320 -86 -416 %! 192 777 1286 -83 126 %! 171 788 1309 -84 153 %! 171 810 1307 -84 173 %! 174 758 1297 -85 182 %! 188 910 1329 -1036 236 %! 189 944 1344 -1669 245 %! 196 859 1330 -1668 249 %! 190 797 1320 -1669 231 %! 191 784 1341 -1665 248 %! 184 737 1335 -1656 240 %! 182 733 1342 -1655 245 %! 182 696 1350 -1653 253 %! 185 614 1341 -1658 236 %! 188 708 1370 -1654 238 %! 191 729 1366 -1655 237 %! 183 714 1380 -1651 262 %! 186 695 1378 -1651 257 %! 189 758 1387 -1649 276 %! 164 572 1364 -1649 257 %! 163 587 1343 -1650 246 %! 161 683 1353 -1658 261 %! 160 572 1359 -1654 261 %! 160 607 1355 -1651 267 %! 158 580 1349 -1655 273 %! 161 631 1362 -1652 272 %! 160 706 1382 -1657 298 %! 161 601 1356 -1654 277 %! 159 570 1354 -1653 274 %! 158 547 1347 -1657 261 %! 158 582 1332 -1657 257 %! 157 570 1340 -1657 267 %! 154 556 1343 -1657 268 %! 157 537 1345 -1657 -425 %! 158 555 1331 -1653 -581 %! 158 551 1315 -1654 -643 %! 159 590 1322 -1656 -687 %! 160 566 1315 -1657 -737 %! 160 553 1315 -1653 -767 %! 161 644 1327 -1396 -731 %! 159 640 1335 -577 -639 %! 161 726 1334 -577 -730 %! 175 729 1310 -573 -711 %! 175 854 1330 -576 -690 %! 161 729 1313 -573 -636 %! 176 706 1314 -583 -745 ]; %! %! %! Y = [ 117 129 -47 %! 113 141 -42 %! 83 150 -37 %! 144 174 -3 %! 156 196 19 %! 174 192 6 %! 171 193 3 %! 169 224 14 %! 155 211 14 %! 137 175 4 %! 130 165 13 %! 145 172 21 %! 128 173 30 %! 119 194 30 %! 127 188 11 %! 147 176 5 %! 162 183 19 %! 173 178 14 %! 131 161 4 %! 112 139 -11 %! 91 133 -10 %! 75 127 -7 %! 80 115 -10 %! 87 132 6 %! 51 151 6 %! 28 167 3 %! 22 167 -3 %! 9 171 4 %! 7 158 6 %! -11 155 5 %! -87 149 -8 %! -153 146 -12 %! -184 116 -20 %! -167 118 8 %! -563 78 -4 %! -694 -3 -155 %! -1000 -39 -168 %! -1101 -28 -135 %! -1080 55 6 %! -1075 162 89 %! -1035 254 117 %! -894 329 148 %! -747 409 218 %! -774 416 231 %! -781 361 189 %! -825 288 171 %! -897 220 166 %! -960 175 169 %! -994 157 196 %! -1009 202 237 %! -1003 267 258 %! -981 326 267 %! -947 361 276 %! -921 369 280 %! -894 369 292 %! -876 359 302 %! -869 356 318 %! -857 334 316 %! -857 336 320 %! -868 331 323 %! -873 316 324 %! -870 301 326 %! -867 294 323 %! -878 293 322 %! -877 300 309 %! -891 317 233 %! -919 308 160 %! -919 296 111 %! -898 304 94 %! -884 336 101 %! -876 344 92 %! -877 347 77 %! -859 354 79 %! -852 368 84 %! -852 362 79 %! -846 337 67 %! -846 322 71 %! -836 325 82 %! -831 330 81 %! -848 332 84 %! -834 334 89 %! -830 314 87 %! -838 322 93 %! -859 313 73 %! -864 317 59 %! -852 318 48 %! -850 313 59 %! -858 327 65 %! -849 332 82 %! -822 332 77 %! -721 377 121 %! -517 492 193 %! -306 596 226 %! -117 683 244 %! 22 716 212 %! 87 684 157 %! 151 642 129 %! 198 599 105 %! 233 549 93 %! 244 512 83 %! 255 487 70 %! 247 453 49 %! 225 437 32 %! 175 410 4 %! 105 386 -12 %! 86 379 -35 %! 101 382 -16 %! 111 412 3 %! 158 451 31 %! 217 486 37 %! 259 504 29 %! 301 503 24 %! 317 495 7 %! 354 472 -3 %! 361 441 -8 %! 368 413 -32 %! 362 381 -34 %! 354 370 -42 %! 310 360 -36 %! 318 369 -36 %! 313 372 -36 %! 298 377 -28 %! 311 341 -34 %! 339 292 -33 %! 362 223 -52 %! 380 161 -73 %! 397 113 -90 %! 402 64 -117 %! 398 30 -136 %! 396 12 -148 %! 390 -22 -189 %! 377 -47 -211 %! 366 -54 -227 %! 331 -52 -240 %! 227 -74 -289 %! 168 -117 -355 %! 149 -104 -357 %! 10 -101 -344 %! -77 -115 -374 %! -116 -88 -330 %! -106 -53 -223 %! -82 -4 -98 %! -99 5 -40 %! -89 22 3 %! -79 34 26 %! -74 34 42 %! -101 28 61 %! -102 28 73 %! -124 22 74 %! -125 19 90 %! -132 9 100 %! -167 8 102 %! -161 7 90 %! -185 19 105 %! -207 29 110 %! -295 24 159 %! -361 17 293 %! -411 12 436 %! -401 48 540 %! -441 86 601 %! -475 118 553 %! -448 146 472 %! -401 172 428 %! -366 197 400 %! -304 231 391 %! -252 253 387 %! -183 261 399 %! -123 255 405 %! -107 247 397 %! -76 228 392 %! -71 205 389 %! -66 192 381 %! -92 171 371 %! -40 178 356 %! -15 185 353 %! -27 170 359 %! 15 173 351 %! 23 175 356 %! 44 178 338 %! 47 171 323 %! 4 153 294 %! 23 141 295 %! 39 141 306 %! 52 139 313 %! 53 142 313 %! 69 142 312 %! 74 136 306 %! 196 164 277 %! 347 208 133 %! 461 230 -58 %! 526 204 -251 %! 581 161 -385 %! 588 119 -458 %! 549 63 -528 %! 497 5 -550 %! 420 -20 -498 %! 327 -49 -464 %! 198 -78 -438 %! 154 -80 -409 %! 130 -60 -377 ]; %! %! DAT = iddata (Y, U, 1228.8); %! %! [SYS, X0] = moen4 (DAT, "s", 10, "n", 8, "rcond", 0.0, "tol", -1.0); %! %! Ae = [ 0.9811 0.0574 0.3270 0.0003 0.0358 0.0403 -0.1366 0.0276 %! 0.1043 0.7634 -0.1308 0.2252 0.0203 -0.0117 -0.2328 -0.2516 %! -0.0612 0.1437 0.8378 -0.2400 0.0367 -0.3205 -0.0367 -0.0978 %! -0.0213 0.0149 0.0706 0.8902 0.2415 -0.1329 0.3080 -0.0143 %! -0.0076 0.0680 0.0557 -0.0469 0.7084 0.2852 0.2565 0.1508 %! -0.0016 0.0603 0.0071 -0.0448 -0.0870 0.8608 0.1495 -0.1148 %! 0.0046 0.0120 0.0040 -0.0272 0.0117 -0.0654 0.8348 -0.4557 %! -0.0058 0.0133 -0.0112 -0.0171 -0.0353 -0.0892 0.3328 0.7650 ]; %! %! Ce = [ -0.1635 0.6294 0.1157 -0.2129 0.0812 -0.0238 0.0745 0.2027 %! 0.1775 0.1736 -0.2839 -0.0750 -0.4986 -0.1254 0.3740 0.1598 %! 0.2012 -0.0045 -0.4447 -0.3289 0.4767 -0.3377 0.0200 -0.0978 ]; %! %! Be = [ -0.0436 0.1911 -0.1345 0.0409 0.2828 %! 0.5541 -0.4223 0.0001 -0.0151 -0.1778 %! -0.3644 0.2868 0.0754 -0.0457 -0.1691 %! -0.0372 0.1270 -0.0772 -0.0714 -0.0255 %! -0.1251 -0.0021 0.1094 0.0576 0.2121 %! -0.1372 0.1139 0.0030 0.0141 0.0493 %! -0.0187 0.0712 -0.0042 0.0075 -0.0221 %! -0.0834 0.0509 0.0322 -0.0089 0.0009 ]; %! %! De = [ 0.1581 -0.3111 -0.0350 0.0179 -0.1403 %! -0.0037 -0.0461 -0.1177 0.0092 -0.0242 %! 0.0476 -0.0237 -0.0159 0.0174 0.0464 ]; %! %!assert (abs (SYS.A), abs (Ae), 1e-4); %!assert (abs (SYS.B), abs (Be), 1e-4); %!assert (abs (SYS.C), abs (Ce), 1e-4); %!assert (abs (SYS.D), abs (De), 1e-4); ## [96-007] Data of a CD-player arm %!shared SYS, Ae, Be, Ce, De %! UY = [ 0.0531 -0.0313 0.0495 0.0342 %! 0.0526 -0.0328 0.0525 0.0342 %! 0.0504 -0.0482 0.0485 0.0567 %! 0.0423 -0.1048 0.0285 0.1542 %! 0.0269 -0.1515 0.0555 0.1192 %! 0.0072 -0.1404 0.0635 0.2067 %! -0.0091 -0.0936 0.1885 -0.0558 %! -0.0272 -0.0770 0.1145 0.2292 %! -0.0602 -0.0720 0.0985 0.2767 %! -0.0962 -0.1012 0.2485 -0.0733 %! -0.1266 -0.2732 0.2705 0.3017 %! -0.1548 -0.4999 0.3675 0.2567 %! -0.1831 -0.6836 0.3665 0.3492 %! -0.2225 -0.8477 0.3145 0.7492 %! -0.2638 -0.9533 0.4615 0.4067 %! -0.2865 -0.9926 0.5025 0.4917 %! -0.2877 -0.9519 0.5355 0.5942 %! -0.2652 -0.8100 0.5835 0.2917 %! -0.2229 -0.6546 0.4355 0.4417 %! -0.1690 -0.4802 0.3175 0.2667 %! -0.1022 -0.3147 0.2265 0.1067 %! -0.0262 -0.2181 0.1655 0.1717 %! 0.0508 -0.1214 0.1295 0.0167 %! 0.1181 0.0027 -0.0375 0.0642 %! 0.1639 0.1290 -0.2015 0.0767 %! 0.1915 0.2491 -0.2445 -0.0933 %! 0.2141 0.3501 -0.2085 -0.1833 %! 0.2383 0.4384 -0.2055 -0.2083 %! 0.2591 0.5325 -0.2915 -0.1533 %! 0.2720 0.6505 -0.3725 -0.2108 %! 0.2855 0.7959 -0.3975 -0.4083 %! 0.3036 0.9837 -0.4585 -0.4258 %! 0.3180 1.2078 -0.5175 -0.4108 %! 0.3278 1.4001 -0.4415 -0.6958 %! 0.3333 1.5244 -0.4445 -0.7183 %! 0.3201 1.6123 -0.6065 -0.4658 %! 0.2920 1.6983 -0.5645 -0.8133 %! 0.2720 1.7388 -0.3875 -1.0058 %! 0.2556 1.6854 -0.3775 -0.6208 %! 0.2279 1.5986 -0.4115 -0.6783 %! 0.1954 1.4945 -0.4215 -0.7808 %! 0.1667 1.3238 -0.3475 -0.6483 %! 0.1384 1.0917 -0.2985 -0.5033 %! 0.1039 0.8539 -0.2995 -0.3083 %! 0.0697 0.6475 -0.1155 -0.4483 %! 0.0391 0.4233 -0.0315 -0.2033 %! 0.0027 0.2235 -0.0775 0.0317 %! -0.0330 0.0966 0.0155 -0.2458 %! -0.0606 -0.0817 0.1125 0.0542 %! -0.0827 -0.3000 0.2685 0.1242 %! -0.0939 -0.4675 0.2865 0.0642 %! -0.0950 -0.5866 0.1685 0.3717 %! -0.0840 -0.6702 0.2735 0.1017 %! -0.0611 -0.7676 0.2505 0.2867 %! -0.0432 -0.7891 0.1245 0.5342 %! -0.0289 -0.6736 0.1945 0.0842 %! -0.0126 -0.5516 0.1885 0.2392 %! 0.0003 -0.4055 0.1795 0.1967 %! 0.0159 -0.2097 0.1605 -0.1208 %! 0.0348 -0.0408 0.0925 0.0217 %! 0.0506 0.1208 0.1395 -0.1408 %! 0.0627 0.3078 0.0795 -0.2183 %! 0.0723 0.4773 0.0175 -0.3158 %! 0.0792 0.5095 0.0215 -0.3533 %! 0.0752 0.4314 0.0085 -0.1533 %! 0.0590 0.4053 0.0425 -0.2683 %! 0.0378 0.4174 0.0245 -0.2658 %! 0.0103 0.4410 0.0295 -0.1783 %! -0.0258 0.4940 0.0905 -0.2633 %! -0.0652 0.4837 0.1385 -0.2758 %! -0.1035 0.3611 0.1805 -0.2258 %! -0.1429 0.1376 0.2035 -0.0383 %! -0.1847 -0.2251 0.3095 0.0167 %! -0.2340 -0.7017 0.3245 0.3692 %! -0.3036 -1.1050 0.3355 0.7517 %! -0.3827 -1.3696 0.5995 0.4917 %! -0.4557 -1.6423 0.7655 0.7017 %! -0.5309 -1.9082 0.7975 1.1367 %! -0.6018 -2.0464 0.9805 0.8317 %! -0.6458 -2.0844 1.1445 0.7692 %! -0.6613 -2.0949 1.2125 0.9217 %! -0.6589 -2.0794 1.1595 0.9317 %! -0.6399 -1.9906 1.1525 0.7942 %! -0.6009 -1.8632 1.1895 0.6867 %! -0.5490 -1.7523 1.0765 0.7617 %! -0.4912 -1.6356 0.9615 0.6342 %! -0.4272 -1.5132 0.8845 0.5817 %! -0.3567 -1.3945 0.8295 0.5517 %! -0.2766 -1.3215 0.7085 0.4392 %! -0.1868 -1.3317 0.5175 0.4792 %! -0.0971 -1.3609 0.3765 0.5317 %! -0.0192 -1.3473 0.2125 0.6367 %! 0.0466 -1.2796 0.0715 0.5192 %! 0.1089 -1.2096 -0.0385 0.4117 %! 0.1644 -1.1582 -0.1425 0.5742 %! 0.2056 -1.0522 -0.2015 0.4542 %! 0.2331 -0.9216 -0.3305 0.3817 %! 0.2478 -0.8330 -0.4485 0.4142 %! 0.2518 -0.7613 -0.4545 0.3017 %! 0.2470 -0.7081 -0.4315 0.2892 %! 0.2276 -0.6960 -0.4605 0.3442 %! 0.1865 -0.7076 -0.5155 0.3692 %! 0.1253 -0.7396 -0.4485 0.3242 %! 0.0472 -0.8146 -0.3375 0.4292 %! -0.0473 -0.8961 -0.2235 0.4967 %! -0.1521 -0.9436 -0.0515 0.4692 %! -0.2577 -0.9516 0.1265 0.5667 %! -0.3484 -0.8765 0.3575 0.3617 %! -0.4103 -0.7632 0.4915 0.2542 %! -0.4523 -0.6816 0.4825 0.4667 %! -0.4835 -0.5694 0.5725 0.2892 %! -0.4953 -0.4093 0.6665 0.1417 %! -0.4786 -0.2418 0.7055 0.0917 %! -0.4345 -0.0829 0.6615 0.0192 %! -0.3719 0.0565 0.5455 0.0617 %! -0.2949 0.2073 0.4655 -0.0808 %! -0.2016 0.3566 0.3255 -0.1433 %! -0.0946 0.3950 0.1765 -0.1383 %! 0.0180 0.2803 0.0325 -0.1083 %! 0.1239 0.1148 -0.1245 0.0392 %! 0.2137 -0.0357 -0.2785 0.0642 %! 0.2851 -0.1776 -0.4545 0.1267 %! 0.3386 -0.2838 -0.5135 0.1392 %! 0.3724 -0.3703 -0.5395 0.1992 %! 0.3804 -0.4262 -0.5995 0.2692 %! 0.3606 -0.3882 -0.6205 0.1717 %! 0.3143 -0.2657 -0.6005 0.1667 %! 0.2456 -0.0453 -0.4925 0.0092 %! 0.1633 0.2726 -0.3835 -0.1683 %! 0.0757 0.6140 -0.2705 -0.3183 %! -0.0088 0.8954 -0.1175 -0.5333 %! -0.0827 1.0590 -0.0395 -0.5358 %! -0.1377 1.1857 0.0545 -0.7408 %! -0.1707 1.3715 0.0725 -0.7933 %! -0.1911 1.5865 0.0695 -0.7758 %! -0.2027 1.7486 0.1445 -0.9933 %! -0.2042 1.8322 0.1265 -0.8733 %! -0.1901 1.9362 0.2065 -1.0933 %! -0.1550 2.0756 0.2185 -1.2008 %! -0.1150 2.2175 0.0835 -0.9258 %! -0.0782 2.4161 0.1225 -1.3158 %! -0.0382 2.5968 0.0995 -1.3508 %! 0.0009 2.6974 0.0685 -1.2508 %! 0.0392 2.7343 0.0955 -1.4683 %! 0.0755 2.6634 0.0165 -1.2133 %! 0.1076 2.5477 0.0225 -1.2458 %! 0.1431 2.4412 -0.0305 -1.2558 %! 0.1844 2.3062 -0.1165 -1.0908 %! 0.2260 2.1947 -0.1755 -1.1083 %! 0.2587 2.1622 -0.2835 -0.9508 %! 0.2798 2.1829 -0.3055 -1.0508 %! 0.2967 2.2071 -0.3375 -1.1083 %! 0.3114 2.2195 -0.3515 -1.0408 %! 0.3235 2.2042 -0.3295 -1.1133 %! 0.3315 2.0932 -0.3775 -0.9583 %! 0.3365 1.8795 -0.3815 -0.9233 %! 0.3446 1.6517 -0.4025 -0.8583 %! 0.3561 1.4464 -0.4285 -0.7183 %! 0.3619 1.2298 -0.4995 -0.5708 %! 0.3524 1.0610 -0.6035 -0.3683 %! 0.3355 0.9913 -0.5375 -0.5383 %! 0.3266 0.9109 -0.4835 -0.5358 %! 0.3218 0.7731 -0.5415 -0.2683 %! 0.3110 0.6149 -0.5715 -0.2408 %! 0.2969 0.4368 -0.5525 -0.1883 %! 0.2860 0.2525 -0.4825 -0.1483 %! 0.2740 0.0546 -0.5115 0.0767 %! 0.2521 -0.1300 -0.5075 0.2217 %! 0.2261 -0.2571 -0.3805 0.1317 %! 0.2065 -0.3238 -0.3155 0.1867 %! 0.1931 -0.2987 -0.3305 0.1942 %! 0.1808 -0.2329 -0.3565 0.2092 %! 0.1728 -0.2060 -0.2655 0.1117 %! 0.1778 -0.1696 -0.1935 0.0042 %! 0.1933 -0.0814 -0.2805 0.1292 %! 0.2136 0.0401 -0.3195 0.0017 %! 0.2433 0.1480 -0.3015 -0.1308 %! 0.2830 0.1864 -0.3395 -0.1183 %! 0.3231 0.1801 -0.5095 -0.0508 %! 0.3548 0.1985 -0.6255 -0.0283 %! 0.3828 0.2212 -0.5505 -0.2358 %! 0.4071 0.2130 -0.6185 -0.0833 %! 0.4161 0.2078 -0.7505 -0.0233 %! 0.4070 0.1729 -0.7735 -0.1208 %! 0.3820 0.0580 -0.7255 0.0892 %! 0.3472 -0.1245 -0.6195 0.0492 %! 0.3091 -0.3883 -0.6545 0.2617 %! 0.2664 -0.6903 -0.6355 0.5492 %! 0.2262 -1.0013 -0.4655 0.4792 %! 0.1930 -1.3467 -0.4925 0.8692 %! 0.1636 -1.6417 -0.5175 0.9717 %! 0.1427 -1.8525 -0.4415 0.9067 %! 0.1266 -2.0179 -0.4015 1.2292 %! 0.1118 -2.1203 -0.3485 1.0842 %! 0.1018 -2.1909 -0.3845 1.0817 %! 0.0912 -2.2222 -0.3635 1.2492 %! 0.0774 -2.1564 -0.2115 0.9892 %! 0.0614 -2.0300 -0.1995 1.0467 %! 0.0438 -1.8759 -0.1595 0.9517 %! 0.0385 -1.7096 0.0155 0.5367 %! 0.0457 -1.5773 0.0455 0.6742 %! 0.0461 -1.4479 -0.0415 0.7242 %! 0.0371 -1.2867 -0.0325 0.3842 %! 0.0259 -1.1849 0.0445 0.4142 %! 0.0133 -1.1751 0.1485 0.4092 %! -0.0023 -1.1735 0.1255 0.3992 %! -0.0234 -1.1757 0.0705 0.5492 %! -0.0427 -1.2288 0.1825 0.3642 %! -0.0546 -1.3379 0.1545 0.5367 %! -0.0693 -1.4457 0.0385 0.8092 %! -0.0862 -1.4768 0.0965 0.6067 %! -0.0960 -1.4632 0.1635 0.6542 %! -0.1013 -1.4416 0.1445 0.7117 %! -0.1098 -1.3817 0.0555 0.6417 %! -0.1307 -1.2943 0.0305 0.7067 %! -0.1687 -1.1853 0.1585 0.6017 %! -0.2203 -1.0350 0.2505 0.5317 %! -0.2805 -0.8197 0.3175 0.4542 %! -0.3427 -0.5362 0.4545 0.3067 %! -0.3950 -0.2783 0.6245 0.1217 %! -0.4257 -0.1265 0.6745 -0.0083 %! -0.4347 -0.0352 0.6115 0.0642 %! -0.4273 0.0849 0.6455 -0.0608 %! -0.4080 0.2669 0.6375 -0.0983 %! -0.3808 0.4593 0.5595 -0.1183 %! -0.3378 0.6035 0.5465 -0.4158 %! -0.2776 0.6956 0.4895 -0.3233 %! -0.2139 0.7767 0.4145 -0.2483 %! -0.1485 0.8759 0.3105 -0.5033 %! -0.0798 0.9496 0.1465 -0.4033 %! -0.0138 0.9867 0.1065 -0.4133 %! 0.0481 1.0353 0.0825 -0.5458 %! 0.1014 1.0584 -0.0285 -0.3958 %! 0.1417 1.0195 -0.1245 -0.3508 %! 0.1810 0.9643 -0.1175 -0.5058 %! 0.2304 0.8987 -0.1325 -0.4633 %! 0.2784 0.8223 -0.3155 -0.1833 %! 0.3158 0.7741 -0.4165 -0.3058 %! 0.3477 0.7241 -0.4475 -0.3608 %! 0.3700 0.6195 -0.5515 -0.0458 %! 0.3809 0.5139 -0.5905 -0.1633 %! 0.3896 0.4472 -0.5885 -0.1983 %! 0.3935 0.3697 -0.5985 0.0417 %! 0.3879 0.3296 -0.6335 -0.0308 %! 0.3781 0.3844 -0.6795 -0.1183 %! 0.3661 0.4741 -0.6555 -0.0908 %! 0.3493 0.6039 -0.5805 -0.2358 %! 0.3240 0.7004 -0.5635 -0.2533 %! 0.2858 0.6189 -0.5785 -0.0933 %! 0.2432 0.4427 -0.4435 -0.2383 %! 0.2074 0.2354 -0.3355 -0.1458 %! 0.1672 0.0128 -0.4075 0.2967 %! 0.1183 -0.1060 -0.3275 0.0767 %! 0.0720 -0.1569 -0.2185 0.0792 %! 0.0281 -0.1837 -0.1755 0.2942 %! -0.0110 -0.1465 -0.0965 0.0092 %! -0.0430 -0.1124 -0.1035 0.1892 %! -0.0694 -0.0907 0.0345 0.1267 %! -0.0822 -0.0402 0.1395 -0.0883 %! -0.0840 0.0366 0.0545 0.1267 %! -0.0813 0.1044 0.1125 -0.0758 %! -0.0744 0.0936 0.1615 -0.0783 %! -0.0736 0.0579 0.1215 0.1442 %! -0.0809 0.0546 0.1445 -0.0458 %! -0.0869 0.0402 0.1665 -0.0033 %! -0.0882 0.0025 0.2585 0.0167 %! -0.0891 -0.0893 0.2335 0.0617 %! -0.0989 -0.2016 0.1095 0.2567 %! -0.1157 -0.2898 0.1755 0.1517 %! -0.1279 -0.4203 0.2505 0.2192 %! -0.1321 -0.5523 0.2425 0.2892 %! -0.1353 -0.6380 0.1515 0.3892 %! -0.1451 -0.7266 0.0955 0.5492 %! -0.1517 -0.8032 0.1775 0.3517 %! -0.1402 -0.8444 0.1445 0.4217 %! -0.1103 -0.8473 0.0825 0.4467 %! -0.0677 -0.8711 0.0725 0.3342 %! -0.0286 -0.9469 -0.0805 0.6842 %! 0.0023 -0.9579 -0.1245 0.4242 %! 0.0381 -0.9479 -0.1135 0.2117 %! 0.0664 -0.9993 -0.2065 0.7217 %! 0.0742 -1.0048 -0.1365 0.4642 %! 0.0735 -1.0174 -0.0945 0.2817 %! 0.0619 -1.1243 -0.1855 0.6692 %! 0.0339 -1.2704 -0.1235 0.5067 %! -0.0041 -1.4761 -0.0475 0.6292 %! -0.0512 -1.6788 0.0215 0.8992 %! -0.0973 -1.7716 0.1645 0.6467 %! -0.1341 -1.8061 0.1835 0.7942 %! -0.1695 -1.7935 0.1665 0.9717 %! -0.1988 -1.7171 0.2675 0.6517 %! -0.2107 -1.6368 0.3285 0.6417 %! -0.2119 -1.5489 0.3375 0.7267 %! -0.2123 -1.4181 0.3615 0.5617 %! -0.2172 -1.2714 0.3455 0.5342 %! -0.2319 -1.0671 0.3335 0.4817 %! -0.2558 -0.7123 0.3665 0.2467 %! -0.2838 -0.2669 0.4385 0.0517 %! -0.3123 0.1531 0.5195 -0.1483 %! -0.3390 0.5226 0.5215 -0.3008 %! -0.3576 0.8555 0.4985 -0.4808 %! -0.3583 1.1684 0.5345 -0.7233 %! -0.3436 1.4304 0.5085 -0.6883 %! -0.3233 1.6427 0.4625 -0.7483 %! -0.2932 1.8196 0.4685 -1.0258 %! -0.2534 1.8815 0.3835 -0.8933 %! -0.2164 1.8872 0.2795 -0.7883 %! -0.1816 1.9679 0.2695 -0.9983 %! -0.1423 2.0907 0.2485 -1.0408 %! -0.1081 2.2150 0.1715 -0.9933 %! -0.0923 2.2823 0.0785 -0.9883 %! -0.0926 2.2505 0.0865 -1.0533 %! -0.0974 2.2151 0.1815 -1.1133 %! -0.1033 2.1746 0.2105 -1.0133 %! -0.1130 2.0369 0.1895 -0.9683 %! -0.1261 1.8613 0.1905 -0.8858 %! -0.1377 1.7249 0.2185 -0.8358 %! -0.1403 1.5793 0.2285 -0.8208 %! -0.1310 1.4183 0.2345 -0.7683 %! -0.1180 1.2871 0.2165 -0.6133 %! -0.1102 1.1936 0.1955 -0.5383 %! -0.1015 1.1864 0.2075 -0.7283 %! -0.0886 1.2592 0.1275 -0.5833 %! -0.0749 1.3410 0.1485 -0.5658 %! -0.0470 1.4143 0.2895 -0.9458 %! -0.0031 1.4604 0.1465 -0.6733 %! 0.0402 1.5447 -0.0435 -0.6083 %! 0.0876 1.6959 -0.0645 -1.0383 %! 0.1412 1.7869 -0.1525 -0.8083 %! 0.1883 1.8245 -0.2255 -0.8208 %! 0.2252 1.8679 -0.3275 -0.9933 %! 0.2467 1.9045 -0.4395 -0.8208 %! 0.2469 1.9533 -0.4565 -0.9058 %! 0.2301 1.9731 -0.5135 -0.9308 %! 0.2034 1.9264 -0.4865 -0.9233 %! 0.1679 1.8261 -0.4125 -0.8458 %! 0.1191 1.7087 -0.3975 -0.7083 %! 0.0578 1.6395 -0.3765 -0.7758 %! -0.0120 1.5785 -0.3385 -0.6708 %! -0.0860 1.4653 -0.1925 -0.6008 %! -0.1515 1.3092 0.0135 -0.6958 %! -0.2008 1.0958 0.0985 -0.5583 %! -0.2418 0.8423 0.0795 -0.3258 %! -0.2793 0.5898 0.1115 -0.2908 %! -0.3053 0.3063 0.2445 -0.3208 %! -0.3232 -0.0402 0.2745 0.0467 %! -0.3460 -0.3642 0.2835 0.3317 %! -0.3602 -0.6486 0.4845 0.0767 %! -0.3510 -1.0138 0.5625 0.3067 %! -0.3365 -1.3782 0.4035 0.8092 %! -0.3285 -1.6311 0.3775 0.7192 %! -0.3177 -1.8760 0.4785 0.7742 %! -0.3025 -2.0846 0.5275 1.0067 %! -0.2861 -2.1696 0.4745 1.0042 %! -0.2632 -2.1988 0.4385 0.9742 %! -0.2280 -2.2098 0.4705 0.9567 %! -0.1835 -2.2121 0.3835 1.0492 %! -0.1343 -2.2346 0.2365 1.0167 %! -0.0799 -2.2940 0.2005 0.9267 %! -0.0296 -2.4018 0.1485 1.1567 %! 0.0054 -2.5377 0.0265 1.2892 %! 0.0379 -2.6953 -0.0205 1.1017 %! 0.0850 -2.8762 -0.0155 1.1667 %! 0.1365 -3.0209 -0.1045 1.4792 %! 0.1758 -3.0638 -0.2865 1.4992 %! 0.2069 -3.0199 -0.3975 1.3217 %! 0.2399 -2.9711 -0.3785 1.2517 %! 0.2687 -2.9543 -0.4125 1.3817 %! 0.2813 -2.9447 -0.4615 1.3142 %! 0.2737 -2.9591 -0.4575 1.2217 %! 0.2385 -3.0112 -0.4585 1.4592 %! 0.1778 -3.0317 -0.3535 1.3667 %! 0.1115 -2.9655 -0.1805 1.1367 %! 0.0469 -2.8414 -0.0535 1.2642 %! -0.0254 -2.7009 0.0305 1.2717 %! -0.1049 -2.5296 0.0665 1.1567 %! -0.1749 -2.2903 0.2005 0.8842 %! -0.2261 -2.0294 0.3485 0.7692 %! -0.2709 -1.8409 0.3965 0.9217 %! -0.3160 -1.7059 0.4925 0.6592 %! -0.3553 -1.5587 0.5645 0.5542 %! -0.3887 -1.4311 0.6065 0.6467 %! -0.4132 -1.3691 0.6915 0.4717 %! -0.4202 -1.3607 0.7465 0.4992 %! -0.4079 -1.3743 0.8165 0.5242 %! -0.3790 -1.3781 0.7885 0.4892 %! -0.3393 -1.3442 0.6355 0.5792 %! -0.2925 -1.2459 0.5555 0.4817 %! -0.2378 -1.1148 0.5695 0.3467 %! -0.1856 -0.9931 0.4825 0.4667 %! -0.1486 -0.8185 0.3805 0.3767 %! -0.1216 -0.5874 0.3885 0.0717 %! -0.1015 -0.3821 0.3235 0.1667 %! -0.0898 -0.1778 0.2855 0.0667 %! -0.0712 0.0346 0.3355 -0.2858 %! -0.0379 0.2085 0.3175 -0.2483 %! -0.0027 0.3772 0.2065 -0.2333 %! 0.0236 0.5655 0.0175 -0.3458 %! 0.0438 0.7329 -0.0445 -0.4733 %! 0.0635 0.8253 0.0225 -0.5433 %! 0.0801 0.8649 0.0045 -0.4633 %! 0.0903 0.9308 -0.0115 -0.6133 %! 0.0901 1.0208 -0.0875 -0.5258 %! 0.0744 1.1279 -0.1185 -0.4933 %! 0.0558 1.2299 -0.0115 -0.8233 %! 0.0408 1.2587 -0.0005 -0.6233 %! 0.0191 1.2574 0.0385 -0.5433 %! -0.0063 1.2484 0.0785 -0.7158 %! -0.0255 1.1921 0.0205 -0.5933 %! -0.0336 1.1108 0.0425 -0.6808 %! -0.0357 0.9984 0.0135 -0.4883 %! -0.0415 0.8908 0.0155 -0.3583 %! -0.0416 0.8448 0.0965 -0.6158 %! -0.0291 0.8026 0.0265 -0.4233 %! -0.0135 0.7560 -0.0525 -0.2958 %! 0.0064 0.7406 -0.0195 -0.4958 %! 0.0347 0.7367 -0.0165 -0.4208 %! 0.0633 0.7939 -0.0945 -0.3233 %! 0.0912 0.9579 -0.1195 -0.5658 %! 0.1225 1.1168 -0.1215 -0.6283 %! 0.1480 1.1866 -0.2055 -0.4508 %! 0.1662 1.2186 -0.1875 -0.7083 %! 0.1851 1.2276 -0.2065 -0.7058 %! 0.1996 1.2388 -0.2605 -0.4633 %! 0.2131 1.3063 -0.2195 -0.7858 %! 0.2323 1.3793 -0.3485 -0.6333 %! 0.2558 1.4242 -0.3975 -0.6083 %! 0.2937 1.4094 -0.3555 -0.9133 %! 0.3392 1.2810 -0.5475 -0.4583 %! 0.3757 1.1218 -0.6745 -0.4483 %! 0.4103 0.9746 -0.7225 -0.6508 %! 0.4441 0.8090 -0.8085 -0.2808 %! 0.4710 0.6612 -0.8195 -0.3258 %! 0.4931 0.5291 -0.8975 -0.3208 %! 0.5052 0.4500 -0.9735 -0.1608 %! 0.4969 0.4825 -1.0065 -0.2058 %! 0.4640 0.5571 -1.0395 -0.1783 %! 0.4134 0.6007 -0.9225 -0.3358 %! 0.3525 0.5695 -0.7705 -0.2733 %! 0.2808 0.4708 -0.6625 -0.1683 %! 0.2038 0.3850 -0.5825 -0.2383 %! 0.1330 0.3329 -0.4905 -0.1808 %! 0.0800 0.2964 -0.3315 -0.2783 %! 0.0499 0.2154 -0.2585 -0.2158 %! 0.0356 0.0501 -0.2945 0.0017 %! 0.0345 -0.1225 -0.2425 -0.1633 %! 0.0424 -0.2963 -0.2555 0.0417 %! 0.0454 -0.4329 -0.3185 0.2467 %! 0.0459 -0.4041 -0.2455 -0.0408 %! 0.0501 -0.2796 -0.1785 0.0867 %! 0.0562 -0.1394 -0.1245 0.0392 %! 0.0708 0.0026 -0.1095 -0.2233 %! 0.0921 0.1000 -0.2235 0.0242 %! 0.1173 0.2755 -0.1985 -0.2208 %! 0.1574 0.5562 -0.1505 -0.5558 %! 0.2034 0.7663 -0.3005 -0.3233 %! 0.2345 0.8696 -0.3995 -0.4508 %! 0.2466 0.8681 -0.4305 -0.5183 %! 0.2426 0.8135 -0.4365 -0.4033 %! 0.2294 0.8375 -0.3955 -0.5383 %! 0.2117 0.9195 -0.3865 -0.4633 %! 0.1889 1.0417 -0.3415 -0.5258 %! 0.1615 1.2338 -0.3415 -0.6758 %! 0.1260 1.4168 -0.4065 -0.6508 %! 0.0810 1.5369 -0.3505 -0.7708 %! 0.0342 1.5810 -0.1615 -0.8558 %! -0.0091 1.5314 -0.0225 -0.8083 %! -0.0503 1.4241 -0.0285 -0.7283 %! -0.0876 1.2739 0.0045 -0.7208 %! -0.1141 1.0825 0.0965 -0.7008 %! -0.1308 0.9130 0.1345 -0.5608 %! -0.1446 0.7945 0.1385 -0.5108 %! -0.1608 0.7183 0.1425 -0.4833 %! -0.1846 0.7185 0.1855 -0.4508 %! -0.2169 0.7908 0.2355 -0.5133 %! -0.2502 0.8577 0.2985 -0.5833 %! -0.2747 0.8571 0.4505 -0.6633 %! -0.2911 0.7965 0.5145 -0.5358 %! -0.3055 0.7070 0.5125 -0.5008 %! -0.3153 0.5460 0.5205 -0.5583 %! -0.3242 0.2987 0.4875 -0.2383 %! -0.3392 0.0706 0.5795 -0.1583 %! -0.3528 -0.0710 0.6875 -0.2433 %! -0.3649 -0.1759 0.6485 0.0217 %! -0.3827 -0.3188 0.6335 0.1142 %! -0.3987 -0.5033 0.7005 0.0142 %! -0.4061 -0.6932 0.7185 0.2192 %! -0.4107 -0.8736 0.7015 0.4317 %! -0.4089 -1.0089 0.7395 0.3192 %! -0.3942 -1.1045 0.7075 0.4167 %! -0.3738 -1.2074 0.5835 0.6192 %! -0.3490 -1.3199 0.5075 0.5067 %! -0.3125 -1.4543 0.4785 0.5492 %! -0.2685 -1.5904 0.4425 0.6667 %! -0.2324 -1.6779 0.2805 0.7667 %! -0.2177 -1.7222 0.1215 0.8817 %! -0.2177 -1.7124 0.2035 0.6517 %! -0.2191 -1.6440 0.2945 0.6017 %! -0.2262 -1.5442 0.2725 0.7342 %! -0.2446 -1.4283 0.2725 0.5967 %! -0.2675 -1.3620 0.3145 0.5392 %! -0.2852 -1.3847 0.3815 0.5242 %! -0.2915 -1.4221 0.3885 0.5442 %! -0.2859 -1.4241 0.3955 0.6142 %! -0.2662 -1.4117 0.4345 0.5817 %! -0.2261 -1.4121 0.3795 0.5092 %! -0.1681 -1.4674 0.1875 0.5967 %! -0.1067 -1.5856 0.0025 0.7967 %! -0.0484 -1.7123 0.0105 0.7017 %! 0.0055 -1.8149 -0.0275 0.7892 %! 0.0441 -1.8852 -0.1755 1.0267 %! 0.0662 -1.9856 -0.1775 0.8417 %! 0.0769 -2.2193 -0.1235 1.0092 %! 0.0690 -2.4788 -0.1095 1.2917 %! 0.0433 -2.5547 -0.1075 1.1892 %! 0.0121 -2.4502 -0.0515 1.1292 %! -0.0163 -2.3174 0.0695 1.0167 %! -0.0431 -2.2535 0.0355 1.0567 %! -0.0690 -2.2237 -0.0085 1.0267 %! -0.0830 -2.1826 0.0905 0.8217 %! -0.0802 -2.1610 0.1245 0.9717 %! -0.0644 -2.1314 0.0895 0.8942 %! -0.0389 -2.0083 -0.0235 0.7767 %! -0.0144 -1.7807 -0.0925 0.8117 %! 0.0039 -1.5045 -0.0555 0.5167 %! 0.0181 -1.2185 -0.0975 0.3967 %! 0.0257 -0.9135 -0.1055 0.2917 %! 0.0253 -0.5901 -0.0405 0.0192 %! 0.0119 -0.2804 -0.0405 -0.0208 %! -0.0223 0.0284 -0.0525 -0.1183 %! -0.0722 0.3668 0.0375 -0.3658 %! -0.1248 0.6931 0.2185 -0.5433 %! -0.1762 0.9826 0.3255 -0.5983 %! -0.2279 1.2387 0.3505 -0.7408 %! -0.2801 1.4081 0.3595 -0.7783 %! -0.3313 1.4824 0.4385 -0.7633 %! -0.3714 1.4586 0.5705 -0.8483 %! -0.3940 1.3294 0.5915 -0.6908 %! -0.4058 1.2356 0.5915 -0.5733 %! -0.4124 1.2560 0.5965 -0.6108 %! -0.4123 1.3012 0.5405 -0.5908 %! -0.3988 1.2608 0.5535 -0.6983 %! -0.3752 1.0714 0.5255 -0.4733 %! -0.3563 0.8491 0.5215 -0.2658 %! -0.3428 0.6776 0.5805 -0.4083 %! -0.3337 0.4690 0.4925 -0.1158 %! -0.3337 0.2571 0.5005 0.0117 %! -0.3316 0.0998 0.6345 -0.1208 %! -0.3140 -0.0275 0.7025 -0.0458 %! -0.2848 -0.1338 0.6455 0.0317 %! -0.2575 -0.2528 0.4625 0.1967 %! -0.2342 -0.3685 0.4365 0.0917 %! -0.2083 -0.4770 0.4665 0.0767 %! -0.1875 -0.6038 0.3695 0.3942 %! -0.1767 -0.6891 0.3645 0.2767 %! -0.1647 -0.7274 0.4225 0.1342 %! -0.1496 -0.7568 0.3685 0.2642 %! -0.1427 -0.8146 0.2315 0.3767 %! -0.1438 -0.9273 0.2285 0.3517 %! -0.1384 -1.0284 0.4105 0.2117 %! -0.1249 -1.1491 0.4185 0.4717 %! -0.1147 -1.3249 0.2665 0.6917 %! -0.0989 -1.4528 0.2795 0.4867 %! -0.0619 -1.5434 0.3525 0.5667 %! -0.0081 -1.6087 0.2615 0.7242 %! 0.0507 -1.6067 0.0855 0.6692 %! 0.1051 -1.5640 -0.0405 0.7317 %! 0.1482 -1.4225 -0.0795 0.6492 %! 0.1804 -1.1516 -0.1505 0.4267 %! 0.2027 -0.8082 -0.2635 0.3492 %! 0.2135 -0.4165 -0.2375 0.1192 %! 0.2148 -0.0383 -0.1745 -0.1108 %! 0.2076 0.3144 -0.3045 -0.1433 %! 0.2004 0.6244 -0.3415 -0.4108 %! 0.2110 0.7909 -0.2105 -0.6583 %! 0.2361 0.8213 -0.2295 -0.3833 %! 0.2602 0.7620 -0.3495 -0.3533 %! 0.2841 0.6144 -0.4395 -0.4783 %! 0.3056 0.4209 -0.4835 -0.1683 %! 0.3176 0.2084 -0.4535 -0.0783 %! 0.3250 -0.0407 -0.4695 -0.0783 %! 0.3287 -0.3328 -0.5005 0.1792 %! 0.3187 -0.6266 -0.5085 0.4117 %! 0.2961 -0.8894 -0.5195 0.4617 %! 0.2740 -1.0959 -0.4995 0.4492 %! 0.2542 -1.1684 -0.4795 0.6467 %! 0.2321 -1.0800 -0.3895 0.5542 %! 0.2110 -0.8808 -0.3495 0.3067 %! 0.1900 -0.6415 -0.3965 0.3167 %! 0.1682 -0.4753 -0.3005 0.0467 %! 0.1458 -0.4414 -0.2055 0.0917 %! 0.1163 -0.4386 -0.1425 0.2617 %! 0.0891 -0.3733 0.0045 -0.0933 %! 0.0760 -0.3217 0.0415 -0.0408 %! 0.0669 -0.3010 -0.0135 0.1467 %! 0.0560 -0.2167 -0.0305 -0.0708 %! 0.0476 -0.1060 -0.0405 -0.0458 %! 0.0458 -0.0230 0.0035 -0.1208 %! 0.0540 0.0155 -0.0085 -0.1733 %! 0.0668 -0.0147 -0.1005 0.0117 %! 0.0792 -0.0398 -0.1425 -0.0508 %! 0.0933 -0.0347 -0.1735 -0.0508 %! 0.1089 -0.0374 -0.2045 -0.0283 %! 0.1195 -0.0380 -0.2625 0.0292 %! 0.1195 -0.0179 -0.2725 0.0267 %! 0.1110 0.0673 -0.2335 -0.1483 %! 0.0916 0.2159 -0.2865 -0.0383 %! 0.0596 0.3960 -0.2405 -0.1733 %! 0.0267 0.6076 -0.1025 -0.4333 %! -0.0017 0.8519 -0.0395 -0.3883 %! -0.0248 1.1211 0.0015 -0.6533 %! -0.0414 1.3389 -0.0415 -0.6833 %! -0.0505 1.4805 -0.0145 -0.7208 %! -0.0418 1.5473 0.0785 -1.0008 %! -0.0182 1.5086 -0.0465 -0.7158 %! 0.0047 1.4786 -0.1455 -0.6858 %! 0.0286 1.4618 -0.1135 -0.8933 %! 0.0544 1.3645 -0.1525 -0.6733 %! 0.0735 1.2553 -0.1945 -0.6583 %! 0.0804 1.1186 -0.2285 -0.5433 %! 0.0768 0.9336 -0.1345 -0.4483 %! 0.0700 0.7648 -0.0355 -0.4333 %! 0.0642 0.5964 -0.0615 -0.2583 %! 0.0608 0.4208 -0.0495 -0.2258 %! 0.0602 0.2307 -0.0155 -0.0408 %! 0.0659 0.0514 0.0435 -0.0433 %! 0.0820 -0.0945 0.0105 -0.0583 %! 0.0956 -0.2256 -0.1235 0.2667 %! 0.0948 -0.2807 -0.1055 0.1717 %! 0.0868 -0.2945 -0.0515 0.0367 %! 0.0754 -0.3494 -0.0825 0.1942 %! 0.0574 -0.4033 -0.0815 0.1867 %! 0.0357 -0.4587 -0.0115 0.1717 %! 0.0156 -0.5346 0.0505 0.1867 %! -0.0023 -0.6237 0.0135 0.3017 %! -0.0165 -0.7228 0.0525 0.2717 %! -0.0230 -0.7928 0.1605 0.2417 %! -0.0284 -0.8219 0.1615 0.4642 %! -0.0327 -0.8036 0.1625 0.2917 %! -0.0210 -0.7466 0.1915 0.1567 %! 0.0111 -0.7242 0.2005 0.3042 %! 0.0592 -0.7573 0.1465 0.2667 %! 0.1231 -0.7844 -0.0095 0.2467 %! 0.1999 -0.7794 -0.1375 0.2692 %! 0.2804 -0.7513 -0.2825 0.3117 %! 0.3560 -0.6774 -0.4315 0.2467 %! 0.4198 -0.5854 -0.5625 0.2242 %! 0.4650 -0.5277 -0.6685 0.3242 %! 0.4975 -0.4888 -0.6675 0.1642 %! 0.5329 -0.4856 -0.6705 0.0967 %! 0.5714 -0.5286 -0.7535 0.2292 %! 0.5974 -0.5528 -0.8835 0.3717 %! 0.6036 -0.5237 -0.9285 0.3542 %! 0.6018 -0.4513 -0.8545 0.1417 %! 0.5995 -0.3414 -0.8395 0.2067 %! 0.5916 -0.2010 -0.8205 0.1942 %! 0.5754 -0.0626 -0.8065 0.0892 %! 0.5542 0.0470 -0.8155 0.0967 %! 0.5375 0.0963 -0.7775 -0.0808 %! 0.5306 0.0637 -0.8045 0.0367 %! 0.5296 0.0065 -0.7795 0.0492 %! 0.5319 -0.0821 -0.7785 -0.0208 %! 0.5239 -0.2330 -0.9285 0.2767 %! 0.4911 -0.3534 -0.9875 0.2492 %! 0.4368 -0.4041 -0.9135 0.2767 %! 0.3774 -0.4396 -0.6965 0.2317 %! 0.3299 -0.5189 -0.5085 0.1617 %! 0.2946 -0.5876 -0.5285 0.4617 %! 0.2732 -0.5118 -0.4495 0.1917 %! 0.2735 -0.3676 -0.4335 0.1092 %! 0.2897 -0.2488 -0.4875 0.2317 %! 0.3188 -0.1107 -0.4445 -0.0758 %! 0.3580 0.0310 -0.4655 0.0367 %! 0.3999 0.2099 -0.4825 -0.0933 %! 0.4416 0.4537 -0.6305 -0.2708 %! 0.4758 0.6744 -0.8075 -0.1533 %! 0.5033 0.8167 -0.7235 -0.5058 %! 0.5249 0.8812 -0.7385 -0.3783 %! 0.5302 0.9226 -0.8115 -0.2858 %! 0.5213 0.9572 -0.8195 -0.5783 %! 0.4967 0.9528 -0.8725 -0.2483 %! 0.4535 0.9563 -0.7685 -0.3808 %! 0.4022 0.9923 -0.6685 -0.5258 %! 0.3435 1.0462 -0.6205 -0.2433 %! 0.2816 1.1181 -0.4365 -0.5508 %! 0.2276 1.1811 -0.3685 -0.5858 %! 0.1752 1.2622 -0.3765 -0.4508 %! 0.1188 1.4145 -0.3025 -0.6808 %! 0.0624 1.5738 -0.1975 -0.7083 %! 0.0118 1.6516 -0.0645 -0.8358 %! -0.0333 1.6513 -0.0625 -0.8383 %! -0.0826 1.5769 -0.1065 -0.6158 %! -0.1318 1.4702 0.0125 -0.7833 %! -0.1670 1.4121 0.1115 -0.7908 %! -0.1917 1.3594 0.0985 -0.5433 %! -0.2125 1.2994 0.1295 -0.6108 %! -0.2249 1.2767 0.2415 -0.6833 %! -0.2319 1.2354 0.2795 -0.5283 %! -0.2430 1.1914 0.2665 -0.4958 %! -0.2598 1.1599 0.3615 -0.5608 %! -0.2818 1.0552 0.4915 -0.4008 %! -0.3076 0.8983 0.5765 -0.2833 %! -0.3210 0.7477 0.6535 -0.4083 %! -0.3101 0.6146 0.6765 -0.3383 %! -0.2900 0.5163 0.5625 0.0067 %! -0.2723 0.4557 0.4575 -0.0683 %! -0.2465 0.3869 0.4035 -0.2233 %! -0.2099 0.2499 0.3495 0.0092 %! -0.1739 0.1037 0.3055 0.0267 %! -0.1512 0.0119 0.1695 0.1517 %! -0.1484 -0.0519 0.0805 0.2842 %! -0.1510 -0.0699 0.2045 -0.0533 %! -0.1495 -0.0575 0.2485 0.0517 %! -0.1572 -0.0494 0.2205 0.2867 %! -0.1715 -0.0612 0.3355 -0.0658 %! -0.1810 -0.1391 0.3715 0.0442 %! -0.1916 -0.2317 0.3255 0.2367 %! -0.2045 -0.3066 0.3375 0.0842 %! -0.2152 -0.4015 0.3625 0.2167 %! -0.2228 -0.5104 0.4315 0.2292 %! -0.2264 -0.6669 0.3695 0.2817 %! -0.2322 -0.8304 0.2035 0.5217 %! -0.2424 -0.9202 0.2115 0.4992 %! -0.2438 -0.9549 0.3045 0.3742 %! -0.2288 -0.9232 0.2735 0.3792 %! -0.2076 -0.8649 0.1505 0.5142 %! -0.1881 -0.8226 0.1345 0.4142 %! -0.1670 -0.7409 0.1615 0.2617 %! -0.1451 -0.6425 0.1015 0.3617 %! -0.1230 -0.5638 0.1275 0.1917 %! -0.0984 -0.5363 0.2195 0.1492 %! -0.0803 -0.5953 0.1945 0.3917 %! -0.0729 -0.6555 0.1085 0.3267 %! -0.0632 -0.6865 0.1145 0.1717 %! -0.0489 -0.7249 0.1485 0.3442 %! -0.0439 -0.7028 0.0995 0.4717 %! -0.0451 -0.6042 0.0625 0.2117 %! -0.0406 -0.5269 0.0245 0.1367 %! -0.0383 -0.4894 -0.0435 0.3392 %! -0.0485 -0.4438 -0.0455 0.2067 %! -0.0679 -0.4331 -0.0395 0.2042 %! -0.0895 -0.4768 0.0445 0.2217 %! -0.1077 -0.4958 0.1265 0.0817 %! -0.1329 -0.4951 0.0275 0.3717 %! -0.1737 -0.5221 0.0445 0.3217 %! -0.2132 -0.5739 0.2165 0.0742 %! -0.2403 -0.6589 0.2895 0.3267 %! -0.2575 -0.7592 0.3165 0.3442 %! -0.2639 -0.8295 0.2885 0.3417 %! -0.2594 -0.8865 0.2775 0.3992 %! -0.2440 -0.9962 0.2605 0.3367 %! -0.2267 -1.1301 0.0985 0.6167 %! -0.2183 -1.1910 0.0655 0.6417 %! -0.2123 -1.1804 0.1735 0.4192 %! -0.2063 -1.1308 0.1145 0.5367 %! -0.2103 -1.0522 -0.0135 0.6292 %! -0.2177 -0.9723 0.0595 0.3867 %! -0.2085 -0.8974 0.2295 0.2317 %! -0.1816 -0.8520 0.1905 0.4017 %! -0.1471 -0.8541 0.0715 0.3917 %! -0.1048 -0.8622 0.0615 0.2592 %! -0.0576 -0.8463 0.0615 0.3767 %! -0.0136 -0.8192 -0.0185 0.3392 %! 0.0235 -0.7788 -0.1415 0.3292 %! 0.0551 -0.7021 -0.1095 0.2867 %! 0.0856 -0.6467 -0.0675 0.1667 %! 0.1120 -0.6449 -0.2705 0.3217 %! 0.1297 -0.6291 -0.4115 0.2392 %! 0.1433 -0.6098 -0.3755 0.1167 %! 0.1493 -0.5649 -0.4015 0.2967 %! 0.1413 -0.4200 -0.4795 0.1742 %! 0.1251 -0.2349 -0.5045 -0.0333 %! 0.1035 -0.0723 -0.4195 -0.0283 %! 0.0714 0.0871 -0.3485 -0.0533 %! 0.0300 0.2866 -0.3345 -0.1883 %! -0.0115 0.5233 -0.2235 -0.3508 %! -0.0474 0.7916 -0.0725 -0.4258 %! -0.0765 1.1562 -0.0105 -0.6133 %! -0.0981 1.5229 -0.0365 -0.7958 %! -0.1162 1.7436 -0.0585 -0.7983 %! -0.1316 1.8822 0.0025 -0.9008 %! -0.1349 1.9916 0.0605 -1.0358 %! -0.1202 2.0391 0.0065 -1.0233 %! -0.0932 2.0123 -0.0625 -0.9483 %! -0.0609 1.9210 -0.0805 -0.9583 %! -0.0293 1.8059 -0.1785 -0.8458 %! -0.0052 1.7253 -0.2625 -0.7733 %! 0.0091 1.7317 -0.2195 -0.8883 %! 0.0063 1.7910 -0.1925 -0.7333 %! -0.0175 1.8525 -0.1535 -0.7433 %! -0.0460 1.9114 -0.0405 -1.0208 %! -0.0656 1.9435 0.0365 -0.9383 %! -0.0793 1.9967 0.1075 -0.9433 %! -0.0890 2.0655 0.1275 -1.0708 %! -0.0971 2.0547 0.0985 -0.9808 %! -0.1056 1.9936 0.1425 -1.0508 %! -0.1133 1.9289 0.1445 -1.0233 %! -0.1202 1.8841 0.1445 -1.0283 %! -0.1278 1.8653 0.1605 -1.1133 %! -0.1456 1.8273 0.0905 -0.9433 %! -0.1796 1.7730 0.0805 -0.9833 %! -0.2191 1.6584 0.1575 -1.0358 %! -0.2544 1.4743 0.2565 -0.9083 %! -0.2854 1.3089 0.3565 -0.9083 %! -0.3208 1.1494 0.3295 -0.6758 %! -0.3642 0.9927 0.3655 -0.6683 %! -0.4052 0.8263 0.5005 -0.7658 %! -0.4443 0.5780 0.5355 -0.4133 %! -0.4891 0.3103 0.6355 -0.3558 %! -0.5373 0.1107 0.7275 -0.2808 %! -0.5849 0.0334 0.7645 -0.1008 %! -0.6212 0.1017 0.8415 -0.3308 %! -0.6375 0.2141 0.8055 -0.2608 %! -0.6349 0.3187 0.8195 -0.2683 %! -0.6100 0.4173 0.8785 -0.5558 %! -0.5747 0.4404 0.6735 -0.2058 %! -0.5453 0.3679 0.5525 -0.1733 %! -0.5116 0.2496 0.6265 -0.4183 %! -0.4674 0.1314 0.6485 -0.1408 %! -0.4214 0.0316 0.6365 -0.1358 %! -0.3811 -0.0349 0.5035 -0.0783 %! -0.3498 -0.0504 0.4445 0.0092 %! -0.3165 -0.0772 0.5055 -0.2258 %! -0.2736 -0.1911 0.4325 -0.0983 %! -0.2333 -0.3845 0.3495 0.0917 %! -0.2116 -0.6080 0.2855 0.1917 %! -0.2140 -0.7873 0.2095 0.3642 %! -0.2278 -0.9055 0.1795 0.2642 %! -0.2368 -1.0029 0.1665 0.3317 %! -0.2350 -1.0601 0.2255 0.3617 %! -0.2232 -1.0793 0.2145 0.2592 %! -0.2166 -1.0744 -0.0305 0.4942 %! -0.2288 -0.9663 -0.1795 0.4292 %! -0.2482 -0.7630 -0.0595 0.1117 %! -0.2598 -0.5967 0.1215 0.0967 %! -0.2618 -0.4925 0.2025 0.0492 %! -0.2590 -0.4010 0.1605 0.0367 %! -0.2550 -0.3328 0.1185 0.0867 %! -0.2431 -0.2946 0.1605 -0.0808 %! -0.2166 -0.2787 0.1645 -0.0383 %! -0.1817 -0.3122 0.1755 0.1067 %! -0.1408 -0.4076 0.2485 -0.0058 %! -0.0955 -0.5121 0.1335 0.1192 %! -0.0580 -0.5406 -0.0965 0.2292 %! -0.0371 -0.4996 -0.1945 0.1292 %! -0.0357 -0.4631 -0.1495 0.2142 %! -0.0497 -0.4436 -0.0065 0.0717 %! -0.0737 -0.4519 -0.0165 0.0992 %! -0.1119 -0.4687 -0.0865 0.2892 %! -0.1561 -0.4543 0.0375 -0.0208 %! -0.1950 -0.4876 0.1145 0.0817 %! -0.2343 -0.5827 0.1375 0.2967 %! -0.2695 -0.6924 0.2895 0.0017 %! -0.2983 -0.8815 0.3295 0.3067 %! -0.3325 -1.1117 0.3055 0.5492 %! -0.3656 -1.2956 0.3705 0.2967 %! -0.3906 -1.4858 0.4285 0.5417 %! -0.4198 -1.6773 0.4905 0.8242 %! -0.4544 -1.8112 0.5555 0.6917 %! -0.4817 -1.9092 0.5635 0.7317 %! -0.4984 -1.9624 0.5865 0.8517 %! -0.5052 -1.9578 0.6725 0.7392 %! -0.5043 -1.9750 0.6675 0.8467 %! -0.5005 -2.0244 0.6645 0.8942 %! -0.4895 -2.0492 0.7325 0.7292 %! -0.4713 -2.0865 0.6905 0.9517 %! -0.4485 -2.1329 0.7045 0.9417 %! -0.4108 -2.1318 0.7525 0.7467 %! -0.3562 -2.1064 0.6665 0.9792 %! -0.2897 -2.0510 0.5845 0.8717 %! -0.2097 -1.9414 0.4475 0.7142 %! -0.1240 -1.8255 0.2975 0.8092 %! -0.0458 -1.7261 0.1825 0.7217 %! 0.0176 -1.5932 0.0225 0.7017 %! 0.0662 -1.4228 -0.0535 0.5917 %! 0.1049 -1.3030 -0.0845 0.4792 %! 0.1335 -1.2418 -0.1745 0.5192 %! 0.1483 -1.1914 -0.2385 0.4542 %! 0.1466 -1.1992 -0.2885 0.5092 %! 0.1282 -1.2520 -0.2755 0.5542 %! 0.0992 -1.2430 -0.2205 0.5267 %! 0.0699 -1.1510 -0.1345 0.4517 %! 0.0478 -1.0541 -0.0455 0.3867 %! 0.0322 -0.9852 -0.0195 0.4267 %! 0.0207 -0.8964 0.0005 0.2817 %! 0.0094 -0.8166 -0.0155 0.3042 %! -0.0108 -0.7687 0.0055 0.3492 %! -0.0421 -0.6628 0.0855 0.1892 %! -0.0816 -0.5166 0.1335 0.2067 %! -0.1240 -0.4531 0.1985 0.1517 %! -0.1587 -0.4955 0.2175 0.1317 %! -0.1765 -0.5726 0.2375 0.2242 %! -0.1697 -0.6386 0.2575 0.1567 %! -0.1396 -0.6945 0.1815 0.2542 %! -0.0986 -0.7609 0.0965 0.3417 %! -0.0577 -0.8911 0.0185 0.4417 %! -0.0179 -1.0590 0.0145 0.4917 %! 0.0260 -1.1811 0.0785 0.4217 %! 0.0683 -1.2444 0.0305 0.6217 %! 0.0949 -1.1920 -0.0365 0.6392 %! 0.1073 -1.0508 -0.0215 0.4067 %! 0.1155 -0.9975 -0.0025 0.4092 %! 0.1214 -1.0199 0.0115 0.4092 %! 0.1192 -0.9884 0.0035 0.4167 %! 0.1011 -0.8955 -0.0175 0.4642 %! 0.0706 -0.7734 -0.0055 0.3317 %! 0.0435 -0.6574 0.0385 0.1417 %! 0.0254 -0.5681 0.0495 0.1792 %! 0.0072 -0.4466 0.0325 0.2567 %! -0.0080 -0.2535 0.0875 -0.0533 %! -0.0115 -0.0873 0.0575 -0.1208 %! -0.0112 -0.0130 -0.0395 0.0817 %! -0.0089 0.0369 0.0145 -0.1758 %! 0.0030 0.0533 0.0625 -0.1708 %! 0.0194 -0.0025 0.0185 0.0467 %! 0.0374 -0.0338 -0.0445 -0.0783 %! 0.0607 0.0134 -0.0985 -0.0633 %! 0.0869 0.0666 -0.1175 -0.0358 %! 0.1116 0.1083 -0.1945 -0.0683 %! 0.1329 0.1776 -0.2645 -0.0883 %! 0.1515 0.2269 -0.2495 -0.1808 %! 0.1643 0.2195 -0.3015 -0.0533 %! 0.1677 0.2351 -0.3745 -0.0333 %! 0.1713 0.2860 -0.3445 -0.2508 %! 0.1813 0.2769 -0.3195 -0.1283 %! 0.1898 0.1852 -0.3195 -0.0083 %! 0.1937 0.0800 -0.3415 -0.0683 %! 0.1960 0.0125 -0.3745 0.0767 %! 0.2058 -0.0679 -0.2675 -0.0358 %! 0.2313 -0.2067 -0.2205 -0.0433 %! 0.2604 -0.2975 -0.3315 0.2442 %! 0.2806 -0.2723 -0.3645 0.0692 %! 0.2919 -0.2029 -0.3955 0.0117 %! 0.2899 -0.0986 -0.4325 0.0517 %! 0.2731 0.0407 -0.3855 -0.1783 %! 0.2420 0.1252 -0.3585 -0.1008 %! 0.1990 0.1201 -0.2645 -0.1433 %! 0.1551 0.0612 -0.2165 -0.2383 %! 0.1148 -0.0002 -0.2735 -0.0133 %! 0.0804 -0.0214 -0.2015 -0.1033 %! 0.0626 -0.0484 -0.0865 -0.2383 %! 0.0590 -0.1404 -0.1335 0.0367 %! 0.0604 -0.2218 -0.2005 0.0567 %! 0.0731 -0.2085 -0.1645 -0.1383 %! 0.0997 -0.1695 -0.1575 -0.0058 %! 0.1295 -0.1504 -0.2145 0.0242 %! 0.1556 -0.1147 -0.2605 -0.0783 %! 0.1749 -0.1237 -0.2505 0.0192 %! 0.1857 -0.2165 -0.2005 -0.0008 %! 0.1859 -0.2956 -0.2755 0.1167 %! 0.1726 -0.2906 -0.3025 0.1892 %! 0.1556 -0.2457 -0.1625 -0.0583 %! 0.1417 -0.2264 -0.1545 0.0942 %! 0.1269 -0.2038 -0.1905 0.0842 %! 0.1129 -0.1753 -0.1785 -0.0658 %! 0.0992 -0.1663 -0.1525 0.1442 %! 0.0875 -0.1169 -0.0275 -0.1183 %! 0.0790 -0.0582 -0.0545 -0.0283 %! 0.0663 -0.0174 -0.0605 0.0767 %! 0.0562 0.0665 0.0935 -0.3133 %! 0.0531 0.1512 0.0685 -0.1408 %! 0.0461 0.2151 0.0115 -0.1208 %! 0.0374 0.2488 0.0475 -0.3083 %! 0.0370 0.2115 0.1245 -0.2358 %! 0.0492 0.1377 0.1595 -0.3158 %! 0.0650 0.0180 -0.0255 -0.0283 %! 0.0750 -0.1500 -0.0805 0.0317 %! 0.0891 -0.3396 0.0055 -0.0733 %! 0.1133 -0.5416 -0.0395 0.1742 %! 0.1401 -0.7139 -0.1485 0.2417 %! 0.1603 -0.9009 -0.2485 0.4742 %! 0.1780 -1.1065 -0.1815 0.4742 %! 0.2099 -1.2319 -0.1665 0.3667 %! 0.2553 -1.2788 -0.3375 0.6817 %! 0.3076 -1.2666 -0.3695 0.5117 %! 0.3685 -1.2250 -0.3785 0.4217 %! 0.4305 -1.1970 -0.5295 0.5917 %! 0.4856 -1.1524 -0.7095 0.4792 %! 0.5373 -1.1079 -0.7375 0.4042 %! 0.5897 -1.0977 -0.6755 0.3842 %! 0.6362 -1.0851 -0.8095 0.4792 %! 0.6699 -1.0496 -0.9775 0.4067 %! 0.6894 -0.9976 -1.0365 0.3367 %! 0.6901 -0.9060 -1.0275 0.4142 %! 0.6719 -0.7677 -0.9995 0.2142 %! 0.6371 -0.6139 -1.0315 0.2267 %! 0.5849 -0.4463 -0.9335 0.1717 %! 0.5219 -0.2783 -0.7995 0.0167 %! 0.4573 -0.1421 -0.7785 0.0567 %! 0.4020 -0.0451 -0.6615 -0.1833 %! 0.3559 0.0389 -0.5805 -0.0758 %! 0.3050 0.2057 -0.5415 0.0042 %! 0.2521 0.4271 -0.4775 -0.3708 %! 0.2065 0.5643 -0.4755 -0.3033 %! 0.1686 0.6331 -0.3475 -0.3783 %! 0.1358 0.6722 -0.2895 -0.3708 %! 0.1037 0.6797 -0.3195 -0.2408 %! 0.0778 0.6585 -0.2575 -0.4533 %! 0.0624 0.5950 -0.2515 -0.2558 %! 0.0517 0.5407 -0.2285 -0.1908 %! 0.0473 0.5332 -0.1815 -0.3383 %! 0.0465 0.6105 -0.1795 -0.2558 %! 0.0382 0.7590 -0.1585 -0.3383 %! 0.0154 0.8366 -0.1745 -0.3208 %! -0.0179 0.8604 -0.0875 -0.3733 %! -0.0470 0.8872 0.1005 -0.5033 %! -0.0626 0.8752 0.1875 -0.3783 %! -0.0602 0.8813 0.2065 -0.5008 %! -0.0402 0.8863 0.1245 -0.4583 %! -0.0124 0.8391 0.0235 -0.3433 %! 0.0192 0.7919 -0.0165 -0.4783 %! 0.0554 0.7361 -0.0555 -0.3733 %! 0.0929 0.6751 -0.0475 -0.3683 %! 0.1304 0.6184 -0.0965 -0.3933 %! 0.1639 0.5700 -0.2405 -0.2308 %! 0.1967 0.6093 -0.2875 -0.4483 %! 0.2338 0.7093 -0.3125 -0.4933 %! 0.2645 0.8005 -0.3845 -0.3158 %! 0.2869 0.8915 -0.3885 -0.5983 %! 0.3091 0.9385 -0.4365 -0.6158 %! 0.3251 0.9464 -0.5195 -0.4483 %! 0.3311 0.9586 -0.5255 -0.6083 %! 0.3313 0.9269 -0.5245 -0.5558 %! 0.3224 0.8837 -0.5195 -0.4508 %! 0.3000 0.9051 -0.5285 -0.4733 %! 0.2679 0.9557 -0.5235 -0.4883 %! 0.2360 1.0002 -0.4485 -0.6033 %! 0.2076 0.9830 -0.4105 -0.5083 %! 0.1760 0.8692 -0.4285 -0.3633 %! 0.1395 0.7508 -0.4055 -0.3758 %! 0.1052 0.6499 -0.3275 -0.3558 %! 0.0774 0.5699 -0.2655 -0.3033 %! 0.0517 0.6138 -0.2495 -0.3008 %! 0.0213 0.7287 -0.1975 -0.3383 %! -0.0184 0.7977 -0.1255 -0.3633 %! -0.0668 0.8589 -0.0445 -0.4283 %! -0.1192 0.9156 0.0605 -0.5133 %! -0.1747 0.9026 0.1815 -0.4483 %! -0.2323 0.8126 0.3325 -0.4383 %! -0.2839 0.6700 0.4405 -0.4483 %! -0.3233 0.5204 0.4685 -0.3258 %! -0.3485 0.4290 0.5085 -0.3033 %! -0.3550 0.4219 0.5735 -0.3533 %! -0.3426 0.4701 0.5585 -0.3108 %! -0.3172 0.5523 0.4785 -0.3108 %! -0.2791 0.6432 0.4565 -0.4733 %! -0.2315 0.6805 0.4115 -0.4533 %! -0.1894 0.6917 0.2995 -0.3708 %! -0.1591 0.7669 0.2685 -0.5033 %! -0.1352 0.8755 0.2685 -0.5383 %! -0.1136 0.9348 0.2695 -0.5183 %! -0.0901 0.9419 0.2325 -0.6033 %! -0.0643 0.9225 0.1315 -0.5033 %! -0.0407 0.8522 0.0945 -0.4833 %! -0.0225 0.7295 0.0265 -0.4108 %! -0.0137 0.6081 -0.0405 -0.3008 %! -0.0151 0.4741 -0.0395 -0.2883 %! -0.0274 0.3231 -0.0395 -0.1408 %! -0.0520 0.2134 -0.0215 -0.0958 %! -0.0864 0.1254 -0.0295 -0.0233 %! -0.1263 0.0256 0.0055 0.0442 %! -0.1653 -0.0778 0.0825 0.0617 %! -0.1988 -0.1712 0.1285 0.1667 %! -0.2238 -0.2647 0.1945 0.1292 %! -0.2412 -0.3741 0.1865 0.2617 %! -0.2549 -0.4006 0.1935 0.3267 %! -0.2573 -0.3240 0.2765 0.0892 %! -0.2433 -0.2505 0.2645 0.1417 %! -0.2231 -0.1534 0.2415 0.1142 %! -0.2069 -0.0410 0.2085 0.0342 %! -0.2000 0.0238 0.1645 0.0642 %! -0.2001 0.0544 0.1985 -0.0808 %! -0.2037 -0.0099 0.2065 0.0317 %! -0.2104 -0.1788 0.2635 0.1042 %! -0.2160 -0.3564 0.2935 0.1217 %! -0.2186 -0.5024 0.2335 0.2917 %! -0.2186 -0.5871 0.1745 0.3217 %! -0.2108 -0.5916 0.1735 0.2567 %! -0.1901 -0.5845 0.2265 0.1692 %! -0.1679 -0.6241 0.1165 0.4067 %! -0.1573 -0.6377 0.0325 0.3817 %! -0.1530 -0.6189 0.0765 0.1342 %! -0.1569 -0.6339 0.0265 0.4667 %! -0.1734 -0.6051 0.0855 0.3067 %! -0.1951 -0.5319 0.1385 0.1567 %! -0.2237 -0.4474 0.1275 0.3742 %! -0.2571 -0.3082 0.2075 0.0317 %! -0.2879 -0.2211 0.1775 0.0992 %! -0.3188 -0.1793 0.1985 0.1917 %! -0.3429 -0.0881 0.3475 -0.1158 %! -0.3541 -0.0297 0.3825 -0.0333 %! -0.3618 0.0134 0.3265 0.0117 %! -0.3716 0.0785 0.2805 -0.0033 %! -0.3719 0.0910 0.3965 -0.1883 %! -0.3546 0.0535 0.4395 -0.1708 %! -0.3345 0.0320 0.3055 0.1492 %! -0.3160 0.0821 0.3255 -0.1533 %! -0.2892 0.1773 0.3235 -0.2208 %! -0.2587 0.2922 0.2255 -0.0133 %! -0.2214 0.4446 0.2095 -0.3608 %! -0.1671 0.5549 0.2005 -0.4208 %! -0.1050 0.5845 0.1275 -0.2683 %! -0.0486 0.5966 -0.0525 -0.3083 %! 0.0032 0.5948 -0.1795 -0.4133 %! 0.0516 0.5806 -0.2155 -0.3683 %! 0.0863 0.5808 -0.2465 -0.2633 %! 0.1019 0.5570 -0.2995 -0.3258 %! 0.1007 0.4895 -0.3555 -0.2833 %! 0.0823 0.4475 -0.3445 -0.2083 %! 0.0486 0.4517 -0.3085 -0.1983 %! 0.0106 0.4461 -0.2495 -0.2433 %! -0.0204 0.3968 -0.1525 -0.2433 %! -0.0427 0.3374 -0.1145 -0.1308 %! -0.0570 0.3425 -0.1035 -0.2283 %! -0.0662 0.4094 -0.1785 -0.1583 %! -0.0749 0.4980 -0.1755 -0.1458 %! -0.0767 0.5981 -0.0495 -0.3908 %! -0.0689 0.6888 -0.0465 -0.2683 %! -0.0577 0.7881 -0.0605 -0.3833 %! -0.0469 0.8619 -0.1055 -0.4733 %! -0.0422 0.8321 -0.1485 -0.3083 %! -0.0387 0.7466 -0.0695 -0.5408 %! -0.0309 0.6634 -0.0885 -0.3933 %! -0.0268 0.5794 -0.0795 -0.2908 %! -0.0307 0.5002 -0.0445 -0.4458 %! -0.0508 0.4760 -0.1505 -0.1608 %! -0.0894 0.5679 -0.0875 -0.3133 %! -0.1283 0.7078 0.1175 -0.6508 %! -0.1663 0.7919 0.1845 -0.3833 %! -0.2133 0.8265 0.2345 -0.4483 %! -0.2599 0.8374 0.3125 -0.6758 %! -0.2969 0.8162 0.3435 -0.4808 %! -0.3210 0.7420 0.4165 -0.5058 %! -0.3215 0.5730 0.4795 -0.5808 %! -0.3013 0.2979 0.4085 -0.1883 %! -0.2680 0.0143 0.3635 -0.1408 %! -0.2172 -0.1931 0.2695 -0.1933 %! -0.1568 -0.3482 0.0795 0.1542 %! -0.0990 -0.5055 0.0565 0.0917 %! -0.0475 -0.7178 -0.0035 0.2642 %! -0.0025 -0.9341 -0.0295 0.3692 %! 0.0428 -1.0834 -0.0325 0.2067 %! 0.0814 -1.2097 -0.2115 0.6567 %! 0.1090 -1.3175 -0.1885 0.4517 %! 0.1336 -1.4127 -0.1795 0.3767 %! 0.1445 -1.5065 -0.3165 0.8592 %! 0.1424 -1.5894 -0.2155 0.4442 %! 0.1379 -1.7094 -0.2005 0.5842 %! 0.1194 -1.7628 -0.2475 0.9192 %! 0.0871 -1.6474 -0.1865 0.5092 %! 0.0541 -1.4894 -0.1495 0.4967 %! 0.0205 -1.3807 -0.0495 0.4817 %! -0.0189 -1.3325 -0.0365 0.4492 %! -0.0630 -1.3452 -0.0175 0.4417 %! -0.1055 -1.4067 0.0835 0.4317 %! -0.1473 -1.4711 0.1405 0.6042 %! -0.1886 -1.4663 0.2055 0.4892 %! -0.2252 -1.4405 0.2435 0.5292 %! -0.2545 -1.4530 0.3345 0.5367 %! -0.2726 -1.4790 0.3915 0.4667 %! -0.2804 -1.5291 0.3805 0.6292 %! -0.2800 -1.6064 0.4145 0.5967 %! -0.2711 -1.6577 0.4125 0.6642 %! -0.2543 -1.6503 0.4445 0.6142 %! -0.2332 -1.6413 0.3945 0.6267 %! -0.2186 -1.6598 0.2995 0.8167 %! -0.2105 -1.6905 0.3655 0.6167 %! -0.2016 -1.7773 0.3605 0.6767 %! -0.1990 -1.8767 0.3035 0.8442 %! -0.2109 -1.9359 0.2905 0.8542 %! -0.2309 -1.9978 0.3625 0.8267 %! -0.2464 -2.0725 0.4795 0.7117 %! -0.2598 -2.1566 0.3915 1.0417 %! -0.2729 -2.2060 0.4415 0.9092 %! 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0.2233 -0.7484 -0.3445 0.0942 %! 0.1902 -0.5707 -0.2465 0.1017 %! 0.1574 -0.4626 -0.2065 0.2217 %! 0.1245 -0.3397 -0.1795 -0.0258 %! 0.0987 -0.1692 -0.2115 0.0592 %! 0.0892 0.0685 -0.1245 -0.2233 %! 0.1028 0.3453 -0.1275 -0.3658 %! 0.1273 0.5826 -0.2405 -0.2433 %! 0.1497 0.7764 -0.2575 -0.4733 %! 0.1646 0.9231 -0.3065 -0.4433 %! 0.1718 1.0326 -0.3265 -0.4683 %! 0.1809 1.1361 -0.2795 -0.6558 %! 0.1953 1.2657 -0.2505 -0.6283 %! 0.2050 1.4464 -0.2775 -0.6108 %! 0.2068 1.6188 -0.3545 -0.7183 %! 0.2126 1.7093 -0.3495 -0.9183 %! 0.2281 1.7044 -0.3355 -0.8058 %! 0.2457 1.6703 -0.3325 -0.7608 %! 0.2611 1.6585 -0.3575 -0.9008 %! 0.2661 1.6143 -0.4705 -0.6558 %! 0.2578 1.5015 -0.4585 -0.7583 %! 0.2451 1.2956 -0.4175 -0.8033 %! 0.2207 1.0253 -0.4485 -0.3858 %! 0.1768 0.8074 -0.3655 -0.4483 %! 0.1220 0.6616 -0.2795 -0.4508 %! 0.0620 0.5891 -0.2285 -0.3108 %! 0.0002 0.5812 -0.1485 -0.4083 %! -0.0594 0.5578 -0.0655 -0.3083 %! -0.1114 0.5396 0.0895 -0.3433 %! -0.1434 0.5741 0.2125 -0.4708 %! -0.1458 0.6506 0.2315 -0.4908 %! 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0.1359 -1.2722 -0.1825 0.5567 %! 0.1069 -1.0629 -0.1565 0.3892 %! 0.0833 -0.7969 -0.0815 0.2242 %! 0.0679 -0.5100 -0.0695 0.1892 %! 0.0615 -0.2389 -0.1075 -0.0033 %! 0.0664 0.0303 -0.1485 -0.0908 %! 0.0822 0.3079 -0.1295 -0.1783 %! 0.1140 0.5658 -0.1465 -0.4283 %! 0.1627 0.7941 -0.3005 -0.5208 %! 0.2134 0.9634 -0.4675 -0.4783 %! 0.2515 1.0687 -0.5295 -0.5583 %! 0.2738 1.1300 -0.5445 -0.6233 %! 0.2803 1.1322 -0.5615 -0.6283 %! 0.2668 1.0979 -0.5675 -0.5358 %! 0.2311 1.0801 -0.4795 -0.5458 %! 0.1799 1.0888 -0.3715 -0.6558 %! 0.1121 1.1266 -0.4035 -0.4358 %! 0.0311 1.2269 -0.3055 -0.5658 %! -0.0402 1.3304 -0.0415 -0.9458 %! -0.0970 1.3157 0.0015 -0.6333 %! -0.1536 1.2545 -0.0375 -0.5383 %! -0.2054 1.2245 0.0305 -0.7933 %! -0.2445 1.1415 0.1235 -0.5958 %! -0.2725 1.0068 0.2105 -0.5658 %! -0.2878 0.8388 0.2305 -0.5408 %! -0.2910 0.6395 0.2735 -0.3933 %! -0.2874 0.5035 0.3115 -0.3133 %! -0.2779 0.4273 0.2875 -0.2908 %! -0.2564 0.3242 0.2965 -0.3508 %! -0.2262 0.1788 0.3125 -0.1683 %! -0.1995 0.0612 0.3315 -0.0883 %! 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0.1629 -0.3218 -0.1705 0.1792 %! 0.2469 -0.1316 -0.2225 -0.0458 %! 0.3188 0.0698 -0.4895 0.0292 %! 0.3657 0.2711 -0.6915 -0.0558 %! 0.3947 0.3881 -0.6545 -0.3083 %! 0.4166 0.3752 -0.6155 -0.2233 %! 0.4322 0.3157 -0.6895 -0.1933 %! 0.4374 0.2851 -0.8475 -0.1083 %! 0.4346 0.2856 -0.8755 -0.1283 %! 0.4354 0.2982 -0.8335 -0.3133 %! 0.4374 0.3127 -0.9455 -0.0808 %! 0.4289 0.3315 -0.9695 -0.0958 %! 0.4117 0.3484 -0.8695 -0.2483 %! 0.3875 0.3695 -0.8635 -0.1233 %! 0.3534 0.4046 -0.8965 -0.2008 %! 0.3109 0.4103 -0.8595 -0.2033 %! 0.2621 0.3735 -0.6865 -0.1533 %! 0.2091 0.3471 -0.5615 -0.1958 %! 0.1487 0.3500 -0.6085 -0.0883 %! 0.0755 0.4095 -0.5965 -0.0558 %! -0.0007 0.5566 -0.3735 -0.2433 %! -0.0624 0.7375 -0.1545 -0.3708 %! -0.1055 0.9106 -0.1155 -0.3558 %! -0.1340 1.0728 -0.0775 -0.4608 %! -0.1476 1.2037 -0.0005 -0.5433 %! -0.1445 1.3093 0.0225 -0.5708 %! -0.1205 1.4124 0.0125 -0.6983 %! -0.0755 1.5512 -0.0005 -0.7383 %! -0.0197 1.7541 -0.0045 -0.7633 %! 0.0390 1.9343 -0.0825 -0.9433 %! 0.0935 1.9883 -0.2615 -0.9758 %! 0.1276 1.9447 -0.3705 -0.8358 %! 0.1302 1.8773 -0.3225 -0.8033 %! 0.1103 1.7874 -0.2095 -0.9333 %! 0.0776 1.6447 -0.2025 -0.8183 %! 0.0259 1.5027 -0.2675 -0.5433 %! -0.0390 1.4181 -0.1615 -0.7108 %! -0.0957 1.2941 -0.0355 -0.7858 %! -0.1442 1.0659 -0.0665 -0.4158 %! -0.1955 0.8406 -0.0395 -0.3733 %! -0.2470 0.6814 0.0665 -0.4008 %! -0.2951 0.5478 0.1495 -0.2533 %! -0.3391 0.4030 0.1875 -0.2308 %! -0.3757 0.2509 0.2445 -0.1508 %! -0.3997 0.0973 0.4195 -0.1908 %! -0.4133 -0.1007 0.4435 0.0092 %! -0.4258 -0.3112 0.3315 0.2217 %! -0.4290 -0.4816 0.4055 0.0417 %! -0.4116 -0.6439 0.5125 0.1667 %! -0.3855 -0.7518 0.4715 0.3892 %! -0.3658 -0.8248 0.3845 0.3667 %! -0.3557 -0.9450 0.3525 0.4867 %! -0.3478 -1.0281 0.4715 0.4367 %! -0.3283 -1.0926 0.5655 0.3717 %! -0.2970 -1.2113 0.4865 0.5767 %! -0.2629 -1.3095 0.4455 0.5692 %! -0.2277 -1.4162 0.4285 0.5542 %! -0.1940 -1.5722 0.2955 0.7242 %! -0.1658 -1.7209 0.1985 0.7442 %! -0.1448 -1.8574 0.2275 0.7892 %! -0.1375 -1.9598 0.2405 0.9342 %! -0.1496 -1.9831 0.1605 0.9617 %! 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0.3223 -0.9466 -0.4315 0.3642 %! 0.2962 -1.0383 -0.3285 0.3042 %! 0.2693 -1.0274 -0.3485 0.3917 %! 0.2392 -0.9591 -0.3755 0.3742 %! 0.2038 -0.9085 -0.3265 0.3042 %! 0.1593 -0.8824 -0.2615 0.4042 %! 0.1061 -0.8394 -0.1675 0.3567 %! 0.0545 -0.7459 -0.0505 0.2442 %! 0.0122 -0.6037 0.0655 0.1592 %! -0.0254 -0.4611 0.0855 0.2192 %! -0.0626 -0.3247 0.0695 0.1617 %! -0.0875 -0.1777 0.1505 -0.1158 %! -0.0905 -0.0481 0.2245 -0.0983 %! -0.0766 0.0802 0.2345 -0.1633 %! -0.0549 0.2306 0.1405 -0.2558 %! -0.0366 0.3284 0.0435 -0.1458 %! -0.0207 0.3329 0.1075 -0.3483 %! 0.0026 0.3131 0.1475 -0.3983 %! 0.0262 0.3154 0.1045 -0.1858 %! 0.0400 0.3280 0.0795 -0.2658 %! 0.0476 0.3332 0.0685 -0.3658 %! 0.0513 0.3067 0.0155 -0.3033 %! 0.0436 0.2422 -0.0525 -0.2133 %! 0.0191 0.1745 -0.0125 -0.1783 %! -0.0154 0.1173 0.1125 -0.2333 %! -0.0520 0.0566 0.1405 -0.1883 %! -0.0912 -0.0241 0.1125 -0.1083 %! -0.1337 -0.1957 0.1615 -0.0583 %! -0.1775 -0.4680 0.2625 0.0967 %! -0.2182 -0.7258 0.3555 0.1792 %! -0.2523 -0.9660 0.3795 0.3267 %! 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-0.0919 -0.6882 0.2665 0.1967 %! -0.0507 -0.5098 0.2545 0.0167 %! -0.0053 -0.3457 0.2025 -0.0408 %! 0.0384 -0.2101 0.0915 -0.0308 %! 0.0784 -0.1111 0.0025 -0.1683 %! 0.1181 -0.0751 -0.0865 -0.1308 %! 0.1605 -0.0730 -0.1175 -0.2033 %! 0.2059 -0.1001 -0.2135 -0.2008 %! 0.2462 -0.2077 -0.3615 -0.0183 %! 0.2772 -0.4270 -0.4225 -0.0558 %! 0.3007 -0.7541 -0.4925 0.1717 %! 0.3146 -1.1303 -0.5105 0.3567 %! 0.3186 -1.4819 -0.5215 0.4617 %! 0.3091 -1.7927 -0.5565 0.7917 %! 0.2890 -2.0763 -0.4785 0.7642 %! 0.2692 -2.3198 -0.4585 0.8617 %! 0.2490 -2.4531 -0.4775 1.1192 %! 0.2292 -2.4601 -0.4055 0.9467 %! 0.2131 -2.4495 -0.3885 0.9992 %! 0.1952 -2.4696 -0.3775 1.0642 %! 0.1710 -2.4586 -0.3765 1.0317 %! 0.1456 -2.4272 -0.2785 0.9392 %! 0.1230 -2.4762 -0.1635 0.9317 %! 0.0921 -2.5929 -0.1965 1.2292 %! 0.0478 -2.6509 -0.1055 1.0467 %! -0.0025 -2.6497 0.0055 1.0742 %! -0.0578 -2.6410 0.0905 1.2692 %! -0.1096 -2.5957 0.2545 0.9617 %! -0.1513 -2.5087 0.2895 1.1817 %! -0.1795 -2.3898 0.4275 1.0492 %! -0.1794 -2.2767 0.5375 0.6942 %! -0.1570 -2.1483 0.3505 1.0492 %! -0.1248 -1.8914 0.3095 0.7342 %! -0.0796 -1.5767 0.2835 0.4517 %! -0.0317 -1.3301 0.1355 0.6567 %! 0.0092 -1.1046 0.0485 0.3492 %! 0.0457 -0.8055 -0.0545 0.2217 %! 0.0755 -0.4035 -0.0825 0.1117 %! 0.0943 0.0059 -0.1495 -0.0883 %! 0.1011 0.3715 -0.2115 -0.2533 %! 0.0943 0.7130 -0.1945 -0.4308 %! 0.0694 1.0155 -0.1785 -0.3908 %! 0.0356 1.3196 -0.0615 -0.7733 %! 0.0046 1.5734 -0.0275 -0.8458 %! -0.0314 1.7202 -0.0395 -0.6133 %! -0.0681 1.8521 0.1035 -1.0408 %! -0.0942 1.9659 0.1055 -0.9358 %! -0.1141 2.0471 0.0675 -0.8083 %! -0.1209 2.0915 0.1535 -1.1308 %! -0.1063 2.0434 0.1825 -1.0608 %! -0.0888 1.9430 0.0735 -0.8483 %! -0.0915 1.8424 -0.1075 -0.6683 %! -0.1083 1.8053 -0.0255 -0.9108 %! -0.1202 1.8339 0.1335 -0.9908 %! -0.1316 1.8332 0.1095 -0.6858 %! -0.1505 1.8459 0.1255 -0.9133 %! -0.1761 1.8897 0.1255 -0.8433 %! -0.2072 1.8891 0.1965 -0.7508 %! -0.2260 1.8554 0.3535 -1.1083 %! -0.2255 1.8173 0.3455 -0.8533 %! -0.2197 1.7742 0.3785 -0.7258 %! -0.2068 1.6928 0.4225 -0.9683 %! -0.1823 1.5929 0.2655 -0.7583 %! -0.1528 1.5525 0.1635 -0.6833 %! -0.1153 1.5768 0.1865 -0.8233 %! -0.0639 1.6492 0.1845 -0.8233 %! -0.0036 1.7389 0.0385 -0.7983 %! 0.0568 1.8252 -0.1795 -0.8408 %! 0.1135 1.9303 -0.3065 -0.9008 %! 0.1663 2.0578 -0.3395 -0.9508 %! 0.2171 2.2149 -0.3865 -1.1108 %! 0.2609 2.3688 -0.5255 -1.0583 %! 0.2903 2.4811 -0.5965 -1.0783 %! 0.3116 2.5456 -0.5985 -1.2933 %! 0.3302 2.5368 -0.6915 -1.2033 %! 0.3386 2.4999 -0.7635 -1.1133 %! 0.3349 2.4705 -0.7265 -1.1508 %! 0.3272 2.3935 -0.6665 -1.1583 %! 0.3200 2.2373 -0.7235 -1.0883 %! 0.3069 2.0341 -0.8575 -0.8508 %! 0.2852 1.8725 -0.7935 -0.9233 %! 0.2581 1.7540 -0.7195 -0.8208 %! 0.2203 1.6180 -0.7705 -0.5283 %! 0.1816 1.4957 -0.6725 -0.8258 %! 0.1549 1.3512 -0.5665 -0.7058 %! 0.1282 1.1388 -0.5585 -0.2658 %! 0.1008 0.9536 -0.4685 -0.5208 %! 0.0874 0.8015 -0.3885 -0.4808 %! 0.0840 0.6058 -0.3185 -0.1583 %! 0.0854 0.4115 -0.2425 -0.2633 %! 0.0948 0.2596 -0.2405 -0.1983 %! 0.1089 0.1118 -0.2375 -0.0158 %! 0.1282 -0.0454 -0.1845 -0.0983 %! 0.1559 -0.2334 -0.2045 -0.0283 %! 0.1806 -0.4327 -0.3215 0.2567 %! 0.1940 -0.5798 -0.3475 0.2092 %! 0.2039 -0.6991 -0.3195 0.1717 %! 0.2138 -0.7922 -0.3635 0.3517 %! 0.2221 -0.8503 -0.4145 0.3267 %! 0.2299 -0.9164 -0.4665 0.3792 %! 0.2400 -0.9682 -0.4465 0.3692 %! 0.2545 -1.0512 -0.4625 0.3767 %! 0.2670 -1.2018 -0.5555 0.6217 %! 0.2737 -1.3174 -0.5085 0.5317 %! 0.2764 -1.4090 -0.4815 0.6092 %! 0.2693 -1.5248 -0.5265 0.8217 %! 0.2531 -1.6471 -0.4865 0.6892 %! 0.2321 -1.8015 -0.4525 0.8867 %! 0.2056 -1.9496 -0.3925 1.0067 %! 0.1804 -2.0230 -0.3695 0.8967 %! 0.1599 -1.9852 -0.4155 1.0367 %! 0.1436 -1.8214 -0.3455 0.8392 %! 0.1329 -1.6423 -0.3275 0.7317 %! 0.1199 -1.5459 -0.3995 0.8667 %! 0.1026 -1.4838 -0.3475 0.6542 %! 0.0885 -1.4278 -0.2525 0.5967 %! 0.0761 -1.3787 -0.2145 0.6842 %! 0.0645 -1.2595 -0.1895 0.5417 %! 0.0588 -1.0485 -0.1635 0.3892 %! 0.0552 -0.8090 -0.1595 0.4317 %! 0.0479 -0.5480 -0.1485 0.2867 %! 0.0420 -0.3196 -0.1045 0.0467 %! 0.0378 -0.2052 -0.0815 0.0967 %! 0.0258 -0.1789 -0.0605 0.0942 %! 0.0006 -0.2476 -0.0715 0.1042 %! -0.0359 -0.4128 -0.0735 0.2417 %! -0.0730 -0.6137 0.0265 0.2317 %! -0.0993 -0.8314 0.0895 0.3617 %! -0.1153 -0.9895 0.0605 0.5417 %! -0.1249 -1.0287 -0.0115 0.6092 %! -0.1250 -1.0177 -0.0275 0.5817 %! -0.1093 -0.9871 0.0055 0.4892 %! -0.0832 -0.9130 -0.0565 0.6292 %! -0.0526 -0.8006 -0.0585 0.5317 %! -0.0077 -0.7273 0.0065 0.2742 %! 0.0553 -0.7710 -0.0665 0.4267 %! 0.1270 -0.9222 -0.1935 0.4617 %! 0.1985 -1.1668 -0.3345 0.6017 %! 0.2688 -1.4234 -0.3835 0.6767 %! 0.3422 -1.5956 -0.4405 0.5992 %! 0.4068 -1.7452 -0.6875 0.9367 %! 0.4443 -1.9072 -0.8405 0.9867 %! 0.4553 -2.0635 -0.8345 0.9092 %! 0.4476 -2.2464 -0.8135 1.0567 %! 0.4252 -2.4329 -0.7825 1.0967 %! 0.3876 -2.5541 -0.7555 1.2767 %! 0.3383 -2.5291 -0.6055 1.1367 %! 0.2853 -2.3477 -0.5295 1.0567 %! 0.2304 -2.0738 -0.5415 1.0692 %! 0.1821 -1.7182 -0.3605 0.5542 %! 0.1429 -1.3468 -0.2425 0.5842 %! 0.1032 -1.0234 -0.2025 0.4742 %! 0.0679 -0.7153 -0.1825 0.0467 %! 0.0414 -0.4370 -0.1845 0.1467 %! 0.0237 -0.2054 -0.0365 -0.1708 %! 0.0103 -0.0543 -0.0845 -0.1233 %! -0.0138 -0.0096 -0.1945 0.1042 %! -0.0442 0.0084 -0.0665 -0.3308 %! -0.0775 0.0274 -0.0545 -0.0058 %! -0.1239 0.0529 -0.0365 0.0892 %! -0.1740 0.1041 0.0725 -0.2208 %! -0.2155 0.1382 0.1855 0.0092 %! -0.2412 0.1550 0.3675 -0.2133 %! -0.2493 0.1734 0.3055 -0.1183 %! -0.2515 0.2194 0.2315 0.0167 %! -0.2507 0.3222 0.3455 -0.2983 %! -0.2498 0.4421 0.3345 -0.1158 %! -0.2534 0.5567 0.3655 -0.2708 %! -0.2559 0.6360 0.3995 -0.3958 %! -0.2582 0.6567 0.4175 -0.2383 %! -0.2604 0.6446 0.4565 -0.4358 %! -0.2618 0.5815 0.3395 -0.2633 %! -0.2665 0.5052 0.3425 -0.2233 %! -0.2708 0.5100 0.4235 -0.3608 %! -0.2728 0.6079 0.3875 -0.2883 %! -0.2742 0.7203 0.3615 -0.4683 %! -0.2795 0.7793 0.3085 -0.3733 %! -0.2924 0.8673 0.3805 -0.4158 %! -0.3020 0.9995 0.5025 -0.6983 %! -0.3037 1.0591 0.4755 -0.5208 %! -0.3040 1.0871 0.5245 -0.5933 %! -0.3049 1.1507 0.5525 -0.6233 %! -0.3058 1.2644 0.5195 -0.6358 %! -0.3004 1.4334 0.4955 -0.8358 %! -0.2875 1.5499 0.4055 -0.6808 %! -0.2652 1.5844 0.4605 -0.8508 %! -0.2284 1.5971 0.3825 -0.8783 %! -0.1892 1.6190 0.1465 -0.5958 %! -0.1537 1.6781 0.1075 -0.9083 %! -0.1213 1.6967 0.0525 -0.7883 %! -0.1047 1.6351 0.0045 -0.5783 %! -0.1042 1.5375 0.0525 -0.8158 %! -0.1170 1.3787 0.0375 -0.5208 %! -0.1449 1.2041 0.1245 -0.4508 %! -0.1783 1.0765 0.1865 -0.4958 %! -0.2074 0.9696 0.2285 -0.3758 %! -0.2269 0.8117 0.3565 -0.4383 %! -0.2434 0.5263 0.3285 -0.0883 %! -0.2634 0.1883 0.3275 0.0292 %! -0.2766 -0.1724 0.4035 -0.0358 %! -0.2855 -0.6067 0.3515 0.5092 %! -0.2971 -0.9636 0.3975 0.5592 %! -0.3002 -1.1742 0.4425 0.4867 %! -0.2883 -1.3005 0.3975 0.6967 %! -0.2630 -1.3348 0.3585 0.5892 %! -0.2244 -1.3593 0.2655 0.6317 %! -0.1732 -1.4072 0.2425 0.5767 %! -0.1119 -1.4491 0.2175 0.5367 %! -0.0519 -1.5377 0.1075 0.7042 %! -0.0045 -1.6482 0.0515 0.6342 %! 0.0222 -1.7353 -0.0215 0.8067 %! 0.0255 -1.7833 -0.0095 0.8492 %! 0.0200 -1.7957 0.0855 0.6017 %! 0.0138 -1.8427 0.0605 0.8717 %! 0.0079 -1.9256 0.0775 0.8342 %! 0.0100 -2.0111 0.0645 0.7467 %! 0.0160 -2.0729 -0.0805 1.0792 %! 0.0231 -2.0740 -0.0975 0.8967 %! 0.0388 -2.0445 -0.1065 0.8617 %! 0.0597 -1.9906 -0.1435 1.0067 %! 0.0807 -1.9170 -0.1825 0.8017 %! 0.0969 -1.8479 -0.3205 0.9292 %! 0.1029 -1.7495 -0.3275 0.8542 %! 0.1043 -1.6604 -0.2355 0.6392 %! 0.1001 -1.6211 -0.2395 0.8142 %! 0.0841 -1.5806 -0.2095 0.7817 %! 0.0630 -1.5319 -0.1415 0.6667 %! 0.0489 -1.5022 -0.0715 0.6367 %! 0.0469 -1.4700 -0.0835 0.6692 %! 0.0566 -1.3234 -0.1335 0.6092 %! 0.0811 -1.0251 -0.0935 0.2617 %! 0.1115 -0.7103 -0.2095 0.3017 %! 0.1288 -0.4227 -0.3555 0.2692 %! 0.1352 -0.1704 -0.2835 -0.2308 %! 0.1320 -0.0594 -0.2505 -0.0183 %! 0.1065 -0.0067 -0.2275 0.0967 %! 0.0680 0.1685 -0.1365 -0.3083 %! 0.0335 0.4148 -0.0715 -0.3258 %! 0.0049 0.5997 -0.0175 -0.3758 %! -0.0195 0.7006 -0.0445 -0.4033 %! -0.0351 0.7637 -0.0225 -0.4233 %! -0.0299 0.8425 0.0905 -0.6383 %! -0.0055 0.9276 0.0615 -0.5283 %! 0.0178 1.0462 -0.0975 -0.3783 %! 0.0368 1.2729 -0.1145 -0.6858 %! 0.0674 1.4969 -0.0265 -0.9283 %! 0.1083 1.6066 -0.1025 -0.7658 %! 0.1448 1.7228 -0.2695 -0.7558 %! 0.1778 1.9169 -0.3095 -1.0808 %! 0.2111 2.1262 -0.3605 -1.0633 %! 0.2372 2.3387 -0.4795 -1.0733 %! 0.2550 2.5757 -0.5415 -1.3358 %! 0.2666 2.7617 -0.4935 -1.4108 %! 0.2636 2.7910 -0.4535 -1.3158 %! 0.2380 2.7246 -0.5335 -1.2708 %! 0.1975 2.6043 -0.4965 -1.3383 %! 0.1524 2.3566 -0.3465 -1.1458 %! 0.1037 2.0830 -0.2385 -0.9383 %! 0.0570 1.8917 -0.1875 -1.0133 %! 0.0150 1.7363 -0.2315 -0.7908 %! -0.0242 1.6378 -0.1455 -0.8158 %! -0.0589 1.5875 -0.0665 -0.9083 %! -0.0979 1.5152 -0.1345 -0.6183 %! -0.1459 1.4203 -0.0325 -0.7233 %! -0.1901 1.3044 0.1315 -0.7433 %! -0.2209 1.1849 0.2195 -0.6183 %! -0.2339 1.0649 0.2275 -0.6983 %! -0.2355 0.9365 0.1425 -0.4533 %! -0.2337 0.8784 0.1855 -0.4658 %! -0.2214 0.8990 0.2485 -0.6558 %! -0.1964 0.9349 0.1745 -0.5258 %! -0.1659 0.9811 0.1785 -0.6333 %! -0.1372 0.9916 0.1725 -0.5783 %! -0.1158 0.9395 0.1425 -0.5083 %! -0.0970 0.8657 0.1345 -0.6183 %! -0.0792 0.8086 0.0965 -0.4358 %! -0.0657 0.8370 0.1385 -0.4958 %! -0.0552 0.9112 0.1155 -0.5783 %! -0.0518 0.9266 -0.0175 -0.4133 %! -0.0560 0.9192 -0.0425 -0.5208 %! -0.0578 0.8932 0.0005 -0.5783 %! -0.0564 0.8007 -0.0235 -0.4083 %! -0.0591 0.6940 -0.0925 -0.3358 %! -0.0655 0.6191 -0.1015 -0.3833 %! -0.0735 0.5441 -0.0995 -0.2783 %! -0.0841 0.4197 -0.1085 -0.1608 %! -0.0892 0.2639 -0.0675 -0.2483 %! -0.0801 0.1240 -0.0085 -0.2133 %! -0.0661 -0.0027 -0.0185 -0.0408 %! -0.0647 -0.0929 -0.1255 0.0367 %! -0.0828 -0.1350 -0.1705 0.0342 %! -0.1111 -0.1750 -0.0195 -0.0983 %! -0.1424 -0.2431 0.1145 -0.0458 %! -0.1823 -0.3343 0.1235 0.1642 %! -0.2282 -0.4034 0.1845 0.0017 %! -0.2684 -0.4463 0.2355 0.0017 %! -0.3042 -0.4858 0.2235 0.1692 %! -0.3374 -0.5038 0.2585 0.0742 %! -0.3619 -0.5260 0.3695 0.0692 %! -0.3751 -0.5692 0.4925 0.0867 %! -0.3804 -0.6271 0.4655 0.1342 %! -0.3838 -0.6946 0.3635 0.2492 %! -0.3849 -0.7012 0.4405 0.1317 %! -0.3802 -0.6456 0.5245 0.1767 %! -0.3685 -0.5710 0.5145 0.1717 %! -0.3432 -0.4960 0.4965 -0.0458 %! -0.3091 -0.5027 0.3875 0.1642 %! -0.2816 -0.5650 0.2855 0.2767 %! -0.2575 -0.6307 0.2745 0.0492 %! -0.2333 -0.7767 0.2115 0.3092 %! -0.2177 -0.9320 0.2115 0.4467 %! -0.2103 -0.9808 0.2295 0.3017 %! -0.2077 -0.9820 0.1235 0.4517 %! -0.2091 -0.9672 0.0925 0.4017 %! -0.2067 -0.9320 0.1505 0.2667 %! -0.1964 -0.9354 0.1875 0.3117 %! -0.1862 -1.0067 0.1385 0.4092 %! -0.1835 -1.1249 0.0845 0.4417 %! -0.1878 -1.2659 0.1145 0.4717 %! -0.1971 -1.4192 0.1755 0.5892 %! -0.2082 -1.5634 0.2345 0.5917 %! -0.2205 -1.6030 0.2205 0.7092 %! -0.2334 -1.4712 0.2515 0.5992 %! -0.2378 -1.2802 0.2915 0.3067 %! -0.2371 -1.1340 0.1805 0.4742 %! -0.2418 -1.0239 0.1895 0.3717 %! -0.2467 -0.9961 0.2855 0.1917 %! -0.2506 -1.0394 0.2625 0.4892 %! -0.2481 -1.0708 0.3205 0.2867 %! -0.2255 -1.1285 0.3115 0.2942 %! -0.1851 -1.2190 0.2515 0.5392 %! -0.1284 -1.2780 0.2205 0.3617 %! -0.0568 -1.3066 0.0485 0.4992 %! 0.0190 -1.3093 -0.0745 0.5842 %! 0.0980 -1.3169 -0.0975 0.3817 %! 0.1790 -1.3629 -0.1905 0.4917 %! 0.2470 -1.3928 -0.3335 0.5942 %! 0.2906 -1.3799 -0.4495 0.5667 %! 0.3119 -1.3872 -0.4345 0.4267 %! 0.3070 -1.4366 -0.4805 0.6267 %! 0.2694 -1.4497 -0.5195 0.7092 %! 0.2157 -1.4180 -0.3445 0.3267 %! 0.1572 -1.4131 -0.2795 0.5917 %! 0.0900 -1.4020 -0.2475 0.6467 %! 0.0252 -1.3726 -0.1455 0.2942 %! -0.0329 -1.3701 -0.1075 0.5992 %! -0.0863 -1.3388 0.0435 0.4792 %! -0.1235 -1.2708 0.1565 0.2142 %! -0.1478 -1.2320 0.1245 0.4817 %! -0.1728 -1.2034 0.1555 0.4392 %! -0.1952 -1.1628 0.2365 0.2667 %! -0.2103 -1.1065 0.2765 0.3317 %! -0.2217 -0.9840 0.3265 0.2767 %! -0.2321 -0.8129 0.3975 0.1317 %! -0.2489 -0.6673 0.3905 0.1867 %! -0.2823 -0.5045 0.3435 0.2217 %! -0.3208 -0.2971 0.4725 -0.1183 %! -0.3473 -0.1283 0.6145 -0.1808 %! -0.3655 0.0017 0.6595 -0.0633 %! -0.3824 0.1148 0.6585 -0.2158 %! -0.3984 0.1791 0.6025 -0.1708 %! -0.4094 0.2068 0.6665 -0.2083 %! -0.4057 0.1834 0.7515 -0.2908 %! -0.3881 0.1013 0.7285 -0.1283 %! -0.3652 0.0076 0.6865 -0.0358 %! -0.3341 -0.0852 0.6455 -0.0683 %! -0.2841 -0.1695 0.5685 -0.0783 %! -0.2157 -0.2379 0.4245 0.0317 %! -0.1411 -0.3204 0.2975 0.1167 %! -0.0709 -0.4622 0.1965 0.1842 %! -0.0090 -0.6683 0.0635 0.2967 %! 0.0471 -0.8651 -0.0275 0.2617 %! 0.0923 -0.9752 -0.1435 0.4992 %! 0.1202 -0.9762 -0.1835 0.5592 %! 0.1442 -0.9324 -0.0975 0.1967 %! 0.1675 -0.9128 -0.2015 0.4142 %! 0.1732 -0.8419 -0.3305 0.5317 %! 0.1625 -0.6695 -0.2815 0.1442 %! 0.1441 -0.4938 -0.2375 0.1892 ]; %! %! DAT = iddata (UY(:, 3:4), UY(:, 1:2)); %! %! [SYS, X0] = moen4 (DAT, "s", 15, "n", 8, "rcond", 0.0, "tol", -1.0); %! %! Ae = [ 0.9893 0.0081 -0.0844 0.0299 0.1262 0.0815 -0.0379 0.1779 %! 0.0076 0.9694 0.1352 0.1793 0.0965 -0.1041 -0.0373 -0.0045 %! 0.0018 -0.0115 0.9413 -0.3450 0.1674 0.0302 -0.1098 -0.0087 %! 0.0036 -0.0197 0.1766 0.6550 -0.1178 0.6168 0.0028 0.0085 %! -0.0010 -0.0032 -0.0507 0.0624 0.9293 -0.0034 0.0229 -0.4511 %! 0.0032 -0.0012 -0.0397 0.1255 0.1578 0.1833 0.7586 0.3125 %! 0.0029 -0.0048 0.0422 0.2343 0.0321 -0.6549 -0.0374 0.2232 %! -0.0036 0.0028 0.0080 -0.2437 0.1074 0.1599 -0.1012 0.2106 ]; %! %! Ce = [ -0.2226 0.2072 -0.3129 -0.3937 0.1722 0.3232 -0.2113 0.2928 %! -0.1680 -0.2205 0.0698 0.4049 0.3210 -0.1143 -0.2451 0.4844 ]; %! %! Be = [ 0.0857 0.0026 %! -0.6433 -0.0314 %! 0.2445 0.0044 %! 1.8942 0.0488 %! 0.0549 0.0151 %! -2.2093 -0.0622 %! -2.5072 -0.0925 %! 0.8189 0.0280 ]; %! %! De = [ -0.4997 0.0451 %! -1.0011 -0.5567 ]; %! %!assert (SYS.A, Ae, 1e-4); %!assert (SYS.B, Be, 1e-4); %!assert (SYS.C, Ce, 1e-4); %!assert (SYS.D, De, 1e-4); control/inst/moesp.m0000644000076500000240000001652612273411257014034 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @var{n}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @var{opt}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} moesp (@var{dat}, @var{n}, @var{opt}, @dots{}) ## Estimate state-space model using @acronym{MOESP} algorithm. ## @acronym{MOESP}: Multivariable Output Error State sPace. ## If no output arguments are given, the singular values are ## plotted on the screen in order to estimate the system order. ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata set containing the measurements, i.e. time-domain signals. ## @item n ## The desired order of the resulting state-space system @var{sys}. ## If not specified, @var{n} is chosen automatically according ## to the singular values and tolerances. ## @item @dots{} ## Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## ## @strong{Outputs} ## @table @var ## @item sys ## Discrete-time state-space model. ## @item x0 ## Initial state vector. If @var{dat} is a multi-experiment dataset, ## @var{x0} becomes a cell vector containing an initial state vector ## for each experiment. ## @item info ## Struct containing additional information. ## @table @var ## @item info.K ## Kalman gain matrix. ## @item info.Q ## State covariance matrix. ## @item info.Ry ## Output covariance matrix. ## @item info.S ## State-output cross-covariance matrix. ## @item info.L ## Noise variance matrix factor. LL'=Ry. ## @end table ## @end table ## ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'n' ## The desired order of the resulting state-space system @var{sys}. ## @var{s} > @var{n} > 0. ## ## @item 's' ## The number of block rows @var{s} in the input and output ## block Hankel matrices to be processed. @var{s} > 0. ## In the MOESP theory, @var{s} should be larger than @var{n}, ## the estimated dimension of state vector. ## ## @item 'alg', 'algorithm' ## Specifies the algorithm for computing the triangular ## factor R, as follows: ## @table @var ## @item 'C' ## Cholesky algorithm applied to the correlation ## matrix of the input-output data. Default method. ## @item 'F' ## Fast QR algorithm. ## @item 'Q' ## QR algorithm applied to the concatenated block ## Hankel matrices. ## @end table ## ## @item 'tol' ## Absolute tolerance used for determining an estimate of ## the system order. If @var{tol} >= 0, the estimate is ## indicated by the index of the last singular value greater ## than or equal to @var{tol}. (Singular values less than @var{tol} ## are considered as zero.) When @var{tol} = 0, an internally ## computed default value, @var{tol} = @var{s}*@var{eps}*SV(1), is used, ## where SV(1) is the maximal singular value, and @var{eps} is ## the relative machine precision. ## When @var{tol} < 0, the estimate is indicated by the ## index of the singular value that has the largest ## logarithmic gap to its successor. Default value is 0. ## ## @item 'rcond' ## The tolerance to be used for estimating the rank of ## matrices. If the user sets @var{rcond} > 0, the given value ## of @var{rcond} is used as a lower bound for the reciprocal ## condition number; an m-by-n matrix whose estimated ## condition number is less than 1/@var{rcond} is considered to ## be of full rank. If the user sets @var{rcond} <= 0, then an ## implicitly computed, default tolerance, defined by ## @var{rcond} = m*n*@var{eps}, is used instead, where @var{eps} is the ## relative machine precision. Default value is 0. ## ## @item 'confirm' ## Specifies whether or not the user's confirmation of the ## system order estimate is desired, as follows: ## @table @var ## @item true ## User's confirmation. ## @item false ## No confirmation. Default value. ## @end table ## ## @item 'noiseinput' ## The desired type of noise input channels. ## @table @var ## @item 'n' ## No error inputs. Default value. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k $$ ## $$ y_k = C x_k + D u_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] ## y[k] = C x[k] + D u[k] ## @end example ## @end ifnottex ## ## @item 'e' ## Return @var{sys} as a (p-by-m+p) state-space model with ## both measured input channels u and noise channels e ## with covariance matrix @var{Ry}. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k + K e_k $$ ## $$ y_k = C x_k + D u_k + e_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] + K e[k] ## y[k] = C x[k] + D u[k] + e[k] ## @end example ## @end ifnottex ## ## @item 'v' ## Return @var{sys} as a (p-by-m+p) state-space model with ## both measured input channels u and white noise channels v ## with identity covariance matrix. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k + K L v_k $$ ## $$ y_k = C x_k + D u_k + L v_k $$ ## $$ e = L v, \\ L L^T = R_y $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] + K L v[k] ## y[k] = C x[k] + D u[k] + L v[k] ## e = L v, L L' = Ry ## @end example ## @end ifnottex ## ## @item 'k' ## Return @var{sys} as a Kalman predictor for simulation. ## @iftex ## @tex ## $$ \\widehat{x}_{k+1} = A \\widehat{x}_k + B u_k + K (y_k - \\widehat{y}_k) $$ ## $$ \\widehat{y}_k = C \\widehat{x}_k + D u_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ^ ^ ^ ## x[k+1] = A x[k] + B u[k] + K(y[k] - y[k]) ## ^ ^ ## y[k] = C x[k] + D u[k] ## @end example ## @end ifnottex ## ## @iftex ## @tex ## $$ \\widehat{x}_{k+1} = (A-KC) \\widehat{x}_k + (B-KD) u_k + K y_k $$ ## $$ \\widehat{y}_k = C \\widehat{x}_k + D u_k + 0 y_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ^ ^ ## x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k] ## ^ ^ ## y[k] = C x[k] + D u[k] + 0 y[k] ## @end example ## @end ifnottex ## @end table ## @end table ## ## ## @strong{Algorithm}@* ## Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function [sys, x0, info] = moesp (varargin) if (nargin == 0) print_usage (); endif if (nargout == 0) __slicot_identification__ ("moesp", nargout, varargin{:}); else [sys, x0, info] = __slicot_identification__ ("moesp", nargout, varargin{:}); endif endfunction control/inst/n4sid.m0000644000076500000240000001656212273411257013732 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @var{n}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @var{opt}, @dots{}) ## @deftypefnx {Function File} {[@var{sys}, @var{x0}, @var{info}] =} n4sid (@var{dat}, @var{n}, @var{opt}, @dots{}) ## Estimate state-space model using @acronym{N4SID} algorithm. ## @acronym{N4SID}: Numerical algorithm for Subspace State Space System IDentification. ## If no output arguments are given, the singular values are ## plotted on the screen in order to estimate the system order. ## ## @strong{Inputs} ## @table @var ## @item dat ## iddata set containing the measurements, i.e. time-domain signals. ## @item n ## The desired order of the resulting state-space system @var{sys}. ## If not specified, @var{n} is chosen automatically according ## to the singular values and tolerances. ## @item @dots{} ## Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## ## @strong{Outputs} ## @table @var ## @item sys ## Discrete-time state-space model. ## @item x0 ## Initial state vector. If @var{dat} is a multi-experiment dataset, ## @var{x0} becomes a cell vector containing an initial state vector ## for each experiment. ## @item info ## Struct containing additional information. ## @table @var ## @item info.K ## Kalman gain matrix. ## @item info.Q ## State covariance matrix. ## @item info.Ry ## Output covariance matrix. ## @item info.S ## State-output cross-covariance matrix. ## @item info.L ## Noise variance matrix factor. LL'=Ry. ## @end table ## @end table ## ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'n' ## The desired order of the resulting state-space system @var{sys}. ## @var{s} > @var{n} > 0. ## ## @item 's' ## The number of block rows @var{s} in the input and output ## block Hankel matrices to be processed. @var{s} > 0. ## In the MOESP theory, @var{s} should be larger than @var{n}, ## the estimated dimension of state vector. ## ## @item 'alg', 'algorithm' ## Specifies the algorithm for computing the triangular ## factor R, as follows: ## @table @var ## @item 'C' ## Cholesky algorithm applied to the correlation ## matrix of the input-output data. Default method. ## @item 'F' ## Fast QR algorithm. ## @item 'Q' ## QR algorithm applied to the concatenated block ## Hankel matrices. ## @end table ## ## @item 'tol' ## Absolute tolerance used for determining an estimate of ## the system order. If @var{tol} >= 0, the estimate is ## indicated by the index of the last singular value greater ## than or equal to @var{tol}. (Singular values less than @var{tol} ## are considered as zero.) When @var{tol} = 0, an internally ## computed default value, @var{tol} = @var{s}*@var{eps}*SV(1), is used, ## where SV(1) is the maximal singular value, and @var{eps} is ## the relative machine precision. ## When @var{tol} < 0, the estimate is indicated by the ## index of the singular value that has the largest ## logarithmic gap to its successor. Default value is 0. ## ## @item 'rcond' ## The tolerance to be used for estimating the rank of ## matrices. If the user sets @var{rcond} > 0, the given value ## of @var{rcond} is used as a lower bound for the reciprocal ## condition number; an m-by-n matrix whose estimated ## condition number is less than 1/@var{rcond} is considered to ## be of full rank. If the user sets @var{rcond} <= 0, then an ## implicitly computed, default tolerance, defined by ## @var{rcond} = m*n*@var{eps}, is used instead, where @var{eps} is the ## relative machine precision. Default value is 0. ## ## @item 'confirm' ## Specifies whether or not the user's confirmation of the ## system order estimate is desired, as follows: ## @table @var ## @item true ## User's confirmation. ## @item false ## No confirmation. Default value. ## @end table ## ## @item 'noiseinput' ## The desired type of noise input channels. ## @table @var ## @item 'n' ## No error inputs. Default value. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k $$ ## $$ y_k = C x_k + D u_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] ## y[k] = C x[k] + D u[k] ## @end example ## @end ifnottex ## ## @item 'e' ## Return @var{sys} as a (p-by-m+p) state-space model with ## both measured input channels u and noise channels e ## with covariance matrix @var{Ry}. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k + K e_k $$ ## $$ y_k = C x_k + D u_k + e_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] + K e[k] ## y[k] = C x[k] + D u[k] + e[k] ## @end example ## @end ifnottex ## ## @item 'v' ## Return @var{sys} as a (p-by-m+p) state-space model with ## both measured input channels u and white noise channels v ## with identity covariance matrix. ## @iftex ## @tex ## $$ x_{k+1} = A x_k + B u_k + K L v_k $$ ## $$ y_k = C x_k + D u_k + L v_k $$ ## $$ e = L v, \\ L L^T = R_y $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## x[k+1] = A x[k] + B u[k] + K L v[k] ## y[k] = C x[k] + D u[k] + L v[k] ## e = L v, L L' = Ry ## @end example ## @end ifnottex ## ## @item 'k' ## Return @var{sys} as a Kalman predictor for simulation. ## @iftex ## @tex ## $$ \\widehat{x}_{k+1} = A \\widehat{x}_k + B u_k + K (y_k - \\widehat{y}_k) $$ ## $$ \\widehat{y}_k = C \\widehat{x}_k + D u_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ^ ^ ^ ## x[k+1] = A x[k] + B u[k] + K(y[k] - y[k]) ## ^ ^ ## y[k] = C x[k] + D u[k] ## @end example ## @end ifnottex ## ## @iftex ## @tex ## $$ \\widehat{x}_{k+1} = (A-KC) \\widehat{x}_k + (B-KD) u_k + K y_k $$ ## $$ \\widehat{y}_k = C \\widehat{x}_k + D u_k + 0 y_k $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ^ ^ ## x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k] ## ^ ^ ## y[k] = C x[k] + D u[k] + 0 y[k] ## @end example ## @end ifnottex ## @end table ## @end table ## ## ## @strong{Algorithm}@* ## Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function [sys, x0, info] = n4sid (varargin) if (nargin == 0) print_usage (); endif if (nargout == 0) __slicot_identification__ ("n4sid", nargout, varargin{:}); else [sys, x0, info] = __slicot_identification__ ("n4sid", nargout, varargin{:}); endif endfunction control/inst/ncfsyn.m0000644000076500000240000004761612273411257014215 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} ncfsyn (@var{G}, @var{W1}, @var{W2}, @var{factor}) ## Loop shaping H-infinity synthesis. Compute positive feedback controller using ## the McFarlane/Glover loop shaping design procedure [1]. ## Using a precompensator @var{W1} and/or a postcompensator @var{W2}, the singular values ## of the nominal plant @var{G} are shaped to give a desired open-loop shape. ## The nominal plant @var{G} and shaping functions @var{W1}, @var{W2} are combined to ## form the shaped plant, @var{Gs} where @code{Gs = W2 G W1}. ## We assume that @var{W1} and @var{W2} are such that @var{Gs} contains no hidden modes. ## It is relatively easy to approximate the closed-loop requirements by the following ## open-loop objectives [2]: ## @enumerate ## @item For @emph{disturbance rejection} make ## @iftex ## @tex ## $\\underline{\\sigma}(W_2 G W_1)$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## large; valid for frequencies at which ## @iftex ## @tex ## $\\underline{\\sigma}(G_S) \\gg 1$. ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## @item For @emph{noise attenuation} make ## @iftex ## @tex ## $\\overline{\\sigma}(W_2 G W_1)$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## small; valid for frequencies at which ## @iftex ## @tex ## $\\overline{\\sigma}(G_S) \\ll 1$. ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## @item For @emph{reference tracking} make ## @iftex ## @tex ## $\\underline{\\sigma}(W_2 G W_1)$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## large; valid for frequencies at which ## @iftex ## @tex ## $\\underline{\\sigma}(G_S) \\gg 1$. ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## @item For @emph{robust stability} to a multiplicative output perturbation ## @iftex ## @tex ## $G_p = (I + \\Delta) G$, make ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## @iftex ## @tex ## $\\overline{\\sigma}(W_2 G W_1)$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ## @end example ## @end ifnottex ## small; valid for frequencies at which ## @iftex ## @tex ## $\\overline{\\sigma}(G_S) \\ll 1$. ## @end tex ## @end iftex ## @ifnottex ## @example ## . ## @end example ## @end ifnottex ## @end enumerate ## Then a stabilizing controller @var{Ks} is synthesized for shaped plant @var{Gs}. ## The final positive feedback controller @var{K} is then constructed by combining ## the ## @iftex ## @tex ## $H_{\\infty}$ ## @end tex ## @end iftex ## @ifnottex ## @example ## H-infinity ## @end example ## @end ifnottex ## controller @var{Ks} with the shaping functions @var{W1} and @var{W2} ## such that @code{K = W1 Ks W2}. ## In [1] is stated further that the given robust stabilization objective can be ## interpreted as a ## @iftex ## @tex ## $H_{\\infty}$ ## @end tex ## @end iftex ## @ifnottex ## @example ## H-infinity ## @end example ## @end ifnottex ## problem formulation of minimizing the ## @iftex ## @tex ## $H_{\\infty}$ ## @end tex ## @end iftex ## @ifnottex ## @example ## H-infinity ## @end example ## @end ifnottex ## norm of the frequency weighted gain from disturbances on the plant input and output ## to the controller input and output as follows: ## @iftex ## @tex ## $$ \\underset{K}{\\min} \\, || N(K) ||_{\\infty}, $$ ## $$ N = | W_{1}^{-1}; W_2 G | \\ (I - K G)^{-1} \\ | W_1, \\ G W_{2}^{-1} | $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 -1 -1 ## min || N(K) || , N = | W1 | (I - K G) | W1 G W2 | ## K oo | W2 G | ## @end example ## @end ifnottex ## @iftex ## @tex ## ## @end tex ## @end iftex ## ## @code{[K, N] = ncfsyn (G, W1, W2, f)} ## The function @command{ncfsyn} - the somewhat cryptic name stands ## for @emph{normalized coprime factorization synthesis} - allows the specification of ## an additional argument, factor @var{f}. Default value @code{f = 1} implies that an ## optimal controller is required, whereas @code{f > 1} implies that a suboptimal ## controller is required, achieving a performance that is @var{f} times less than optimal. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model of plant. ## @item W1 ## @acronym{LTI} model of precompensator. Model must be SISO or of appropriate size. ## An identity matrix is taken if @var{W1} is not specified or if an empty model ## @code{[]} is passed. ## @item W2 ## @acronym{LTI} model of postcompensator. Model must be SISO or of appropriate size. ## An identity matrix is taken if @var{W2} is not specified or if an empty model ## @code{[]} is passed. ## @item factor ## @code{factor = 1} implies that an optimal controller is required. ## @code{factor > 1} implies that a suboptimal controller is required, ## achieving a performance that is @var{factor} times less than optimal. ## Default value is 1. ## @end table ## ## @strong{Outputs} ## @table @var ## @item K ## State-space model of the H-infinity loop-shaping controller. ## Note that @var{K} is a @emph{positive} feedback controller. ## @item N ## State-space model of the closed loop depicted below. ## @item info ## Structure containing additional information. ## @item info.gamma ## L-infinity norm of @var{N}. @code{gamma = norm (N, inf)}. ## @item info.emax ## Nugap robustness. @code{emax = inv (gamma)}. ## @item info.Gs ## Shaped plant. @code{Gs = W2 * G * W1}. ## @item info.Ks ## Controller for shaped plant. @code{Ks = ncfsyn (Gs)}. ## @item info.rcond ## Estimates of the reciprocal condition numbers of the Riccati equations ## and a few other things. For details, see the description of the ## corresponding SLICOT routine. ## @end table ## ## @strong{Block Diagram of N} ## @example ## @group ## ## ^ z1 ^ z2 ## | | ## w1 + | +--------+ | +--------+ ## ----->(+)---+-->| Ks |----+--->(+)---->| Gs |----+ ## ^ + +--------+ ^ +--------+ | ## | w2 | | ## | | ## +-------------------------------------------------+ ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB10ID, SB10KD and SB10ZD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## ## @strong{References}@* ## [1] D. McFarlane and K. Glover, ## @cite{A Loop Shaping Design Procedure Using H-infinity Synthesis}, ## IEEE Transactions on Automatic Control, Vol. 37, No. 6, June 1992.@* ## [2] S. Skogestad and I. Postlethwaite, ## @cite{Multivariable Feedback Control: Analysis and Design: ## Second Edition}. Wiley, Chichester, England, 2005.@* ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: July 2011 ## Version: 0.2 function [K, varargout] = ncfsyn (G, W1 = [], W2 = [], factor = 1.0) if (nargin == 0 || nargin > 4) print_usage (); endif if (! isa (G, "lti")) error ("ncfsyn: first argument must be an LTI system"); endif if (! is_real_scalar (factor) || factor < 1.0) error ("ncfsyn: fourth argument invalid"); endif [p, m] = size (G); W1 = __adjust_weighting__ (W1, m); W2 = __adjust_weighting__ (W2, p); Gs = W2 * G * W1; # shaped plant [a, b, c, d, tsam] = ssdata (Gs); ## synthesis if (isct (Gs)) # continuous-time [ak, bk, ck, dk, rcond] = __sl_sb10id__ (a, b, c, d, factor); elseif (any (d(:))) # discrete-time, d != 0 [ak, bk, ck, dk, rcond] = __sl_sb10zd__ (a, b, c, d, factor, 0.0); else # discrete-time, d == 0 [ak, bk, ck, dk, rcond] = __sl_sb10kd__ (a, b, c, factor); endif ## controller Ks = ss (ak, bk, ck, dk, tsam); K = W1 * Ks * W2; if (nargout > 1) ## FIXME: is this really the same thing as the dark side does? N = append (eye (p), Ks, Gs); M = [zeros(p,p), zeros(p,m), eye(p); eye(p), zeros(p,m), zeros(p,p); zeros(m,p), eye(m), zeros(m,p)]; in_idx = [1:p, 2*p+(1:m)]; out_idx = 1:p+m; N = mconnect (N, M, in_idx, out_idx); varargout{1} = N; if (nargout > 2) gamma = norm (N, inf); varargout{2} = gamma; if (nargout > 3) varargout{3} = struct ("gamma", gamma, "emax", inv (gamma), "Gs", Gs, "Ks", Ks, "rcond", rcond); endif endif endif endfunction function W = __adjust_weighting__ (W, s) if (isempty (W)) W = ss (eye (s)); else W = ss (W); ## if (! isstable (W)) ## error ("ncfsyn: %s must be stable", inputname (1)); ## endif ## if (! isminimumphase (W)) ## error ("ncfsyn: %s must be minimum-phase", inputname (1)); ## endif [p, m] = size (W); if (m == s && p == s) # model is of correct size return; elseif (m == 1 && p == 1) # model is SISO tmp = W; for k = 2 : s W = append (W, tmp); # stack SISO model s times endfor else # model is invalid error ("ncfsyn: %s must have 1 or %d inputs and outputs", inputname (1), s); endif endif endfunction ## continuous-time case, direct access to sb10id %!shared AK, BK, CK, DK, RCOND, AKe, BKe, CKe, DKe, RCONDe %! A = [ -1.0 0.0 4.0 5.0 -3.0 -2.0 %! -2.0 4.0 -7.0 -2.0 0.0 3.0 %! -6.0 9.0 -5.0 0.0 2.0 -1.0 %! -8.0 4.0 7.0 -1.0 -3.0 0.0 %! 2.0 5.0 8.0 -9.0 1.0 -4.0 %! 3.0 -5.0 8.0 0.0 2.0 -6.0 ]; %! %! B = [ -3.0 -4.0 %! 2.0 0.0 %! -5.0 -7.0 %! 4.0 -6.0 %! -3.0 9.0 %! 1.0 -2.0 ]; %! %! C = [ 1.0 -1.0 2.0 -4.0 0.0 -3.0 %! -3.0 0.0 5.0 -1.0 1.0 1.0 %! -7.0 5.0 0.0 -8.0 2.0 -2.0 ]; %! %! D = [ 1.0 -2.0 %! 0.0 4.0 %! 5.0 -3.0 ]; %! %! FACTOR = 1.0; %! %! [AK, BK, CK, DK, RCOND] = __sl_sb10id__ (A, B, C, D, FACTOR); %! %! AKe = [ -39.0671 9.9293 22.2322 -27.4113 43.8655 %! -6.6117 3.0006 11.0878 -11.4130 15.4269 %! 33.6805 -6.6934 -23.9953 14.1438 -33.4358 %! -32.3191 9.7316 25.4033 -24.0473 42.0517 %! -44.1655 18.7767 34.8873 -42.4369 50.8437 ]; %! %! BKe = [ -10.2905 -16.5382 -10.9782 %! -4.3598 -8.7525 -5.1447 %! 6.5962 1.8975 6.2316 %! -9.8770 -14.7041 -11.8778 %! -9.6726 -22.7309 -18.2692 ]; %! %! CKe = [ -0.6647 -0.0599 -1.0376 0.5619 1.7297 %! -8.4202 3.9573 7.3094 -7.6283 10.6768 ]; %! %! DKe = [ 0.8466 0.4979 -0.6993 %! -1.2226 -4.8689 -4.5056 ]; %! %! RCONDe = [ 0.13861D-01 0.90541D-02 ].'; %! %!assert (AK, AKe, 1e-4); %!assert (BK, BKe, 1e-4); %!assert (CK, CKe, 1e-4); %!assert (DK, DKe, 1e-4); %!assert (RCOND, RCONDe, 1e-4); ## continuous-time case %!shared AK, BK, CK, DK, RCOND, AKe, BKe, CKe, DKe, RCONDe %! A = [ -1.0 0.0 4.0 5.0 -3.0 -2.0 %! -2.0 4.0 -7.0 -2.0 0.0 3.0 %! -6.0 9.0 -5.0 0.0 2.0 -1.0 %! -8.0 4.0 7.0 -1.0 -3.0 0.0 %! 2.0 5.0 8.0 -9.0 1.0 -4.0 %! 3.0 -5.0 8.0 0.0 2.0 -6.0 ]; %! %! B = [ -3.0 -4.0 %! 2.0 0.0 %! -5.0 -7.0 %! 4.0 -6.0 %! -3.0 9.0 %! 1.0 -2.0 ]; %! %! C = [ 1.0 -1.0 2.0 -4.0 0.0 -3.0 %! -3.0 0.0 5.0 -1.0 1.0 1.0 %! -7.0 5.0 0.0 -8.0 2.0 -2.0 ]; %! %! D = [ 1.0 -2.0 %! 0.0 4.0 %! 5.0 -3.0 ]; %! %! FACTOR = 1.0; %! %! G = ss (A, B, C, D); %! K = ncfsyn (G, [], [], FACTOR); %! [AK, BK, CK, DK] = ssdata (K); %! %! AKe = [ -39.0671 9.9293 22.2322 -27.4113 43.8655 %! -6.6117 3.0006 11.0878 -11.4130 15.4269 %! 33.6805 -6.6934 -23.9953 14.1438 -33.4358 %! -32.3191 9.7316 25.4033 -24.0473 42.0517 %! -44.1655 18.7767 34.8873 -42.4369 50.8437 ]; %! %! BKe = [ -10.2905 -16.5382 -10.9782 %! -4.3598 -8.7525 -5.1447 %! 6.5962 1.8975 6.2316 %! -9.8770 -14.7041 -11.8778 %! -9.6726 -22.7309 -18.2692 ]; %! %! CKe = [ -0.6647 -0.0599 -1.0376 0.5619 1.7297 %! -8.4202 3.9573 7.3094 -7.6283 10.6768 ]; %! %! DKe = [ 0.8466 0.4979 -0.6993 %! -1.2226 -4.8689 -4.5056 ]; %! %! RCONDe = [ 0.13861D-01 0.90541D-02 ]; %! %!assert (AK, AKe, 1e-4); %!assert (BK, BKe, 1e-4); %!assert (CK, CKe, 1e-4); %!assert (DK, DKe, 1e-4); ## discrete-time case D==0, direct access to sb10kd %!shared AK, BK, CK, DK, RCOND, AKe, BKe, CKe, DKe, RCONDe %! A = [ 0.2 0.0 0.3 0.0 -0.3 -0.1 %! -0.3 0.2 -0.4 -0.3 0.0 0.0 %! -0.1 0.1 -0.1 0.0 0.0 -0.3 %! 0.1 0.0 0.0 -0.1 -0.1 0.0 %! 0.0 0.3 0.6 0.2 0.1 -0.4 %! 0.2 -0.4 0.0 0.0 0.2 -0.2 ]; %! %! B = [ -1.0 -2.0 %! 1.0 3.0 %! -3.0 -4.0 %! 1.0 -2.0 %! 0.0 1.0 %! 1.0 5.0 ]; %! %! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0 %! -3.0 0.0 1.0 -1.0 1.0 -1.0 ]; %! %! FACTOR = 1.1; %! %! [AK, BK, CK, DK, RCOND] = __sl_sb10kd__ (A, B, C, FACTOR); %! %! AKe = [ 0.0337 0.0222 0.0858 0.1264 -0.1872 0.1547 %! 0.4457 0.0668 -0.2255 -0.3204 -0.4548 -0.0691 %! -0.2419 -0.2506 -0.0982 -0.1321 -0.0130 -0.0838 %! -0.4402 0.3654 -0.0335 -0.2444 0.6366 -0.6469 %! -0.3623 0.3854 0.4162 0.4502 0.0065 0.1261 %! -0.0121 -0.4377 0.0604 0.2265 -0.3389 0.4542 ]; %! %! BKe = [ 0.0931 -0.0269 %! -0.0872 0.1599 %! 0.0956 -0.1469 %! -0.1728 0.0129 %! 0.2022 -0.1154 %! 0.2419 -0.1737 ]; %! %! CKe = [ -0.3677 0.2188 0.0403 -0.0854 0.3564 -0.3535 %! 0.1624 -0.0708 0.0058 0.0606 -0.2163 0.1802 ]; %! %! DKe = [ -0.0857 -0.0246 %! 0.0460 0.0074 ]; %! %! RCONDe = [ 0.11269D-01 0.17596D-01 0.18225D+00 0.75968D-03 ].'; %! %!assert (AK, AKe, 1e-4); %!assert (BK, BKe, 1e-4); %!assert (CK, CKe, 1e-4); %!assert (DK, DKe, 1e-4); %!assert (RCOND, RCONDe, 1e-4); ## discrete-time case D==0 %!shared AK, BK, CK, DK, RCOND, AKe, BKe, CKe, DKe, RCONDe %! A = [ 0.2 0.0 0.3 0.0 -0.3 -0.1 %! -0.3 0.2 -0.4 -0.3 0.0 0.0 %! -0.1 0.1 -0.1 0.0 0.0 -0.3 %! 0.1 0.0 0.0 -0.1 -0.1 0.0 %! 0.0 0.3 0.6 0.2 0.1 -0.4 %! 0.2 -0.4 0.0 0.0 0.2 -0.2 ]; %! %! B = [ -1.0 -2.0 %! 1.0 3.0 %! -3.0 -4.0 %! 1.0 -2.0 %! 0.0 1.0 %! 1.0 5.0 ]; %! %! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0 %! -3.0 0.0 1.0 -1.0 1.0 -1.0 ]; %! %! FACTOR = 1.1; %! %! G = ss (A, B, C, [], 1); # value of sampling time doesn't matter %! K = ncfsyn (G, [], [], FACTOR); %! [AK, BK, CK, DK] = ssdata (K); %! %! AKe = [ 0.0337 0.0222 0.0858 0.1264 -0.1872 0.1547 %! 0.4457 0.0668 -0.2255 -0.3204 -0.4548 -0.0691 %! -0.2419 -0.2506 -0.0982 -0.1321 -0.0130 -0.0838 %! -0.4402 0.3654 -0.0335 -0.2444 0.6366 -0.6469 %! -0.3623 0.3854 0.4162 0.4502 0.0065 0.1261 %! -0.0121 -0.4377 0.0604 0.2265 -0.3389 0.4542 ]; %! %! BKe = [ 0.0931 -0.0269 %! -0.0872 0.1599 %! 0.0956 -0.1469 %! -0.1728 0.0129 %! 0.2022 -0.1154 %! 0.2419 -0.1737 ]; %! %! CKe = [ -0.3677 0.2188 0.0403 -0.0854 0.3564 -0.3535 %! 0.1624 -0.0708 0.0058 0.0606 -0.2163 0.1802 ]; %! %! DKe = [ -0.0857 -0.0246 %! 0.0460 0.0074 ]; %! %! RCONDe = [ 0.11269D-01 0.17596D-01 0.18225D+00 0.75968D-03 ].'; %! %!assert (AK, AKe, 1e-4); %!assert (BK, BKe, 1e-4); %!assert (CK, CKe, 1e-4); %!assert (DK, DKe, 1e-4); ## discrete-time case D!=0, direct access to sb10zd %!shared AK, BK, CK, DK, RCOND, AKe, BKe, CKe, DKe, RCONDe %! A = [ 0.2 0.0 3.0 0.0 -0.3 -0.1 %! -3.0 0.2 -0.4 -0.3 0.0 0.0 %! -0.1 0.1 -1.0 0.0 0.0 -3.0 %! 1.0 0.0 0.0 -1.0 -1.0 0.0 %! 0.0 0.3 0.6 2.0 0.1 -0.4 %! 0.2 -4.0 0.0 0.0 0.2 -2.0 ]; %! %! B = [ -1.0 -2.0 %! 1.0 3.0 %! -3.0 -4.0 %! 1.0 -2.0 %! 0.0 1.0 %! 1.0 5.0 ]; %! %! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0 %! -3.0 0.0 1.0 -1.0 1.0 -1.0 %! 2.0 4.0 -3.0 0.0 5.0 1.0 ]; %! %! D = [ 10.0 -6.0 %! -7.0 8.0 %! 2.0 -4.0 ]; %! %! FACTOR = 1.1; %! %! [AK, BK, CK, DK, RCOND] = __sl_sb10zd__ (A, B, C, D, FACTOR, 0.0); %! %! AKe = [ 1.0128 0.5101 -0.1546 1.1300 3.3759 0.4911 %! -2.1257 -1.4517 -0.4486 0.3493 -1.5506 -1.4296 %! -1.0930 -0.6026 -0.1344 0.2253 -1.5625 -0.6762 %! 0.3207 0.1698 0.2376 -1.1781 -0.8705 0.2896 %! 0.5017 0.9006 0.0668 2.3613 0.2049 0.3703 %! 1.0787 0.6703 0.2783 -0.7213 0.4918 0.7435 ]; %! %! BKe = [ 0.4132 0.3112 -0.8077 %! 0.2140 0.4253 0.1811 %! -0.0710 0.0807 0.3558 %! -0.0121 -0.2019 0.0249 %! 0.1047 0.1399 -0.0457 %! -0.2542 -0.3472 0.0523 ]; %! %! CKe = [ -0.0372 -0.0456 -0.0040 0.0962 -0.2059 -0.0571 %! 0.1999 0.2994 0.1335 -0.0251 -0.3108 0.2048 ]; %! %! DKe = [ 0.0629 -0.0022 0.0363 %! -0.0228 0.0195 0.0600 ]; %! %! RCONDe = [ 0.27949D-03 0.66679D-03 0.45677D-01 0.23433D-07 0.68495D-01 0.76854D-01 ].'; %! %!assert (AK, AKe, 1e-4); %!assert (BK, BKe, 1e-4); %!assert (CK, CKe, 1e-4); %!assert (DK, DKe, 1e-4); %!assert (RCOND, RCONDe, 1e-4); ## discrete-time case D!=0 %!shared AK, BK, CK, DK, RCOND, AKe, BKe, CKe, DKe, RCONDe %! A = [ 0.2 0.0 3.0 0.0 -0.3 -0.1 %! -3.0 0.2 -0.4 -0.3 0.0 0.0 %! -0.1 0.1 -1.0 0.0 0.0 -3.0 %! 1.0 0.0 0.0 -1.0 -1.0 0.0 %! 0.0 0.3 0.6 2.0 0.1 -0.4 %! 0.2 -4.0 0.0 0.0 0.2 -2.0 ]; %! %! B = [ -1.0 -2.0 %! 1.0 3.0 %! -3.0 -4.0 %! 1.0 -2.0 %! 0.0 1.0 %! 1.0 5.0 ]; %! %! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0 %! -3.0 0.0 1.0 -1.0 1.0 -1.0 %! 2.0 4.0 -3.0 0.0 5.0 1.0 ]; %! %! D = [ 10.0 -6.0 %! -7.0 8.0 %! 2.0 -4.0 ]; %! %! FACTOR = 1.1; %! %! G = ss (A, B, C, D, 1); # value of sampling time doesn't matter %! K = ncfsyn (G, [], [], FACTOR); %! [AK, BK, CK, DK] = ssdata (K); %! %! AKe = [ 1.0128 0.5101 -0.1546 1.1300 3.3759 0.4911 %! -2.1257 -1.4517 -0.4486 0.3493 -1.5506 -1.4296 %! -1.0930 -0.6026 -0.1344 0.2253 -1.5625 -0.6762 %! 0.3207 0.1698 0.2376 -1.1781 -0.8705 0.2896 %! 0.5017 0.9006 0.0668 2.3613 0.2049 0.3703 %! 1.0787 0.6703 0.2783 -0.7213 0.4918 0.7435 ]; %! %! BKe = [ 0.4132 0.3112 -0.8077 %! 0.2140 0.4253 0.1811 %! -0.0710 0.0807 0.3558 %! -0.0121 -0.2019 0.0249 %! 0.1047 0.1399 -0.0457 %! -0.2542 -0.3472 0.0523 ]; %! %! CKe = [ -0.0372 -0.0456 -0.0040 0.0962 -0.2059 -0.0571 %! 0.1999 0.2994 0.1335 -0.0251 -0.3108 0.2048 ]; %! %! DKe = [ 0.0629 -0.0022 0.0363 %! -0.0228 0.0195 0.0600 ]; %! %! RCONDe = [ 0.27949D-03 0.66679D-03 0.45677D-01 0.23433D-07 0.68495D-01 0.76854D-01 ].'; %! %!assert (AK, AKe, 1e-4); %!assert (BK, BKe, 1e-4); %!assert (CK, CKe, 1e-4); %!assert (DK, DKe, 1e-4); control/inst/nichols.m0000644000076500000240000000730312273411257014341 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} nichols (@var{sys}) ## @deftypefnx {Function File} {} nichols (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx {Function File} {} nichols (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) ## @deftypefnx {Function File} {} nichols (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} nichols (@var{sys}) ## @deftypefnx {Function File} {[@var{mag}, @var{pha}, @var{w}] =} nichols (@var{sys}, @var{w}) ## Nichols chart of frequency response. If no output arguments are given, ## the response is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. Must be a single-input and single-output (SISO) system. ## @item w ## Optional vector of frequency values. If @var{w} is not specified, ## it is calculated by the zeros and poles of the system. ## Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, ## where @var{wmin} and @var{wmax} denote minimum and maximum frequencies ## in rad/s. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item mag ## Vector of magnitude. Has length of frequency vector @var{w}. ## @item pha ## Vector of phase. Has length of frequency vector @var{w}. ## @item w ## Vector of frequency values used. ## @end table ## ## @seealso{bode, nyquist, sigma} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.5 function [mag_r, pha_r, w_r] = nichols (varargin) if (nargin == 0) print_usage (); endif [H, w] = __frequency_response__ (varargin, false, "ext"); H = cellfun (@reshape, H, {[]}, {1}, "uniformoutput", false); mag = cellfun (@abs, H, "uniformoutput", false); pha = cellfun (@(H) unwrap (arg (H)) * 180 / pi, H, "uniformoutput", false); if (! nargout) mag_db = cellfun (@(mag) 20 * log10 (mag), mag, "uniformoutput", false); tmp = cellfun (@isa, varargin, {"lti"}); sys_idx = find (tmp); tmp = cellfun (@ischar, varargin); style_idx = find (tmp); len = numel (H); plot_args = {}; legend_args = cell (len, 1); for k = 1:len if (k == len) lim = nargin; else lim = sys_idx(k+1); endif style = varargin(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); plot_args = cat (2, plot_args, pha(k), mag_db(k), style); try legend_args{k} = inputname(sys_idx(k)); catch legend_args{k} = ""; end_try_catch endfor plot (plot_args{:}) axis ("tight") xlim (__axis_margin__ (xlim)) ylim (__axis_margin__ (ylim)) grid ("on") title ("Nichols Chart") xlabel ("Phase [deg]") ylabel ("Magnitude [dB]") legend (legend_args) else mag_r = mag{1}; pha_r = pha{1}; w_r = w{1}; endif endfunction control/inst/nyquist.m0000644000076500000240000001035412273411257014416 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} nyquist (@var{sys}) ## @deftypefnx {Function File} {} nyquist (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx {Function File} {} nyquist (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) ## @deftypefnx {Function File} {} nyquist (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx {Function File} {[@var{re}, @var{im}, @var{w}] =} nyquist (@var{sys}) ## @deftypefnx {Function File} {[@var{re}, @var{im}, @var{w}] =} nyquist (@var{sys}, @var{w}) ## Nyquist diagram of frequency response. If no output arguments are given, ## the response is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. Must be a single-input and single-output (SISO) system. ## @item w ## Optional vector of frequency values. If @var{w} is not specified, ## it is calculated by the zeros and poles of the system. ## Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, ## where @var{wmin} and @var{wmax} denote minimum and maximum frequencies ## in rad/s. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item re ## Vector of real parts. Has length of frequency vector @var{w}. ## @item im ## Vector of imaginary parts. Has length of frequency vector @var{w}. ## @item w ## Vector of frequency values used. ## @end table ## ## @seealso{bode, nichols, sigma} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.4 function [re_r, im_r, w_r] = nyquist (varargin) if (nargin == 0) print_usage (); endif [H, w] = __frequency_response__ (varargin, false, "ext"); H = cellfun (@reshape, H, {[]}, {1}, "uniformoutput", false); re = cellfun (@real, H, "uniformoutput", false); im = cellfun (@imag, H, "uniformoutput", false); if (! nargout) tmp = cellfun (@isa, varargin, {"lti"}); sys_idx = find (tmp); tmp = cellfun (@ischar, varargin); style_idx = find (tmp); len = numel (H); pos_args = {}; neg_args = {}; legend_args = cell (len, 1); colororder = get (gca, "colororder"); rc = rows (colororder); for k = 1:len col = colororder(1+rem (k-1, rc), :); if (k == len) lim = nargin; else lim = sys_idx(k+1); endif style = varargin(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); if (isempty (style)) pos_args = cat (2, pos_args, re{k}, im{k}, {"-", "color", col}); neg_args = cat (2, neg_args, re{k}, -im{k}, {"-.", "color", col}); else pos_args = cat (2, pos_args, re{k}, im{k}, style); neg_args = cat (2, neg_args, re{k}, -im{k}, style); endif try legend_args{k} = inputname(sys_idx(k)); catch legend_args{k} = ""; end_try_catch endfor ## FIXME: pos_args = cat (2, pos_args, re{k}, im{k}, {"-", "color", col}, style); ## doesn't work! it would be nice to have default arguments that can be ## (partially) overwritten by user-specified plot styles. h = plot (pos_args{:}, neg_args{:}); axis ("tight") xlim (__axis_margin__ (xlim)) ylim (__axis_margin__ (ylim)) grid ("on") title ("Nyquist Diagram") xlabel ("Real Axis") ylabel ("Imaginary Axis") legend (h(1:len), legend_args) else re_r = re{1}; im_r = im{1}; w_r = w{1}; endif endfunction control/inst/obsv.m0000644000076500000240000000416712273411257013660 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## Copyright (C) 2009 Luca Favatella ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{ob} =} obsv (@var{sys}) ## @deftypefnx {Function File} {@var{ob} =} obsv (@var{a}, @var{c}) ## Return observability matrix. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item a ## State matrix (n-by-n). ## @item c ## Measurement matrix (p-by-n). ## @end table ## ## @strong{Outputs} ## @table @var ## @item ob ## Observability matrix. ## @end table ## ## @strong{Equation} ## @iftex ## @tex ## $$ O_b = \\left[ \\matrix{ C \\cr ## CA \\cr ## CA^2 \\cr ## \\vdots \\cr ## CA^{n-1} } \\right ] $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## @group ## | C | ## | CA | ## Ob = | CA^2 | ## | ... | ## | CA^(n-1) | ## @end group ## @end example ## @end ifnottex ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function ob = obsv (a, c) if (nargin == 1) # obsv (sys) ob = ctrb (a.').'; # transpose is overloaded for lti models elseif (nargin == 2) # obsv (a, c) ob = ctrb (a.', c.').'; # size checked inside else print_usage (); endif endfunction %!assert (obsv ([1, 0; 0, -0.5], [8, 8]), [8, 8; 8, -4]); control/inst/obsvf.m0000644000076500000240000000516312273411257014023 0ustar lukasstaff## Copyright (C) 2010 Benjamin Fernandez ## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} obsvf (@var{sys}) ## @deftypefnx{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} obsvf (@var{sys}, @var{tol}) ## @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} obsvf (@var{A}, @var{B}, @var{C}) ## @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} obsvf (@var{A}, @var{B}, @var{C}, @var{TOL}) ## If Ob=obsv(A,C) has rank r <= n = SIZE(A,1), then there is a ## similarity transformation Tc such that To = [t1;t2] where t1 is c ## and t2 is orthogonal to t1 ## ## @example ## @group ## Abar = To \\ A * To , Bbar = To \\ B , Cbar = C * To ## @end group ## @end example ## ## and the transformed system has the form ## ## @example ## @group ## | Ao 0 | | Bo | ## Abar = |----------|, Bbar = | --- |, Cbar = [Co | 0 ]. ## | A21 Ano| | Bno | ## @end group ## @end example ## ## where (Ao,Bo) is observable, and Co(sI-Ao)^(-1)Bo = C(sI-A)^(-1)B. And ## system is detectable if Ano has no eigenvalues in the right ## half plane. The last output K is a vector of length n containing the ## number of observable states. ## @end deftypefn ## Author: Benjamin Fernandez ## Created: 2010-05-02 ## Version: 0.1 function [ac, bc, cc, z, ncont] = obsvf (a, b = [], c, tol = []) if (nargin < 1 || nargin > 4) print_usage (); endif if (isa (a, "lti")) if (nargin > 2) print_usage (); endif [ac, bc, cc] = ctrbf (a.', b); # [sysbar, z, ncont] = ctrbf (sys.', tol); ac = ac.'; z = ncont = []; else if (nargin < 3) print_usage (); endif [ac, tmp, cc, z, ncont] = ctrbf (a.', c.', b.', tol); ac = ac.'; bc = cc.'; cc = tmp.'; endif endfunction control/inst/options.m0000644000076500000240000000445512273411257014402 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{opt} =} options (@var{'key1'}, @var{value1}, @var{'key2'}, @var{value2}, @dots{}) ## Create options struct @var{opt} from a number of key and value pairs. ## For use with order reduction and system identification functions. ## Option structs are a way to avoid typing the same key and value pairs ## over and over again. ## ## @strong{Inputs} ## @table @var ## @item key, property ## The name of the property. ## @item value ## The value of the property. ## @end table ## ## @strong{Outputs} ## @table @var ## @item opt ## Struct with fields for each key. ## @end table ## ## @strong{Example} ## @example ## @group ## octave:1> opt = options ("method", "spa", "tol", 1e-6) ## opt = ## ## scalar structure containing the fields: ## ## method = spa ## tol = 1.0000e-06 ## ## @end group ## @end example ## @example ## @group ## octave:2> save filename opt ## octave:3> # save the struct 'opt' to file 'filename' for later use ## octave:4> load filename ## octave:5> # load struct 'opt' from file 'filename' ## @end group ## @end example ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function opt = options (varargin) if (nargin == 0) print_usage (); endif if (rem (nargin, 2)) error ("options: properties and values must come in pairs"); endif ## alternative: opt = struct (varargin{:}); key = reshape (varargin(1:2:end-1), [], 1); val = reshape (varargin(2:2:end), [], 1); opt = cell2struct (val, key, 1); opt = orderfields (opt); endfunction control/inst/optiPID.m0000644000076500000240000001671412273411257014220 0ustar lukasstaff%% -*- texinfo -*- %% Numerical optimization of a PID controller using an objective function. %% The objective function is located in the file @command{optiPIDfun}. %% Type @code{which optiPID} to locate, @code{edit optiPID} to open %% and simply @code{optiPID} to run the example file. %% In this example called @code{optiPID}, loosely based on [1], it is assumed %% that the plant %% @iftex %% @tex %% $$ P(s) = {1 \\over (s^{2} + s + 1)\\ (s + 1)^{4}} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% 1 %% P(s) = ----------------------- %% (s^2 + s + 1) (s + 1)^4 %% @end example %% @end ifnottex %% is controlled by a PID controller with second-order roll-off %% @iftex %% @tex %% $$ C(s) = k_p \\ (1 + {1 \\over T_i \\ s} + T_d \\ s) \\ {1 \\over (\\tau \\ s + 1)^{2}} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% 1 1 %% C(s) = Kp (1 + ---- + Td s) ------------- %% Ti s (tau s + 1)^2 %% @end example %% @end ifnottex %% in the usual negative feedback structure %% @iftex %% @tex %% $$ T(s) = {L(s) \\over 1 + L(s)} = {P(s) \\ C(s) \\over 1 + P(s) \\ C(s)} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% L(s) P(s) C(s) %% T(s) = -------- = ------------- %% 1 + L(s) 1 + P(s) C(s) %% @end example %% @end ifnottex %% The plant P(s) is of higher order but benign. The initial values for the %% controller parameters %% @iftex %% @tex %% $k_p$, $T_i$ and $T_d$ %% @end tex %% @end iftex %% @ifnottex %% Kp, Ti and Td %% @end ifnottex %% are obtained by applying the %% Astroem and Haegglund rules [2]. These values are to be improved using a %% numerical optimization as shown below. %% As with all numerical methods, this approach can never guarantee that a %% proposed solution is a global minimum. Therefore, good initial guesses for %% the parameters to be optimized are very important. %% The Octave function @code{fminsearch} minimizes the objective function @var{J}, %% which is chosen to be %% @iftex %% @tex %% $$ J(k_p, T_i, T_d) = \\mu_1 \\cdot \\int_0^{\\infty} \\! t \\ |e(t)| \\ dt \\ + \\ \\mu_2 \\cdot (|| y(t) ||_{\\infty} - 1) \\ + \\ \\mu_3 \\cdot ||S(jw)||_{\\infty} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% inf %% J(Kp, Ti, Td) = mu1 INT t |e(t)| dt + mu2 (||y(t)|| - 1) + mu3 ||S(jw)|| %% 0 inf inf %% @end example %% @end ifnottex %% This particular objective function penalizes the integral of time-weighted absolute error %% @iftex %% @tex %% $$ ITAE = \\int_0^{\\infty} \\! t \\ |e(t)| \\ dt $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% inf %% ITAE = INT t |e(t)| dt %% 0 %% @end example %% @end ifnottex %% and the maximum overshoot %% @iftex %% @tex %% $$ y_{max} - 1 = || y(t) ||_{\\infty} - 1 $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% y - 1 = ||y(t)|| - 1 %% max inf %% @end example %% @end ifnottex %% to a unity reference step %% @iftex %% @tex %% $r(t) = \\varepsilon (t)$ %% @end tex %% @end iftex %% in the time domain. In the frequency domain, the sensitivity %% @iftex %% @tex %% $$ M_s = ||S(jw)||_{\\infty} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% Ms = ||S(jw)|| %% inf %% @end example %% @end ifnottex %% is minimized for good robustness, where S(jw) denotes the @emph{sensitivity} transfer function %% @iftex %% @tex %% $$ S(s) = {1 \\over 1 + L(s)} = {1 \\over 1 + P(s) \\ C(s)} $$ %% @end tex %% @end iftex %% @ifnottex %% @example %% 1 1 %% S(s) = -------- = ------------- %% 1 + L(s) 1 + P(s) C(s) %% @end example %% @end ifnottex %% The constants %% @iftex %% @tex %% $\\mu_1$, $\\mu_2$ and $\\mu_3$ %% @end tex %% @end iftex %% @ifnottex %% mu1, mu2 and mu3 %% @end ifnottex %% are @emph{relative weighting factors} or @guillemetleft{}tuning knobs@guillemetright{} %% which reflect the importance of the different design goals. Varying these factors %% corresponds to changing the emphasis from, say, high performance to good robustness. %% The main advantage of this approach is the possibility to explore the tradeoffs of %% the design problem in a systematic way. %% In a first approach, all three design objectives are weigthed equally. %% In subsequent iterations, the parameters %% @iftex %% @tex %% $\\mu_1 = 1$, $\\mu_2 = 10$ and $\\mu_3 = 20$ %% @end tex %% @end iftex %% @ifnottex %% mu1 = 1, mu2 = 10 and mu3 = 20 %% @end ifnottex %% are found to yield satisfactory closed-loop performance. This controller results %% in a system with virtually no overshoot and a phase margin of 64 degrees. %% %% @*@strong{References}@* %% [1] Guzzella, L. %% @cite{Analysis and Design of SISO Control Systems}, %% VDF Hochschulverlag, ETH Zurich, 2007@* %% [2] Astroem, K. and Haegglund, T. %% @cite{PID Controllers: Theory, Design and Tuning}, %% Second Edition, %% Instrument Society of America, 1995 % =============================================================================== % optiPID Lukas Reichlin July 2009 % =============================================================================== % Numerical Optimization of an A/H PID Controller % Required OCTAVE Packages: control, optim (and its dependencies) % Required MATLAB Toolboxes: Control, Optimization % =============================================================================== % Tabula Rasa clear all, close all, clc; % Global Variables global P t dt mu_1 mu_2 mu_3 % Plant numP = [1]; denP = conv ([1, 1, 1], [1, 4, 6, 4, 1]); P = tf (numP, denP); % Relative Weighting Factors: PLAY AROUND WITH THESE! mu_1 = 1; % Minimize ITAE Criterion mu_2 = 10; % Minimize Max Overshoot mu_3 = 20; % Minimize Sensitivity Ms % Simulation Settings: PLANT-DEPENDENT! t_sim = 30; % Simulation Time [s] dt = 0.05; % Sampling Time [s] t = 0 : dt : t_sim; % Time Vector [s] % A/H PID Controller: Ms = 2.0 [gamma, phi, w_gamma, w_phi] = margin (P); ku = gamma; Tu = 2*pi / w_gamma; kappa = inv (dcgain (P)); disp ('optiPID: Astrom/Hagglund PID controller parameters:'); kp_AH = ku * 0.72 * exp ( -1.60 * kappa + 1.20 * kappa^2 ) Ti_AH = Tu * 0.59 * exp ( -1.30 * kappa + 0.38 * kappa^2 ) Td_AH = Tu * 0.15 * exp ( -1.40 * kappa + 0.56 * kappa^2 ) C_AH = optiPIDctrl (kp_AH, Ti_AH, Td_AH); % Initial Values C_par_0 = [kp_AH; Ti_AH; Td_AH]; % Optimization if (exist ('fminsearch')) warning ('optiPID: optimization starts, please be patient ...'); else error ('optiPID: please load/install optim package to proceed'); end C_par_opt = fminsearch (@optiPIDfun, C_par_0); % Optimized Controller disp ('optiPID: optimized PID controller parameters:'); kp_opt = C_par_opt(1) Ti_opt = C_par_opt(2) Td_opt = C_par_opt(3) C_opt = optiPIDctrl (kp_opt, Ti_opt, Td_opt); % Open Loop L_AH = P * C_AH; L_opt = P * C_opt; % Closed Loop T_AH = feedback (L_AH, 1); T_opt = feedback (L_opt, 1); % A Posteriori Stability Check disp ('optiPID: closed-loop stability check:'); st_AH = isstable (T_AH) st_opt = isstable (T_opt) % Stability Margins disp ('optiPID: gain margin gamma [-] and phase margin phi [deg]:'); [gamma_AH, phi_AH] = margin (L_AH) [gamma_opt, phi_opt] = margin (L_opt) % Plot Step Response figure (1) step (T_AH, 'b', T_opt, 'r', t) legend ('Astroem/Haegglund PID', 'Optimized PID', 'Location', 'SouthEast') % =============================================================================== control/inst/optiPIDctrl.m0000644000076500000240000000122212273411257015071 0ustar lukasstaff% =============================================================================== % optiPIDctrl Lukas Reichlin February 2012 % =============================================================================== % Return PID controller with roll-off for given parameters Kp, Ti and Td. % =============================================================================== function C = optiPIDctrl (Kp, Ti, Td) tau = Td / 10; % roll-off num = Kp * [Ti*Td, Ti, 1]; den = conv ([Ti, 0], [tau^2, 2*tau, 1]); C = tf (num, den); end % =============================================================================== control/inst/optiPIDfun.m0000644000076500000240000000310312273411257014715 0ustar lukasstaff% =============================================================================== % optiPIDfun Lukas Reichlin July 2009 % =============================================================================== % Objective Function % Reference: Guzzella, L. (2007) Analysis and Synthesis of SISO Control Systems. % vdf Hochschulverlag, Zurich % =============================================================================== function J = optiPIDfun (C_par) % Global Variables global P t dt mu_1 mu_2 mu_3 % Function Argument -> Controller Parameters kp = C_par(1); Ti = C_par(2); Td = C_par(3); % PID Controller with Roll-Off C = optiPIDctrl (kp, Ti, Td); % Open Loop L = P * C; % Sum Block: e = r - y SUM = ss ([1, -1]); % Matlab converts to SS (and back) for MIMO TF connections % Group Sum Block and Open Loop SUML = append (SUM, L); % Build System Interconnections CM = [3, 1; % Controller Input with Sum Block Output 2, 2]; % Sum Block Negative Input with Plant Output inputs = [1]; % Input 1: reference r(t) outputs = [1, 2]; % Output 1: error e(t), Output 2: output y(t) SUML = connect (SUML, CM, inputs, outputs); % Simulation [y, t_y] = step (SUML, t); % ITAE Criterion itae = dt * (t_y.' * abs (y(:, 1))); % Sensitivity S = inv (1 + L); Ms = norm (S, inf); % Objective Function J = mu_1 * itae + mu_2 * (max (y(:, 2)) - 1) + mu_3 * Ms; end % function % =============================================================================== control/inst/place.m0000644000076500000240000001406612273411257013772 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{f} =} place (@var{sys}, @var{p}) ## @deftypefnx {Function File} {@var{f} =} place (@var{a}, @var{b}, @var{p}) ## @deftypefnx {Function File} {[@var{f}, @var{info}] =} place (@var{sys}, @var{p}, @var{alpha}) ## @deftypefnx {Function File} {[@var{f}, @var{info}] =} place (@var{a}, @var{b}, @var{p}, @var{alpha}) ## Pole assignment for a given matrix pair (@var{A},@var{B}) such that @code{p = eig (A-B*F)}. ## If parameter @var{alpha} is specified, poles with real parts (continuous-time) ## or moduli (discrete-time) below @var{alpha} are left untouched. ## ## @strong{Inputs} ## @table @var ## @item sys ## Continuous- or discrete-time @acronym{LTI} system. ## @item a ## State matrix (n-by-n) of a continuous-time system. ## @item b ## Input matrix (n-by-m) of a continuous-time system. ## @item p ## Desired eigenvalues of the closed-loop system state-matrix @var{A-B*F}. ## @code{length (p) <= rows (A)}. ## @item alpha ## Specifies the maximum admissible value, either for real ## parts or for moduli, of the eigenvalues of @var{A} which will ## not be modified by the eigenvalue assignment algorithm. ## @code{alpha >= 0} for discrete-time systems. ## @end table ## ## @strong{Outputs} ## @table @var ## @item f ## State feedback gain matrix. ## @item info ## Structure containing additional information. ## @item info.nfp ## The number of fixed poles, i.e. eigenvalues of @var{A} having ## real parts less than @var{alpha}, or moduli less than @var{alpha}. ## These eigenvalues are not modified by @command{place}. ## @item info.nap ## The number of assigned eigenvalues. @code{nap = n-nfp-nup}. ## @item info.nup ## The number of uncontrollable eigenvalues detected by the ## eigenvalue assignment algorithm. ## @item info.z ## The orthogonal matrix @var{z} reduces the closed-loop ## system state matrix @code{A + B*F} to upper real Schur form. ## Note the positive sign in @code{A + B*F}. ## @end table ## ## @strong{Note} ## @example ## Place is also suitable to design estimator gains: ## @group ## L = place (A.', C.', p).' ## L = place (sys.', p).' # useful for discrete-time systems ## @end group ## @end example ## ## @strong{Algorithm}@* ## Uses SLICOT SB01BD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Special thanks to Peter Benner from TU Chemnitz for his advice. ## Author: Lukas Reichlin ## Created: December 2009 ## Version: 0.5 function [f, info] = place (a, b, p = [], alpha = [], tol = []) if (nargin < 2 || nargin > 5) print_usage (); endif if (isa (a, "lti")) # place (sys, p), place (sys, p, alpha), place (sys, p, alpha, tol) if (nargin > 4) # nargin < 2 already tested print_usage (); endif tol = alpha; alpha = p; p = b; sys = a; [a, b] = ssdata (sys); # descriptor matrice e should be regular discrete = ! isct (sys); # treat tsam = -2 as continuous system else # place (a, b, p), place (a, b, p, alpha), place (a, b, p, alpha, tol) if (nargin < 3) # nargin > 5 already tested print_usage (); endif if (! is_real_square_matrix (a) || ! is_real_matrix (b) || rows (a) != rows (b)) error ("place: matrices a and b not conformal"); endif discrete = 0; # assume continuous system endif if (! isnumeric (p) || ! isvector (p) || isempty (p)) # p could be complex error ("place: p must be a vector"); endif p = sort (reshape (p, [], 1)); # complex conjugate pairs must appear together wr = real (p); wi = imag (p); n = rows (a); # number of states np = length (p); # number of given eigenvalues if (np > n) error ("place: at most %d eigenvalues can be assigned for the given matrix a (%dx%d)", n, n, n); endif if (isempty (alpha)) if (discrete) alpha = 0; else alpha = - norm (a, inf); endif endif if (isempty (tol)) tol = 0; endif [f, nfp, nap, nup, z] = __sl_sb01bd__ (a, b, wr, wi, discrete, alpha, tol); f = -f; # A + B*F --> A - B*F info = struct ("nfp", nfp, "nap", nap, "nup", nup, "z", z); endfunction ## Test from "legacy" control package 1.0.* %!shared A, B, C, P, Kexpected %! A = [0, 1; 3, 2]; %! B = [0; 1]; %! C = [2, 1]; # C is needed for ss; it doesn't matter what the value of C is %! P = [-1, -0.5]; %! Kexpected = [3.5, 3.5]; %!assert (place (ss (A, B, C), P), Kexpected, 2*eps); %!assert (place (A, B, P), Kexpected, 2*eps); ## FIXME: Test from SLICOT example SB01BD fails with 4 eigenvalues in P %!shared F, F_exp, ev_ol, ev_cl %! A = [-6.8000 0.0000 -207.0000 0.0000 %! 1.0000 0.0000 0.0000 0.0000 %! 43.2000 0.0000 0.0000 -4.2000 %! 0.0000 0.0000 1.0000 0.0000]; %! %! B = [ 5.6400 0.0000 %! 0.0000 0.0000 %! 0.0000 1.1800 %! 0.0000 0.0000]; %! %! P = [-0.5000 + 0.1500i %! -0.5000 - 0.1500i]; #%! -2.0000 + 0.0000i #%! -0.4000 + 0.0000i]; %! %! ALPHA = -0.4; %! TOL = 1e-8; %! %! F = place (A, B, P, ALPHA, TOL); %! %! F_exp = - [-0.0876 -4.2138 0.0837 -18.1412 %! -0.0233 18.2483 -0.4259 -4.8120]; %! %! ev_ol = sort (eig (A)); %! ev_cl = sort (eig (A - B*F)); %! %!assert (F, F_exp, 1e-4); %!assert (ev_ol(3:4), ev_cl(3:4), 1e-4); control/inst/pzmap.m0000644000076500000240000000705012273411257014030 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} pzmap (@var{sys}) ## @deftypefnx {Function File} {} pzmap (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx {Function File} {} pzmap (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx {Function File} {[@var{p}, @var{z}] =} pzmap (@var{sys}) ## Plot the poles and zeros of an LTI system in the complex plane. ## If no output arguments are given, the result is plotted on the screen. ## Otherwise, the poles and zeros are computed and returned. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item p ## Poles of @var{sys}. ## @item z ## Transmission zeros of @var{sys}. ## @end table ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2009 ## Version: 0.2 function [pol_r, zer_r] = pzmap (varargin) if (nargin == 0) print_usage (); endif sys_idx = cellfun (@isa, varargin, {"lti"}); # look for LTI models pol = cellfun (@pole, varargin(sys_idx), "uniformoutput", false); zer = cellfun (@zero, varargin(sys_idx), "uniformoutput", false); if (! nargout) pol_re = cellfun (@real, pol, "uniformoutput", false); pol_im = cellfun (@imag, pol, "uniformoutput", false); zer_re = cellfun (@real, zer, "uniformoutput", false); zer_im = cellfun (@imag, zer, "uniformoutput", false); sys_idx = find (sys_idx); tmp = cellfun (@ischar, varargin); style_idx = find (tmp); len = numel (pol); pol_args = {}; zer_args = {}; legend_args = cell (len, 1); colororder = get (gca, "colororder"); rc = rows (colororder); for k = 1 : len col = colororder(1+rem (k-1, rc), :); if (k == len) lim = nargin; else lim = sys_idx(k+1); endif style = varargin(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); if (isempty (style)) pol_args = cat (2, pol_args, pol_re{k}, pol_im{k}, {"x", "color", col}); zer_args = cat (2, zer_args, zer_re{k}, zer_im{k}, {"o", "color", col}); else pol_args = cat (2, pol_args, pol_re{k}, pol_im{k}, style); zer_args = cat (2, zer_args, zer_re{k}, zer_im{k}, style); endif try legend_args{k} = inputname(sys_idx(k)); catch legend_args{k} = ""; end_try_catch endfor ## FIXME: try to combine "x", "o" and style for custom colors h = plot (pol_args{:}, zer_args{:}); grid ("on") title ("Pole-Zero Map") xlabel ("Real Axis") ylabel ("Imaginary Axis") legend (h(1:len), legend_args) else pol_r = pol{1}; zer_r = zer{1}; endif endfunction control/inst/ramp.m0000644000076500000240000000704312273411257013642 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {} ramp (@var{sys}) ## @deftypefnx{Function File} {} ramp (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx{Function File} {} ramp (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx{Function File} {} ramp (@var{sys1}, @dots{}, @var{t}) ## @deftypefnx{Function File} {} ramp (@var{sys1}, @dots{}, @var{tfinal}) ## @deftypefnx{Function File} {} ramp (@var{sys1}, @dots{}, @var{tfinal}, @var{dt}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}, @var{t}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}, @var{tfinal}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} ramp (@var{sys}, @var{tfinal}, @var{dt}) ## Ramp response of @acronym{LTI} system. ## If no output arguments are given, the response is printed on the screen. ## @iftex ## @tex ## $$ r(t) = t \\, \\cdot \\, h(t) $$ ## @end tex ## @end iftex ## @ifnottex ## ## @example ## r(t) = t * h(t) ## @end example ## ## @end ifnottex ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item t ## Time vector. Should be evenly spaced. If not specified, it is calculated by ## the poles of the system to reflect adequately the response transients. ## @item tfinal ## Optional simulation horizon. If not specified, it is calculated by ## the poles of the system to reflect adequately the response transients. ## @item dt ## Optional sampling time. Be sure to choose it small enough to capture transient ## phenomena. If not specified, it is calculated by the poles of the system. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item y ## Output response array. Has as many rows as time samples (length of t) ## and as many columns as outputs. ## @item t ## Time row vector. ## @item x ## State trajectories array. Has @code{length (t)} rows and as many columns as states. ## @end table ## ## @seealso{impulse, initial, lsim, step} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2012 ## Version: 0.1 function [y_r, t_r, x_r] = ramp (varargin) if (nargin == 0) print_usage (); endif if (nargout) sysname = {}; else sys_idx = find (cellfun (@isa, varargin, {"lti"})); len = length (sys_idx); sysname = cell (len, 1); for k = 1 : len try sysname{k} = inputname(sys_idx(k)); catch sysname{k} = ""; end_try_catch endfor endif [y, t, x] = __time_response__ ("ramp", varargin, sysname, ! nargout); if (nargout) y_r = y{1}; t_r = t{1}; x_r = x{1}; endif endfunction control/inst/rlocus.m0000644000076500000240000002336312273411257014215 0ustar lukasstaff## Copyright (C) 1996, 2000, 2004, 2005, 2006, 2007 ## Auburn University. All rights reserved. ## ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; see the file COPYING. If not, see ## . ## -*- texinfo -*- ## @deftypefn {Function File} {} rlocus (@var{sys}) ## @deftypefnx {Function File} {[@var{rldata}, @var{k}] =} rlocus (@var{sys}, @var{increment}, @var{min_k}, @var{max_k}) ## Display root locus plot of the specified @acronym{SISO} system. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. Must be a single-input and single-output (SISO) system. ## @item increment ## The increment used in computing gain values. ## @item min_k ## Minimum value of @var{k}. ## @item max_k ## Maximum value of @var{k}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item rldata ## Data points plotted: in column 1 real values, in column 2 the imaginary values. ## @item k ## Gains for real axis break points. ## @end table ## ## @strong{Block Diagram} ## @example ## @group ## u + +---+ +------+ y ## ------>(+)----->| k |----->| SISO |-------+-------> ## ^ - +---+ +------+ | ## | | ## +---------------------------------+ ## @end group ## @end example ## @end deftypefn ## Author: David Clem ## Author: R. Bruce Tenison ## Updated by Kristi McGowan July 1996 for intelligent gain selection ## Updated by John Ingram July 1996 for systems ## Adapted-By: Lukas Reichlin ## Date: December 2009 ## Version: 0.4 ## TODO: Improve compatibility function [rldata_r, k_break, rlpol, gvec, real_ax_pts] = rlocus (sys, increment, min_k, max_k) ## TODO: multiplot feature: rlocus (sys1, "b", sys2, "r", ...) if (nargin < 1 || nargin > 4) print_usage (); endif if (! isa (sys, "lti") || ! issiso (sys)) error ("rlocus: first argument must be a SISO LTI model"); endif ## Convert the input to a transfer function if necessary [num, den] = tfdata (sys, "vector"); # extract numerator/denominator polynomials lnum = length (num); lden = length (den); ## equalize length of num, den polynomials ## TODO: handle case lnum > lden (non-proper models) if (lden < 2) error ("rlocus: system has no poles"); elseif (lnum < lden) num = [zeros(1,lden-lnum), num]; # so that derivative is shortened by one endif olpol = roots (den); olzer = roots (num); nas = lden - lnum; # number of asymptotes maxk = 0; if (nas > 0) cas = (sum (olpol) - sum (olzer)) / nas; angles = (2*[1:nas]-1)*pi/nas; ## printf("rlocus: there are %d asymptotes centered at %f\n", nas, cas); else cas = angles = []; maxk = 100*den(1)/num(1); endif ## compute real axis break points and corresponding gains dnum = polyder (num); dden = polyder (den); brkp = conv (den, dnum) - conv (num, dden); real_ax_pts = roots (brkp); real_ax_pts = real_ax_pts(find (imag (real_ax_pts) == 0)); k_break = -polyval (den, real_ax_pts) ./ polyval (num, real_ax_pts); idx = find (k_break >= 0); k_break = k_break(idx); real_ax_pts = real_ax_pts(idx); if (! isempty (k_break)) maxk = max (max (k_break), maxk); endif if (nas == 0) maxk = max (1, 2*maxk); # get at least some root locus else ## get distance from breakpoints, poles, and zeros to center of asymptotes dmax = 3*max (abs ([vec(olzer); vec(olpol); vec(real_ax_pts)] - cas)); if (dmax == 0) dmax = 1; endif ## get gain for dmax along each asymptote, adjust maxk if necessary svals = cas + dmax * exp (j*angles); kvals = -polyval (den, svals) ./ polyval (num, svals); maxk = max (maxk, max (real (kvals))); endif ## check for input arguments: if (nargin > 2) mink = min_k; else mink = 0; endif if (nargin > 3) maxk = max_k; endif if (nargin > 1) if (increment <= 0) error ("rlocus: increment must be positive"); else ngain = fix ((maxk-mink)/increment); endif else ngain = 30; endif ## vector of gains ngain = max (30, ngain); gvec = linspace (mink, maxk, ngain); if (length (k_break)) gvec = sort ([gvec, reshape(k_break, 1, [])]); endif ## Find the open loop zeros and the initial poles rlzer = roots (num); ## update num to be the same length as den lnum = length (num); if (lnum < lden) num = [zeros(1,lden - lnum),num]; endif ## compute preliminary pole sets nroots = lden - 1; for ii = 1:ngain gain = gvec(ii); rlpol(1:nroots,ii) = vec(sort (roots (den + gain*num))); endfor ## set smoothing tolerance smtolx = 0.01*(max (max (real (rlpol))) - min (min (real (rlpol)))); smtoly = 0.01*(max (max (imag (rlpol))) - min (min (imag (rlpol)))); smtol = max (smtolx, smtoly); ## sort according to nearest-neighbor rlpol = sort_roots (rlpol, smtolx, smtoly); done = (nargin == 4); # perform a smoothness check while (! done && ngain < 1000) done = 1 ; # assume done dp = abs (diff (rlpol.')).'; maxdp = max (dp); ## search for poles whose neighbors are distant if (lden == 2) idx = find (dp > smtol); else idx = find (maxdp > smtol); endif for ii = 1:length(idx) i1 = idx(ii); g1 = gvec(i1); p1 = rlpol(:,i1); i2 = idx(ii)+1; g2 = gvec(i2); p2 = rlpol(:,i2); ## isolate poles in p1, p2 if (max (abs (p2-p1)) > smtol) newg = linspace (g1, g2, 5); newg = newg(2:4); gvec = [gvec,newg]; done = 0; # need to process new gains endif endfor ## process new gain values ngain1 = length (gvec); for ii = (ngain+1):ngain1 gain = gvec(ii); rlpol(1:nroots,ii) = vec(sort (roots (den + gain*num))); endfor [gvec, idx] = sort (gvec); rlpol = rlpol(:,idx); ngain = length (gvec); ## sort according to nearest-neighbor rlpol = sort_roots (rlpol, smtolx, smtoly); endwhile rldata = rlpol; ## Plot the data if (nargout == 0) rlpolv = vec(rlpol); axdata = [real(rlpolv), imag(rlpolv); real(olzer), imag(olzer)]; axlim = __axis_limits__ (axdata); rldata = [real(rlpolv), imag(rlpolv) ]; %inname = get (sys, "inname"); %outname = get (sys, "outname"); ## build plot command args pole by pole n_rlpol = rows (rlpol); nelts = n_rlpol+1; if (! isempty (rlzer)) nelts++; endif ## add asymptotes n_A = length (olpol) - length (olzer); if (n_A > 0) nelts += n_A; endif args = cell (3, nelts); kk = 0; ## asymptotes first if (n_A > 0) len_A = 2*max (abs (axlim)); sigma_A = (sum(olpol) - sum(olzer))/n_A; for i_A=0:n_A-1 phi_A = pi*(2*i_A + 1)/n_A; args{1,++kk} = [sigma_A sigma_A+len_A*cos(phi_A)]; args{2,kk} = [0 len_A*sin(phi_A)]; if (i_A == 1) args{3,kk} = "k--;asymptotes;"; else args{3,kk} = "k--"; endif endfor endif ## locus next for ii = 1:rows(rlpol) args{1,++kk} = real (rlpol (ii,:)); args{2,kk} = imag (rlpol (ii,:)); if (ii == 1) args{3,kk} = "b-;locus;"; else args{3,kk} = "b-"; endif endfor ## poles and zeros last args{1,++kk} = real (olpol); args{2,kk} = imag (olpol); args{3,kk} = "rx;open loop poles;"; if (! isempty (rlzer)) args{1,++kk} = real (rlzer); args{2,kk} = imag (rlzer); args{3,kk} = "go;zeros;"; endif set (gcf,"visible","off"); hplt = plot (args{:}); set (hplt(kk--), "markersize", 2); if (! isempty (rlzer)) set (hplt(kk--), "markersize", 2); endif for ii = 1:rows(rlpol) set (hplt(kk--), "linewidth", 2); endfor legend ("boxon", 2); grid ("on"); axis (axlim); title (["Root Locus of ", inputname(1)]); xlabel (sprintf ("Real Axis gain = [%g, %g]", gvec(1), gvec(ngain))); ylabel ("Imaginary Axis"); set (gcf (), "visible", "on"); else rldata_r = rldata; endif endfunction function rlpol = sort_roots (rlpol, tolx, toly) ## no point sorting of you've only got one pole! if (rows (rlpol) == 1) return; endif ## reorder entries in each column of rlpol to be by their nearest-neighbors rlpol dp = diff (rlpol.').'; drp = max (real (dp)); dip = max (imag (dp)); idx = find (drp > tolx | dip > toly); if (isempty (idx)) return; endif [np, ng] = size (rlpol); # num poles, num gains for jj = idx vals = rlpol(:,[jj,jj+1]); jdx = (jj+1):ng; for ii = 1:rows(rlpol-1) rdx = ii:np; dval = abs (rlpol(rdx,jj+1)-rlpol(ii,jj)); mindist = min (dval); sidx = min (find (dval == mindist)) + ii - 1; if (sidx != ii) c1 = norm (diff(vals.')); [vals(ii,2), vals(sidx,2)] = swap (vals(ii,2), vals(sidx,2)); c2 = norm (diff (vals.')); if (c1 > c2) ## perform the swap [rlpol(ii,jdx), rlpol(sidx,jdx)] = swap (rlpol(ii,jdx), rlpol(sidx,jdx)); vals = rlpol(:,[jj,jj+1]); endif endif endfor endfor endfunction function [b, a] = swap (a, b) endfunction control/inst/sensitivity.m0000644000076500000240000001072312273411257015274 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Ms}, @var{ws}] =} sensitivity (@var{L}) ## @deftypefnx{Function File} {[@var{Ms}, @var{ws}] =} sensitivity (@var{P}, @var{C}) ## @deftypefnx{Function File} {[@var{Ms}, @var{ws}] =} sensitivity (@var{P}, @var{C1}, @var{C2}, @dots{}) ## Return sensitivity margin @var{Ms}. ## The quantity @var{Ms} is simply the inverse of the shortest ## distance from the Nyquist curve to the critical point -1. ## Reasonable values of @var{Ms} are in the range from 1.3 to 2. ## @iftex ## @tex ## $$ M_s = ||S(j\\omega)||_{\\infty} $$ ## @end tex ## @end iftex ## @ifnottex ## ## @example ## Ms = ||S(jw)|| ## inf ## @end example ## ## @end ifnottex ## If no output arguments are given, the critical distance 1/Ms ## is plotted on a Nyquist diagram. ## In contrast to gain and phase margin as computed by command ## @command{margin}, the sensitivity @var{Ms} is a more robust ## criterion to assess the stability of a feedback system. ## ## @strong{Inputs} ## @table @var ## @item L ## Open loop transfer function. ## @var{L} can be any type of @acronym{LTI} system, but it must be square. ## @item P ## Plant model. Any type of @acronym{LTI} system. ## @item C ## Controller model. Any type of @acronym{LTI} system. ## @item C1, C2, @dots{} ## If several controllers are specified, command @command{sensitivity} ## computes the sensitivity @var{Ms} for each of them in combination ## with plant @var{P}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Ms ## Sensitivity margin @var{Ms} as defined in [1]. ## Scalar value. ## If several controllers are specified, @var{Ms} becomes ## a row vector with as many entries as controllers. ## @item ws ## The frequency [rad/s] corresponding to the sensitivity peak. ## Scalar value. ## If several controllers are specified, @var{ws} becomes ## a row vector with as many entries as controllers. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT AB13DD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## to calculate the infinity norm of the sensitivity function. ## ## @strong{References}@* ## [1] Astr@"om, K. and H@"agglund, T. (1995) ## PID Controllers: ## Theory, Design and Tuning, ## Second Edition. ## Instrument Society of America. ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: August 2012 ## Version: 0.1 function [ret, ws] = sensitivity (G, varargin) if (nargin == 0) print_usage (); elseif (nargin == 1) # L := G L = G; I = eye (size (L)); S = feedback (I, L); # S = inv (I + L), S = feedback (I, L*-I, "+") [Ms, ws] = norm (S, inf); else # P := G, C := varargin L = cellfun (@(C) G*C, varargin, "uniformoutput", false); I = cellfun (@(L) eye (size (L)), L, "uniformoutput", false); S = cellfun (@feedback, I, L, "uniformoutput", false); [Ms, ws] = cellfun (@(S) norm (S, inf), S); endif if (nargout == 0) ## TODO: don't show entire Nyquist curve if critical distance becomes small on plot if (length (Ms) > 1) error ("sensitivity: plotting only works for a single controller"); endif if (iscell (L)) L = L{1}; endif if (! issiso (L)) error ("sensitivity: Nyquist plot requires SISO systems"); endif [H, w] = __frequency_response__ (L, false, "ext"); H = H(:); re = real (H); im = imag (H); Hs = freqresp (L, ws); res = real (Hs); ims = imag (Hs); plot (re, im, "b", [-1, res], [0, ims], "r") axis ("equal") xlim (__axis_margin__ (xlim)) ylim (__axis_margin__ (ylim)) grid ("on") title (sprintf ("Sensitivity Ms = %g (at %g rad/s)", Ms, ws)) xlabel ("Real Axis") ylabel ("Imaginary Axis") else ret = Ms; endif endfunction control/inst/sigma.m0000644000076500000240000001107212273411257014000 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {} sigma (@var{sys}) ## @deftypefnx {Function File} {} sigma (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx {Function File} {} sigma (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}, @var{w}) ## @deftypefnx {Function File} {} sigma (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}) ## @deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{w}) ## Singular values of frequency response. If no output arguments are given, ## the singular value plot is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} system. Multiple inputs and/or outputs (MIMO systems) make practical sense. ## @item w ## Optional vector of frequency values. If @var{w} is not specified, ## it is calculated by the zeros and poles of the system. ## Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range, ## where @var{wmin} and @var{wmax} denote minimum and maximum frequencies ## in rad/s. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sv ## Array of singular values. For a system with m inputs and p outputs, the array sv ## has @code{min (m, p)} rows and as many columns as frequency points @code{length (w)}. ## The singular values at the frequency @code{w(k)} are given by @code{sv(:,k)}. ## @item w ## Vector of frequency values used. ## @end table ## ## @seealso{bodemag, svd} ## @end deftypefn ## Author: Lukas Reichlin ## Created: May 2009 ## Version: 0.7 function [sv_r, w_r] = sigma (varargin) if (nargin == 0) print_usage (); endif [H, w] = __frequency_response__ (varargin, true, "std", true); sv = cellfun (@(H) cellfun (@svd, H, "uniformoutput", false), H, "uniformoutput", false); sv = cellfun (@(sv) horzcat (sv{:}), sv, "uniformoutput", false); if (! nargout) # plot the information ## convert to dB for plotting sv_db = cellfun (@(sv) 20 * log10 (sv), sv, "uniformoutput", false); tmp = cellfun (@isa, varargin, {"lti"}); sys_idx = find (tmp); tmp = cellfun (@ischar, varargin); style_idx = find (tmp); len = numel (H); plot_args = {}; legend_args = cell (len, 1); colororder = get (gca, "colororder"); rc = rows (colororder); for k = 1:len col = colororder(1+rem (k-1, rc), :); if (k == len) lim = nargin; else lim = sys_idx(k+1); endif style = varargin(style_idx(style_idx > sys_idx(k) & style_idx <= lim)); if (isempty (style)) plot_args = cat (2, plot_args, w(k), sv_db(k), {"-", "color", col}); else plot_args = cat (2, plot_args, w(k), sv_db(k), style); endif try legend_args{k} = inputname(sys_idx(k)); # watch out for sigma (lticell{:}) catch legend_args{k} = ""; end_try_catch endfor ## adjust line colors in legend idx = horzcat (1, cellfun (@rows, sv_db)(1:end-1)); idx = cumsum (idx); ## plot results h = semilogx (plot_args{:}); axis ("tight") ylim (__axis_margin__ (ylim)) grid ("on") title ("Singular Values") xlabel ("Frequency [rad/s]") ylabel ("Singular Values [dB]") legend (h(idx), legend_args) else # return values sv_r = sv{1}; w_r = reshape (w{1}, [], 1); endif endfunction %!shared sv_exp, w_exp, sv_obs, w_obs %! A = [1, 2; 3, 4]; %! B = [5, 6; 7, 8]; %! C = [4, 3; 2, 1]; %! D = [8, 7; 6, 5]; %! w = [2, 3, 4]; %! sv_exp = [7.9176, 8.6275, 9.4393; %! 0.6985, 0.6086, 0.5195]; %! w_exp = [2; 3; 4]; %! [sv_obs, w_obs] = sigma (ss (A, B, C, D), w); %!assert (sv_obs, sv_exp, 1e-4); %!assert (w_obs, w_exp, 1e-4); control/inst/spaconred.m0000644000076500000240000001724112273411257014662 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{ncr}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} spaconred (@var{G}, @var{K}, @var{ncr}, @var{opt}, @dots{}) ## ## Controller reduction by frequency-weighted Singular Perturbation Approximation (SPA). ## Given a plant @var{G} and a stabilizing controller @var{K}, determine a reduced ## order controller @var{Kr} such that the closed-loop system is stable and closed-loop ## performance is retained. ## ## The algorithm tries to minimize the frequency-weighted error ## @iftex ## @tex ## $$ || V \\ (K - K_r) \\ W ||_{\\infty} = \\min $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ||V (K-Kr) W|| = min ## inf ## @end example ## @end ifnottex ## where @var{V} and @var{W} denote output and input weightings. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model of the plant. ## It has m inputs, p outputs and n states. ## @item K ## @acronym{LTI} model of the controller. ## It has p inputs, m outputs and nc states. ## @item ncr ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Kr ## State-space model of reduced order controller. ## @item info ## Struct containing additional information. ## @table @var ## @item info.ncr ## The order of the obtained reduced order controller @var{Kr}. ## @item info.ncs ## The order of the alpha-stable part of original controller @var{K}. ## @item info.hsvc ## The Hankel singular values of the alpha-stable part of @var{K}. ## The @var{ncs} Hankel singular values are ordered decreasingly. ## @end table ## @end table ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'ncr' ## The desired order of the resulting reduced order controller @var{Kr}. ## If not specified, @var{ncr} is chosen automatically such that states with ## Hankel singular values @var{info.hsvc} > @var{tol1} are retained. ## ## @item 'method' ## Order reduction approach to be used as follows: ## @table @var ## @item 'sr', 's' ## Use the square-root Singular Perturbation Approximation method. ## @item 'bfsr', 'p' ## Use the balancing-free square-root Singular Perturbation Approximation method. Default method. ## @end table ## ## @item 'weight' ## Specifies the type of frequency-weighting as follows: ## @table @var ## @item 'none' ## No weightings are used (V = I, W = I). ## ## @item 'left', 'output' ## Use stability enforcing left (output) weighting ## @iftex ## @tex ## $$ V = (I - G K)^{-1} G, \\qquad W = I $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 ## V = (I-G*K) *G , W = I ## @end example ## @end ifnottex ## ## @item 'right', 'input' ## Use stability enforcing right (input) weighting ## @iftex ## @tex ## $$ V = I, \\qquad W = (I - G K)^{-1} G $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 ## V = I , W = (I-G*K) *G ## @end example ## @end ifnottex ## ## @item 'both', 'performance' ## Use stability and performance enforcing weightings ## @iftex ## @tex ## $$ V = (I - G K)^{-1} G, \\qquad W = (I - G K)^{-1} $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## -1 -1 ## V = (I-G*K) *G , W = (I-G*K) ## @end example ## @end ifnottex ## Default value. ## @end table ## ## @item 'feedback' ## Specifies whether @var{K} is a positive or negative feedback controller: ## @table @var ## @item '+' ## Use positive feedback controller. Default value. ## @item '-' ## Use negative feedback controller. ## @end table ## ## @item 'alpha' ## Specifies the ALPHA-stability boundary for the eigenvalues ## of the state dynamics matrix @var{K.A}. For a continuous-time ## controller, ALPHA <= 0 is the boundary value for ## the real parts of eigenvalues, while for a discrete-time ## controller, 0 <= ALPHA <= 1 represents the ## boundary value for the moduli of eigenvalues. ## The ALPHA-stability domain does not include the boundary. ## Default value is 0 for continuous-time controllers and ## 1 for discrete-time controllers. ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced controller. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsvc(1), where c lies in the interval [0.00001, 0.001]. ## Default value is info.ncs*eps*info.hsvc(1). ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the ALPHA-stable part of the given ## controller. TOL2 <= TOL1. ## If not specified, ncs*eps*info.hsvc(1) is chosen. ## ## @item 'gram-ctrb' ## Specifies the choice of frequency-weighted controllability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to standard Enns' method [1]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified Enns' method of [2]. ## @end table ## ## @item 'gram-obsv' ## Specifies the choice of frequency-weighted observability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to standard Enns' method [1]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified Enns' method of [2]. ## @end table ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on @var{G} and @var{K} prior to order reduction. ## Default value is false if both @code{G.scaled == true, K.scaled == true} ## and true otherwise. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## @strong{Algorithm}@* ## Uses SLICOT SB16AD by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: December 2011 ## Version: 0.1 function [Kr, info] = spaconred (varargin) [Kr, info] = __conred_sb16ad__ ("spa", varargin{:}); endfunction ## TODO: add a test control/inst/spamodred.m0000644000076500000240000002035012273411257014655 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @var{opt}, @dots{}) ## ## Model order reduction by frequency weighted Singular Perturbation Approximation (SPA). ## The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order ## @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} ## approximates the one from original system @var{G}. ## ## SPA is an absolute error method which tries to minimize ## @iftex ## @tex ## $$ || G - G_r ||_{\\infty} = \\min $$ ## $$ || V \\ (G - G_r) \\ W ||_{\\infty} = \\min $$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ||G-Gr|| = min ## inf ## ## ||V (G-Gr) W|| = min ## inf ## @end example ## @end ifnottex ## where @var{V} and @var{W} denote output and input weightings. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model to be reduced. ## @item nr ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Gr ## Reduced order state-space model. ## @item info ## Struct containing additional information. ## @table @var ## @item info.n ## The order of the original system @var{G}. ## @item info.ns ## The order of the @var{alpha}-stable subsystem of the original system @var{G}. ## @item info.hsv ## The Hankel singular values of the @var{alpha}-stable part of ## the original system @var{G}, ordered decreasingly. ## @item info.nu ## The order of the @var{alpha}-unstable subsystem of both the original ## system @var{G} and the reduced-order system @var{Gr}. ## @item info.nr ## The order of the obtained reduced order system @var{Gr}. ## @end table ## @end table ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'nr' ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically such that states with ## Hankel singular values @var{info.hsv} > @var{tol1} are retained. ## ## @item 'left', 'output' ## @acronym{LTI} model of the left/output frequency weighting @var{V}. ## Default value is an identity matrix. ## ## @item 'right', 'input' ## @acronym{LTI} model of the right/input frequency weighting @var{W}. ## Default value is an identity matrix. ## ## @item 'method' ## Approximation method for the L-infinity norm to be used as follows: ## @table @var ## @item 'sr', 's' ## Use the square-root Singular Perturbation Approximation method. ## @item 'bfsr', 'p' ## Use the balancing-free square-root Singular Perturbation Approximation method. Default method. ## @end table ## ## @item 'alpha' ## Specifies the ALPHA-stability boundary for the eigenvalues ## of the state dynamics matrix @var{G.A}. For a continuous-time ## system, ALPHA <= 0 is the boundary value for ## the real parts of eigenvalues, while for a discrete-time ## system, 0 <= ALPHA <= 1 represents the ## boundary value for the moduli of eigenvalues. ## The ALPHA-stability domain does not include the boundary. ## Default value is 0 for continuous-time systems and ## 1 for discrete-time systems. ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced model. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. ## Default value is info.ns*eps*info.hsv(1). ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the ALPHA-stable part of the given ## model. TOL2 <= TOL1. ## If not specified, ns*eps*info.hsv(1) is chosen. ## ## @item 'gram-ctrb' ## Specifies the choice of frequency-weighted controllability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to a combination method [4] ## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified combination method of [4]. ## @end table ## ## @item 'gram-obsv' ## Specifies the choice of frequency-weighted observability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to a combination method [4] ## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified combination method of [4]. ## @end table ## ## @item 'alpha-ctrb' ## Combination method parameter for defining the ## frequency-weighted controllability Grammian. ## abs(alphac) <= 1. ## If alphac = 0, the choice of ## Grammian corresponds to the method of Enns [1], while if ## alphac = 1, the choice of Grammian corresponds ## to the method of Lin and Chiu [2,3]. ## Default value is 0. ## ## @item 'alpha-obsv' ## Combination method parameter for defining the ## frequency-weighted observability Grammian. ## abs(alphao) <= 1. ## If alphao = 0, the choice of ## Grammian corresponds to the method of Enns [1], while if ## alphao = 1, the choice of Grammian corresponds ## to the method of Lin and Chiu [2,3]. ## Default value is 0. ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on system @var{G} prior to order reduction. ## Default value is true if @code{G.scaled == false} and ## false if @code{G.scaled == true}. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## ## @strong{References}@* ## [1] Enns, D. ## @cite{Model reduction with balanced realizations: An error bound ## and a frequency weighted generalization}. ## Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984. ## ## [2] Lin, C.-A. and Chiu, T.-Y. ## @cite{Model reduction via frequency-weighted balanced realization}. ## Control Theory and Advanced Technology, vol. 8, ## pp. 341-351, 1992. ## ## [3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. ## @cite{New results on frequency weighted balanced reduction ## technique}. ## Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995. ## ## [4] Varga, A. and Anderson, B.D.O. ## @cite{Square-root balancing-free methods for the frequency-weighted ## balancing related model reduction}. ## (report in preparation) ## ## ## @strong{Algorithm}@* ## Uses SLICOT AB09ID by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function [Gr, info] = spamodred (varargin) [Gr, info] = __modred_ab09id__ ("spa", varargin{:}); endfunction ## TODO: add a test control/inst/step.m0000644000076500000240000000657012273411257013662 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {} step (@var{sys}) ## @deftypefnx{Function File} {} step (@var{sys1}, @var{sys2}, @dots{}, @var{sysN}) ## @deftypefnx{Function File} {} step (@var{sys1}, @var{'style1'}, @dots{}, @var{sysN}, @var{'styleN'}) ## @deftypefnx{Function File} {} step (@var{sys1}, @dots{}, @var{t}) ## @deftypefnx{Function File} {} step (@var{sys1}, @dots{}, @var{tfinal}) ## @deftypefnx{Function File} {} step (@var{sys1}, @dots{}, @var{tfinal}, @var{dt}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{t}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{tfinal}) ## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} step (@var{sys}, @var{tfinal}, @var{dt}) ## Step response of @acronym{LTI} system. ## If no output arguments are given, the response is printed on the screen. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model. ## @item t ## Time vector. Should be evenly spaced. If not specified, it is calculated by ## the poles of the system to reflect adequately the response transients. ## @item tfinal ## Optional simulation horizon. If not specified, it is calculated by ## the poles of the system to reflect adequately the response transients. ## @item dt ## Optional sampling time. Be sure to choose it small enough to capture transient ## phenomena. If not specified, it is calculated by the poles of the system. ## @item 'style' ## Line style and color, e.g. 'r' for a solid red line or '-.k' for a dash-dotted ## black line. See @command{help plot} for details. ## @end table ## ## @strong{Outputs} ## @table @var ## @item y ## Output response array. Has as many rows as time samples (length of t) ## and as many columns as outputs. ## @item t ## Time row vector. ## @item x ## State trajectories array. Has @code{length (t)} rows and as many columns as states. ## @end table ## ## @seealso{impulse, initial, lsim} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2009 ## Version: 0.2 function [y_r, t_r, x_r] = step (varargin) if (nargin == 0) print_usage (); endif if (nargout) sysname = {}; else sys_idx = find (cellfun (@isa, varargin, {"lti"})); len = length (sys_idx); sysname = cell (len, 1); for k = 1 : len try sysname{k} = inputname(sys_idx(k)); catch sysname{k} = ""; end_try_catch endfor endif [y, t, x] = __time_response__ ("step", varargin, sysname, ! nargout); if (nargout) y_r = y{1}; t_r = t{1}; x_r = x{1}; endif endfunction control/inst/strseq.m0000644000076500000240000000256712273411257014232 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{strvec} =} strseq (@var{str}, @var{idx}) ## Return a cell vector of indexed strings by appending the indices @var{idx} ## to the string @var{str}. ## ## @example ## strseq ("x", 1:3) = @{"x1"; "x2"; "x3"@} ## strseq ("u", [1, 2, 5]) = @{"u1"; "u2"; "u5"@} ## @end example ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.3 function strvec = strseq (str, idx) if (nargin != 2 || ! ischar (str) || ! isnumeric (idx)) print_usage (); endif strvec = arrayfun (@(x) sprintf ("%s%d", str, x), idx(:), "uniformoutput", false); endfunction control/inst/sumblk.m0000644000076500000240000000650212273411257014177 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {@var{S} =} sumblk (@var{formula}) ## @deftypefnx{Function File} {@var{S} =} sumblk (@var{formula}, @var{n}) ## Create summing junction @var{S} from string @var{formula} ## for name-based interconnections. ## ## @strong{Inputs} ## @table @var ## @item formula ## String containing the formula of the summing junction, ## e.g. @code{e = r - y + d} ## @item n ## Signal size. Default value is 1. ## @end table ## ## @strong{Outputs} ## @table @var ## @item S ## State-space model of the summing junction. ## @end table ## ## @strong{Example} ## @example ## @group ## octave:1> S = sumblk ('e = r - y + d') ## ## S.d = ## r y d ## e 1 -1 1 ## ## Static gain. ## octave:2> S = sumblk ('e = r - y + d', 2) ## ## S.d = ## r1 r2 y1 y2 d1 d2 ## e1 1 0 -1 0 1 0 ## e2 0 1 0 -1 0 1 ## ## Static gain. ## @end group ## @end example ## @seealso{connect} ## @end deftypefn ## Author: Lukas Reichlin ## Created: October 2013 ## Version: 0.1 function s = sumblk (formula, n = 1) if (nargin == 0 || nargin > 2) print_usage (); endif if (! ischar (formula)) error ("sumblk: require string as first argument"); endif if (! is_real_scalar (n)) error ("sumblk: require integer as second argument"); endif if (length (strfind (formula, "=")) != 1) error ("sumblk: formula requires exactly one '='"); endif ## if the first signal has no sign, add a plus if (isempty (regexp (formula, "=\\s*[+-]", "once"))) formula = regexprep (formula, "=", "=+", "once"); endif ## extract operators, remove "=" from formula operator = regexp (formula, "[=+-]", "match"); if (! strcmp (operator{1}, "=")) error ("sumblk: formula has misplaced '='"); endif operator = operator(2:end); formula = formula(formula != "="); ## extract signal names signal = regexp (formula, "\\s*[+-]\\s*", "split"); if (any (cellfun (@isempty, signal))) error ("sumblk: formula is missing some input/output names"); endif outname = signal(1); inname = signal(2:end); signs = ones (1, numel (inname)); signs(strcmp (operator, "-")) = -1; if (n != 1) outname = strseq (outname{1}, 1:n); tmp = cellfun (@strseq, inname, {1:n}, "uniformoutput", false); inname = vertcat (tmp{:}); endif d = kron (signs, eye (n)); s = ss (d); s = set (s, "inname", inname, "outname", outname); ## NOTE: the dark side returns a tf, but i prefer an ss model ## because in general, transfer functions and mimo ## interconnections don't mix well endfunction control/inst/test_control.m0000644000076500000240000000612112273411257015416 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Script File} {} test_control ## Execute all available tests at once. ## The Octave control package is based on the @uref{http://www.slicot.org, @acronym{SLICOT}} library. ## @acronym{SLICOT} needs @acronym{BLAS} and @acronym{LAPACK} libraries which are also prerequisites ## for Octave itself. ## In case of failing tests, it is highly recommended to use ## @uref{http://www.netlib.org/blas/, Netlib's reference @acronym{BLAS}} and ## @uref{http://www.netlib.org/lapack/, @acronym{LAPACK}} ## for building Octave. Using @acronym{ATLAS} may lead to sign changes ## in some entries of the state-space matrices. ## In general, these sign changes are not 'wrong' and can be regarded as ## the result of state transformations. Such state transformations ## (but not input/output transformations) have no influence on the ## input-output behaviour of the system. For better numerics, ## the control package uses such transformations by default when ## calculating the frequency responses and a few other things. ## However, arguments like the Hankel singular Values (@acronym{HSV}) must not change. ## Differing @acronym{HSV}s and failing algorithms are known for using Framework Accelerate ## from Mac OS X 10.7. ## @end deftypefn ## Author: Lukas Reichlin ## Created: May 2010 ## Version: 0.5 ## test collection test ltimodels ## LTI methods test @lti/c2d test @lti/connect test @lti/d2c test @lti/d2d test @lti/feedback test @lti/horzcat test @lti/inv test @lti/minreal test @lti/mtimes test @lti/norm test @lti/plus test @lti/prescale test @lti/sminreal test @lti/subsref test @lti/zero ## robust control test h2syn test hinfsyn test ncfsyn ## ARE solvers test care test dare test kalman ## Lyapunov test covar test dlyap ## test dlyapchol # TODO: add tests test gram test lyap test lyapchol ## model order reduction test bstmodred test btamodred test hnamodred ## test spamodred # TODO: create test case ## controller order reduction test btaconred test cfconred test fwcfconred ## test spaconred # TODO: create test case ## various oct-files test ctrbf test hsvd test place ## various m-files test db2mag test ctrb test filt test initial test issample test mag2db test margin test obsv test sigma test thiran test __dss2ss__ ## identification test @iddata/iddata test @iddata/cat test @iddata/detrend test @iddata/fft test fitfrd test moen4 control/inst/tfpoly2str.m0000644000076500000240000000461112273411257015031 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{str} =} tfpoly2str (@var{p}) ## @deftypefnx {Function File} {@var{str} =} tfpoly2str (@var{p}, @var{tfvar}) ## Return the string of a polynomial with string @var{tfvar} as variable. ## @end deftypefn ## Author: Lukas Reichlin ## Created: May 2012 ## Version: 0.1 function str = tfpoly2str (p, tfvar = "x") ## TODO: simplify this ugly code str = ""; lp = numel (p); if (lp > 0) # first element (lowest order) idx = find (p); # first non-zero element if (isempty (idx)) str = "0"; return; else idx = idx(1); endif a = p(idx); if (a < 0) cs = "-"; else cs = ""; endif if (idx == 1) str = [cs, num2str(abs (a), 4)]; else if (abs (a) == 1) str = [cs, __variable__(tfvar, idx-1)]; else str = [cs, __coefficient__(a), " ", __variable__(tfvar, idx-1)]; endif endif if (lp > idx) # remaining elements of higher order for k = idx+1 : lp a = p(k); if (a != 0) if (a < 0) cs = " - "; else cs = " + "; endif if (abs (a) == 1) str = [str, cs, __variable__(tfvar, k-1)]; else str = [str, cs, __coefficient__(a), " ", __variable__(tfvar, k-1)]; endif endif endfor endif endif endfunction function str = __coefficient__ (a) b = abs (a); if (b == 1) str = ""; else str = num2str (b, 4); endif endfunction function str = __variable__ (tfvar, n) str = [tfvar, "^-", num2str(n)]; endfunction control/inst/tfpolyones.m0000644000076500000240000000177612273411257015114 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Return (pxm) cell of tfpoly([1]). For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function ret = tfpolyones (p, m) ret = cell (p, m); one = tfpoly ([1]); ret(:) = {one}; endfunction control/inst/tfpolyzeros.m0000644000076500000240000000200112273411257015270 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## Return (pxm) cell of tfpoly([0]). For internal use only. ## Author: Lukas Reichlin ## Created: September 2009 ## Version: 0.1 function ret = tfpolyzeros (p, m) ret = cell (p, m); zero = tfpoly ([0]); ret(:) = {zero}; endfunction control/inst/thiran.m0000644000076500000240000000735512273411257014176 0ustar lukasstaff## Copyright (C) 2013 Thomas Vasileiou ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{sys} =} thiran (@var{tau}, @var{tsam}) ## Approximation of continuous-time delay using a discrete-time ## allpass Thiran filter. ## ## Thiran filters can approximate continuous-time delays that are ## non-integer multiples of the sampling time (fractional delays). ## This approximation gives a better matching of the phase shift ## between the continuous- and the discrete-time system. ## If there is no fractional part in the delay, then the standard ## discrete-time delay representation is used. ## ## @strong{Inputs} ## @table @var ## @item tau ## A continuous-time delay, given in time units (seconds). ## @item tsam ## The sampling time of the resulting Thiran filter. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Transfer function model of the resulting filter. ## The order of the filter is determined automatically. ## @end table ## ## @strong{Example} ## @example ## @group ## octave:1> sys = thiran (1.33, 0.5) ## ## Transfer function 'sys' from input 'u1' to output ... ## ## 0.003859 z^3 - 0.03947 z^2 + 0.2787 z + 1 ## y1: ----------------------------------------- ## z^3 + 0.2787 z^2 - 0.03947 z + 0.003859 ## ## Sampling time: 0.5 s ## Discrete-time model. ## @end group ## @end example ## @example ## @group ## octave:2> sys = thiran (1, 0.5) ## ## Transfer function 'sys' from input 'u1' to output ... ## ## 1 ## y1: --- ## z^2 ## ## Sampling time: 0.5 s ## Discrete-time model. ## @end group ## @end example ## ## @seealso{absorbdelay, pade} ## @end deftypefn ## Author: Thomas Vasileiou ## Created: January 2013 ## Version: 0.1 function sys = thiran (del, tsam) ## check args if (nargin != 2) print_usage (); endif if (! issample (del, 0)) error ("thiran: the delay parameter 'tau' must be a non-negative scalar."); endif if (! issample (tsam)) error ("thiran: the second parameter 'tsam' is not a valid sampling time."); endif if (del == 0) sys = tf (1); return; endif ## find fractional and discrete delay N = floor (del/tsam + eps); # put eps or sometimes it misses d = del - N*tsam; ## check if we do need a thiran filter if (d/tsam < eps) sys = tf (1, [1, zeros(1, N)], tsam); else ## make filter order ~ del to ensure stability N = N + 1; # order of the filter d = del/tsam; tmp = ((d-N+(0:N).') * ones (1,N)) ./ (d-N + bsxfun (@plus, 1:N, (0:N).')); a = horzcat (1, (-1).^(1:N) .* bincoeff (N, 1:N) .* prod (tmp)); sys = tf (fliplr (a), a, tsam); endif endfunction %!shared num, den, expc %! expc = [1, 0.5294, -0.04813, 0.004159]; %! sys = thiran (2.4, 1); %! [num, den] = tfdata (sys, "vector"); %!assert (den, expc, 1e-4); %!assert (num, fliplr (expc), 1e-4); %!shared num, den %! sys = thiran (0.5, 0.1); %! [num, den] = tfdata (sys, "vector"); %!assert (den, [1, 0, 0, 0, 0, 0], eps); %!assert (num, 1, eps); %!error (thiran (-1, 1)); %!error (thiran (1, -1)); %!error (thiran ([1 2 3], 1)); control/inst/WestlandLynx.m0000644000076500000240000001067312273411257015342 0ustar lukasstaff## -*- texinfo -*- ## @deftypefn{Function File} {@var{sys} =} WestlandLynx () ## Model of the Westland Lynx Helicopter about hover. ## @example ## @group ## INPUTS ## main rotor collective ## longitudinal cyclic ## lateral cyclic ## tail rotor collective ## @end group ## @end example ## @example ## @group ## STATES ## pitch attitude theta [rad] ## roll attitude phi [rad] ## roll rate (body-axis) p [rad/s] ## pitch rate (body-axis) q [rad/s] ## yaw rate xi [rad/s] ## forward velocity v_x [ft/s] ## lateral velocity v_y [ft/s] ## vertical velocity v_z [ft/s] ## @end group ## @end example ## @example ## @group ## OUTPUTS ## heave velocity H_dot [ft/s] ## pitch attitude theta [rad] ## roll attitude phi [rad] ## heading rate psi_dot [rad/s] ## roll rate p [rad/s] ## pitch rate q [rad/s] ## @end group ## @end example ## ## @strong{References}@* ## [1] Skogestad, S. and Postlethwaite I. (2005) ## @cite{Multivariable Feedback Control: Analysis and Design: ## Second Edition}. Wiley. ## @url{http://www.nt.ntnu.no/users/skoge/book/2nd_edition/matlab_m/matfiles.html} ## ## @end deftypefn ## Author: Lukas Reichlin ## Created: January 2010 ## Version: 0.1 function sys = WestlandLynx () if (nargin) print_usage (); endif a01 = [ 0 0 0 0.99857378005981; 0 0 1.00000000000000 -0.00318221934140; 0 0 -11.57049560546880 -2.54463768005371; 0 0 0.43935656547546 -1.99818229675293; 0 0 -2.04089546203613 -0.45899915695190; -32.10360717773440 0 -0.50335502624512 2.29785919189453; 0.10216116905212 32.05783081054690 -2.34721755981445 -0.50361156463623; -1.91097259521484 1.71382904052734 -0.00400543212891 -0.05741119384766]; a02 = [ 0.05338427424431 0 0 0; 0.05952465534210 0 0 0; -0.06360262632370 0.10678052902222 -0.09491866827011 0.00710757449269; 0 0.01665188372135 0.01846204698086 -0.00118747074157; -0.73502779006958 0.01925575733185 -0.00459562242031 0.00212036073208; 0 -0.02121581137180 -0.02116791903973 0.01581159234047; 0.83494758605957 0.02122657001019 -0.03787973523140 0.00035400385968; 0 0.01398963481188 -0.00090675335377 -0.29051351547241]; a0 = [a01 a02]; b0 = [ 0 0 0 0; 0 0 0 0; 0.12433505058289 0.08278584480286 -2.75247764587402 -0.01788876950741; -0.03635892271996 0.47509527206421 0.01429074257612 0; 0.30449151992798 0.01495801657438 -0.49651837348938 -0.20674192905426; 0.28773546218872 -0.54450607299805 -0.01637935638428 0; -0.01907348632812 0.01636743545532 -0.54453611373901 0.23484230041504; -4.82063293457031 -0.00038146972656 0 0]; c0 = [ 0 0 0 0 0 0.0595 0.05329 -0.9968; 1.0 0 0 0 0 0 0 0; 0 1.0 0 0 0 0 0 0; 0 0 0 -0.05348 1.0 0 0 0; 0 0 1.0 0 0 0 0 0; 0 0 0 1.0 0 0 0 0]; d0 = zeros (6, 4); inname = {"main rotor collective", "longitudinal cyclic", "lateral cyclic", "tail rotor collective"}; stname = {"theta", "phi", "p", "q", "xi", "v_x", "v_y", "v_z"}; outname = {"H_dot", "theta", "phi", "psi_dot", "p", "q"}; sys = ss (a0, b0, c0, d0, "inname", inname, "stname", stname, "outname", outname); endfunction control/inst/zpk.m0000644000076500000240000000676412273411257013520 0ustar lukasstaff## Copyright (C) 2009-2014 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## LTI Syncope is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{s} =} zpk (@var{"s"}) ## @deftypefnx {Function File} {@var{z} =} zpk (@var{"z"}, @var{tsam}) ## @deftypefnx {Function File} {@var{sys} =} zpk (@var{sys}) ## @deftypefnx {Function File} {@var{sys} =} zpk (@var{k}) ## @deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @dots{}) ## @deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @var{tsam}, @dots{}) ## @deftypefnx {Function File} {@var{sys} =} zpk (@var{z}, @var{p}, @var{k}, @var{tsam}, @dots{}) ## Create transfer function model from zero-pole-gain data. ## This is just a stop-gap compatibility wrapper since zpk ## models are not yet implemented. ## ## @strong{Inputs} ## @table @var ## @item sys ## @acronym{LTI} model to be converted to transfer function. ## @item z ## Cell of vectors containing the zeros for each channel. ## z@{i,j@} contains the zeros from input j to output i. ## In the SISO case, a single vector is accepted as well. ## @item p ## Cell of vectors containing the poles for each channel. ## p@{i,j@} contains the poles from input j to output i. ## In the SISO case, a single vector is accepted as well. ## @item k ## Matrix containing the gains for each channel. ## k(i,j) contains the gain from input j to output i. ## @item tsam ## Sampling time in seconds. If @var{tsam} is not specified, ## a continuous-time model is assumed. ## @item @dots{} ## Optional pairs of properties and values. ## Type @command{set (tf)} for more information. ## @end table ## ## @strong{Outputs} ## @table @var ## @item sys ## Transfer function model. ## @end table ## ## @seealso{tf, ss, dss, frd} ## @end deftypefn ## Author: Lukas Reichlin ## Created: September 2011 ## Version: 0.1 function sys = zpk (z = {}, p = {}, k = [], varargin) switch (nargin) case 0 # sys = zpk () sys = tf (); return; case 1 # sys = zpk (sys), sys = zpk (k), s = zpk ("s") if (isa (z, "lti") || is_real_matrix (z) || ischar (z)) sys = tf (z); return; else print_usage (); endif case 2 # z = zpk ("z", tsam) if (ischar (z) && issample (p, -1)) sys = tf (z, p); return; else print_usage (); endif otherwise # sys = zpk (z, p, k, ...) if (! iscell (z)) z = {z}; endif if (! iscell (p)) p = {p}; endif if (! size_equal (z, p, k)) error ("zpk: arguments z, p and k must have equal dimensions"); endif num = cellfun (@(zer, gain) real (gain * poly (zer)), z, num2cell (k), "uniformoutput", false); den = cellfun (@(pol) real (poly (pol)), p, "uniformoutput", false); sys = tf (num, den, varargin{:}); endswitch endfunction control/NEWS0000644000076500000240000006342412273411256012253 0ustar lukasstaffSummary of important user-visible changes for releases of the control package =============================================================================== control-2.6.2 Release Date: 2014-02-02 Release Manager: Lukas Reichlin =============================================================================== ** hinfsyn, mixsyn By default, hinfsyn and mixsyn compute now an optimal instead of a suboptimal controller. Furthermore, Matlab compatibility is improved if hinfsyn/mixsyn is called with more than three/four input arguments. (Thanks to Thomas Vasileiou for the gamma iteration) The old behavior (sub-optimal controller) can be restored by the expressions hinfsyn (P, nmeas, ncont, 'method', 'sub') mixsyn (G, W1, W2, W3, 'method', 'sub') ** h2syn, hinfsyn, mixsyn, ncfsyn For Matlab compatibility, the third return argument is now norm "gamma". The info struct is now returned as fourth argument. [K, N, gamma, info] = *syn (G, …) ** mktito New function to partition plant for robust control. "mktito" is also used by function "augw". If a plant is partitioned that way, one can omit the inputs "nmeas" and "ncon" when calling the functions "hinfsyn" and "h2syn". ** ss The conversion of realizable descriptor systems to regular state-space form is now possible, even when the descriptor matrix "E" is singular. (Thanks to Thomas Vasileiou and Bruce Minaker) =============================================================================== control-2.6.1 Release Date: 2013-12-10 Release Manager: Lukas Reichlin =============================================================================== ** LTI models Raise an error if the user tries to create an ingroup or outgroup with indices larger than the number of inputs m or outputs p. ** thiran New function to approximate a continuous-time delay by a discrete-time allpass Thiran filter. (Thanks to Thomas Vasileiou and Spencer Jackson) ** end "end" indexing for LTI and IDDATA objects works now as expected. =============================================================================== control-2.6.0 Release Date: 2013-10-24 Release Manager: Lukas Reichlin =============================================================================== ** LTI models -- Support for name-based channel selection in various commands. Example: sys = Boeing707, sys('pitch', {'rudder', 'thrust'}) instead of sys(2, [2, 1]). -- New properties "ingroup" and "outgroup" to pool several channels: sys.ingroup.controls = [3, 1], sys.outgroup.measurements = [1, 2] sys('measurements', 'controls') instead of sys([1, 2], [3, 1]). -- Support names instead of indices in commands like "feedback" and "xperm". ** connect Support name-based interconnections. This means that the LTI models passed to "connect" will be interconnected automatically if they have their inputs and outputs named accordingly. ** sumblk New function to build summing junctions for name-based interconnections by "connect". Example: Sum = sumblk ('e = r - y + n') ** tf State-space to transfer function conversions of static gains, e.g. "tf (ss (5))", no longer error out. (Thanks to Thomas Vasileiou) ** IDDATA identification datasets Support for name-based output, input and experiment selection similar to LTI models. =============================================================================== control-2.4.5 Release Date: 2013-10-07 Release Manager: Lukas Reichlin =============================================================================== ** Compatibility to GNU Octave 3.8 Replaced line continuation marker "\" by "..." to avoid deprecated syntax warning. (Thanks to John W. Eaton) ** db2mag, mag2db New functions to convert Decibels to Magnitude and vice versa. ** d2d New function for resampling of discrete-time models. =============================================================================== control-2.4.4 Release Date: 2013-08-11 Release Manager: Lukas Reichlin =============================================================================== ** zero -- Fix documentation. Function "zero" computes the invariant zeros, not the transmission zeros when it is called with one input argument. -- Added new options to compute the system, transmission, input decoupling and output decoupling zeros besides the already existing invariant zeros. Examples: z = zero (sys, 'system'), z = zero (sys, 'transmission') -- Return an "info" struct as third output argument. It contains additional rank information as well as Kronecker indices and the numbers of infinite elementary divisors. Example: [z, k, info] = zero (sys) (Special thanks to Ferdinand Svaricek) ** resample The "resample" function for "iddata" identification datasets now handles unspecified sampling times (tsam = -1) correctly. ** Time response functions (impulse, initial, lsim, ramp, step) Fix a thinko introduced with control-2.4.3. In some cases, several responses were mistakingly plotted on the same subplot. (Thanks to Thomas Vasileiou) ** Various documentation enhancements and fixes. =============================================================================== control-2.4.3 Release Date: 2013-07-12 Release Manager: Lukas Reichlin =============================================================================== ** resample The resample function for iddata sets now adjusts the sampling time correctly after resampling. ** Time response functions (impulse, initial, lsim, ramp, step) Don't use subplot for SISO models. This allows the user to plot different responses on a single figure by calling subplot. (Thanks to Matthias Meier) ** h2syn, hinfsyn, mixsyn, ncfsyn Third return argument is now an info struct. [K, N, info] = *syn (G, …) ** optiPID Beef up documentation. =============================================================================== control-2.4.2 Release Date: 2013-02-16 Release Manager: Lukas Reichlin =============================================================================== ** Transfer function to state-space conversion (Thanks to Thomas Vasileiou): -- Fix crash. -- Fix bug if denominators are zero. ** minreal Fix crash in minreal for state-space systems. (Thanks to Thomas Vasileiou) ** sigma The plot types 1, 2, 3 were no longer supported since the introduction of multi-system plots in control-2.3.54. The help text is now updated accordingly and cruft is removed from the codebase. ** mixsyn, ncfsyn, Madievski Beef up help text. =============================================================================== control-2.4.1 Release Date: 2012-11-01 Release Manager: Lukas Reichlin =============================================================================== ** impulse, initial, lsim, step These time response functions now error out correctly when the user tries to plot multiple systems in one window with mismatching numbers of inputs and outputs. The problem was caused by Octave's built-in function "size_equal" which gave false positives for LTI objects. The problem is solved by overloading the built-in function with a new LTI method "size_equal". ** ramp Added a new time domain analysis function "ramp" which plots/computes the system's response to a ramp signal. ** c2d, d2c Added matched pole/zero discretization method "matched". Usage: Gd = c2d (Gc, Ts, 'matched'), Gc = d2c (Gd, 'matched') ** doc/control.pdf Revised PDF manual. There is now a function index at the end of the manual. Improved docstrings in several functions. =============================================================================== control-2.4.0 Release Date: 2012-09-26 Release Manager: Lukas Reichlin =============================================================================== ** Multiplot feature for time-domain plotting: impulse lsim initial step Supported are now function calls like impulse (sys1, sys2, ...), step (sys1, 'b', sys2, 'r', ...) lsim (sys1, '-k', sys2, '--k', u), initial (sys, sys2, x0) The multiplot feature for frequency-domain plotting has already been introduced with control-2.3.54. ** Anderson, Madievski, MDSSystem, optiPID Updated example scripts to use new multiplot feature. ** sensitivity Fixed a problem with plotting, introduced with control-2.3.54. ** doc/control.pdf Updated PDF manual. =============================================================================== control-2.3.54 Release Date: 2012-09-15 Release Manager: Lukas Reichlin =============================================================================== ** Multiplot feature for frequency-domain plotting: bode nichols pzmap bodemag nyquist sigma The functions above support calls like sigma (sys1, sys2, ...), nyquist (sys1, 'b', sys2, 'r', ...), bode (sys1, '-k', sys2, '-.k', sys3, ':k', {wmin, wmax}). Time domain functions are not yet multiplot ready, but they will follow in control-2.4.0. ** plot A selection of experiments from iddata identification datasets can be plotted by plot (dat, exp) instead of plot (dat(:,:,:,exp)). ** sensitivity Fixed a problem where an error was raised about an undefined function "issiso". ** All SLICOT function names have now leading and trailing underscores to emphasize their private nature. =============================================================================== control-2.3.53 Release Date: 2012-08-27 Release Manager: Lukas Reichlin =============================================================================== ** Added new functions for system identification, including: arx iddata ifft moen4 detrend nkshift moesp fft plot n4sid filter resample ** sensitivity Added new function to compute and plot the sensitivity margin Ms. ** feedback Fixed an argument check which caused false positive error messages. It was a copy-paste mistake affecting non-square systems. (Thanks to Tony Olivo) =============================================================================== control-2.3.52 Release Date: 2012-06-25 Release Manager: Lukas Reichlin =============================================================================== ** Fixed a silly mistake in MIMO transfer function to state-space conversion. The bug has been introduced with control-2.3.51. (Thanks to Jim Rawlings for providing the test case) ** dlqe, lqe Added new functions for linear quadratic estimators. (Thanks to Megan Zagrobelny) ** Upon request of the Debian maintainers, the tex-files control.tex and functions.texi for generating control.pdf are included in the doc folder, next to control.pdf. Note that functions.texi is generated automatically by the scripts in the control/devel folder and the package generate_html. They are not included in the control package and can be found on the OctaveForge SVN server. ** Minor improvements in various help texts. =============================================================================== control-2.3.51 Release Date: 2012-06-03 Release Manager: Lukas Reichlin =============================================================================== ** filt, filtdata, tf -- Added function "filt" to specify disrete-time transfer functions in DSP format, i.e. z^-1. -- Added function "filtdata" to return any type of discrete-time LTI model in DSP format. -- tf models have a new property "inv". To display a discrete-time TF sys in z^-1, set sys.inv=true. In order to switch to z, set sys.inv=false. "filt" sets property "inv" to true (z^-1) by default, while "tf" uses false (z) as default value. ** ctranspose Conjugate transpose or pertransposition of LTI objects. Used by Octave for "sys'". For a transfer-function matrix G, G' denotes the conjugate of G given by G.'(-s) for a continuous-time system or G.'(1/z) for a discrete-time system. The frequency response of the pertransposition of G is the Hermitian (conjugate) transpose of G(jw), i.e. freqresp (G', w) = freqresp (G, w)'. WARNING: Do NOT use this for dual problems, use the transpose "sys.'" (note the dot) instead. ** test_control Add a few remarks to the help text regarding the severity of failing tests. ** Makefile fixed to work with non-standard linker options e.g on Apple. ** The conversion to state-space of multi-input transfer functions with common row denominators is now handled more efficiently. =============================================================================== control-2.3.50 Release Date: 2012-03-06 Release Manager: Lukas Reichlin =============================================================================== ** Added new functions for frequency-weighted model and controller order reduction: bstmodred btaconred btamodred cfconred hnamodred fwcfconred spamodred spaconred ** Anderson, Madievski -- Added two examples for controller reduction. The m-files are named after the authors of the corresponding papers. ** fitfrd -- Added function to fit frequency response data with a state-space model. ** set -- The set command doesn't need a return argument anymore in order to save the modified values. set (sys, "key", value) is now equivalent to sys = set (sys, "key", value). ** Require Octave version 3.6.0 or better. (The frequency response plotting commands have been simplified. They now use the fixed "axis tight" command. This is a first step towards multiple systems in one plot, e.g. bode (sys1, sys2, sys3). Furthermore, the code takes advantage of the new "arrayfun" function which became a faster oct-file instead of an m-file) ** Revised package installation and cleanup efforts under the hood. The new solution compiles the SLICOT library in a less barbaric way and creates only a single oct-file containing all the SLICOT routines. This also brings along faster compile times. (Special thanks to Hans Buchmann, Carlo De Falco and Michael Goffioul for their advice) ** doc/control.pdf -- Extended PDF manual. =============================================================================== control-2.2.5 Release Date: 2012-02-09 Release Manager: Lukas Reichlin =============================================================================== ** Improved Matlab compatibility for frequency response commands. It is now possible to specify a frequency range. Example: bode (sys, {wmin, wmax}) where wmin and wmax denote frequencies in rad/s. ** margin -- Fixed a variable name such that discrete-time models are plotted without an error. (Thanks to Renato Caldas) =============================================================================== control-2.2.4 Release Date: 2012-01-07 Release Manager: Lukas Reichlin =============================================================================== ** Compatibility with Octave 3.6.0. (The makefile must specify the libraries that mkoctfile needs to link. Thanks to Marco Atzeri and Carlo De Falco) ** ctrbf, obsvf -- Added new functions to compute controllable and observable block Hessenberg realizations based on SLICOT TB01UD. (Thanks to Benjamin Fernandez and Alexandre Felipe) =============================================================================== control-2.2.3 Release Date: 2011-12-07 Release Manager: Lukas Reichlin =============================================================================== ** Improved performance when computing the frequency response of transfer function models by orders of magnitude. (I realized that "polyval" can evaluate a polynomial at several values at once in a vectorized manner.) ** bode, bodemag, nichols, sigma -- Fixed a hang when plotting pairs of purely imaginary poles or zeros. The hang was caused by nonsensical (Inf, NaN) axis values. (Reported by Laurent Tissier) ** Use single instead of double quotes when displaying names of TF and FRD models. =============================================================================== control-2.2.2 Release Date: 2011-12-04 Release Manager: Lukas Reichlin =============================================================================== ** Improved error messages for place, ARE solvers and robust control commands. ** minreal, ss -- Fixed a crash for descriptor models. minreal for dss models as well as conversion from non-proper transfer functions to descriptor state-space models should work now as expected. ** ss -- Revised default tolerance for transfer function to state-space conversion by SLICOT TD04AD. ** Better performance when computing the frequency response of SISO transfer function models. ** Reorganized tests. Most tests have been moved from "ltimodels" to the files of the LTI methods being tested. All available tests can be executed by "test_control". ** The NEWS file is handled correctly when using Octave version 3.6. Type "news("control")" or "news control" to display the news (3.6 only). =============================================================================== control-2.2.1 Release Date: 2011-10-24 Release Manager: Lukas Reichlin =============================================================================== ** tf -- Fixed a nasty bug that prevented shortening of numerator and denominator polynomials consisting solely of zeros. -- MIMO support for descriptor state-space to transfer function conversion. Usage: tf_sys = tf (dss_sys) -- MIMO support for interconnections of non-proper transfer functions via internal conversion to state-space. ** ss -- Support conversion from non-proper transfer function to descriptor state- space. Usage: dss_sys = ss (tf_sys) ** c2d, d2c -- Support bilinear transformation of descriptor state-space models. ** inv -- Support the inversion of MIMO transfer functions. Inverses of 2x2 TFs are computed directly, larger models are computed internally in state-space. ** place -- Return the number of fixed, assigned and uncontrollable poles in a single "info" struct instead of three individual output arguments. ** rlocus -- Clarify usage statement in help string. -- Check whether system is SISO. ** MDSSystem -- Display bode plots of controllers instead of singular value plots of the closed loops. ** hsvd -- Added option "alpha" to specify the alpha-stability boundary for the eigenvalues of the state dynamics matrix A. ** isctrb, isobsv -- Return number of controllable/observable states as a second output argument. ** doc/control.pdf -- Added preface to PDF manual. =============================================================================== control-2.2.0 Release Date: 2011-09-26 Release Manager: Lukas Reichlin =============================================================================== ** ss -- Transfer function to state-space conversion uses now SLICOT TD04AD. Conversion of MIMO models is now supported. Usage: ss_sys = ss (tf_sys) ** tf -- Support for interconnections of MIMO transfer functions. This is done by an internal conversion to a minimal state-space representation. With the current tf2ss and ss2tf conversions, only proper transfer function are supported. This limitation does not exist for SISO transfer functions. -- Fixed a cellfun statement that caused problems on MinGW32 builds and possibly some others. (Reported by Bernhard Weller) ** pole, zero -- Computation of poles and zeros of MIMO transfer functions is now possible via conversion to state-space. Please note that the state-space realization of SLICOT TD04AD is a minimal one. Therefore certain poles and zeros might be missing. ** zpk, zpkdata -- Included wrappers that create transfer function models from zero-pole-gain data (zpk) and zero-pole-gain data from lti models (zpkdata). They are stop-gap measures for compatibility until ZPK models are implemented. ** tfdata -- "vector" option added. For SISO models, it returns numerator and denominator directly as column vectors instead of cells containing a single column vector. ** doc/control.pdf -- Revised PDF manual. =============================================================================== control-2.1.55 Release Date: 2011-09-07 Release Manager: Lukas Reichlin =============================================================================== ** c2d -- Support for "tustin" and "prewarp" method added. -- Transfer functions are now supported via the state-space methods. -- Improved Texinfo string. ** d2c -- Discrete to continuous-time conversion added. However, support is limited to the zero-order hold, tustin and pre-warping methods. ** Conversion from descriptor to regular state-space is now performed by SLICOT routine SB10JD. Better numerical results are to be expected over the previous naive inversion formula. This conversion is used internally for ssdata and some other functions. =============================================================================== control-2.1.54 Release Date: 2011-08-22 Release Manager: Lukas Reichlin =============================================================================== ** tf -- State-space to transfer function conversion uses now SLICOT TB04BD. Conversion of MIMO models is now supported. Usage: tf_sys = tf (ss_sys) -- Display an empty line between title and numerator for better readability. -- Display whether model is static, continuous- or discrete-time. ** A PDF manual is included for the first time. It is located inside the "doc" folder. It has been generated automatically from the Texinfo help strings and is not yet completely sorted out. =============================================================================== control-2.1.53 Release Date: 2011-08-08 Release Manager: Lukas Reichlin =============================================================================== ** ncfsyn -- Added support for McFarlane/Glover loop shaping design procedure. "ncfsyn" stands for Normalized Coprime Factor Synthesis. ** MDSSystem -- Added example script which demonstrates the usage of the robust control commands "mixsyn" and "ncfsyn". ** Texinfo help strings of several functions have been extended, although documentation still leaves a lot to be desired. =============================================================================== control-2.1.52 Release Date: 2011-07-27 Release Manager: Lukas Reichlin =============================================================================== ** hsvd -- Use scaling unless state-space model property "scaled" is set to true. ** norm -- Use scaling for computation of L-infinity norm unless state-space model property "scaled" is set to true. ** minreal -- Use scaling for state-space and descriptor state-space models unless property "scaled" is set to true. -- More accurate results are to be expected for descriptor state-space models by performing only those reduction phases where effective order reduction occurs. This is achieved by saving the system matrices before each phase and restoring them if no order reduction took place. ** zero -- Use scaling for state-space and descriptor state-space models unless property "scaled" is set to true. ** frdata -- The frequency response is now returned correctly as an array and not as a vector, unless the "vector" option is set and the system is single-input single-output. -- Added help text. =============================================================================== control-2.1.51 Release Date: 2011-07-21 Release Manager: Lukas Reichlin =============================================================================== ** frd -- Support for Frequency Response Data (frd) measurement "models". =============================================================================== control-2.1.50 Release Date: 2011-07-06 Release Manager: Lukas Reichlin =============================================================================== ** ss -- Support for property "scaled". By default, it is set to "false". ** prescale -- Scaling for state-space models (SLICOT TB01ID) and descriptor models (SLICOT TG01AD). ** freqresp -- Scale state-space models using @lti/prescale.m if property "scaled" is set to "false". Frequency response commands now perform automatic scaling unless model property "scaled" is set to "true". =============================================================================== control-2.0.2 Release Date: 2011-03-18 Release Manager: Lukas Reichlin =============================================================================== ** lsim -- Fixed a logical error that refused valid initial state vectors. It was due to a thinko introduced with the changes in control-2.0.1. (Thanks to Rob Frohne) =============================================================================== control-2.0.1 Release Date: 2011-03-06 Release Manager: Lukas Reichlin =============================================================================== ** lsim -- Support time vectors not starting at zero. (Thanks to Rob Frohne) -- Improved help text. ** zero -- The gain of descriptor state-space models is now computed correctly. (fingers crossed) =============================================================================== control-2.0.0 Release Date: 2011-02-08 Release Manager: Lukas Reichlin =============================================================================== ** First official release. =============================================================================== control/src/0000755000076500000240000000000012273411257012333 5ustar lukasstaffcontrol/src/__control_slicot_functions__.cc0000644000076500000240000000724212273411257020570 0ustar lukasstaff#include "sl_ab08nd.cc" // transmission zeros of state-space models #include "sl_ab13dd.cc" // L-infinity norm #include "sl_sb10hd.cc" // H-2 controller synthesis - continuous-time #include "sl_sb10ed.cc" // H-2 controller synthesis - discrete-time #include "sl_ab13bd.cc" // H-2 norm #include "sl_sb01bd.cc" // Pole assignment #include "sl_sb10fd.cc" // H-infinity controller synthesis - continuous-time #include "sl_sb10dd.cc" // H-infinity controller synthesis - discrete-time #include "sl_sb03md.cc" // Lyapunov equations #include "sl_sb04md.cc" // Sylvester equations - continuous-time #include "sl_sb04qd.cc" // Sylvester equations - discrete-time #include "sl_sg03ad.cc" // generalized Lyapunov equations #include "sl_sb02od.cc" // algebraic Riccati equations #include "sl_ab13ad.cc" // Hankel singular values #include "sl_ab01od.cc" // staircase form using orthogonal transformations #include "sl_tb01pd.cc" // minimal realization of state-space models #include "sl_sb03od.cc" // Cholesky factor of Lyapunov equations #include "sl_sg03bd.cc" // Cholesky factor of generalized Lyapunov equations #include "sl_ag08bd.cc" // transmission zeros of descriptor state-space models #include "sl_tg01jd.cc" // minimal realization of descriptor state-space models #include "sl_tg01hd.cc" // controllability staircase form of descriptor state-space models #include "sl_tg01id.cc" // observability staircase form of descriptor state-space models #include "sl_sg02ad.cc" // solution of algebraic Riccati equations for descriptor systems #include "sl_tg04bx.cc" // gain of descriptor state-space models #include "sl_tb01id.cc" // scaling of state-space models #include "sl_tg01ad.cc" // scaling of descriptor state-space models #include "sl_sb10id.cc" // H-infinity loop shaping - continuous-time #include "sl_sb10kd.cc" // H-infinity loop shaping - discrete-time - strictly proper case #include "sl_sb10zd.cc" // H-infinity loop shaping - discrete-time - proper case #include "sl_tb04bd.cc" // State-space to transfer function conversion #include "sl_ab04md.cc" // bilinear transformation #include "sl_sb10jd.cc" // descriptor to regular state-space conversion #include "sl_td04ad.cc" // transfer function to state-space conversion #include "sl_tb01ud.cc" // controllable block Hessenberg realization #include "sl_ab09hd.cc" // balanced stochastic truncation model reduction #include "sl_ab09id.cc" // balanced truncation & singular perturbation approximation model reduction #include "sl_ab09jd.cc" // hankel-norm approximation model reduction #include "sl_sb16ad.cc" // balanced truncation & singular perturbation approximation controller reduction #include "sl_sb16bd.cc" // coprime factorization state-feedback controller reduction #include "sl_sb16cd.cc" // frequency-weighted coprime factorization state-feedback controller reduction #include "sl_sb10yd.cc" // fit state-space model to frequency response data #include "sl_ident.cc" // system identification #include "sl_ib01cd.cc" // compute initial state vector #include "sl_ib01ad.cc" // compute singular values // #include "sl_are.cc" // solve ARE with Schur vector approach and scaling #include "sl_tg01fd.cc" // orthogonal reduction of dss to a SVD-like coordinate form #include "sl_sb10ad.cc" // H-infinity optimal controller using modified Glover's and Doyle's formulas (continuous-time) // stub function to avoid gen_doc_cache warning upon package installation DEFUN_DLD (__control_slicot_functions__, args, nargout, "-*- texinfo -*-\n\ Slicot Release 5.0\n\ No argument checking.\n\ For internal use only.") { octave_value_list retval; error ("__control_slicot_functions__: for internal use only"); return retval; } control/src/common.cc0000644000076500000240000000435512273411257014141 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Common code for oct-files. Author: Lukas Reichlin Created: April 2010 Version: 0.3 */ #include int max (int a, int b) { if (a > b) return a; else return b; } int max (int a, int b, int c) { return max (max (a, b), c); } int max (int a, int b, int c, int d) { return max (max (a, b), max (c, d)); } int max (int a, int b, int c, int d, int e) { return max (max (a, b, c, d), e); } int min (int a, int b) { if (a < b) return a; else return b; } void error_msg (const char name[], int index, int max, const char* msg[]) { if (index == 0) return; if (index < 0) error ("%s: the %d-th argument had an invalid value", name, index); else if (index <= max) error ("%s: %s", name, msg[index]); else error ("%s: unknown error, info = %d", name, index); } void warning_msg (const char name[], int index, int max, const char* msg[]) { if (index == 0) return; if (index > 0 && index <= max) warning ("%s: %s", name, msg[index]); else warning ("%s: unknown warning, iwarn = %d", name, index); } void warning_msg (const char name[], int index, int max, const char* msg[], int offset) { if (index == 0) return; if (index > 0 && index <= max) warning ("%s: %s", name, msg[index]); else if (index > offset) warning ("%s: %d+%d: %d %s", name, offset, index-offset, index-offset, msg[max+1]); else warning ("%s: unknown warning, iwarn = %d", name, index); } control/src/common.h0000644000076500000240000000235312273411257013777 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Common code for oct-files. Author: Lukas Reichlin Created: February 2012 Version: 0.1 */ #ifndef COMMON_H #define COMMON_H int max (int a, int b); int max (int a, int b, int c); int max (int a, int b, int c, int d); int max (int a, int b, int c, int d, int e); int min (int a, int b); void error_msg (const char name[], int index, int max, const char* msg[]); void warning_msg (const char name[], int index, int max, const char* msg[]); void warning_msg (const char name[], int index, int max, const char* msg[], int offset); #endif control/src/is_matrix.cc0000644000076500000240000000317512273411257014647 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Return true if argument is a real matrix. Author: Lukas Reichlin Created: June 2012 Version: 0.1 */ #include DEFUN_DLD (is_matrix, args, nargout, "-*- texinfo -*-\n\ @deftypefn {Loadable Function} {} is_matrix (@var{a}, @dots{})\n\ Return true if argument is a matrix.\n\ @var{[]} is a valid matrix.\n\ Avoid nasty stuff like @code{true = isreal (\"a\")}\n\ @seealso{is_real_matrix, is_real_square_matrix, is_real_vector, is_real_scalar}\n\ @end deftypefn") { octave_value retval = true; int nargin = args.length (); if (nargin == 0) { print_usage (); } else { for (int i = 0; i < nargin; i++) { if (args(i).ndims () != 2 || ! args(i).is_numeric_type () || ! (args(i).is_complex_type () || args(i).is_real_type ())) { retval = false; break; } } } return retval; } control/src/is_real_matrix.cc0000644000076500000240000000314112273411257015643 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Return true if argument is a real matrix. Author: Lukas Reichlin Created: September 2010 Version: 0.1 */ #include DEFUN_DLD (is_real_matrix, args, nargout, "-*- texinfo -*-\n\ @deftypefn {Loadable Function} {} is_real_matrix (@var{a}, @dots{})\n\ Return true if argument is a real matrix.\n\ @var{[]} is a valid matrix.\n\ Avoid nasty stuff like @code{true = isreal (\"a\")}\n\ @seealso{is_real_square_matrix, is_real_vector, is_real_scalar}\n\ @end deftypefn") { octave_value retval = true; int nargin = args.length (); if (nargin == 0) { print_usage (); } else { for (int i = 0; i < nargin; i++) { if (args(i).ndims () != 2 || ! args(i).is_numeric_type () || ! args(i).is_real_type ()) { retval = false; break; } } } return retval; } control/src/is_real_scalar.cc0000644000076500000240000000315512273411257015611 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Return true if argument is a real scalar. Author: Lukas Reichlin Created: September 2010 Version: 0.1 */ #include DEFUN_DLD (is_real_scalar, args, nargout, "-*- texinfo -*-\n\ @deftypefn {Loadable Function} {} is_real_scalar (@var{a}, @dots{})\n\ Return true if argument is a real scalar.\n\ @var{[]} is not a valid scalar.\n\ Avoid nasty stuff like @code{true = isreal (\"a\")}\n\ @seealso{is_real_matrix, is_real_vector}\n\ @end deftypefn") { octave_value retval = true; int nargin = args.length (); if (nargin == 0) { print_usage (); } else { for (int i = 0; i < nargin; i++) { if (args(i).ndims () != 2 || ! args(i).is_scalar_type () || ! args(i).is_numeric_type () || ! args(i).is_real_type ()) { retval = false; break; } } } return retval; } control/src/is_real_square_matrix.cc0000644000076500000240000000324612273411257017231 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Return true if argument is a real square matrix. Author: Lukas Reichlin Created: September 2010 Version: 0.1 */ #include DEFUN_DLD (is_real_square_matrix, args, nargout, "-*- texinfo -*-\n\ @deftypefn {Loadable Function} {} is_real_square_matrix (@var{a}, @dots{})\n\ Return true if argument is a real square matrix.\n\ @var{[]} is a valid square matrix.\n\ Avoid nasty stuff like @code{true = isreal (\"a\")}\n\ @seealso{is_real_matrix, is_real_vector, is_real_scalar}\n\ @end deftypefn") { octave_value retval = true; int nargin = args.length (); if (nargin == 0) { print_usage (); } else { for (int i = 0; i < nargin; i++) { if (args(i).ndims () != 2 || args(i).rows () != args(i).columns () || ! args(i).is_numeric_type () || ! args(i).is_real_type ()) { retval = false; break; } } } return retval; } control/src/is_real_vector.cc0000644000076500000240000000323412273411257015644 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Return true if argument is a real vector. Author: Lukas Reichlin Created: September 2010 Version: 0.1 */ #include DEFUN_DLD (is_real_vector, args, nargout, "-*- texinfo -*-\n\ @deftypefn {Loadable Function} {} is_real_vector (@var{a}, @dots{})\n\ Return true if argument is a real vector.\n\ @var{[]} is not a valid vector.\n\ Avoid nasty stuff like @code{true = isreal (\"a\")}\n\ @seealso{is_real_square_matrix, is_real_matrix, is_real_scalar}\n\ @end deftypefn") { octave_value retval = true; int nargin = args.length (); if (nargin == 0) { print_usage (); } else { for (int i = 0; i < nargin; i++) { if (args(i).ndims () != 2 || ! (args(i).rows () == 1 || args(i).columns () == 1) || ! args(i).is_numeric_type () || ! args(i).is_real_type ()) { retval = false; break; } } } return retval; } control/src/Makefile0000644000076500000240000000315512273411257013777 0ustar lukasstaffMKOCTFILE ?= mkoctfile ifndef LAPACK_LIBS LAPACK_LIBS := $(shell $(MKOCTFILE) -p LAPACK_LIBS) endif ifndef BLAS_LIBS BLAS_LIBS := $(shell $(MKOCTFILE) -p BLAS_LIBS) endif ifndef FLIBS FLIBS := $(shell $(MKOCTFILE) -p FLIBS) endif LFLAGS := $(shell $(MKOCTFILE) -p LFLAGS) $(LAPACK_LIBS) $(BLAS_LIBS) $(FLIBS) all: __control_slicot_functions__.oct \ is_real_scalar.oct \ is_real_vector.oct \ is_real_matrix.oct \ is_real_square_matrix.oct \ is_matrix.oct # TODO: Private oct-files for control package. # unpack and compile SLICOT library # Note that TG04BX is a custom routine. # It has the extension .fortran such that # it is not deleted by rm *.f when using # the developer makefile makefile_control.m slicotlibrary.a: slicot.tar.gz tar -xzf slicot.tar.gz mkdir sltmp mv slicot/src/*.f ./sltmp mv slicot/src_aux/*.f ./sltmp cp TG04BX.fortran ./sltmp/TG04BX.f cd sltmp; $(MKOCTFILE) -c *.f ar -rc slicotlibrary.a ./sltmp/*.o rm -rf sltmp slicot # slicot functions __control_slicot_functions__.oct: __control_slicot_functions__.cc slicotlibrary.a LFLAGS="$(LFLAGS)" \ $(MKOCTFILE) __control_slicot_functions__.cc common.cc slicotlibrary.a # helpers is_real_scalar.oct: is_real_scalar.cc $(MKOCTFILE) is_real_scalar.cc is_real_vector.oct: is_real_vector.cc $(MKOCTFILE) is_real_vector.cc is_real_matrix.oct: is_real_matrix.cc $(MKOCTFILE) is_real_matrix.cc is_real_square_matrix.oct: is_real_square_matrix.cc $(MKOCTFILE) is_real_square_matrix.cc is_matrix.oct: is_matrix.cc $(MKOCTFILE) is_matrix.cc clean: rm -rf *.o core octave-core *.oct *~ *.f slicot sltmp realclean: clean rm -rf *.acontrol/src/readme0000644000076500000240000000624412273411257013521 0ustar lukasstaffSLICOT Library Root Directory ----------------------------- SLICOT - Subroutine Library In COntrol Theory - is a general purpose basic mathematical library for control theoretical computations. The library provides tools to perform essential system analysis and synthesis tasks. The main emphasis in SLICOT is on numerical reliability of implemented algorithms and the numerical robustness and efficiency of routines. Providing algorithmic flexibility and the use of rigorous implementation and documentation standards are other SLICOT features. The SLICOT Library is available as standard Fortran 77 code in double precision. Each user-callable subroutine for control computations is accompanied by an example program which illustrates the use of the subroutine and can act as a template for the user's own routines. The SLICOT Library is organized by chapters, sections and subsections. The following chapters are currently included: A : Analysis Routines B : Benchmark and Test Problems D : Data Analysis F : Filtering I : Identification M : Mathematical Routines N : Nonlinear Systems (not yet available, except for some auxiliary routines for Wiener systems) S : Synthesis Routines T : Transformation Routines U : Utility Routines SLICOT Library Root Directory contains few, basic files for the SLICOT Library distribution and generation. When distributed, SLICOT software comes with several filled-in subdirectories (benchmark_data, doc, examples, examples77, src, and src_aux), and five files in this root directory: - this file, readme, - the file Installation.txt, describing the SLICOT software installation, - the main SLICOT Library documentation index, libindex.html, and - two template files for building the object library and executable programs, make.inc and makefile, - GNU GENERAL PUBLIC LICENSE Version 2 text file. The last two files might need few changes for being adapted to the specific platform used. Details about installing/updating the SLICOT software are given in the file Installation.txt. After software installation, this directory will also contain the library file slicot.a or slicot.lib, for Unix or Windows platforms, respectively. The library file could then be linked in applications programs, as usual. Specific examples are contained in the directories examples and examples77. The on-line documentation of the SLICOT user's callable routines is accessible via the main SLICOT Library documentation index, libindex.html. This file also contains a link to the documentation of the lower-level, support routines. The SLICOT Library is built on LAPACK (Linear Algebra PACKage) and BLAS (Basic Linear Algebra Subprograms) collections. Therefore, these packages should be available on the platform used. Basic References: 1. P. Benner, V. Mehrmann, V. Sima, S. Van Huffel, and A. Varga, "SLICOT - A Subroutine Library in Systems and Control Theory", Applied and Computational Control, Signals, and Circuits (Birkhauser), Vol. 1, Ch. 10, pp. 505-546, 1999. 2. S. Van Huffel, V. Sima, A. Varga, S. Hammarling, and F. Delebecque, "Development of High Performance Numerical Software for Control", IEEE Control Systems Magazine, Vol. 24, Nr. 1, Feb., pp. 60-76, 2004. control/src/sl_ab01od.cc0000644000076500000240000000724212273411257014413 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Staircase controllability form. Uses SLICOT AB01OD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: August 2010 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (ab01od, AB01OD) (char& STAGES, char& JOBU, char& JOBV, int& N, int& M, double* A, int& LDA, double* B, int& LDB, double* U, int& LDU, double* V, int& LDV, int& NCONT, int& INDCON, int* KSTAIR, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_ab01od__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab01od__, args, nargout, "-*- texinfo -*-\n\ Slicot AB01OD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 3) { print_usage (); } else { // arguments in char stages = 'F'; char jobu = 'I'; char jobv = 'N'; // not referenced because stages = 'F' Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); double tol = args(2).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int lda = max (1, n); int ldb = max (1, n); int ldu = max (1, n); int ldv = 1; // arguments out Matrix u (ldu, n); double* v = 0; // not referenced because stages = 'F' int ncont; int indcon; OCTAVE_LOCAL_BUFFER (int, kstair, n); // workspace int ldwork = max (1, n + max (n, 3*m)); OCTAVE_LOCAL_BUFFER (int, iwork, m); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info; // SLICOT routine AB01OD F77_XFCN (ab01od, AB01OD, (stages, jobu, jobv, n, m, a.fortran_vec (), lda, b.fortran_vec (), ldb, u.fortran_vec (), ldu, v, ldv, ncont, indcon, kstair, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("__sl_ab01od__: exception in SLICOT subroutine AB01OD"); if (info != 0) error ("__sl_ab01od__: AB01OD returned info = %d", info); // resize a.resize (n, n); b.resize (n, m); u.resize (n, n); // return values retval(0) = a; retval(1) = b; retval(2) = u; retval(3) = octave_value (ncont); } return retval; } control/src/sl_ab04md.cc0000644000076500000240000000673712273411257014424 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Discrete-time <--> continuous-time systems conversion by a bilinear transformation. Uses SLICOT AB04MD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (ab04md, AB04MD) (char& TYPE, int& N, int& M, int& P, double& ALPHA, double& BETA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_ab04md__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab04md__, args, nargout, "-*- texinfo -*-\n\ Slicot AB04MD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 7) { print_usage (); } else { // arguments in char type; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); double alpha = args(4).double_value (); double beta = args(5).double_value (); int discrete = args(6).int_value (); if (discrete == 0) type = 'C'; else type = 'D'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd = max (1, p); // workspace int ldwork = max (1, n); OCTAVE_LOCAL_BUFFER (int, iwork, n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine AB04MD F77_XFCN (ab04md, AB04MD, (type, n, m, p, alpha, beta, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("__sl_ab04md__: exception in SLICOT subroutine AB04MD"); if (info != 0) error ("__sl_ab04md__: AB04MD returned info = %d", info); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; } return retval; } control/src/sl_ab08nd.cc0000644000076500000240000001623112273411257014417 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Transmission zeros of state-space models. Uses SLICOT AB08ND by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2009 Version: 0.6 */ #include #include #include "common.h" #include #include extern "C" { int F77_FUNC (ab08nd, AB08ND) (char& EQUIL, int& N, int& M, int& P, const double* A, int& LDA, const double* B, int& LDB, const double* C, int& LDC, const double* D, int& LDD, int& NU, int& RANK, int& DINFZ, int& NKROR, int& NKROL, int* INFZ, int* KRONR, int* KRONL, double* AF, int& LDAF, double* BF, int& LDBF, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); int F77_FUNC (dggev, DGGEV) (char& JOBVL, char& JOBVR, int& N, double* AF, int& LDAF, double* BF, int& LDBF, double* ALPHAR, double* ALPHAI, double* BETA, double* VL, int& LDVL, double* VR, int& LDVR, double* WORK, int& LWORK, int& INFO); } // PKG_ADD: autoload ("__sl_ab08nd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab08nd__, args, nargout, "-*- texinfo -*-\n\ Slicot AB08ND Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in char equil; const Matrix a = args(0).matrix_value (); const Matrix b = args(1).matrix_value (); const Matrix c = args(2).matrix_value (); const Matrix d = args(3).matrix_value (); const int scaled = args(4).int_value (); if (scaled == 0) equil = 'S'; else equil = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); int ldd = max (1, d.rows ()); // arguments out int nu; int rank; int dinfz; int nkror; int nkrol; int ldaf = max (1, n + m); int ldbf = max (1, n + p); OCTAVE_LOCAL_BUFFER (int, infz, n); OCTAVE_LOCAL_BUFFER (int, kronr, 1 + max (n, m)); OCTAVE_LOCAL_BUFFER (int, kronl, 1 + max (n, p)); OCTAVE_LOCAL_BUFFER (double, af, ldaf * (n + min (p, m))); OCTAVE_LOCAL_BUFFER (double, bf, ldbf * (n + m)); // workspace int s = max (m, p); int ldwork = max (s, n) + max (3*s-1, n+s); OCTAVE_LOCAL_BUFFER (int, iwork, s); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // tolerance double tol = 0; // AB08ND uses DLAMCH for default tolerance // SLICOT routine AB08ND F77_XFCN (ab08nd, AB08ND, (equil, n, m, p, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, nu, rank, dinfz, nkror, nkrol, infz, kronr, kronl, af, ldaf, bf, ldbf, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("ss: zero: __sl_ab08nd__: exception in SLICOT subroutine AB08ND"); if (info != 0) error ("ss: zero: __sl_ab08nd__: AB08ND returned info = %d", info); // DGGEV Part char jobvl = 'N'; char jobvr = 'N'; double* vl = 0; // not referenced because jobvl = 'N' int ldvl = 1; double* vr = 0; // not referenced because jobvr = 'N' int ldvr = 1; int lwork = max (1, 8*nu); OCTAVE_LOCAL_BUFFER (double, work, lwork); ColumnVector alphar (nu); ColumnVector alphai (nu); ColumnVector beta (nu); int info2; F77_XFCN (dggev, DGGEV, (jobvl, jobvr, nu, af, ldaf, bf, ldbf, alphar.fortran_vec (), alphai.fortran_vec (), beta.fortran_vec (), vl, ldvl, vr, ldvr, work, lwork, info2)); if (f77_exception_encountered) error ("ss: zero: __sl_ab08nd__: exception in LAPACK subroutine DGGEV"); if (info2 != 0) error ("ss: zero: __sl_ab08nd__: DGGEV returned info = %d", info2); // calculate gain octave_value gain = Matrix (0, 0);; if (m == 1 && p == 1) { if (nu < n) gain = c * xpow (a, double (n-1-nu)) * b; else gain = d; } // assemble complex vector - adapted from DEFUN complex in data.cc ColumnVector zeror (nu); ColumnVector zeroi (nu); zeror = quotient (alphar, beta); zeroi = quotient (alphai, beta); ComplexColumnVector zero (nu, Complex ()); for (octave_idx_type i = 0; i < nu; i++) zero.xelem (i) = Complex (zeror(i), zeroi(i)); // prepare additional outputs for info struct RowVector infzr (dinfz); RowVector kronrr (nkror); RowVector kronlr (nkrol); for (octave_idx_type i = 0; i < dinfz; i++) infzr.xelem (i) = infz[i]; for (octave_idx_type i = 0; i < nkror; i++) kronrr.xelem (i) = kronr[i]; for (octave_idx_type i = 0; i < nkrol; i++) kronlr.xelem (i) = kronl[i]; // return values retval(0) = zero; retval(1) = gain; retval(2) = octave_value (rank); retval(3) = infzr; retval(4) = kronrr; retval(5) = kronlr; } return retval; } control/src/sl_ab09hd.cc0000644000076500000240000001666212273411257014422 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Model reduction based on balanced stochastic truncation method. Uses SLICOT AB09HD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: October 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (ab09hd, AB09HD) (char& DICO, char& JOB, char& EQUIL, char& ORDSEL, int& N, int& M, int& P, int& NR, double& ALPHA, double& BETA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, int& NS, double* HSV, double& TOL1, double& TOL2, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_ab09hd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab09hd__, args, nargout, "-*- texinfo -*-\n\ Slicot AB09HD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 13) { print_usage (); } else { // arguments in char dico; char job; char equil; char ordsel; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); const int idico = args(4).int_value (); const int iequil = args(5).int_value (); const int ijob = args(6).int_value (); int nr = args(7).int_value (); const int iordsel = args(8).int_value (); double alpha = args(9).double_value (); double beta = args(10).double_value (); double tol1 = args(11).double_value (); double tol2 = args(12).double_value (); switch (ijob) { case 0: job = 'B'; break; case 1: job = 'F'; break; case 2: job = 'S'; break; case 3: job = 'P'; break; default: error ("__sl_ab09hd__: argument job invalid"); } if (idico == 0) dico = 'C'; else dico = 'D'; if (iequil == 0) equil = 'S'; else equil = 'N'; if (iordsel == 0) ordsel = 'F'; else ordsel = 'A'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd = max (1, p); // arguments out int ns; ColumnVector hsv (n); // workspace int liwork = max (1, 2*n); int mb; if (beta == 0) mb = m; else mb = m + p; int ldwork = 2*n*n + mb*(n+p) + max (2, n*(max (n,mb,p)+5), 2*n*p + max (p*(mb+2), 10*n*(n+1))); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicators int iwarn = 0; int info = 0; // SLICOT routine AB09HD F77_XFCN (ab09hd, AB09HD, (dico, job, equil, ordsel, n, m, p, nr, alpha, beta, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ns, hsv.fortran_vec (), tol1, tol2, iwork, dwork, ldwork, bwork, iwarn, info)); if (f77_exception_encountered) error ("bstmodred: exception in SLICOT subroutine AB09HD"); static const char* err_msg[] = { "0: OK", "1: the computation of the ordered real Schur form of A " "failed", "2: the reduction of the Hamiltonian matrix to real " "Schur form failed", "3: the reordering of the real Schur form of the " "Hamiltonian matrix failed", "4: the Hamiltonian matrix has less than N stable " "eigenvalues", "5: the coefficient matrix U11 in the linear system " "X*U11 = U21 to determine X is singular to working " "precision", "6: BETA = 0 and D has not a maximal row rank", "7: the computation of Hankel singular values failed", "8: the separation of the ALPHA-stable/unstable diagonal " "blocks failed because of very close eigenvalues", "9: the resulting order of reduced stable part is less " "than the number of unstable zeros of the stable " "part"}; static const char* warn_msg[] = { "0: OK", "1: with ORDSEL = 'F', the selected order NR is greater " "than NSMIN, the sum of the order of the " "ALPHA-unstable part and the order of a minimal " "realization of the ALPHA-stable part of the given " "system; in this case, the resulting NR is set equal " "to NSMIN.", "2: with ORDSEL = 'F', the selected order NR corresponds " "to repeated singular values for the ALPHA-stable " "part, which are neither all included nor all " "excluded from the reduced model; in this case, the " "resulting NR is automatically decreased to exclude " "all repeated singular values.", "3: with ORDSEL = 'F', the selected order NR is less " "than the order of the ALPHA-unstable part of the " "given system; in this case NR is set equal to the " "order of the ALPHA-unstable part."}; error_msg ("bstmodred", info, 9, err_msg); warning_msg ("bstmodred", iwarn, 3, warn_msg); // resize a.resize (nr, nr); b.resize (nr, m); c.resize (p, nr); hsv.resize (ns); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; retval(4) = octave_value (nr); retval(5) = hsv; retval(6) = octave_value (ns); } return retval; } control/src/sl_ab09id.cc0000644000076500000240000003044012273411257014411 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Model reduction based on Balance & Truncate (B&T) or Singular Perturbation Approximation (SPA) method. Uses SLICOT AB09ID by courtesy of NICONET e.V. Author: Lukas Reichlin Created: October 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (ab09id, AB09ID) (char& DICO, char& JOBC, char& JOBO, char& JOB, char& WEIGHT, char& EQUIL, char& ORDSEL, int& N, int& M, int& P, int& NV, int& PV, int& NW, int& MW, int& NR, double& ALPHA, double& ALPHAC, double& ALPHAO, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AV, int& LDAV, double* BV, int& LDBV, double* CV, int& LDCV, double* DV, int& LDDV, double* AW, int& LDAW, double* BW, int& LDBW, double* CW, int& LDCW, double* DW, int& LDDW, int& NS, double* HSV, double& TOL1, double& TOL2, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_ab09id__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab09id__, args, nargout, "-*- texinfo -*-\n\ Slicot AB09ID Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 25) { print_usage (); } else { // arguments in char dico; char jobc; char jobo; char job; char weight; char equil; char ordsel; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); const int idico = args(4).int_value (); const int iequil = args(5).int_value (); int nr = args(6).int_value (); const int iordsel = args(7).int_value (); double alpha = args(8).double_value (); const int ijob = args(9).int_value (); Matrix av = args(10).matrix_value (); Matrix bv = args(11).matrix_value (); Matrix cv = args(12).matrix_value (); Matrix dv = args(13).matrix_value (); Matrix aw = args(14).matrix_value (); Matrix bw = args(15).matrix_value (); Matrix cw = args(16).matrix_value (); Matrix dw = args(17).matrix_value (); const int iweight = args(18).int_value (); const int ijobc = args(19).int_value (); double alphac = args(20).double_value (); const int ijobo = args(21).int_value (); double alphao = args(22).double_value (); double tol1 = args(23).double_value (); double tol2 = args(24).double_value (); if (idico == 0) dico = 'C'; else dico = 'D'; if (iequil == 0) equil = 'S'; else equil = 'N'; if (iordsel == 0) ordsel = 'F'; else ordsel = 'A'; if (ijobc == 0) jobc = 'S'; else jobc = 'E'; if (ijobo == 0) jobo = 'S'; else jobo = 'E'; switch (ijob) { case 0: job = 'B'; break; case 1: job = 'F'; break; case 2: job = 'S'; break; case 3: job = 'P'; break; default: error ("__sl_ab09id__: argument job invalid"); } switch (iweight) { case 0: weight = 'N'; break; case 1: weight = 'L'; break; case 2: weight = 'R'; break; case 3: weight = 'B'; break; default: error ("__sl_ab09id__: argument weight invalid"); } int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int nv = av.rows (); int pv = cv.rows (); int nw = aw.rows (); int mw = bw.columns (); int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd = max (1, p); int ldav = max (1, nv); int ldbv = max (1, nv); int ldcv = max (1, pv); int lddv = max (1, pv); int ldaw = max (1, nw); int ldbw = max (1, nw); int ldcw = max (1, m); int lddw = max (1, m); // arguments out int ns; ColumnVector hsv (n); // workspace int liwork; int liwrk1; int liwrk2; int liwrk3; switch (job) { case 'B': liwrk1 = 0; break; case 'F': liwrk1 = n; break; default: liwrk1 = 2*n; } if (nv == 0 || weight == 'R' || weight == 'N') liwrk2 = 0; else liwrk2 = nv + max (p, pv); if (nw == 0 || weight == 'L' || weight == 'N') liwrk3 = 0; else liwrk3 = nw + max (m, mw); liwork = max (3, liwrk1, liwrk2, liwrk3); int ldwork; int lminl; int lrcf; int lminr; int llcf; int lleft; int lright; if (nw == 0 || weight == 'L' || weight == 'N') { lrcf = 0; lminr = 0; } else { lrcf = mw*(nw+mw) + max (nw*(nw+5), mw*(mw+2), 4*mw, 4*m); if (m == mw) lminr = nw + max (nw, 3*m); else lminr = 2*nw*max (m, mw) + nw + max (nw, 3*m, 3*mw); } llcf = pv*(nv+pv) + pv*nv + max (nv*(nv+5), pv*(pv+2), 4*pv, 4*p); if (nv == 0 || weight == 'R' || weight == 'N') lminl = 0; else if (p == pv) lminl = max (llcf, nv + max (nv, 3*p)); else lminl = max (p, pv) * (2*nv + max (p, pv)) + max (llcf, nv + max (nv, 3*p, 3*pv)); if (pv == 0 || weight == 'R' || weight == 'N') lleft = n*(p+5); else lleft = (n+nv) * (n + nv + max (n+nv, pv) + 5); if (mw == 0 || weight == 'L' || weight == 'N') lright = n*(m+5); else lright = (n+nw) * (n + nw + max (n+nw, mw) + 5); ldwork = max (lminl, lminr, lrcf, 2*n*n + max (1, lleft, lright, 2*n*n+5*n, n*max (m, p))); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int iwarn = 0; int info = 0; // SLICOT routine AB09ID F77_XFCN (ab09id, AB09ID, (dico, jobc, jobo, job, weight, equil, ordsel, n, m, p, nv, pv, nw, mw, nr, alpha, alphac, alphao, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, av.fortran_vec (), ldav, bv.fortran_vec (), ldbv, cv.fortran_vec (), ldcv, dv.fortran_vec (), lddv, aw.fortran_vec (), ldaw, bw.fortran_vec (), ldbw, cw.fortran_vec (), ldcw, dw.fortran_vec (), lddw, ns, hsv.fortran_vec (), tol1, tol2, iwork, dwork, ldwork, iwarn, info)); if (f77_exception_encountered) error ("modred: exception in SLICOT subroutine AB09ID"); static const char* err_msg[] = { "0: OK", "1: the computation of the ordered real Schur form of A " "failed", "2: the separation of the ALPHA-stable/unstable " "diagonal blocks failed because of very close " "eigenvalues", "3: the reduction to a real Schur form of the state " "matrix of a minimal realization of V failed", "4: a failure was detected during the ordering of the " "real Schur form of the state matrix of a minimal " "realization of V or in the iterative process to " "compute a left coprime factorization with inner " "denominator", "5: if DICO = 'C' and the matrix AV has an observable " "eigenvalue on the imaginary axis, or DICO = 'D' and " "AV has an observable eigenvalue on the unit circle", "6: the reduction to a real Schur form of the state " "matrix of a minimal realization of W failed", "7: a failure was detected during the ordering of the " "real Schur form of the state matrix of a minimal " "realization of W or in the iterative process to " "compute a right coprime factorization with inner " "denominator", "8: if DICO = 'C' and the matrix AW has a controllable " "eigenvalue on the imaginary axis, or DICO = 'D' and " "AW has a controllable eigenvalue on the unit circle", "9: the computation of eigenvalues failed", "10: the computation of Hankel singular values failed"}; static const char* warn_msg[] = { "0: OK", "1: with ORDSEL = 'F', the selected order NR is greater " "than NSMIN, the sum of the order of the " "ALPHA-unstable part and the order of a minimal " "realization of the ALPHA-stable part of the given " "system; in this case, the resulting NR is set equal " "to NSMIN.", "2: with ORDSEL = 'F', the selected order NR corresponds " "to repeated singular values for the ALPHA-stable " "part, which are neither all included nor all " "excluded from the reduced model; in this case, the " "resulting NR is automatically decreased to exclude " "all repeated singular values.", "3: with ORDSEL = 'F', the selected order NR is less " "than the order of the ALPHA-unstable part of the " "given system; in this case NR is set equal to the " "order of the ALPHA-unstable part.", /* 10+%d: %d */ "violations of the numerical stability condition " "occured during the assignment of eigenvalues in the " "SLICOT Library routines SB08CD and/or SB08DD."}; error_msg ("modred", info, 10, err_msg); warning_msg ("modred", iwarn, 3, warn_msg, 10); // resize a.resize (nr, nr); b.resize (nr, m); c.resize (p, nr); hsv.resize (ns); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; retval(4) = octave_value (nr); retval(5) = hsv; retval(6) = octave_value (ns); } return retval; } control/src/sl_ab09jd.cc0000644000076500000240000003032612273411257014415 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Model reduction based on Hankel-norm approximation method. Uses SLICOT AB09JD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: July 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (ab09jd, AB09JD) (char& JOBV, char& JOBW, char& JOBINV, char& DICO, char& EQUIL, char& ORDSEL, int& N, int& NV, int& NW, int& M, int& P, int& NR, double& ALPHA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AV, int& LDAV, double* BV, int& LDBV, double* CV, int& LDCV, double* DV, int& LDDV, double* AW, int& LDAW, double* BW, int& LDBW, double* CW, int& LDCW, double* DW, int& LDDW, int& NS, double* HSV, double& TOL1, double& TOL2, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_ab09jd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab09jd__, args, nargout, "-*- texinfo -*-\n\ Slicot AB09JD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 22) { print_usage (); } else { // arguments in char jobv; char jobw; char jobinv; char dico; char equil; char ordsel; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); const int idico = args(4).int_value (); const int iequil = args(5).int_value (); int nr = args(6).int_value (); const int iordsel = args(7).int_value (); double alpha = args(8).double_value (); const int ijobv = args(9).int_value (); Matrix av = args(10).matrix_value (); Matrix bv = args(11).matrix_value (); Matrix cv = args(12).matrix_value (); Matrix dv = args(13).matrix_value (); const int ijobw = args(14).int_value (); Matrix aw = args(15).matrix_value (); Matrix bw = args(16).matrix_value (); Matrix cw = args(17).matrix_value (); Matrix dw = args(18).matrix_value (); const int ijobinv = args(19).int_value (); double tol1 = args(20).double_value (); double tol2 = args(21).double_value (); switch (ijobv) { case 0: jobv = 'N'; break; case 1: jobv = 'V'; break; case 2: jobv = 'I'; break; case 3: jobv = 'C'; break; case 4: jobv = 'R'; break; default: error ("__sl_ab09jd__: argument jobv invalid"); } switch (ijobw) { case 0: jobw = 'N'; break; case 1: jobw = 'W'; break; case 2: jobw = 'I'; break; case 3: jobw = 'C'; break; case 4: jobw = 'R'; break; default: error ("__sl_ab09jd__: argument jobw invalid"); } switch (ijobinv) { case 0: jobinv = 'N'; break; case 1: jobinv = 'I'; break; case 2: jobinv = 'A'; break; default: error ("__sl_ab09jd__: argument jobinv invalid"); } if (idico == 0) dico = 'C'; else dico = 'D'; if (iequil == 0) equil = 'S'; else equil = 'N'; if (iordsel == 0) ordsel = 'F'; else ordsel = 'A'; int n = a.rows (); // n: number of states int nv = av.rows (); int nw = aw.rows (); int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd = max (1, p); int ldav = max (1, nv); int ldbv = max (1, nv); int ldcv = max (1, p); int lddv = max (1, p); int ldaw = max (1, nw); int ldbw = max (1, nw); int ldcw = max (1, m); int lddw = max (1, m); // arguments out int ns; ColumnVector hsv (n); // workspace int liwork; int tmpc; int tmpd; if (jobv == 'N') tmpc = 0; else tmpc = max (2*p, nv+p+n+6, 2*nv+p+2); if (jobw == 'N') tmpd = 0; else tmpd = max (2*m, nw+m+n+6, 2*nw+m+2); if (dico == 'C') liwork = max (1, m, tmpc, tmpd); else liwork = max (1, n, m, tmpc, tmpd); int ldwork; int nvp = nv + p; int nwm = nw + m; int ldw1; int ldw2; int ldw3 = n*(2*n + max (n, m, p) + 5) + n*(n+1)/2; int ldw4 = n*(m+p+2) + 2*m*p + min (n, m) + max (3*m+1, min (n, m) + p); if (jobv == 'N') { ldw1 = 0; } else { ldw1 = 2*nvp*(nvp+p) + p*p + max (2*nvp*nvp + max (11*nvp+16, p*nvp), nvp*n + max (nvp*n+n*n, p*n, p*m)); } if (jobw == 'N') { ldw2 = 0; } else { ldw2 = 2*nwm*(nwm+m) + m*m + max (2*nwm*nwm + max (11*nwm+16, m*nwm), nwm*n + max (nwm*n+n*n, m*n, p*m)); } ldwork = max (ldw1, ldw2, ldw3, ldw4); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int iwarn = 0; int info = 0; // SLICOT routine AB09JD F77_XFCN (ab09jd, AB09JD, (jobv, jobw, jobinv, dico, equil, ordsel, n, nv, nw, m, p, nr, alpha, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, av.fortran_vec (), ldav, bv.fortran_vec (), ldbv, cv.fortran_vec (), ldcv, dv.fortran_vec (), lddv, aw.fortran_vec (), ldaw, bw.fortran_vec (), ldbw, cw.fortran_vec (), ldcw, dw.fortran_vec (), lddw, ns, hsv.fortran_vec (), tol1, tol2, iwork, dwork, ldwork, iwarn, info)); if (f77_exception_encountered) error ("hnamodred: exception in SLICOT subroutine AB09JD"); static const char* err_msg[] = { "0: OK", "1: the computation of the ordered real Schur form of A " "failed", "2: the separation of the ALPHA-stable/unstable " "diagonal blocks failed because of very close eigenvalues", "3: the reduction of AV to a real Schur form failed", "4: the reduction of AW to a real Schur form failed", "5: the reduction to generalized Schur form of the " "descriptor pair corresponding to the inverse of V " "failed", "6: the reduction to generalized Schur form of the " "descriptor pair corresponding to the inverse of W " "failed", "7: the computation of Hankel singular values failed", "8: the computation of stable projection in the " "Hankel-norm approximation algorithm failed", "9: the order of computed stable projection in the " "Hankel-norm approximation algorithm differs " "from the order of Hankel-norm approximation", "10: the reduction of AV-BV*inv(DV)*CV to a " "real Schur form failed", "11: the reduction of AW-BW*inv(DW)*CW to a " "real Schur form failed", "12: the solution of the Sylvester equation failed " "because the poles of V (if JOBV = 'V') or of " "conj(V) (if JOBV = 'C') are not distinct from " "the poles of G1 (see METHOD)", "13: the solution of the Sylvester equation failed " "because the poles of W (if JOBW = 'W') or of " "conj(W) (if JOBW = 'C') are not distinct from " "the poles of G1 (see METHOD)", "14: the solution of the Sylvester equation failed " "because the zeros of V (if JOBV = 'I') or of " "conj(V) (if JOBV = 'R') are not distinct from " "the poles of G1sr (see METHOD)", "15: the solution of the Sylvester equation failed " "because the zeros of W (if JOBW = 'I') or of " "conj(W) (if JOBW = 'R') are not distinct from " "the poles of G1sr (see METHOD)", "16: the solution of the generalized Sylvester system " "failed because the zeros of V (if JOBV = 'I') or " "of conj(V) (if JOBV = 'R') are not distinct from " "the poles of G1sr (see METHOD)", "17: the solution of the generalized Sylvester system " "failed because the zeros of W (if JOBW = 'I') or " "of conj(W) (if JOBW = 'R') are not distinct from " "the poles of G1sr (see METHOD)", "18: op(V) is not antistable", "19: op(W) is not antistable", "20: V is not invertible", "21: W is not invertible"}; static const char* warn_msg[] = { "0: OK", "1: with ORDSEL = 'F', the selected order NR is greater " "than NSMIN, the sum of the order of the " "ALPHA-unstable part and the order of a minimal " "realization of the ALPHA-stable part of the given " "system. In this case, the resulting NR is set equal " "to NSMIN.", "2: with ORDSEL = 'F', the selected order NR is less " "than the order of the ALPHA-unstable part of the " "given system. In this case NR is set equal to the " "order of the ALPHA-unstable part."}; error_msg ("hnamodred", info, 21, err_msg); warning_msg ("hnamodred", iwarn, 2, warn_msg); // resize a.resize (nr, nr); b.resize (nr, m); c.resize (p, nr); hsv.resize (ns); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; retval(4) = octave_value (nr); retval(5) = hsv; retval(6) = octave_value (ns); } return retval; } control/src/sl_ab13ad.cc0000644000076500000240000000713112273411257014375 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Hankel singular values. Uses SLICOT AB13AD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: January 2010 Version: 0.3 */ #include #include #include "common.h" extern "C" { int F77_FUNC (ab13ad, AB13AD) (char& DICO, char& EQUIL, int& N, int& M, int& P, double& ALPHA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, int& NS, double* HSV, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_ab13ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab13ad__, args, nargout, "-*- texinfo -*-\n\ Slicot AB13AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 6) { print_usage (); } else { // arguments in char dico; char equil; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); int discrete = args(3).int_value (); double alpha = args(4).double_value (); const int scaled = args(5).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; if (scaled == 0) equil = 'S'; else equil = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); // arguments out int ns = 0; ColumnVector hsv (n); // workspace int ldwork = max (1, n*(max (n, m, p) + 5) + n*(n+1)/2); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info = 0; // SLICOT routine AB13AD F77_XFCN (ab13ad, AB13AD, (dico, equil, n, m, p, alpha, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, ns, hsv.fortran_vec (), dwork, ldwork, info)); if (f77_exception_encountered) error ("hsvd: __sl_ab13ad__: exception in SLICOT subroutine AB13AD"); if (info != 0) error ("hsvd: __sl_ab13ad__: AB13AD returned info = %d", info); // resize hsv.resize (ns); // return values retval(0) = hsv; retval(1) = octave_value (ns); } return retval; } control/src/sl_ab13bd.cc0000644000076500000240000000747612273411257014412 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . H-2 norm of a SS model. Uses SLICOT AB13BD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2009 Version: 0.4 */ #include #include #include "common.h" extern "C" { double F77_FUNC (ab13bd, AB13BD) (char& DICO, char& JOBN, int& N, int& M, int& P, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, int& NQ, double& TOL, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_ab13bd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab13bd__, args, nargout, "-*- texinfo -*-\n\ Slicot AB13BD Release 5.\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in char dico; char jobn = 'H'; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); int discrete = args(4).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); int ldd = max (1, d.rows ()); // arguments out double norm; int nq; // tolerance double tol = 0; // workspace int ldwork = max (1, m*(n+m) + max (n*(n+5), m*(m+2), 4*p ), n*(max (n, p) + 4 ) + min (n, p)); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int iwarn; int info; // SLICOT routine AB13BD norm = F77_FUNC (ab13bd, AB13BD) (dico, jobn, n, m, p, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, nq, tol, dwork, ldwork, iwarn, info); if (f77_exception_encountered) error ("lti: norm: __sl_ab13bd__: exception in SLICOT subroutine AB13BD"); if (info != 0) error ("lti: norm: __sl_ab13bd__: AB13BD returned info = %d", info); if (iwarn != 0) warning ("lti: norm: __sl_ab13bd__: AB13BD returned iwarn = %d", iwarn); // return value retval(0) = octave_value (norm); } return retval; } control/src/sl_ab13dd.cc0000644000076500000240000001151312273411257014377 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . L-infinity norm of a SS model. Uses SLICOT AB13DD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2009 Version: 0.4 */ #include #include #include #include "common.h" extern "C" { int F77_FUNC (ab13dd, AB13DD) (char& DICO, char& JOBE, char& EQUIL, char& JOBD, int& N, int& M, int& P, double* FPEAK, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* GPEAK, double& TOL, int* IWORK, double* DWORK, int& LDWORK, Complex* CWORK, int& LCWORK, int& INFO); } // PKG_ADD: autoload ("__sl_ab13dd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ab13dd__, args, nargout, "-*- texinfo -*-\n\ Slicot AB13DD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 9) { print_usage (); } else { // arguments in char dico; char jobe; char equil; char jobd = 'D'; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); Matrix d = args(4).matrix_value (); int discrete = args(5).int_value (); int descriptor = args(6).int_value (); double tol = args(7).double_value (); const int scaled = args(8).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; if (descriptor == 0) jobe = 'I'; else jobe = 'G'; if (scaled == 0) equil = 'S'; else equil = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int lde = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd = max (1, p); ColumnVector fpeak (2); ColumnVector gpeak (2); fpeak(0) = 0; fpeak(1) = 1; // workspace int ldwork = max (1, 15*n*n + p*p + m*m + (6*n+3)*(p+m) + 4*p*m + n*m + 22*n + 7*min(p,m)); int lcwork = max (1, (n+m)*(n+p) + 2*min(p,m) + max(p,m)); OCTAVE_LOCAL_BUFFER (int, iwork, n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (Complex, cwork, lcwork); // error indicator int info; // SLICOT routine AB13DD F77_XFCN (ab13dd, AB13DD, (dico, jobe, equil, jobd, n, m, p, fpeak.fortran_vec (), a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, gpeak.fortran_vec (), tol, iwork, dwork, ldwork, cwork, lcwork, info)); if (f77_exception_encountered) error ("lti: norm: __sl_ab13dd__: exception in SLICOT subroutine AB13DD"); static const char* err_msg[] = { "0: OK", "1: the matrix E is (numerically) singular", "2: the (periodic) QR (or QZ) algorithm for computing " "eigenvalues did not converge", "3: the SVD algorithm for computing singular values did " "not converge", "4: the tolerance is too small and the algorithm did " "not converge"}; error_msg ("__sl_ab13dd__", info, 4, err_msg); // return values retval(0) = fpeak; retval(1) = gpeak; } return retval; } control/src/sl_ag08bd.cc0000644000076500000240000001711012273411257014405 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Transmission zeros of descriptor state-space models. Uses SLICOT AG08BD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2010 Version: 0.4 */ #include #include #include "common.h" #include #include extern "C" { int F77_FUNC (ag08bd, AG08BD) (char& EQUIL, int& L, int& N, int& M, int& P, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, int& NFZ, int& NRANK, int& NIZ, int& DINFZ, int& NKROR, int& NINFE, int& NKROL, int* INFZ, int* KRONR, int* INFE, int* KRONL, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); int F77_FUNC (dggev, DGGEV) (char& JOBVL, char& JOBVR, int& N, double* A, int& LDA, double* B, int& LDB, double* ALPHAR, double* ALPHAI, double* BETA, double* VL, int& LDVL, double* VR, int& LDVR, double* WORK, int& LWORK, int& INFO); } // PKG_ADD: autoload ("__sl_ag08bd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ag08bd__, args, nargout, "-*- texinfo -*-\n\ Slicot AG08BD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 6) { print_usage (); } else { // arguments in char equil; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); Matrix d = args(4).matrix_value (); const int scaled = args(5).int_value (); if (scaled == 0) equil = 'S'; else equil = 'N'; int l = a.rows (); // l: number of states int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, l); int lde = max (1, l); int ldb = max (1, l); if (m == 0) ldb = 1; int ldc = max (1, p); int ldd = max (1, p); // arguments out int nfz; int nrank; int niz; int dinfz; int nkror; int ninfe; int nkrol; OCTAVE_LOCAL_BUFFER (int, infz, n+1); OCTAVE_LOCAL_BUFFER (int, kronr, n+m+1); OCTAVE_LOCAL_BUFFER (int, infe, 1 + min (l+p, n+m)); OCTAVE_LOCAL_BUFFER (int, kronl, l+p+1); // workspace int ldwork = max (l+p, m+n) * (m+n) + max (1, 5 * max (l+p, m+n)); OCTAVE_LOCAL_BUFFER (int, iwork, n + max (1, m)); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // tolerance double tol = 0; // AG08BD uses DLAMCH for default tolerance // SLICOT routine AG08BD F77_XFCN (ag08bd, AG08BD, (equil, l, n, m, p, a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, nfz, nrank, niz, dinfz, nkror, ninfe, nkrol, infz, kronr, infe, kronl, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("dss: zero: __sl_ag08bd__: exception in SLICOT subroutine AG08BD"); if (info != 0) error ("dss: zero: __sl_ag08bd__: AG08BD returned info = %d", info); // DGGEV Part a.resize (nfz, nfz); // Af e.resize (nfz, nfz); // Ef lda = max (1, nfz); lde = max (1, nfz); char jobvl = 'N'; char jobvr = 'N'; ColumnVector alphar (nfz); ColumnVector alphai (nfz); ColumnVector beta (nfz); double* vl = 0; // not referenced because jobvl = 'N' int ldvl = 1; double* vr = 0; // not referenced because jobvr = 'N' int ldvr = 1; int lwork = max (1, 8*nfz); OCTAVE_LOCAL_BUFFER (double, work, lwork); int info2; F77_XFCN (dggev, DGGEV, (jobvl, jobvr, nfz, a.fortran_vec (), lda, e.fortran_vec (), lde, alphar.fortran_vec (), alphai.fortran_vec (), beta.fortran_vec (), vl, ldvl, vr, ldvr, work, lwork, info2)); if (f77_exception_encountered) error ("dss: zero: __sl_ag08bd__: exception in LAPACK subroutine DGGEV"); if (info2 != 0) error ("dss: zero: __sl_ag08bd__: DGGEV returned info = %d", info2); // assemble complex vector - adapted from DEFUN complex in data.cc // LAPACK DGGEV.f says: // // Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) // may easily over- or underflow, and BETA(j) may even be zero. // Thus, the user should avoid naively computing the ratio // alpha/beta. However, ALPHAR and ALPHAI will be always less // than and usually comparable with norm(A) in magnitude, and // BETA always less than and usually comparable with norm(B). // // Since we need the zeros explicitly ... ColumnVector zeror (nfz); ColumnVector zeroi (nfz); zeror = quotient (alphar, beta); zeroi = quotient (alphai, beta); ComplexColumnVector zero (nfz, Complex ()); for (octave_idx_type i = 0; i < nfz; i++) zero.xelem (i) = Complex (zeror(i), zeroi(i)); // prepare additional outputs for info struct RowVector infzr (dinfz); RowVector kronrr (nkror); RowVector kronlr (nkrol); for (octave_idx_type i = 0; i < dinfz; i++) infzr.xelem (i) = infz[i]; for (octave_idx_type i = 0; i < nkror; i++) kronrr.xelem (i) = kronr[i]; for (octave_idx_type i = 0; i < nkrol; i++) kronlr.xelem (i) = kronl[i]; // return values retval(0) = zero; retval(1) = octave_value (nrank); retval(2) = infzr; retval(3) = kronrr; retval(4) = kronlr; } return retval; } control/src/sl_are.cc0000644000076500000240000002102612273411257014110 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Solve algebraic Riccati equation. Uses SLICOT SB02RD and SB02MT by courtesy of NICONET e.V. Author: Lukas Reichlin Created: December 2012 Version: 0.1 */ #include #include #include "common.h" #include extern "C" { int F77_FUNC (sb02mt, SB02MT) (char& JOBG, char& JOBL, char& FACT, char& UPLO, int& N, int& M, double* A, int& LDA, double* B, int& LDB, double* Q, int& LDQ, double* R, int& LDR, double* L, int& LDL, int* IPIV, int& OUFACT, double* G, int& LDG, int* IWORK, double* DWORK, int& LDWORK, int& INFO); int F77_FUNC (sb02rd, SB02RD) (char& JOB, char& DICO, char& HINV, char& TRANA, char& UPLO, char& SCAL, char& SORT, char& FACT, char& LYAPUN, int& N, double* A, int& LDA, double* T, int& LDT, double* V, int& LDV, double* G, int& LDG, double* Q, int& LDQ, double* X, int& LDX, double& SEP, double& RCOND, double& FERR, double* WR, double* WI, double* S, int& LDS, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_are__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_are__, args, nargout, "-*- texinfo -*-\n\ Slicot SB02RD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 7) { print_usage (); } else { // SB02MT // arguments in char dico; char jobg = 'G'; char jobl; char fact = 'N'; char uplo = 'U'; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix q = args(2).matrix_value (); Matrix r = args(3).matrix_value (); Matrix l = args(4).matrix_value (); int discrete = args(5).int_value (); int ijobl = args(6).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; if (ijobl == 0) jobl = 'Z'; else jobl = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int lda = max (1, n); int ldb = max (1, n); int ldq = max (1, n); int ldr = max (1, m); int ldl = max (1, n); // arguments out int ldg = max (1, n); Matrix g (ldg, n); // unused output arguments OCTAVE_LOCAL_BUFFER (int, ipiv, m); int oufact; // workspace OCTAVE_LOCAL_BUFFER (int, iwork_a, m); int ldwork_a = max (2, 3*m, n*m); OCTAVE_LOCAL_BUFFER (double, dwork_a, ldwork_a); // error indicator int info; // SLICOT routine SB02MT F77_XFCN (sb02mt, SB02MT, (jobg, jobl, fact, uplo, n, m, a.fortran_vec (), lda, b.fortran_vec (), ldb, q.fortran_vec (), ldq, r.fortran_vec (), ldr, l.fortran_vec (), ldl, ipiv, oufact, g.fortran_vec (), ldg, iwork_a, dwork_a, ldwork_a, info)); if (f77_exception_encountered) error ("are: __sl_are__: exception in SLICOT subroutine SB02MT"); if (info != 0) { if (info < 0) error ("are: sb02mt: the %d-th argument had an illegal value", info); else if (info == m+1) error ("are: sb02mt: the matrix R is numerically singular"); else error ("are: sb02mt: the %d-th element (1 <= %d <= %d) of the d factor is " "exactly zero; the UdU' (or LdL') factorization has " "been completed, but the block diagonal matrix d is " "exactly singular", info, info, m); } // SB02RD // arguments in char job = 'A'; char hinv = 'D'; char trana = 'N'; char scal = 'G'; char sort = 'S'; char lyapun = 'O'; int ldt = max (1, n); int ldv = max (1, n); int ldx = max (1, n); int lds = max (1, 2*n); // arguments out Matrix x (ldx, n); double sep; double rcond; double ferr; ColumnVector wr (2*n); ColumnVector wi (2*n); // unused output arguments Matrix t (ldt, n); Matrix v (ldv, n); Matrix s (lds, 2*n); // workspace int liwork_b = max (2*n, n*n); OCTAVE_LOCAL_BUFFER (int, iwork_b, liwork_b); int ldwork_b = 5 + max (1, 4*n*n + 8*n); OCTAVE_LOCAL_BUFFER (double, dwork_b, ldwork_b); OCTAVE_LOCAL_BUFFER (bool, bwork_b, 2*n); // SLICOT routine SB02RD F77_XFCN (sb02rd, SB02RD, (job, dico, hinv, trana, uplo, scal, sort, fact, lyapun, n, a.fortran_vec (), lda, t.fortran_vec (), ldt, v.fortran_vec (), ldv, g.fortran_vec (), ldg, q.fortran_vec (), ldq, x.fortran_vec (), ldx, sep, rcond, ferr, wr.fortran_vec (), wi.fortran_vec (), s.fortran_vec (), lds, iwork_b, dwork_b, ldwork_b, bwork_b, info)); static const char* err_msg[] = { "0: OK", "1: matrix A is (numerically) singular in discrete-" "time case", "2: the Hamiltonian or symplectic matrix H cannot be " "reduced to real Schur form", "3: the real Schur form of the Hamiltonian or " "symplectic matrix H cannot be appropriately ordered", "4: the Hamiltonian or symplectic matrix H has less " "than N stable eigenvalues", "5: if the N-th order system of linear algebraic " "equations, from which the solution matrix X would " "be obtained, is singular to working precision", "6: the QR algorithm failed to complete the reduction " "of the matrix Ac to Schur canonical form, T", "7: if T and -T' have some almost equal eigenvalues, if " "DICO = 'C', or T has almost reciprocal eigenvalues, " "if DICO = 'D'; perturbed values were used to solve " "Lyapunov equations, but the matrix T, if given (for " "FACT = 'F'), is unchanged. (This is a warning " "indicator.)"}; error_msg ("are", info, 7, err_msg); // resize x.resize (n, n); wr.resize (n); wi.resize (n); // assemble complex vector - adapted from DEFUN complex in data.cc ComplexColumnVector pole (n, Complex ()); for (octave_idx_type i = 0; i < n; i++) pole.xelem (i) = Complex (wr(i), wi(i)); // return value retval(0) = x; retval(1) = pole; retval(2) = octave_value (ferr); retval(3) = octave_value (rcond); retval(4) = octave_value (sep); } return retval; } control/src/sl_ib01ad.cc0000644000076500000240000003163612273411257014411 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . SLICOT system identification Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: March 2012 Version: 0.1 */ #include #include #include #include "common.h" extern "C" { int F77_FUNC (ib01ad, IB01AD) (char& METH, char& ALG, char& JOBD, char& BATCH, char& CONCT, char& CTRL, int& NOBR, int& M, int& L, int& NSMP, double* U, int& LDU, double* Y, int& LDY, int& N, double* R, int& LDR, double* SV, double& RCOND, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_ib01ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ib01ad__, args, nargout, "-*- texinfo -*-\n\ Slicot IB01AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 10) { print_usage (); } else { //////////////////////////////////////////////////////////////////////////////////// // SLICOT IB01AD - preprocess the input-output data // //////////////////////////////////////////////////////////////////////////////////// // arguments in char meth_a; char meth_b; char alg; char jobd; char batch; char conct; char ctrl = 'N'; const Cell y_cell = args(0).cell_value (); const Cell u_cell = args(1).cell_value (); int nobr = args(2).int_value (); int nuser = args(3).int_value (); const int imeth = args(4).int_value (); const int ialg = args(5).int_value (); const int iconct = args(6).int_value (); const int ictrl = args(7).int_value (); // ignored double rcond = args(8).double_value (); double tol_a = args(9).double_value (); double tol_b = rcond; double tol_c = rcond; switch (imeth) { case 0: meth_a = 'M'; meth_b = 'M'; break; case 1: meth_a = 'N'; meth_b = 'N'; break; case 2: meth_a = 'N'; // no typo here meth_b = 'C'; break; default: error ("__sl_ib01ad__: argument 'meth' invalid"); } switch (ialg) { case 0: alg = 'C'; break; case 1: alg = 'F'; break; case 2: alg = 'Q'; break; default: error ("__sl_ib01ad__: argument 'alg' invalid"); } if (meth_a == 'M') jobd = 'M'; else // meth_a == 'N' jobd = 'N'; // IB01AD.f says: This parameter is not relevant for METH = 'N' if (iconct == 0) conct = 'C'; else conct = 'N'; /* if (ictrl == 0) ctrl = 'C'; else ctrl = 'N'; */ // m and l are equal for all experiments, checked by iddata class int n_exp = y_cell.nelem (); // number of experiments int m = u_cell.elem(0).columns (); // m: number of inputs int l = y_cell.elem(0).columns (); // l: number of outputs int nsmpl = 0; // total number of samples // arguments out int n; int ldr; if (meth_a == 'M' && jobd == 'M') ldr = max (2*(m+l)*nobr, 3*m*nobr); else if (meth_a == 'N' || (meth_a == 'M' && jobd == 'N')) ldr = 2*(m+l)*nobr; else error ("__sl_ib01ad__: could not handle 'ldr' case"); Matrix r (ldr, 2*(m+l)*nobr); ColumnVector sv (l*nobr); // repeat for every experiment in the dataset for (int i = 0; i < n_exp; i++) { if (n_exp == 1) batch = 'O'; // one block only else if (i == 0) batch = 'F'; // first block else if (i == n_exp-1) batch = 'L'; // last block else batch = 'I'; // intermediate block Matrix y = y_cell.elem(i).matrix_value (); Matrix u = u_cell.elem(i).matrix_value (); // y.rows == u.rows is checked by iddata class // int m = u.columns (); // m: number of inputs // int l = y.columns (); // l: number of outputs int nsmp = y.rows (); // nsmp: number of samples in the current experiment nsmpl += nsmp; // nsmpl: total number of samples of all experiments // minimal nsmp size checked by __slicot_identification__.m if (batch == 'O') { if (nsmp < 2*(m+l+1)*nobr - 1) error ("__sl_ident__: require NSMP >= 2*(M+L+1)*NOBR - 1"); } else { if (nsmp < 2*nobr) error ("__sl_ident__: require NSMP >= 2*NOBR"); } int ldu; if (m == 0) ldu = 1; else // m > 0 ldu = nsmp; int ldy = nsmp; // workspace int liwork_a; if (meth_a == 'N') // if METH = 'N' liwork_a = (m+l)*nobr; else if (alg == 'F') // if METH = 'M' and ALG = 'F' liwork_a = m+l; else // if METH = 'M' and ALG = 'C' or 'Q' liwork_a = 0; // TODO: Handle 'k' for DWORK int ldwork_a; int ns = nsmp - 2*nobr + 1; if (alg == 'C') { if (batch == 'F' || batch == 'I') { if (conct == 'C') ldwork_a = (4*nobr-2)*(m+l); else // (conct == 'N') ldwork_a = 1; } else if (meth_a == 'M') // && (batch == 'L' || batch == 'O') { if (conct == 'C' && batch == 'L') ldwork_a = max ((4*nobr-2)*(m+l), 5*l*nobr); else if (jobd == 'M') ldwork_a = max ((2*m-1)*nobr, (m+l)*nobr, 5*l*nobr); else // (jobd == 'N') ldwork_a = 5*l*nobr; } else // meth_b == 'N' && (batch == 'L' || batch == 'O') { ldwork_a = 5*(m+l)*nobr + 1; } } else if (alg == 'F') { if (batch != 'O' && conct == 'C') ldwork_a = (m+l)*2*nobr*(m+l+3); else if (batch == 'F' || batch == 'I') // && conct == 'N' ldwork_a = (m+l)*2*nobr*(m+l+1); else // (batch == 'L' || '0' && conct == 'N') ldwork_a = (m+l)*4*nobr*(m+l+1)+(m+l)*2*nobr; } else // (alg == 'Q') { // int ns = nsmp - 2*nobr + 1; if (ldr >= ns && batch == 'F') { ldwork_a = 4*(m+l)*nobr; } else if (ldr >= ns && batch == 'O') { if (meth_a == 'M') ldwork_a = max (4*(m+l)*nobr, 5*l*nobr); else // (meth == 'N') ldwork_a = 5*(m+l)*nobr + 1; } else if (conct == 'C' && (batch == 'I' || batch == 'L')) { ldwork_a = 4*(nobr+1)*(m+l)*nobr; } else // if ALG = 'Q', (BATCH = 'F' or 'O', and LDR < NS), or (BATCH = 'I' or 'L' and CONCT = 'N') { ldwork_a = 6*(m+l)*nobr; } } /* IB01AD.f Lines 438-445 C FURTHER COMMENTS C C For ALG = 'Q', BATCH = 'O' and LDR < NS, or BATCH <> 'O', the C calculations could be rather inefficient if only minimal workspace C (see argument LDWORK) is provided. It is advisable to provide as C much workspace as possible. Almost optimal efficiency can be C obtained for LDWORK = (NS+2)*(2*(M+L)*NOBR), assuming that the C cache size is large enough to accommodate R, U, Y, and DWORK. */ ldwork_a = max (ldwork_a, (ns+2)*(2*(m+l)*nobr)); /* IB01AD.f Lines 291-195: c the workspace used for alg = 'q' is c ldrwrk*2*(m+l)*nobr + 4*(m+l)*nobr, c where ldrwrk = ldwork/(2*(m+l)*nobr) - 2; recommended c value ldrwrk = ns, assuming a large enough cache size. c for good performance, ldwork should be larger. somehow ldrwrk and ldwork must have been mixed up here */ OCTAVE_LOCAL_BUFFER (int, iwork_a, liwork_a); OCTAVE_LOCAL_BUFFER (double, dwork_a, ldwork_a); // error indicators int iwarn_a = 0; int info_a = 0; // SLICOT routine IB01AD F77_XFCN (ib01ad, IB01AD, (meth_a, alg, jobd, batch, conct, ctrl, nobr, m, l, nsmp, u.fortran_vec (), ldu, y.fortran_vec (), ldy, n, r.fortran_vec (), ldr, sv.fortran_vec (), rcond, tol_a, iwork_a, dwork_a, ldwork_a, iwarn_a, info_a)); if (f77_exception_encountered) error ("ident: exception in SLICOT subroutine IB01AD"); static const char* err_msg[] = { "0: OK", "1: a fast algorithm was requested (ALG = 'C', or 'F') " "in sequential data processing, but it failed; the " "routine can be repeatedly called again using the " "standard QR algorithm", "2: the singular value decomposition (SVD) algorithm did " "not converge"}; static const char* warn_msg[] = { "0: OK", "1: the number of 100 cycles in sequential data " "processing has been exhausted without signaling " "that the last block of data was get; the cycle " "counter was reinitialized", "2: a fast algorithm was requested (ALG = 'C' or 'F'), " "but it failed, and the QR algorithm was then used " "(non-sequential data processing)", "3: all singular values were exactly zero, hence N = 0 " "(both input and output were identically zero)", "4: the least squares problems with coefficient matrix " "U_f, used for computing the weighted oblique " "projection (for METH = 'N'), have a rank-deficient " "coefficient matrix", "5: the least squares problem with coefficient matrix " "r_1 [6], used for computing the weighted oblique " "projection (for METH = 'N'), has a rank-deficient " "coefficient matrix"}; error_msg ("ident: IB01AD", info_a, 2, err_msg); warning_msg ("ident: IB01AD", iwarn_a, 5, warn_msg); } // resize int rs = 2*(m+l)*nobr; r.resize (rs, rs); // return values retval(0) = sv; retval(1) = octave_value (n); } return retval; } control/src/sl_ib01cd.cc0000644000076500000240000001457012273411257014411 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Compute initial state vector x0 Uses IB01CD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: May 2012 Version: 0.1 */ #include #include #include #include "common.h" extern "C" { int F77_FUNC (ib01cd, IB01CD) (char& JOBX0, char& COMUSE, char& JOB, int& N, int& M, int& L, int& NSMP, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* U, int& LDU, double* Y, int& LDY, double* X0, double* V, int& LDV, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_ib01cd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ib01cd__, args, nargout, "-*- texinfo -*-\n\ Slicot IB01CD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 7) { print_usage (); } else { // arguments in char jobx0 = 'X'; char comuse = 'U'; char jobbd = 'D'; const Cell y_cell = args(0).cell_value (); const Cell u_cell = args(1).cell_value (); Matrix a = args(2).matrix_value (); Matrix b = args(3).matrix_value (); Matrix c = args(4).matrix_value (); Matrix d = args(5).matrix_value (); double rcond = args(6).double_value (); double tol_c = rcond; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int l = c.rows (); // l: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, l); int ldd = max (1, l); // m and l are equal for all experiments, checked by iddata class int n_exp = y_cell.nelem (); // number of experiments // arguments out Cell x0_cell (n_exp, 1); // cell of initial state vectors x0 // repeat for every experiment in the dataset // compute individual initial state vector x0 for every experiment for (int i = 0; i < n_exp; i++) { Matrix y = y_cell.elem(i).matrix_value (); Matrix u = u_cell.elem(i).matrix_value (); int nsmp = y.rows (); // nsmp: number of samples int ldv = max (1, n); int ldu; if (m == 0) ldu = 1; else // m > 0 ldu = nsmp; int ldy = nsmp; // arguments out ColumnVector x0 (n); Matrix v (ldv, n); // workspace int liwork_c = n; // if JOBX0 = 'X' and COMUSE <> 'C' int ldwork_c; int t = nsmp; int ldw1_c = 2; int ldw2_c = t*l*(n + 1) + 2*n + max (2*n*n, 4*n); int ldw3_c = n*(n + 1) + 2*n + max (n*l*(n + 1) + 2*n*n + l*n, 4*n); ldwork_c = ldw1_c + n*( n + m + l ) + max (5*n, ldw1_c, min (ldw2_c, ldw3_c)); OCTAVE_LOCAL_BUFFER (int, iwork_c, liwork_c); OCTAVE_LOCAL_BUFFER (double, dwork_c, ldwork_c); // error indicators int iwarn_c = 0; int info_c = 0; // SLICOT routine IB01CD F77_XFCN (ib01cd, IB01CD, (jobx0, comuse, jobbd, n, m, l, nsmp, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, u.fortran_vec (), ldu, y.fortran_vec (), ldy, x0.fortran_vec (), v.fortran_vec (), ldv, tol_c, iwork_c, dwork_c, ldwork_c, iwarn_c, info_c)); if (f77_exception_encountered) error ("__sl_ib01cd__: exception in SLICOT subroutine IB01CD"); static const char* err_msg_c[] = { "0: OK", "1: the QR algorithm failed to compute all the " "eigenvalues of the matrix A (see LAPACK Library " "routine DGEES); the locations DWORK(i), for " "i = g+1:g+N*N, contain the partially converged " "Schur form", "2: the singular value decomposition (SVD) algorithm did " "not converge"}; static const char* warn_msg_c[] = { "0: OK", "1: warning message not specified", "2: warning message not specified", "3: warning message not specified", "4: the least squares problem to be solved has a " "rank-deficient coefficient matrix", "5: warning message not specified", "6: the matrix A is unstable; the estimated x(0) " "and/or B and D could be inaccurate"}; error_msg ("__sl_ib01cd__", info_c, 2, err_msg_c); warning_msg ("__sl_ib01cd__", iwarn_c, 6, warn_msg_c); x0_cell.elem(i) = x0; // add x0 from the current experiment to cell of initial state vectors } // return values retval(0) = x0_cell; } return retval; } control/src/sl_ident.cc0000644000076500000240000006104212273411257014446 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . SLICOT system identification Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: March 2012 Version: 0.1 */ #include #include #include #include "common.h" extern "C" { int F77_FUNC (ib01ad, IB01AD) (char& METH, char& ALG, char& JOBD, char& BATCH, char& CONCT, char& CTRL, int& NOBR, int& M, int& L, int& NSMP, double* U, int& LDU, double* Y, int& LDY, int& N, double* R, int& LDR, double* SV, double& RCOND, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); int F77_FUNC (ib01bd, IB01BD) (char& METH, char& JOB, char& JOBCK, int& NOBR, int& N, int& M, int& L, int& NSMPL, double* R, int& LDR, double* A, int& LDA, double* C, int& LDC, double* B, int& LDB, double* D, int& LDD, double* Q, int& LDQ, double* RY, int& LDRY, double* S, int& LDS, double* K, int& LDK, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& IWARN, int& INFO); int F77_FUNC (ib01cd, IB01CD) (char& JOBX0, char& COMUSE, char& JOB, int& N, int& M, int& L, int& NSMP, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* U, int& LDU, double* Y, int& LDY, double* X0, double* V, int& LDV, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_ident__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_ident__, args, nargout, "-*- texinfo -*-\n\ Slicot IB01AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 10) { print_usage (); } else { //////////////////////////////////////////////////////////////////////////////////// // SLICOT IB01AD - preprocess the input-output data // //////////////////////////////////////////////////////////////////////////////////// // arguments in char meth_a; char meth_b; char alg; char jobd; char batch; char conct; char ctrl; const Cell y_cell = args(0).cell_value (); const Cell u_cell = args(1).cell_value (); int nobr = args(2).int_value (); int nuser = args(3).int_value (); const int imeth = args(4).int_value (); const int ialg = args(5).int_value (); const int iconct = args(6).int_value (); const int ictrl = args(7).int_value (); double rcond = args(8).double_value (); double tol_a = args(9).double_value (); double tol_b = rcond; double tol_c = rcond; switch (imeth) { case 0: meth_a = 'M'; meth_b = 'M'; break; case 1: meth_a = 'N'; meth_b = 'N'; break; case 2: meth_a = 'N'; // no typo here meth_b = 'C'; break; default: error ("__sl_ib01ad__: argument 'meth' invalid"); } switch (ialg) { case 0: alg = 'C'; break; case 1: alg = 'F'; break; case 2: alg = 'Q'; break; default: error ("__sl_ib01ad__: argument 'alg' invalid"); } if (meth_a == 'M') jobd = 'M'; else // meth_a == 'N' jobd = 'N'; // IB01AD.f says: This parameter is not relevant for METH = 'N' if (iconct == 0) conct = 'C'; else conct = 'N'; if (ictrl == 0) ctrl = 'C'; else ctrl = 'N'; // m and l are equal for all experiments, checked by iddata class int n_exp = y_cell.nelem (); // number of experiments int m = u_cell.elem(0).columns (); // m: number of inputs int l = y_cell.elem(0).columns (); // l: number of outputs int nsmpl = 0; // total number of samples // arguments out int n; int ldr; if (meth_a == 'M' && jobd == 'M') ldr = max (2*(m+l)*nobr, 3*m*nobr); else if (meth_a == 'N' || (meth_a == 'M' && jobd == 'N')) ldr = 2*(m+l)*nobr; else error ("__sl_ib01ad__: could not handle 'ldr' case"); Matrix r (ldr, 2*(m+l)*nobr); ColumnVector sv (l*nobr); // repeat for every experiment in the dataset for (int i = 0; i < n_exp; i++) { if (n_exp == 1) batch = 'O'; // one block only else if (i == 0) batch = 'F'; // first block else if (i == n_exp-1) batch = 'L'; // last block else batch = 'I'; // intermediate block Matrix y = y_cell.elem(i).matrix_value (); Matrix u = u_cell.elem(i).matrix_value (); // y.rows == u.rows is checked by iddata class // int m = u.columns (); // m: number of inputs // int l = y.columns (); // l: number of outputs int nsmp = y.rows (); // nsmp: number of samples in the current experiment nsmpl += nsmp; // nsmpl: total number of samples of all experiments // minimal nsmp size checked by __slicot_identification__.m if (batch == 'O') { if (nsmp < 2*(m+l+1)*nobr - 1) error ("__sl_ident__: require NSMP >= 2*(M+L+1)*NOBR - 1"); } else { if (nsmp < 2*nobr) error ("__sl_ident__: require NSMP >= 2*NOBR"); } int ldu; if (m == 0) ldu = 1; else // m > 0 ldu = nsmp; int ldy = nsmp; // workspace int liwork_a; if (meth_a == 'N') // if METH = 'N' liwork_a = (m+l)*nobr; else if (alg == 'F') // if METH = 'M' and ALG = 'F' liwork_a = m+l; else // if METH = 'M' and ALG = 'C' or 'Q' liwork_a = 0; // TODO: Handle 'k' for DWORK int ldwork_a; int ns = nsmp - 2*nobr + 1; if (alg == 'C') { if (batch == 'F' || batch == 'I') { if (conct == 'C') ldwork_a = (4*nobr-2)*(m+l); else // (conct == 'N') ldwork_a = 1; } else if (meth_a == 'M') // && (batch == 'L' || batch == 'O') { if (conct == 'C' && batch == 'L') ldwork_a = max ((4*nobr-2)*(m+l), 5*l*nobr); else if (jobd == 'M') ldwork_a = max ((2*m-1)*nobr, (m+l)*nobr, 5*l*nobr); else // (jobd == 'N') ldwork_a = 5*l*nobr; } else // meth_b == 'N' && (batch == 'L' || batch == 'O') { ldwork_a = 5*(m+l)*nobr + 1; } } else if (alg == 'F') { if (batch != 'O' && conct == 'C') ldwork_a = (m+l)*2*nobr*(m+l+3); else if (batch == 'F' || batch == 'I') // && conct == 'N' ldwork_a = (m+l)*2*nobr*(m+l+1); else // (batch == 'L' || '0' && conct == 'N') ldwork_a = (m+l)*4*nobr*(m+l+1)+(m+l)*2*nobr; } else // (alg == 'Q') { // int ns = nsmp - 2*nobr + 1; if (ldr >= ns && batch == 'F') { ldwork_a = 4*(m+l)*nobr; } else if (ldr >= ns && batch == 'O') { if (meth_a == 'M') ldwork_a = max (4*(m+l)*nobr, 5*l*nobr); else // (meth == 'N') ldwork_a = 5*(m+l)*nobr + 1; } else if (conct == 'C' && (batch == 'I' || batch == 'L')) { ldwork_a = 4*(nobr+1)*(m+l)*nobr; } else // if ALG = 'Q', (BATCH = 'F' or 'O', and LDR < NS), or (BATCH = 'I' or 'L' and CONCT = 'N') { ldwork_a = 6*(m+l)*nobr; } } /* IB01AD.f Lines 438-445 C FURTHER COMMENTS C C For ALG = 'Q', BATCH = 'O' and LDR < NS, or BATCH <> 'O', the C calculations could be rather inefficient if only minimal workspace C (see argument LDWORK) is provided. It is advisable to provide as C much workspace as possible. Almost optimal efficiency can be C obtained for LDWORK = (NS+2)*(2*(M+L)*NOBR), assuming that the C cache size is large enough to accommodate R, U, Y, and DWORK. */ ldwork_a = max (ldwork_a, (ns+2)*(2*(m+l)*nobr)); /* IB01AD.f Lines 291-195: c the workspace used for alg = 'q' is c ldrwrk*2*(m+l)*nobr + 4*(m+l)*nobr, c where ldrwrk = ldwork/(2*(m+l)*nobr) - 2; recommended c value ldrwrk = ns, assuming a large enough cache size. c for good performance, ldwork should be larger. somehow ldrwrk and ldwork must have been mixed up here */ OCTAVE_LOCAL_BUFFER (int, iwork_a, liwork_a); OCTAVE_LOCAL_BUFFER (double, dwork_a, ldwork_a); // error indicators int iwarn_a = 0; int info_a = 0; // SLICOT routine IB01AD F77_XFCN (ib01ad, IB01AD, (meth_a, alg, jobd, batch, conct, ctrl, nobr, m, l, nsmp, u.fortran_vec (), ldu, y.fortran_vec (), ldy, n, r.fortran_vec (), ldr, sv.fortran_vec (), rcond, tol_a, iwork_a, dwork_a, ldwork_a, iwarn_a, info_a)); if (f77_exception_encountered) error ("ident: exception in SLICOT subroutine IB01AD"); static const char* err_msg[] = { "0: OK", "1: a fast algorithm was requested (ALG = 'C', or 'F') " "in sequential data processing, but it failed; the " "routine can be repeatedly called again using the " "standard QR algorithm", "2: the singular value decomposition (SVD) algorithm did " "not converge"}; static const char* warn_msg[] = { "0: OK", "1: the number of 100 cycles in sequential data " "processing has been exhausted without signaling " "that the last block of data was get; the cycle " "counter was reinitialized", "2: a fast algorithm was requested (ALG = 'C' or 'F'), " "but it failed, and the QR algorithm was then used " "(non-sequential data processing)", "3: all singular values were exactly zero, hence N = 0 " "(both input and output were identically zero)", "4: the least squares problems with coefficient matrix " "U_f, used for computing the weighted oblique " "projection (for METH = 'N'), have a rank-deficient " "coefficient matrix", "5: the least squares problem with coefficient matrix " "r_1 [6], used for computing the weighted oblique " "projection (for METH = 'N'), has a rank-deficient " "coefficient matrix"}; error_msg ("ident: IB01AD", info_a, 2, err_msg); warning_msg ("ident: IB01AD", iwarn_a, 5, warn_msg); } // resize int rs = 2*(m+l)*nobr; r.resize (rs, rs); if (nuser > 0) { if (nuser < nobr) { n = nuser; // warning ("ident: nuser (%d) < nobr (%d), n = nuser", nuser, nobr); } else error ("ident: 'nuser' invalid"); } //////////////////////////////////////////////////////////////////////////////////// // SLICOT IB01BD - estimating system matrices, Kalman gain, and covariances // //////////////////////////////////////////////////////////////////////////////////// // arguments in char job = 'A'; char jobck = 'K'; //int nsmpl = nsmp; if (nsmpl < 2*(m+l)*nobr) error ("__sl_ident__: nsmpl (%d) < 2*(m+l)*nobr (%d)", nsmpl, nobr); // arguments out int lda = max (1, n); int ldc = max (1, l); int ldb = max (1, n); int ldd = max (1, l); int ldq = n; // if JOBCK = 'C' or 'K' int ldry = l; // if JOBCK = 'C' or 'K' int lds = n; // if JOBCK = 'C' or 'K' int ldk = n; // if JOBCK = 'K' Matrix a (lda, n); Matrix c (ldc, n); Matrix b (ldb, m); Matrix d (ldd, m); Matrix q (ldq, n); Matrix ry (ldry, l); Matrix s (lds, l); Matrix k (ldk, l); // workspace int liwork_b; int liw1; int liw2; liw1 = max (n, m*nobr+n, l*nobr, m*(n+l)); liw2 = n*n; // if JOBCK = 'K' liwork_b = max (liw1, liw2); int ldwork_b; int ldw1; int ldw2; int ldw3; if (meth_b == 'M') { int ldw1a = max (2*(l*nobr-l)*n+2*n, (l*nobr-l)*n+n*n+7*n); int ldw1b = max (2*(l*nobr-l)*n+n*n+7*n, (l*nobr-l)*n+n+6*m*nobr, (l*nobr-l)*n+n+max (l+m*nobr, l*nobr + max (3*l*nobr+1, m))); ldw1 = max (ldw1a, ldw1b); int aw; if (m == 0 || job == 'C') aw = n + n*n; else aw = 0; ldw2 = l*nobr*n + max ((l*nobr-l)*n+aw+2*n+max(5*n,(2*m+l)*nobr+l), 4*(m*nobr+n)+1, m*nobr+2*n+l ); } else if (meth_b == 'N') { ldw1 = l*nobr*n + max ((l*nobr-l)*n+2*n+(2*m+l)*nobr+l, 2*(l*nobr-l)*n+n*n+8*n, n+4*(m*nobr+n)+1, m*nobr+3*n+l); if (m == 0 || job == 'C') ldw2 = 0; else ldw2 = l*nobr*n+m*nobr*(n+l)*(m*(n+l)+1)+ max ((n+l)*(n+l), 4*m*(n+l)+1); } else // (meth_b == 'C') { int ldw1a = max (2*(l*nobr-l)*n+2*n, (l*nobr-l)*n+n*n+7*n); int ldw1b = l*nobr*n + max ((l*nobr-l)*n+2*n+(2*m+l)*nobr+l, 2*(l*nobr-l)*n+n*n+8*n, n+4*(m*nobr+n)+1, m*nobr+3*n+l); ldw1 = max (ldw1a, ldw1b); ldw2 = l*nobr*n+m*nobr*(n+l)*(m*(n+l)+1)+ max ((n+l)*(n+l), 4*m*(n+l)+1); } ldw3 = max(4*n*n + 2*n*l + l*l + max (3*l, n*l), 14*n*n + 12*n + 5); ldwork_b = max (ldw1, ldw2, ldw3); OCTAVE_LOCAL_BUFFER (int, iwork_b, liwork_b); OCTAVE_LOCAL_BUFFER (double, dwork_b, ldwork_b); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicators int iwarn_b = 0; int info_b = 0; // SLICOT routine IB01BD F77_XFCN (ib01bd, IB01BD, (meth_b, job, jobck, nobr, n, m, l, nsmpl, r.fortran_vec (), ldr, a.fortran_vec (), lda, c.fortran_vec (), ldc, b.fortran_vec (), ldb, d.fortran_vec (), ldd, q.fortran_vec (), ldq, ry.fortran_vec (), ldry, s.fortran_vec (), lds, k.fortran_vec (), ldk, tol_b, iwork_b, dwork_b, ldwork_b, bwork, iwarn_b, info_b)); if (f77_exception_encountered) error ("ident: exception in SLICOT subroutine IB01BD"); static const char* err_msg_b[] = { "0: OK", "1: error message not specified", "2: the singular value decomposition (SVD) algorithm did " "not converge", "3: a singular upper triangular matrix was found", "4: matrix A is (numerically) singular in discrete-" "time case", "5: the Hamiltonian or symplectic matrix H cannot be " "reduced to real Schur form", "6: the real Schur form of the Hamiltonian or " "symplectic matrix H cannot be appropriately ordered", "7: the Hamiltonian or symplectic matrix H has less " "than N stable eigenvalues", "8: the N-th order system of linear algebraic " "equations, from which the solution matrix X would " "be obtained, is singular to working precision", "9: the QR algorithm failed to complete the reduction " "of the matrix Ac to Schur canonical form, T", "10: the QR algorithm did not converge"}; static const char* warn_msg_b[] = { "0: OK", "1: warning message not specified", "2: warning message not specified", "3: warning message not specified", "4: a least squares problem to be solved has a " "rank-deficient coefficient matrix", "5: the computed covariance matrices are too small. " "The problem seems to be a deterministic one; the " "gain matrix is set to zero"}; error_msg ("ident: IB01BD", info_b, 10, err_msg_b); warning_msg ("ident: IB01BD", iwarn_b, 5, warn_msg_b); // resize a.resize (n, n); c.resize (l, n); b.resize (n, m); d.resize (l, m); q.resize (n, n); ry.resize (l, l); s.resize (n, l); k.resize (n, l); //////////////////////////////////////////////////////////////////////////////////// // SLICOT IB01CD - estimating the initial state // //////////////////////////////////////////////////////////////////////////////////// // arguments in char jobx0 = 'X'; char comuse = 'U'; char jobbd = 'D'; // arguments out Cell x0_cell (n_exp, 1); // cell of initial state vectors x0 // repeat for every experiment in the dataset // compute individual initial state vector x0 for every experiment for (int i = 0; i < n_exp; i++) { Matrix y = y_cell.elem(i).matrix_value (); Matrix u = u_cell.elem(i).matrix_value (); int nsmp = y.rows (); // nsmp: number of samples int ldv = max (1, n); int ldu; if (m == 0) ldu = 1; else // m > 0 ldu = nsmp; int ldy = nsmp; // arguments out ColumnVector x0 (n); Matrix v (ldv, n); // workspace int liwork_c = n; // if JOBX0 = 'X' and COMUSE <> 'C' int ldwork_c; int t = nsmp; int ldw1_c = 2; int ldw2_c = t*l*(n + 1) + 2*n + max (2*n*n, 4*n); int ldw3_c = n*(n + 1) + 2*n + max (n*l*(n + 1) + 2*n*n + l*n, 4*n); ldwork_c = ldw1_c + n*( n + m + l ) + max (5*n, ldw1_c, min (ldw2_c, ldw3_c)); OCTAVE_LOCAL_BUFFER (int, iwork_c, liwork_c); OCTAVE_LOCAL_BUFFER (double, dwork_c, ldwork_c); // error indicators int iwarn_c = 0; int info_c = 0; // SLICOT routine IB01CD F77_XFCN (ib01cd, IB01CD, (jobx0, comuse, jobbd, n, m, l, nsmp, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, u.fortran_vec (), ldu, y.fortran_vec (), ldy, x0.fortran_vec (), v.fortran_vec (), ldv, tol_c, iwork_c, dwork_c, ldwork_c, iwarn_c, info_c)); if (f77_exception_encountered) error ("ident: exception in SLICOT subroutine IB01CD"); static const char* err_msg_c[] = { "0: OK", "1: the QR algorithm failed to compute all the " "eigenvalues of the matrix A (see LAPACK Library " "routine DGEES); the locations DWORK(i), for " "i = g+1:g+N*N, contain the partially converged " "Schur form", "2: the singular value decomposition (SVD) algorithm did " "not converge"}; static const char* warn_msg_c[] = { "0: OK", "1: warning message not specified", "2: warning message not specified", "3: warning message not specified", "4: the least squares problem to be solved has a " "rank-deficient coefficient matrix", "5: warning message not specified", "6: the matrix A is unstable; the estimated x(0) " "and/or B and D could be inaccurate"}; error_msg ("ident: IB01CD", info_c, 2, err_msg_c); warning_msg ("ident: IB01CD", iwarn_c, 6, warn_msg_c); x0_cell.elem(i) = x0; // add x0 from the current experiment to cell of initial state vectors } // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; retval(4) = q; retval(5) = ry; retval(6) = s; retval(7) = k; retval(8) = x0_cell; } return retval; } control/src/sl_sb01bd.cc0000644000076500000240000001230212273411257014411 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Pole assignment for a given matrix pair (A,B). Uses SLICOT SB01BD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2009 Version: 0.5 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb01bd, SB01BD) (char& DICO, int& N, int& M, int& NP, double& ALPHA, double* A, int& LDA, double* B, int& LDB, double* WR, double* WI, int& NFP, int& NAP, int& NUP, double* F, int& LDF, double* Z, int& LDZ, double& TOL, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_sb01bd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb01bd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB01BD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 7) { print_usage (); } else { // arguments in char dico; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); ColumnVector wr = args(2).column_vector_value (); ColumnVector wi = args(3).column_vector_value (); int discrete = args(4).int_value (); double alpha = args(5).double_value (); double tol = args(6).double_value (); if (discrete == 1) dico = 'D'; else dico = 'C'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = wr.rows (); int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldf = max (1, m); int ldz = max (1, n); // arguments out int nfp; int nap; int nup; Matrix f (ldf, n); Matrix z (ldz, n); // workspace int ldwork = max (1, 5*m, 5*n, 2*n+4*m); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int iwarn; int info; // SLICOT routine SB01BD F77_XFCN (sb01bd, SB01BD, (dico, n, m, np, alpha, a.fortran_vec (), lda, b.fortran_vec (), ldb, wr.fortran_vec (), wi.fortran_vec (), nfp, nap, nup, f.fortran_vec (), ldf, z.fortran_vec (), ldz, tol, dwork, ldwork, iwarn, info)); if (f77_exception_encountered) error ("place: __sl_sb01bd__: exception in SLICOT subroutine SB01BD"); static const char* err_msg[] = { "0: OK", "1: the reduction of A to a real Schur form failed.", "2: a failure was detected during the ordering of the " "real Schur form of A, or in the iterative process " "for reordering the eigenvalues of Z'*(A + B*F)*Z " "along the diagonal.", "3: the number of eigenvalues to be assigned is less " "than the number of possibly assignable eigenvalues; " "NAP eigenvalues have been properly assigned, " "but some assignable eigenvalues remain unmodified.", "4: an attempt is made to place a complex conjugate " "pair on the location of a real eigenvalue. This " "situation can only appear when N-NFP is odd, " "NP > N-NFP-NUP is even, and for the last real " "eigenvalue to be modified there exists no available " "real eigenvalue to be assigned. However, NAP " "eigenvalues have been already properly assigned."}; static const char* warn_msg[] = { "0: OK", /* 0+%d: %d */ "violations of the numerical stability condition " "NORM(F) <= 100*NORM(A)/NORM(B) occured during the " "assignment of eigenvalues."}; error_msg ("place", info, 4, err_msg); warning_msg ("place", iwarn, 0, warn_msg, 0); // return values retval(0) = f; retval(1) = octave_value (nfp); retval(2) = octave_value (nap); retval(3) = octave_value (nup); retval(4) = z; } return retval; } control/src/sl_sb02od.cc0000644000076500000240000001444112273411257014435 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Solve algebraic Riccati equation. Uses SLICOT SB02OD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: February 2010 Version: 0.4 */ #include #include #include "common.h" #include extern "C" { int F77_FUNC (sb02od, SB02OD) (char& DICO, char& JOBB, char& FACT, char& UPLO, char& JOBL, char& SORT, int& N, int& M, int& P, double* A, int& LDA, double* B, int& LDB, double* Q, int& LDQ, double* R, int& LDR, double* L, int& LDL, double& RCOND, double* X, int& LDX, double* ALFAR, double* ALFAI, double* BETA, double* S, int& LDS, double* T, int& LDT, double* U, int& LDU, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb02od__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb02od__, args, nargout, "-*- texinfo -*-\n\ Slicot SB02OD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 7) { print_usage (); } else { // arguments in char dico; char jobb = 'B'; char fact = 'N'; char uplo = 'U'; char jobl; char sort = 'S'; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix q = args(2).matrix_value (); Matrix r = args(3).matrix_value (); Matrix l = args(4).matrix_value (); int discrete = args(5).int_value (); int ijobl = args(6).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; if (ijobl == 0) jobl = 'Z'; else jobl = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = 0; // p: number of outputs, not used because FACT = 'N' int lda = max (1, n); int ldb = max (1, n); int ldq = max (1, n); int ldr = max (1, m); int ldl = max (1, n); // arguments out double rcond; int ldx = max (1, n); Matrix x (ldx, n); int nu = 2*n; ColumnVector alfar (nu); ColumnVector alfai (nu); ColumnVector beta (nu); int lds = max (1, 2*n + m); Matrix s (lds, lds); // unused output arguments int ldt = max (1, 2*n + m); OCTAVE_LOCAL_BUFFER (double, t, ldt * 2*n); int ldu = max (1, 2*n); OCTAVE_LOCAL_BUFFER (double, u, ldu * 2*n); // tolerance double tol = 0; // use default value // workspace int liwork = max (1, m, 2*n); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); int ldwork = max (7*(2*n + 1) + 16, 16*n, 2*n + m, 3*m); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB02OD F77_XFCN (sb02od, SB02OD, (dico, jobb, fact, uplo, jobl, sort, n, m, p, a.fortran_vec (), lda, b.fortran_vec (), ldb, q.fortran_vec (), ldq, r.fortran_vec (), ldr, l.fortran_vec (), ldl, rcond, x.fortran_vec (), ldx, alfar.fortran_vec (), alfai.fortran_vec (), beta.fortran_vec (), s.fortran_vec (), lds, t, ldt, u, ldu, tol, iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("are: __sl_sb02od__: exception in SLICOT subroutine SB02OD"); static const char* err_msg[] = { "0: OK", "1: the computed extended matrix pencil is singular, " "possibly due to rounding errors", "2: the QZ (or QR) algorithm failed", "3: reordering of the (generalized) eigenvalues " "failed", "4: after reordering, roundoff changed values of " "some complex eigenvalues so that leading eigenvalues " "in the (generalized) Schur form no longer satisfy " "the stability condition; this could also be caused " "due to scaling", "5: the computed dimension of the solution does not " "equal N", "6: a singular matrix was encountered during the " "computation of the solution matrix X"}; error_msg ("are", info, 6, err_msg); // assemble complex vector - adapted from DEFUN complex in data.cc alfar.resize (n); alfai.resize (n); beta.resize (n); ColumnVector poler (n); ColumnVector polei (n); poler = quotient (alfar, beta); polei = quotient (alfai, beta); ComplexColumnVector pole (n, Complex ()); for (octave_idx_type i = 0; i < n; i++) pole.xelem (i) = Complex (poler(i), polei(i)); // return value retval(0) = x; retval(1) = pole; } return retval; } control/src/sl_sb03md.cc0000644000076500000240000000702112273411257014430 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Solution of Lyapunov equations. Uses SLICOT SB03MD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: December 2009 Version: 0.3 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb03md, SB03MD) (char& DICO, char& JOB, char& FACT, char& TRANA, int& N, double* A, int& LDA, double* U, int& LDU, double* C, int& LDC, double& SCALE, double& SEP, double& FERR, double* WR, double* WI, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb03md__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb03md__, args, nargout, "-*- texinfo -*-\n\ Slicot SB03MD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 3) { print_usage (); } else { // arguments in char dico; char job = 'X'; char fact = 'N'; char trana = 'T'; Matrix a = args(0).matrix_value (); Matrix c = args(1).matrix_value (); int discrete = args(2).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; int n = a.rows (); // n: number of states int lda = max (1, n); int ldu = max (1, n); int ldc = max (1, n); // arguments out double scale; double sep = 0; double ferr = 0; Matrix u (ldu, n); ColumnVector wr (n); ColumnVector wi (n); // workspace int* iwork = 0; // not referenced because job = X int ldwork = max (n*n, 3*n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine SB03MD F77_XFCN (sb03md, SB03MD, (dico, job, fact, trana, n, a.fortran_vec (), lda, u.fortran_vec (), ldu, c.fortran_vec (), ldc, scale, sep, ferr, wr.fortran_vec (), wi.fortran_vec (), iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("lyap: __sl_sb03md__: exception in SLICOT subroutine SB03MD"); if (info != 0) error ("lyap: __sl_sb03md__: SB03MD returned info = %d", info); // return values retval(0) = c; retval(1) = octave_value (scale); } return retval; } control/src/sl_sb03od.cc0000644000076500000240000000700012273411257014427 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Square-root solver for Lyapunov equations. Uses SLICOT SB03OD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: January 2010 Version: 0.2 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb03od, SB03OD) (char& DICO, char& FACT, char& TRANS, int& N, int& M, double* A, int& LDA, double* Q, int& LDQ, double* B, int& LDB, double& SCALE, double* WR, double* WI, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb03od__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb03od__, args, nargout, "-*- texinfo -*-\n\ Slicot SB03OD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 3) { print_usage (); } else { // arguments in char dico; char fact = 'N'; char trans = 'N'; // char trans = 'T'; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); int discrete = args(2).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; int n = a.rows (); int m = b.rows (); // int m = b.columns (); int lda = max (1, n); int ldq = max (1, n); int ldb = max (1, n, m); // int ldb = max (1, n); b.resize (ldb, n); // b.resize (ldb, max (m, n)); // arguments out double scale; Matrix q (ldq, n); ColumnVector wr (n); ColumnVector wi (n); // workspace int ldwork = max (1, 4*n + min (m, n)); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine SB03OD F77_XFCN (sb03od, SB03OD, (dico, fact, trans, n, m, a.fortran_vec (), lda, q.fortran_vec (), ldq, b.fortran_vec (), ldb, scale, wr.fortran_vec (), wi.fortran_vec (), dwork, ldwork, info)); if (f77_exception_encountered) error ("lyapchol: __sl_sb03od__: exception in SLICOT subroutine SB03OD"); if (info != 0) error ("lyapchol: __sl_sb03od__: SB03OD returned info = %d", info); // resize b.resize (n, n); // return values retval(0) = b; // b has been overwritten by cholesky factor u retval(1) = octave_value (scale); } return retval; } control/src/sl_sb04md.cc0000644000076500000240000000566012273411257014440 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Solution of continuous-time Sylvester equations. Uses SLICOT SB04MD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: January 2010 Version: 0.2 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb04md, SB04MD) (int& N, int& M, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* Z, int& LDZ, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb04md__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb04md__, args, nargout, "-*- texinfo -*-\n\ Slicot SB04MD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 3) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); int n = a.rows (); int m = b.rows (); int lda = max (1, n); int ldb = max (1, m); int ldc = max (1, n); int ldz = max (1, m); // arguments out Matrix z (ldz, m); // workspace int ldwork = max (1, 2*n*n + 8*n, 5*m, n + m); OCTAVE_LOCAL_BUFFER (int, iwork, 4*n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine SB04MD F77_XFCN (sb04md, SB04MD, (n, m, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, z.fortran_vec (), ldz, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("lyap: __sl_sb04md__: exception in SLICOT subroutine SB04MD"); if (info != 0) error ("lyap: __sl_sb04md__: SB04MD returned info = %d", info); // return values retval(0) = c; } return retval; } control/src/sl_sb04qd.cc0000644000076500000240000000566012273411257014444 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Solution of discrete-time Sylvester equations. Uses SLICOT SB04QD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: January 2010 Version: 0.2 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb04qd, SB04QD) (int& N, int& M, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* Z, int& LDZ, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb04qd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb04qd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB04QD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 3) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); int n = a.rows (); int m = b.rows (); int lda = max (1, n); int ldb = max (1, m); int ldc = max (1, n); int ldz = max (1, m); // arguments out Matrix z (ldz, m); // workspace int ldwork = max (1, 2*n*n + 9*n, 5*m, n + m); OCTAVE_LOCAL_BUFFER (int, iwork, 4*n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine SB04QD F77_XFCN (sb04qd, SB04QD, (n, m, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, z.fortran_vec (), ldz, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("dlyap: __sl_sb04qd__: exception in SLICOT subroutine SB04QD"); if (info != 0) error ("dlyap: __sl_sb04qd__: SB04QD returned info = %d", info); // return values retval(0) = c; } return retval; } control/src/sl_sb10ad.cc0000644000076500000240000002062712273411257014421 0ustar lukasstaff/* Copyright (C) 2014 Thomas Vasileiou This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . H-infinity optimal controller using modified Glover's and Doyle's formulas (continuous-time). Uses SLICOT SB10AD by courtesy of NICONET e.V. Author: Thomas Vasileiou Created: January 2014 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10ad, SB10AD) (int& JOB, int& N, int& M, int& NP, int& NCON, int& NMEAS, double& GAMMA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* AC, int& LDAC, double* BC, int& LDBC, double* CC, int& LDCC, double* DC, int& LDDC, double* RCOND, double& GTOL, double& ACTOL, int* IWORK, int& LIWORK, double* DWORK, int& LDWORK, bool* BWORK, int& LBWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10ad__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 9) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); int ncon = args(4).int_value (); int nmeas = args(5).int_value (); double gamma = args(6).double_value (); double gtol = args(7).double_value (); double actol = args(8).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); int ldd = max (1, d.rows ()); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, ncon); int lddk = max (1, ncon); int ldac = max (1, 2*n); int ldbc = max (1, 2*n); int ldcc = max (1, np-nmeas); int lddc = max (1, np-nmeas); int job = 1; // arguments out Matrix ak (ldak, n); Matrix bk (ldbk, nmeas); Matrix ck (ldck, n); Matrix dk (lddk, nmeas); Matrix ac (ldac, 2*n); Matrix bc (ldbc, m-ncon); Matrix cc (ldcc, 2*n); Matrix dc (lddc, m-ncon); ColumnVector rcond (4); // workspace int m2 = ncon; int m1 = m - m2; int np2 = nmeas; int np1 = np - np2; int nd1 = np1 - m2; int nd2 = m1 - np2; int liwork = max (2*max (n, m-ncon, np-nmeas, ncon, nmeas), n*n); int lw1 = n*m + np*n + np*m + m2*m2 + np2*np2; int lw2 = max ((n + np1 + 1)*(n + m2) + max (3*(n + m2) + n + np1, 5*(n + m2)), (n + np2)*(n + m1 + 1) + max (3*(n + np2) + n + m1, 5*(n + np2)), m2 + np1*np1 + max (np1*max (n, m1), 3*m2 + np1, 5*m2), np2 + m1*m1 + max (max (n, np1)*m1, 3*np2 + m1, 5*np2)); int lw3 = max (nd1*m1 + max (4*min (nd1, m1) + max (nd1,m1), 6*min (nd1, m1)), np1*nd2 + max (4*min (np1, nd2) + max (np1, nd2), 6*min (np1, nd2))); int lw4 = 2*m*m + np*np + 2*m*n + m*np + 2*n*np; int lw5 = 2*n*n + m*n + n*np; int lw6 = max (m*m + max (2*m1, 3*n*n + max (n*m, 10*n*n + 12*n + 5)), np*np + max (2*np1, 3*n*n + max (n*np, 10*n*n + 12*n + 5))); int lw7 = m2*np2 + np2*np2 + m2*m2 + max (nd1*nd1 + max (2*nd1, (nd1 + nd2)*np2), nd2*nd2 + max (2*nd2, nd2*m2), 3*n, n*(2*np2 + m2) + max (2*n*m2, m2*np2 + max (m2*m2 + 3*m2, np2*(2*np2 + m2 + max (np2, n))))); int ldwork = lw1 + max (1, lw2, lw3, lw4, lw5 + max (lw6,lw7)); int lbwork = 2*n; OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, lbwork); // error indicator int info; // SLICOT routine SB10AD F77_XFCN (sb10ad, SB10AD, (job, n, m, np, ncon, nmeas, gamma, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, ac.fortran_vec (), ldac, bc.fortran_vec (), ldbc, cc.fortran_vec (), ldcc, dc.fortran_vec (), lddc, rcond.fortran_vec (), gtol, actol, iwork, liwork, dwork, ldwork, bwork, lbwork, info)); if (f77_exception_encountered) error ("hinfsyn: __sl_sb10ad__: exception in SLICOT subroutine SB10AD"); static const char* err_msg[] = { "0: successful exit", "1: the matrix [A-j*omega*I, B2; C1, D12] had " "not full column rank in respect to the tolerance EPS", "2: the matrix [A-j*omega*I, B1; C2, D21] " "had not full row rank in respect to the tolerance EPS", "3: the matrix D12 had not full column rank in " "respect to the tolerance SQRT(EPS)", "4: the matrix D21 had not full row rank in respect " "to the tolerance SQRT(EPS)", "5: the singular value decomposition (SVD) algorithm " "did not converge (when computing the SVD of one of the matrices " "[A, B2; C1, D12], [A, B1; C2, D21], D12 or D21)", "6: the controller is not admissible (too small value " "of gamma)", "7: the X-Riccati equation was not solved " "successfully (the controller is not admissible or " "there are numerical difficulties)", "8: the Y-Riccati equation was not solved " "successfully (the controller is not admissible or " "there are numerical difficulties)", "9: the determinant of Im2 + Tu*D11HAT*Ty*D22 is " "zero [3]", "10: there was numerical problems when estimating" "the singular values of D1111, D1112, D1111', D1121'", "11: the matrices Inp2 - D22*DK or Im2 - DK*D22" "are singular to working precision", "12: a stabilizing controller cannot be found"}; error_msg ("hinfsyn", info, 12, err_msg); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = ac; retval(5) = bc; retval(6) = cc; retval(7) = dc; retval(8) = gamma; retval(9) = rcond; } return retval; } control/src/sl_sb10dd.cc0000644000076500000240000001402712273411257014421 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . H-infinity (sub)optimal state controller for a discrete-time system. Uses SLICOT SB10DD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: December 2009 Version: 0.5 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10dd, SB10DD) (int& N, int& M, int& NP, int& NCON, int& NMEAS, double& GAMMA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* X, int& LDX, double* Z, int& LDZ, double* RCOND, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10dd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10dd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10DD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 7) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); int ncon = args(4).int_value (); int nmeas = args(5).int_value (); double gamma = args(6).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); int ldd = max (1, d.rows ()); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, ncon); int lddk = max (1, ncon); int ldx = max (1, n); int ldz = max (1, n); double tol = 0; // arguments out Matrix ak (ldak, n); Matrix bk (ldbk, nmeas); Matrix ck (ldck, n); Matrix dk (lddk, nmeas); Matrix x (ldx, n); Matrix z (ldz, n); ColumnVector rcond (8); // workspace int m2 = ncon; int m1 = m - m2; int np1 = np - nmeas; int np2 = nmeas; int liwork = max (2*max (m2, n), m, m2+np2, n*n); int q = max (m1, m2, np1, np2); int ldwork = max ((n+q)*(n+q+6), 13*n*n + m*m + 2*q*q + n*(m+q) + max (m*(m+7*n), 2*q*(8*n+m+2*q)) + 6*n + max (14*n+23, 16*n, 2*n + max (m, 2*q), 3*max (m, 2*q))); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB10DD F77_XFCN (sb10dd, SB10DD, (n, m, np, ncon, nmeas, gamma, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, x.fortran_vec (), ldx, z.fortran_vec (), ldz, rcond.fortran_vec (), tol, iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("hinfsyn: __sl_sb10dd__: exception in SLICOT subroutine SB10DD"); static const char* err_msg[] = { "0: OK", "1: the matrix [A-exp(j*Theta)*I, B2; C1, D12] " "had not full column rank", "2: the matrix | A-exp(j*Theta)*I, B1; C2, D21] " "had not full row rank", "3: the matrix D12 had not full column rank", "4: the matrix D21 had not full row rank", "5: the controller is not admissible " "(too small value of gamma)", "6: the X-Riccati equation was not solved " "successfully (the controller is not admissible or " "there are numerical difficulties)", "7: the Z-Riccati equation was not solved " "successfully (the controller is not admissible or " "there are numerical difficulties)", "8: the matrix Im2 + DKHAT*D22 is singular", "9: the singular value decomposition (SVD) algorithm " "did not converge (when computing the SVD of one of " "the matrices [A, B2; C1, D12], [A, B1; C2, D21], D12 or D21)"}; error_msg ("hinfsyn", info, 9, err_msg); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = rcond; } return retval; } control/src/sl_sb10ed.cc0000644000076500000240000001364412273411257014426 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . H2 optimal state controller for a discrete-time system. Uses SLICOT SB10ED by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2009 Version: 0.5 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10ed, SB10ED) (int& N, int& M, int& NP, int& NCON, int& NMEAS, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* RCOND, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10ed__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10ed__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10ED Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 6) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); int ncon = args(4).int_value (); int nmeas = args(5).int_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); int ldd = max (1, d.rows ()); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, ncon); int lddk = max (1, ncon); double tol = 0; // arguments out Matrix ak (ldak, n); Matrix bk (ldbk, nmeas); Matrix ck (ldck, n); Matrix dk (lddk, nmeas); ColumnVector rcond (7); // workspace int m2 = ncon; int m1 = m - m2; int np1 = np - nmeas; int np2 = nmeas; int q = max (m1, m2, np1, np2); int ldwork = 2*q*(3*q+2*n)+max(1,(n+q)*(n+q+6),q*(q+max(n,q,5)+1), 2*n*n+max(1,14*n*n+6*n+max(14*n+23,16*n), q*(n+q+max(q,3)))); int liwork = max (2*m2, 2*n, n*n, np2); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB10ED F77_XFCN (sb10ed, SB10ED, (n, m, np, ncon, nmeas, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, rcond.fortran_vec (), tol, iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("h2syn: __sl_sb10ed__: exception in SLICOT subroutine SB10ED"); static const char* err_msg[] = { "0: OK", "1: the matrix [A-exp(j*Theta)*I, B2; C1, D12] " "had not full column rank in respect to the tolerance EPS", "2: the matrix [A-exp(j*Theta)*I, B1; C2, D21] " "had not full row rank in respect to the tolerance EPS", "3: the matrix D12 had not full column rank in " "respect to the tolerance TOL", "4: the matrix D21 had not full row rank in respect " "to the tolerance TOL", "5: the singular value decomposition (SVD) algorithm " "did not converge (when computing the SVD of one of the matrices " "[A-I, B2; C1, D12], [A-I, B1; C2, D21], D12 or D21)", "6: the X-Riccati equation was not solved successfully", "7: the matrix Im2 + B2'*X2*B2 is not positive " "definite, or it is numerically singular (with " "respect to the tolerance TOL)", "8: the Y-Riccati equation was not solved successfully", "9: the matrix Ip2 + C2*Y2*C2' is not positive " "definite, or it is numerically singular (with " "respect to the tolerance TOL)", "10: the matrix Im2 + DKHAT*D22 is singular, or its " "estimated condition number is larger than or equal " "to 1/TOL"}; error_msg ("h2syn", info, 10, err_msg); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = rcond; } return retval; } control/src/sl_sb10fd.cc0000644000076500000240000001410412273411257014417 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . H-infinity (sub)optimal state controller for a continuous-time system. Uses SLICOT SB10FD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: December 2009 Version: 0.5 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10fd, SB10FD) (int& N, int& M, int& NP, int& NCON, int& NMEAS, double& GAMMA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* RCOND, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10fd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10fd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10FD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 7) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); int ncon = args(4).int_value (); int nmeas = args(5).int_value (); double gamma = args(6).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); int ldd = max (1, d.rows ()); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, ncon); int lddk = max (1, ncon); double tol = 0; // arguments out Matrix ak (ldak, n); Matrix bk (ldbk, nmeas); Matrix ck (ldck, n); Matrix dk (lddk, nmeas); ColumnVector rcond (4); // workspace int m2 = ncon; int m1 = m - m2; int np1 = np - nmeas; int np2 = nmeas; int liwork = max (2*max (n, m-ncon, np-nmeas, ncon), n*n); int q = max (m1, m2, np1, np2); int ldwork = 2*q*(3*q+2*n) + max (1, (n+q)*(n+q+6), q*(q + max (n, q, 5) + 1), 2*n*(n+2*q) + max (1, 4*q*q + max (2*q, 3*n*n + max (2*n*q, 10*n*n+12*n+5)), q*(3*n + 3*q + max (2*n, 4*q + max (n, q))))); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB10FD F77_XFCN (sb10fd, SB10FD, (n, m, np, ncon, nmeas, gamma, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, rcond.fortran_vec (), tol, iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("hinfsyn: __sl_sb10fd__: exception in SLICOT subroutine SB10FD"); static const char* err_msg[] = { "0: OK", "1: the matrix [A-j*omega*I, B2; C1, D12] had " "not full column rank in respect to the tolerance EPS", "2: the matrix [A-j*omega*I, B1; C2, D21] " "had not full row rank in respect to the tolerance EPS", "3: the matrix D12 had not full column rank in " "respect to the tolerance TOL", "4: the matrix D21 had not full row rank in respect " "to the tolerance TOL", "5: the singular value decomposition (SVD) algorithm " "did not converge (when computing the SVD of one of the matrices " "[A, B2; C1, D12], [A, B1; C2, D21], D12 or D21)", "6: the controller is not admissible (too small value " "of gamma)", "7: the X-Riccati equation was not solved " "successfully (the controller is not admissible or " "there are numerical difficulties)", "8: the Y-Riccati equation was not solved " "successfully (the controller is not admissible or " "there are numerical difficulties)", "9: the determinant of Im2 + Tu*D11HAT*Ty*D22 is " "zero [3]"}; error_msg ("hinfsyn", info, 9, err_msg); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = rcond; } return retval; } control/src/sl_sb10hd.cc0000644000076500000240000001174112273411257014425 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . H2 optimal state controller for a continuous-time system. Uses SLICOT SB10HD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2009 Version: 0.5 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10hd, SB10HD) (int& N, int& M, int& NP, int& NCON, int& NMEAS, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* RCOND, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10hd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10hd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10HD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 6) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); int ncon = args(4).int_value (); int nmeas = args(5).int_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, a.rows ()); int ldb = max (1, b.rows ()); int ldc = max (1, c.rows ()); int ldd = max (1, d.rows ()); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, ncon); int lddk = max (1, ncon); double tol = 0; // arguments out Matrix ak (ldak, n); Matrix bk (ldbk, nmeas); Matrix ck (ldck, n); Matrix dk (lddk, nmeas); ColumnVector rcond (4); // workspace int m2 = ncon; int m1 = m - m2; int np1 = np - nmeas; int np2 = nmeas; int q = max (m1, m2, np1, np2); int ldwork = 2*q*(3*q + 2*n) + max (1, q*(q + max (n, 5) + 1), n*(14*n + 12 + 2*q) + 5); OCTAVE_LOCAL_BUFFER (int, iwork, max (2*n, n*n)); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB10HD F77_XFCN (sb10hd, SB10HD, (n, m, np, ncon, nmeas, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, rcond.fortran_vec (), tol, iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("h2syn: __sl_sb10hd__: exception in SLICOT subroutine SB10HD"); static const char* err_msg[] = { "0: OK", "1: the matrix D12 had not full column rank in " "respect to the tolerance TOL", "2: the matrix D21 had not full row rank in respect " "to the tolerance TOL", "3: the singular value decomposition (SVD) algorithm " "did not converge (when computing the SVD of one of " "the matrices D12 or D21)", "4: the X-Riccati equation was not solved successfully", "5: the Y-Riccati equation was not solved successfully"}; error_msg ("h2syn", info, 5, err_msg); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = rcond; } return retval; } control/src/sl_sb10id.cc0000644000076500000240000001134312273411257014424 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Positive feedback controller for a continuous-time system. Uses SLICOT SB10ID by courtesy of NICONET e.V. Author: Lukas Reichlin Created: July 2011 Version: 0.3 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10id, SB10ID) (int& N, int& M, int& NP, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double& FACTOR, int& NK, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* RCOND, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10id__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10id__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10ID Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); double factor = args(4).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, np); int ldd = max (1, np); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, m); int lddk = max (1, m); // arguments out int nk; Matrix ak (ldak, n); Matrix bk (ldbk, np); Matrix ck (ldck, n); Matrix dk (lddk, np); ColumnVector rcond (2); // workspace int liwork = max (2*n, n*n, m, np); int ldwork = 10*n*n + m*m + np*np + 2*m*n + 2*n*np + 4*n + 5 + max (1, 4*n*n + 8*n); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB10ID F77_XFCN (sb10id, SB10ID, (n, m, np, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, factor, nk, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, rcond.fortran_vec (), iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("ncfsyn: __sl_sb10id__: exception in SLICOT subroutine SB10ID"); static const char* err_msg[] = { "0: OK", "1: the X-Riccati equation is not solved successfully", "2: the Z-Riccati equation is not solved successfully", "3: the iteration to compute eigenvalues or singular " "values failed to converge", "4: the matrix Ip - D*Dk is singular", "5: the matrix Im - Dk*D is singular", "6: the closed-loop system is unstable"}; error_msg ("ncfsyn", info, 6, err_msg); // resizing ak.resize (nk, nk); bk.resize (nk, np); ck.resize (m, nk); dk.resize (m, np); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = rcond; } return retval; } control/src/sl_sb10jd.cc0000644000076500000240000000655112273411257014432 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Convert descriptor state-space system into regular state-space form. Uses SLICOT SB10JD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10jd, SB10JD) (int& N, int& M, int& NP, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* E, int& LDE, int& NSYS, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10jd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10jd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10JD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); Matrix e = args(4).matrix_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, np); int ldd = max (1, np); int lde = max (1, n); // arguments out int nsys; // workspace int ldwork = max (1, 2*n*n + 2*n + n*max (5, n + m + np)); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine SB10JD F77_XFCN (sb10jd, SB10JD, (n, m, np, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, e.fortran_vec (), lde, nsys, dwork, ldwork, info)); if (f77_exception_encountered) error ("__sl_sb10jd__: exception in SLICOT subroutine SB10JD"); if (info != 0) error ("__sl_sb10jd__: SB10JD returned info = %d", info); // resize a.resize (nsys, nsys); b.resize (nsys, m); c.resize (np, nsys); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; } return retval; } control/src/sl_sb10kd.cc0000644000076500000240000001115112273411257014423 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Positive feedback controller for a discrete-time system (D == 0). Uses SLICOT SB10KD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: July 2011 Version: 0.3 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10kd, SB10KD) (int& N, int& M, int& NP, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double& FACTOR, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* RCOND, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10kd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10kd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10KD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 4) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); double factor = args(3).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, np); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, m); int lddk = max (1, m); // arguments out Matrix ak (ldak, n); Matrix bk (ldbk, np); Matrix ck (ldck, n); Matrix dk (lddk, np); ColumnVector rcond (4); // workspace int liwork = 2 * max (n, np+m); int ldwork = 15*n*n + 6*n + max (14*n+23, 16*n, 2*n+np+m, 3*(np+m)) + max (n*n, 11*n*np + 2*m*m + 8*np*np + 8*m*n + 4*m*np + np); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB10KD F77_XFCN (sb10kd, SB10KD, (n, m, np, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, factor, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, rcond.fortran_vec (), iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("ncfsyn: slsb10kd: exception in SLICOT subroutine SB10KD"); static const char* err_msg[] = { "0: OK", "1: the P-Riccati equation is not solved successfully", "2: the Q-Riccati equation is not solved successfully", "3: the X-Riccati equation is not solved successfully", "4: the iteration to compute eigenvalues failed to " "converge", "5: the matrix Rx + Bx'*X*Bx is singular", "6: the closed-loop system is unstable"}; error_msg ("ncfsyn", info, 6, err_msg); // resizing ak.resize (n, n); bk.resize (n, np); ck.resize (m, n); dk.resize (m, np); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = rcond; } return retval; } control/src/sl_sb10yd.cc0000644000076500000240000001042312273411257014442 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Fit FRD with SS model. Uses SLICOT SB10YD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: October 2011 Version: 0.1 */ #include #include #include #include "common.h" extern "C" { int F77_FUNC (sb10yd, SB10YD) (int& DISCFL, int& FLAG, int& LENDAT, double* RFRDAT, double* IFRDAT, double* OMEGA, int& N, double* A, int& LDA, double* B, double* C, double* D, double& TOL, int* IWORK, double* DWORK, int& LDWORK, Complex* ZWORK, int& LZWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10yd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10yd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10YD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 6) { print_usage (); } else { // arguments in Matrix rfrdat = args(0).matrix_value (); Matrix ifrdat = args(1).matrix_value (); Matrix omega = args(2).matrix_value (); int n = args(3).int_value (); int discfl = args(4).int_value (); int flag = args(5).int_value (); int lendat = omega.rows (); // number of frequencies int lda = max (1, n); // arguments out Matrix a (lda, n); Matrix b (n, 1); Matrix c (1, n); Matrix d (1, 1); // workspace int liwork = max (2, 2*n + 1); int ldwork; int lzwork; int hnpts = 2048; int lw1 = 2*lendat + 4*hnpts; int lw2 = lendat + 6*hnpts; int mn = min (2*lendat, 2*n+1); int lw3; int lw4; if (n > 0) { lw3 = 2*lendat*(2*n+1) + max (2*lendat, 2*n+1) + max (mn + 6*n + 4, 2*mn + 1); lzwork = lendat*(2*n+3); } else { lw3 = 4*lendat + 5; lzwork = lendat; } if (flag == 1) lw4 = max (n*n + 5*n, 6*n + 1 + min (1, n)); else lw4 = 0; ldwork = max (2, lw1, lw2, lw3, lw4); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (Complex, zwork, lzwork); // tolerance double tol = 0; // error indicator int info; // SLICOT routine SB10YD F77_XFCN (sb10yd, SB10YD, (discfl, flag, lendat, rfrdat.fortran_vec (), ifrdat.fortran_vec (), omega.fortran_vec (), n, a.fortran_vec (), lda, b.fortran_vec (), c.fortran_vec (), d.fortran_vec (), tol, iwork, dwork, ldwork, zwork, lzwork, info)); if (f77_exception_encountered) error ("fitfrd: __sl_sb10yd__: exception in SLICOT subroutine SB10YD"); if (info != 0) error ("fitfrd: __sl_sb10yd__: SB10YD returned info = %d", info); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; retval(4) = octave_value (n); } return retval; } control/src/sl_sb10zd.cc0000644000076500000240000001210112273411257014436 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Positive feedback controller for a discrete-time system (D != 0). Uses SLICOT SB10ZD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: August 2011 Version: 0.3 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb10zd, SB10ZD) (int& N, int& M, int& NP, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double& FACTOR, double* AK, int& LDAK, double* BK, int& LDBK, double* CK, int& LDCK, double* DK, int& LDDK, double* RCOND, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sb10zd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb10zd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB10ZD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 6) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); double factor = args(4).double_value (); double tol = args(5).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int np = c.rows (); // np: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, np); int ldd = max (1, np); int ldak = max (1, n); int ldbk = max (1, n); int ldck = max (1, m); int lddk = max (1, m); // arguments out Matrix ak (ldak, n); Matrix bk (ldbk, np); Matrix ck (ldck, n); Matrix dk (lddk, np); ColumnVector rcond (6); // workspace int liwork = 2 * max (n, m+np); int ldwork = 16*n*n + 5*m*m + 7*np*np + 6*m*n + 7*m*np + 7*n*np + 6*n + 2*(m + np) + max (14*n+23, 16*n, 2*m-1, 2*np-1); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int info; // SLICOT routine SB10ZD F77_XFCN (sb10zd, SB10ZD, (n, m, np, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, factor, ak.fortran_vec (), ldak, bk.fortran_vec (), ldbk, ck.fortran_vec (), ldck, dk.fortran_vec (), lddk, rcond.fortran_vec (), tol, iwork, dwork, ldwork, bwork, info)); if (f77_exception_encountered) error ("ncfsyn: __sl_sb10zd__: exception in SLICOT subroutine SB10ZD"); static const char* err_msg[] = { "0: OK", "1: the P-Riccati equation is not solved successfully", "2: the Q-Riccati equation is not solved successfully", "3: the iteration to compute eigenvalues or singular " "values failed to converge", "4: the matrix (gamma^2-1)*In - P*Q is singular", "5: the matrix Rx + Bx'*X*Bx is singular", "6: the matrix Ip + D*Dk is singular", "7: the matrix Im + Dk*D is singular", "8: the matrix Ip - D*Dk is singular", "9: the matrix Im - Dk*D is singular", "10: the closed-loop system is unstable"}; error_msg ("ncfsyn", info, 10, err_msg); // resizing ak.resize (n, n); bk.resize (n, np); ck.resize (m, n); dk.resize (m, np); // return values retval(0) = ak; retval(1) = bk; retval(2) = ck; retval(3) = dk; retval(4) = rcond; } return retval; } control/src/sl_sb16ad.cc0000644000076500000240000002302512273411257014422 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Controller reduction based on Balance & Truncate (B&T) or Singular Perturbation Approximation (SPA) method. Uses SLICOT SB16AD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb16ad, SB16AD) (char& DICO, char& JOBC, char& JOBO, char& JOBMR, char& WEIGHT, char& EQUIL, char& ORDSEL, int& N, int& M, int& P, int& NC, int& NCR, double& ALPHA, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* AC, int& LDAC, double* BC, int& LDBC, double* CC, int& LDCC, double* DC, int& LDDC, int& NCS, double* HSVC, double& TOL1, double& TOL2, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_sb16ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb16ad__, args, nargout, "-*- texinfo -*-\n\ Slicot SB16AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 19) { print_usage (); } else { // arguments in char dico; char jobc; char jobo; char jobmr; char weight; char equil; char ordsel; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); const int idico = args(4).int_value (); const int iequil = args(5).int_value (); int ncr = args(6).int_value (); const int iordsel = args(7).int_value (); double alpha = args(8).double_value (); const int ijobmr = args(9).int_value (); Matrix ac = args(10).matrix_value (); Matrix bc = args(11).matrix_value (); Matrix cc = args(12).matrix_value (); Matrix dc = args(13).matrix_value (); const int iweight = args(14).int_value (); const int ijobc = args(15).int_value (); const int ijobo = args(16).int_value (); double tol1 = args(17).double_value (); double tol2 = args(18).double_value (); if (idico == 0) dico = 'C'; else dico = 'D'; if (iequil == 0) equil = 'S'; else equil = 'N'; if (iordsel == 0) ordsel = 'F'; else ordsel = 'A'; if (ijobc == 0) jobc = 'S'; else jobc = 'E'; if (ijobo == 0) jobo = 'S'; else jobo = 'E'; switch (ijobmr) { case 0: jobmr = 'B'; break; case 1: jobmr = 'F'; break; case 2: jobmr = 'S'; break; case 3: jobmr = 'P'; break; default: error ("__sl_sb16ad__: argument jobmr invalid"); } switch (iweight) { case 0: weight = 'N'; break; case 1: weight = 'O'; break; case 2: weight = 'I'; break; case 3: weight = 'P'; break; default: error ("__sl_sb16ad__: argument weight invalid"); } int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int nc = ac.rows (); int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd = max (1, p); int ldac = max (1, nc); int ldbc = max (1, nc); int ldcc = max (1, m); int lddc = max (1, m); // arguments out int ncs; ColumnVector hsvc (n); // workspace int liwork; int liwrk1; int liwrk2; switch (jobmr) { case 'B': liwrk1 = 0; break; case 'F': liwrk1 = nc; break; default: liwrk1 = 2*nc; } if (weight == 'N') liwrk2 = 0; else liwrk2 = 2*(m+p); liwork = max (1, liwrk1, liwrk2); int ldwork; int lfreq; int lsqred; if (weight == 'N') { if (equil == 'N') // if WEIGHT = 'N' and EQUIL = 'N' lfreq = nc*(max (m, p) + 5); else // if WEIGHT = 'N' and EQUIL = 'S' lfreq = max (n, nc*(max (m, p) + 5)); } else // if WEIGHT = 'I' or 'O' or 'P' { lfreq = (n+nc)*(n+nc+2*m+2*p) + max ((n+nc)*(n+nc+max(n+nc,m,p)+7), (m+p)*(m+p+4)); } lsqred = max (1, 2*nc*nc+5*nc); ldwork = 2*nc*nc + max (1, lfreq, lsqred); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int iwarn = 0; int info = 0; // SLICOT routine SB16AD F77_XFCN (sb16ad, SB16AD, (dico, jobc, jobo, jobmr, weight, equil, ordsel, n, m, p, nc, ncr, alpha, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ac.fortran_vec (), ldac, bc.fortran_vec (), ldbc, cc.fortran_vec (), ldcc, dc.fortran_vec (), lddc, ncs, hsvc.fortran_vec (), tol1, tol2, iwork, dwork, ldwork, iwarn, info)); if (f77_exception_encountered) error ("conred: exception in SLICOT subroutine SB16AD"); static const char* err_msg[] = { "0: OK", "1: the closed-loop system is not well-posed; " "its feedthrough matrix is (numerically) singular", "2: the computation of the real Schur form of the " "closed-loop state matrix failed", "3: the closed-loop state matrix is not stable", "4: the solution of a symmetric eigenproblem failed", "5: the computation of the ordered real Schur form " "of Ac failed", "6: the separation of the ALPHA-stable/unstable " "diagonal blocks failed because of very close eigenvalues", "7: the computation of Hankel singular values failed"}; static const char* warn_msg[] = { "0: OK", "1: with ORDSEL = 'F', the selected order NCR is greater " "than NSMIN, the sum of the order of the " "ALPHA-unstable part and the order of a minimal " "realization of the ALPHA-stable part of the given " "controller; in this case, the resulting NCR is set " "equal to NSMIN.", "2: with ORDSEL = 'F', the selected order NCR " "corresponds to repeated singular values for the " "ALPHA-stable part of the controller, which are " "neither all included nor all excluded from the " "reduced model; in this case, the resulting NCR is " "automatically decreased to exclude all repeated " "singular values.", "3: with ORDSEL = 'F', the selected order NCR is less " "than the order of the ALPHA-unstable part of the " "given controller. In this case NCR is set equal to " "the order of the ALPHA-unstable part."}; error_msg ("conred", info, 7, err_msg); warning_msg ("conred", iwarn, 3, warn_msg); // resize ac.resize (ncr, ncr); bc.resize (ncr, p); // p: number of plant outputs cc.resize (m, ncr); // m: number of plant inputs hsvc.resize (ncs); // return values retval(0) = ac; retval(1) = bc; retval(2) = cc; retval(3) = dc; retval(4) = octave_value (ncr); retval(5) = hsvc; retval(6) = octave_value (ncs); } return retval; } control/src/sl_sb16bd.cc0000644000076500000240000001626512273411257014433 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . TODO Uses SLICOT SB16BD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb16bd, SB16BD) (char& DICO, char& JOBD, char& JOBMR, char& JOBCF, char& EQUIL, char& ORDSEL, int& N, int& M, int& P, int& NCR, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* F, int& LDF, double* G, int& LDG, double* DC, int& LDDC, double* HSV, double& TOL1, double& TOL2, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_sb16bd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb16bd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB16BD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 15) { print_usage (); } else { // arguments in char dico; char jobd; char jobmr; char jobcf; char equil; char ordsel; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); const int idico = args(4).int_value (); const int iequil = args(5).int_value (); int ncr = args(6).int_value (); const int iordsel = args(7).int_value (); const int ijobd = args(8).int_value (); const int ijobmr = args(9).int_value (); Matrix f = args(10).matrix_value (); Matrix g = args(11).matrix_value (); const int ijobcf = args(12).int_value (); double tol1 = args(13).double_value (); double tol2 = args(14).double_value (); if (idico == 0) dico = 'C'; else dico = 'D'; if (iequil == 0) equil = 'S'; else equil = 'N'; if (iordsel == 0) ordsel = 'F'; else ordsel = 'A'; if (ijobd == 0) jobd = 'Z'; else jobd = 'D'; if (ijobcf == 0) jobcf = 'L'; else jobcf = 'R'; switch (ijobmr) { case 0: jobmr = 'B'; break; case 1: jobmr = 'F'; break; case 2: jobmr = 'S'; break; case 3: jobmr = 'P'; break; default: error ("__sl_sb16bd__: argument jobmr invalid"); } int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd; if (jobd == 'Z') ldd = 1; else ldd = max (1, p); int ldf = max (1, m); int ldg = max (1, n); int lddc = max (1, m); // arguments out Matrix dc (lddc, p); ColumnVector hsv (n); // workspace int liwork; int pm; int ldwork; int lwr = max (1, n*(2*n+max(n,m+p)+5)+n*(n+1)/2); switch (jobmr) { case 'B': pm = 0; break; case 'F': pm = n; break; default: // if JOBMR = 'S' or 'P' pm = max (1, 2*n); } if (ordsel == 'F' && ncr == n) { liwork = 0; ldwork = p*n; } else if (jobcf == 'L') { liwork = max (pm, m); ldwork = (n+m)*(m+p) + max (lwr, 4*m); } else // if JOBCF = 'R' { liwork = max (pm, p); ldwork = (n+p)*(m+p) + max (lwr, 4*p); } OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int iwarn = 0; int info = 0; // SLICOT routine SB16BD F77_XFCN (sb16bd, SB16BD, (dico, jobd, jobmr, jobcf, equil, ordsel, n, m, p, ncr, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, f.fortran_vec (), ldf, g.fortran_vec (), ldg, dc.fortran_vec (), lddc, hsv.fortran_vec (), tol1, tol2, iwork, dwork, ldwork, iwarn, info)); if (f77_exception_encountered) error ("cfconred: exception in SLICOT subroutine SB16BD"); static const char* err_msg[] = { "0: OK", "1: the reduction of A-L*C to a real Schur form " "failed", "2: the matrix A-L*C is not stable (if DICO = 'C'), " "or not convergent (if DICO = 'D')", "3: the computation of Hankel singular values failed", "4: the reduction of A-B*F to a real Schur form " "failed", "5: the matrix A-B*F is not stable (if DICO = 'C'), " "or not convergent (if DICO = 'D')"}; static const char* warn_msg[] = { "0: OK", "1: with ORDSEL = 'F', the selected order NCR is " "greater than the order of a minimal " "realization of the controller."}; error_msg ("cfconred", info, 5, err_msg); warning_msg ("cfconred", iwarn, 1, warn_msg); // resize a.resize (ncr, ncr); // Ac g.resize (ncr, p); // Bc f.resize (m, ncr); // Cc dc.resize (m, p); // Dc // return values retval(0) = a; retval(1) = g; retval(2) = f; retval(3) = dc; retval(4) = octave_value (ncr); retval(5) = hsv; } return retval; } control/src/sl_sb16cd.cc0000644000076500000240000001511512273411257014425 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . TODO Uses SLICOT SB16CD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: November 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sb16cd, SB16CD) (char& DICO, char& JOBD, char& JOBMR, char& JOBCF, char& ORDSEL, int& N, int& M, int& P, int& NCR, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double* F, int& LDF, double* G, int& LDG, double* HSV, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_sb16cd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sb16cd__, args, nargout, "-*- texinfo -*-\n\ Slicot SB16CD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 13) { print_usage (); } else { // arguments in char dico; char jobd; char jobmr; char jobcf; char ordsel; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); const int idico = args(4).int_value (); int ncr = args(5).int_value (); const int iordsel = args(6).int_value (); const int ijobd = args(7).int_value (); const int ijobmr = args(8).int_value (); Matrix f = args(9).matrix_value (); Matrix g = args(10).matrix_value (); const int ijobcf = args(11).int_value (); double tol = args(12).double_value (); if (idico == 0) dico = 'C'; else dico = 'D'; if (iordsel == 0) ordsel = 'F'; else ordsel = 'A'; if (ijobd == 0) jobd = 'Z'; else jobd = 'D'; if (ijobcf == 0) jobcf = 'L'; else jobcf = 'R'; switch (ijobmr) { case 0: jobmr = 'B'; break; case 1: jobmr = 'F'; break; default: error ("__sl_sb16cd__: argument jobmr invalid"); } int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd; if (jobd == 'Z') ldd = 1; else ldd = max (1, p); int ldf = max (1, m); int ldg = max (1, n); // arguments out ColumnVector hsv (n); // workspace int liwork; if (jobmr == 'B') liwork = 0; else // if JOBMR = 'F' liwork = n; int ldwork; int mp; if (jobcf == 'L') mp = m; else // if JOBCF = 'R' mp = p; ldwork = 2*n*n + max (1, 2*n*n + 5*n, n*max(m,p), n*(n + max(n,mp) + min(n,mp) + 6)); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int iwarn = 0; int info = 0; // SLICOT routine SB16CD F77_XFCN (sb16cd, SB16CD, (dico, jobd, jobmr, jobcf, ordsel, n, m, p, ncr, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, f.fortran_vec (), ldf, g.fortran_vec (), ldg, hsv.fortran_vec (), tol, iwork, dwork, ldwork, iwarn, info)); if (f77_exception_encountered) error ("fwcfconred: exception in SLICOT subroutine SB16CD"); static const char* err_msg[] = { "0: OK", "1: eigenvalue computation failure", "2: the matrix A-L*C is not stable", "3: the matrix A-B*F is not stable", "4: the Lyapunov equation for computing the " "observability Grammian is (nearly) singular", "5: the Lyapunov equation for computing the " "controllability Grammian is (nearly) singular", "6: the computation of Hankel singular values failed"}; static const char* warn_msg[] = { "0: OK", "1: with ORDSEL = 'F', the selected order NCR is " "greater than the order of a minimal realization " "of the controller.", "2: with ORDSEL = 'F', the selected order NCR " "corresponds to repeated singular values, which are " "neither all included nor all excluded from the " "reduced controller. In this case, the resulting NCR " "is set automatically to the largest value such that " "HSV(NCR) > HSV(NCR+1)."}; error_msg ("fwcfconred", info, 6, err_msg); warning_msg ("fwcfconred", iwarn, 2, warn_msg); // resize a.resize (ncr, ncr); // Ac g.resize (ncr, p); // Bc f.resize (m, ncr); // Cc // Dc = 0 // return values retval(0) = a; retval(1) = g; retval(2) = f; retval(3) = octave_value (ncr); retval(4) = hsv; } return retval; } control/src/sl_sg02ad.cc0000644000076500000240000001605112273411257014423 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Solution of algebraic Riccati equations for descriptor systems. Uses SLICOT SG02AD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: October 2010 Version: 0.3 */ #include #include #include "common.h" #include extern "C" { int F77_FUNC (sg02ad, SG02AD) (char& DICO, char& JOBB, char& FACT, char& UPLO, char& JOBL, char& SCAL, char& SORT, char& ACC, int& N, int& M, int& P, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* Q, int& LDQ, double* R, int& LDR, double* L, int& LDL, double& RCONDU, double* X, int& LDX, double* ALFAR, double* ALFAI, double* BETA, double* S, int& LDS, double* T, int& LDT, double* U, int& LDU, double& TOL, int* IWORK, double* DWORK, int& LDWORK, bool* BWORK, int& IWARN, int& INFO); } // PKG_ADD: autoload ("__sl_sg02ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sg02ad__, args, nargout, "-*- texinfo -*-\n\ Slicot SG02AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 8) { print_usage (); } else { // arguments in char dico; char jobb = 'B'; char fact = 'N'; char uplo = 'U'; char jobl; char scal = 'N'; char sort = 'S'; char acc = 'N'; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix q = args(3).matrix_value (); Matrix r = args(4).matrix_value (); Matrix l = args(5).matrix_value (); int discrete = args(6).int_value (); int ijobl = args(7).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; if (ijobl == 0) jobl = 'Z'; else jobl = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = 0; // p: number of outputs, not used because FACT = 'N' int lda = max (1, n); int lde = max (1, n); int ldb = max (1, n); int ldq = max (1, n); int ldr = max (1, m); int ldl = max (1, n); // arguments out double rcondu; int ldx = max (1, n); Matrix x (ldx, n); int nu = 2*n; ColumnVector alfar (nu); ColumnVector alfai (nu); ColumnVector beta (nu); // unused output arguments int lds = max (1, 2*n + m); OCTAVE_LOCAL_BUFFER (double, s, lds * lds); int ldt = max (1, 2*n + m); OCTAVE_LOCAL_BUFFER (double, t, ldt * 2*n); int ldu = max (1, 2*n); OCTAVE_LOCAL_BUFFER (double, u, ldu * 2*n); // tolerance double tol = 0; // use default value // workspace int liwork = max (1, m, 2*n); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); int ldwork = max (7*(2*n + 1) + 16, 16*n, 2*n + m, 3*m); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); OCTAVE_LOCAL_BUFFER (bool, bwork, 2*n); // error indicator int iwarn; int info; // SLICOT routine SG02AD F77_XFCN (sg02ad, SG02AD, (dico, jobb, fact, uplo, jobl, scal, sort, acc, n, m, p, a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, q.fortran_vec (), ldq, r.fortran_vec (), ldr, l.fortran_vec (), ldl, rcondu, x.fortran_vec (), ldx, alfar.fortran_vec (), alfai.fortran_vec (), beta.fortran_vec (), s, lds, t, ldt, u, ldu, tol, iwork, dwork, ldwork, bwork, iwarn, info)); if (f77_exception_encountered) error ("are: __sl_sg02ad__: exception in SLICOT subroutine SG02AD"); static const char* err_msg[] = { "0: OK", "1: the computed extended matrix pencil is singular, " "possibly due to rounding errors", "2: the QZ algorithm failed", "3: reordering of the generalized eigenvalues failed", "4: after reordering, roundoff changed values of " "some complex eigenvalues so that leading eigenvalues " "in the generalized Schur form no longer satisfy the " "stability condition; this could also be caused due " "to scaling", "5: the computed dimension of the solution does not " "equal N", "6: the spectrum is too close to the boundary of " "the stability domain", "7: a singular matrix was encountered during the " "computation of the solution matrix X"}; static const char* warn_msg[] = { "0: OK", "1: solution may be inaccurate due to poor scaling " "or eigenvalues too close to the boundary of the stability domain " "(the imaginary axis, if DICO = 'C', or the unit circle, if DICO = 'D')"}; error_msg ("are", info, 7, err_msg); warning_msg ("are", iwarn, 1, warn_msg); // assemble complex vector - adapted from DEFUN complex in data.cc alfar.resize (n); alfai.resize (n); beta.resize (n); ColumnVector poler (n); ColumnVector polei (n); poler = quotient (alfar, beta); polei = quotient (alfai, beta); ComplexColumnVector pole (n, Complex ()); for (octave_idx_type i = 0; i < n; i++) pole.xelem (i) = Complex (poler(i), polei(i)); // return value retval(0) = x; retval(1) = pole; } return retval; } control/src/sl_sg03ad.cc0000644000076500000240000001003412273411257014417 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Solution of generalized Lyapunov equations. Uses SLICOT SG03AD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: January 2010 Version: 0.2 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sg03ad, SG03AD) (char& DICO, char& JOB, char& FACT, char& TRANS, char& UPLO, int& N, double* A, int& LDA, double* E, int& LDE, double* Q, int& LDQ, double* Z, int& LDZ, double* X, int& LDX, double& SCALE, double& SEP, double& FERR, double* ALPHAR, double* ALPHAI, double* BETA, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sg03ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sg03ad__, args, nargout, "-*- texinfo -*-\n\ Slicot SG03AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 4) { print_usage (); } else { // arguments in char dico; char job = 'X'; char fact = 'N'; char trans = 'T'; char uplo = 'U'; // ?!? Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix x = args(2).matrix_value (); int discrete = args(3).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; int n = a.rows (); // n: number of states int lda = max (1, n); int lde = max (1, n); int ldq = max (1, n); int ldz = max (1, n); int ldx = max (1, n); // arguments out double scale; double sep = 0; double ferr = 0; Matrix q (ldq, n); Matrix z (ldz, n); ColumnVector alphar (n); ColumnVector alphai (n); ColumnVector beta (n); // workspace int* iwork = 0; // not referenced because job = X int ldwork = max (1, 4*n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine SG03AD F77_XFCN (sg03ad, SG03AD, (dico, job, fact, trans, uplo, n, a.fortran_vec (), lda, e.fortran_vec (), lde, q.fortran_vec (), ldq, z.fortran_vec (), ldz, x.fortran_vec (), ldx, scale, sep, ferr, alphar.fortran_vec (), alphai.fortran_vec (), beta.fortran_vec (), iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("lyap: __sl_sg03ad__: exception in SLICOT subroutine SG03AD"); if (info != 0) error ("lyap: __sl_sg03ad__: SG03AD returned info = %d", info); // return values retval(0) = x; retval(1) = octave_value (scale); } return retval; } control/src/sl_sg03bd.cc0000644000076500000240000000747112273411257014433 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Square-root solver for generalized Lyapunov equations. Uses SLICOT SG03BD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2010 Version: 0.2 */ #include #include #include "common.h" extern "C" { int F77_FUNC (sg03bd, SG03BD) (char& DICO, char& FACT, char& TRANS, int& N, int& M, double* A, int& LDA, double* E, int& LDE, double* Q, int& LDQ, double* Z, int& LDZ, double* B, int& LDB, double& SCALE, double* ALPHAR, double* ALPHAI, double* BETA, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_sg03bd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_sg03bd__, args, nargout, "-*- texinfo -*-\n\ Slicot SG03BD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 4) { print_usage (); } else { // arguments in char dico; char fact = 'N'; char trans = 'N'; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); int discrete = args(3).int_value (); if (discrete == 0) dico = 'C'; else dico = 'D'; int n = a.rows (); int m = b.rows (); int lda = max (1, n); int lde = max (1, n); int ldq = max (1, n); int ldz = max (1, n); int ldb = max (1, m, n); int n1 = max (m, n); b.resize (ldb, n1); // arguments out double scale; Matrix q (ldq, n); Matrix z (ldz, n); ColumnVector alphar (n); ColumnVector alphai (n); ColumnVector beta (n); // workspace int ldwork = max (1, 4*n, 6*n-6); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine SG03BD F77_XFCN (sg03bd, SG03BD, (dico, fact, trans, n, m, a.fortran_vec (), lda, e.fortran_vec (), lde, q.fortran_vec (), ldq, z.fortran_vec (), ldz, b.fortran_vec (), ldb, scale, alphar.fortran_vec (), alphai.fortran_vec (), beta.fortran_vec (), dwork, ldwork, info)); if (f77_exception_encountered) error ("lyap: __sl_sg03bd__: exception in SLICOT subroutine SG03BD"); if (info != 0) error ("lyap: __sl_sg03bd__: SG03BD returned info = %d", info); // resize b.resize (n, n); // return values retval(0) = b; // b has been overwritten by cholesky factor u retval(1) = octave_value (scale); } return retval; } control/src/sl_tb01id.cc0000644000076500000240000000575312273411257014435 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Balance state-space model. Uses SLICOT TB01ID by courtesy of NICONET e.V. Author: Lukas Reichlin Created: May 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tb01id, TB01ID) (char& JOB, int& N, int& M, int& P, double& MAXRED, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* SCALE, int& INFO); } // PKG_ADD: autoload ("__sl_tb01id__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tb01id__, args, nargout, "-*- texinfo -*-\n\ Slicot TB01ID Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 4) { print_usage (); } else { // arguments in char job = 'A'; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); double maxred = args(3).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); // arguments out ColumnVector scale (n); // error indicators int info = 0; // SLICOT routine TB01ID F77_XFCN (tb01id, TB01ID, (job, n, m, p, maxred, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, scale.fortran_vec (), info)); if (f77_exception_encountered) error ("ss: prescale: __sl_tb01id__: exception in SLICOT subroutine TB01ID"); if (info != 0) error ("ss: prescale: __sl_tb01id__: TB01ID returned info = %d", info); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = octave_value (maxred); retval(4) = scale; } return retval; } control/src/sl_tb01pd.cc0000644000076500000240000000726512273411257014444 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Minimal realization of state-space models. Uses SLICOT TB01PD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2010 Version: 0.4 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tb01pd, TB01PD) (char& JOB, char& EQUIL, int& N, int& M, int& P, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, int& NR, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tb01pd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tb01pd__, args, nargout, "-*- texinfo -*-\n\ Slicot TB01PD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in char job = 'M'; char equil; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); double tol = args(3).double_value (); const int scaled = args(4).int_value (); if (scaled == 0) equil = 'S'; else equil = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc; if (n == 0) ldc = 1; else ldc = max (1, m, p); b.resize (ldb, max (m, p)); c.resize (ldc, n); // arguments out int nr = 0; // workspace int liwork = n + max (m, p); int ldwork = max (1, n + max (n, 3*m, 3*p)); OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info = 0; // SLICOT routine TB01PD F77_XFCN (tb01pd, TB01PD, (job, equil, n, m, p, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, nr, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("ss: minreal: __sl_tb01pd__: exception in SLICOT subroutine TB01PD"); if (info != 0) error ("ss: minreal: __sl_tb01pd__: TB01PD returned info = %d", info); // resize a.resize (nr, nr); b.resize (nr, m); c.resize (p, nr); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = octave_value (nr); } return retval; } control/src/sl_tb01ud.cc0000644000076500000240000000727212273411257014447 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Orthogonal canonical form. Uses SLICOT TB01UD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: October 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tb01ud, TB01UD) (char& JOBZ, int& N, int& M, int& P, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, int& NCONT, int& INDCON, int* NBLK, double* Z, int& LDZ, double* TAU, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tb01ud__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tb01ud__, args, nargout, "-*- texinfo -*-\n\ Slicot TB01UD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 4) { print_usage (); } else { // arguments in char jobz = 'I'; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); double tol = args(3).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldz = max (1, n); // arguments out Matrix z (ldz, n); int ncont; int indcon; OCTAVE_LOCAL_BUFFER (int, nblk, n); OCTAVE_LOCAL_BUFFER (double, tau, n); // workspace int ldwork = max (1, n, 3*m, p); OCTAVE_LOCAL_BUFFER (int, iwork, m); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info; // SLICOT routine TB01UD F77_XFCN (tb01ud, TB01UD, (jobz, n, m, p, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, ncont, indcon, nblk, z.fortran_vec (), ldz, tau, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("__sl_tb01ud__: exception in SLICOT subroutine TB01UD"); if (info != 0) error ("__sl_tb01ud__: TB01UD returned info = %d", info); // resize a.resize (n, n); b.resize (n, m); c.resize (p, n); z.resize (n, n); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = z; retval(4) = octave_value (ncont); } return retval; } control/src/sl_tb04bd.cc0000644000076500000240000001067412273411257014427 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Transfer matrix of a given state-space representation (A,B,C,D), using the pole-zeros method. Uses SLICOT TB04BD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: October 2010 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tb04bd, TB04BD) (char& JOBD, char& ORDER, char& EQUIL, int& N, int& M, int& P, int& MD, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, int* IGN, int& LDIGN, int* IGD, int& LDIGD, double* GN, double* GD, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tb04bd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tb04bd__, args, nargout, "-*- texinfo -*-\n\ Slicot TB04BD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in char jobd = 'D'; char order = 'D'; char equil; Matrix a = args(0).matrix_value (); Matrix b = args(1).matrix_value (); Matrix c = args(2).matrix_value (); Matrix d = args(3).matrix_value (); const int scaled = args(4).int_value (); if (scaled == 0) equil = 'S'; else equil = 'N'; int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int md = n + 1; int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldd = max (1, p); // arguments out int ldign = max (1, p); int ldigd = max (1, p); int lg = p * m * md; OCTAVE_LOCAL_BUFFER (int, ign, ldign*m); OCTAVE_LOCAL_BUFFER (int, igd, ldigd*m); Matrix ignm (ldign, m); Matrix igdm (ldigd, m); RowVector gn (lg); RowVector gd (lg); // tolerance double tol = 0; // use default value // workspace int ldwork = max (1, n*(n + p) + max (n + max (n, p), n*(2*n + 5))); OCTAVE_LOCAL_BUFFER (int, iwork, n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine TB04BD F77_XFCN (tb04bd, TB04BD, (jobd, order, equil, n, m, p, md, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, ign, ldign, igd, ldigd, gn.fortran_vec (), gd.fortran_vec (), tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("ss2tf: __sl_tb04bd__: exception in SLICOT subroutine TB04BD"); if (info != 0) error ("ss2tf: __sl_tb04bd__: TB04BD returned info = %d", info); for (octave_idx_type i = 0; i < ldign*m; i++) ignm.xelem (i) = ign[i]; for (octave_idx_type i = 0; i < ldigd*m; i++) igdm.xelem (i) = igd[i]; // return values retval(0) = gn; retval(1) = gd; retval(2) = ignm; retval(3) = igdm; retval(4) = octave_value (md); retval(5) = octave_value (p); retval(6) = octave_value (m); } return retval; } control/src/sl_td04ad.cc0000644000076500000240000001045412273411257014424 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Minimal state-space representation (A,B,C,D) for a proper transfer matrix T(s) given as either row or column polynomial vectors over denominator polynomials, possibly with uncancelled common terms. Uses SLICOT TD04AD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: August 2011 Version: 0.2 */ #include #include #include "common.h" extern "C" { int F77_FUNC (td04ad, TD04AD) (char& ROWCOL, int& M, int& P, int* INDEX, double* DCOEFF, int& LDDCOE, double* UCOEFF, int& LDUCO1, int& LDUCO2, int& NR, double* A, int& LDA, double* B, int& LDB, double* C, int& LDC, double* D, int& LDD, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_td04ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_td04ad__, args, nargout, "-*- texinfo -*-\n\ Slicot TD04AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 4) { print_usage (); } else { // arguments in char rowcol = 'R'; NDArray ucoeff = args(0).array_value (); Matrix dcoeff = args(1).matrix_value (); Matrix indexd = args(2).matrix_value (); double tol = args(3).double_value (); int p = ucoeff.rows (); // p: number of outputs int m = ucoeff.columns (); // m: number of inputs int lddcoe = max (1, p); // TODO: handle case ucoeff.rows = 0 int lduco1 = max (1, p); int lduco2 = max (1, m); int n = 0; OCTAVE_LOCAL_BUFFER (int, index, p); for (octave_idx_type i = 0; i < p; i++) { index[i] = indexd.xelem (i); n += index[i]; } // arguments out int nr = max (1, n); // initialize to prevent crash if info != 0 int lda = max (1, n); int ldb = max (1, n); int ldc = max (1, m, p); int ldd = max (1, p); Matrix a (lda, n); Matrix b (ldb, max (m, p)); Matrix c (ldc, n); Matrix d (ldd, m); // workspace int ldwork = max (1, n + max (n, 3*m, 3*p)); OCTAVE_LOCAL_BUFFER (int, iwork, n + max (m, p)); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicator int info; // SLICOT routine TD04AD F77_XFCN (td04ad, TD04AD, (rowcol, m, p, index, dcoeff.fortran_vec (), lddcoe, ucoeff.fortran_vec (), lduco1, lduco2, nr, a.fortran_vec (), lda, b.fortran_vec (), ldb, c.fortran_vec (), ldc, d.fortran_vec (), ldd, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("tf2ss: __sl_td04ad__: exception in SLICOT subroutine TD04AD"); if (info != 0) error ("tf2ss: __sl_td04ad__: TD04AD returned info = %d", info); // resize a.resize (nr, nr); b.resize (nr, m); c.resize (p, nr); d.resize (p, m); // return values retval(0) = a; retval(1) = b; retval(2) = c; retval(3) = d; } return retval; } control/src/sl_tg01ad.cc0000644000076500000240000000657512273411257014435 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Balance descriptor state-space model. Uses SLICOT TG01AD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: June 2011 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tg01ad, TG01AD) (char& JOB, int& L, int& N, int& M, int& P, double& TRESH, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* C, int& LDC, double* LSCALE, double *RSCALE, double* DWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tg01ad__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tg01ad__, args, nargout, "-*- texinfo -*-\n\ Slicot TG01AD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in char job = 'A'; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); double tresh = args(4).double_value (); int l = a.rows (); int n = a.columns (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, l); int lde = max (1, l); int ldb = max (1, l); int ldc = max (1, p); // arguments out ColumnVector lscale (l); ColumnVector rscale (n); // workspace OCTAVE_LOCAL_BUFFER (double, dwork, 3*(l+n)); // error indicators int info = 0; // SLICOT routine TG01AD F77_XFCN (tg01ad, TG01AD, (job, l, n, m, p, tresh, a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, c.fortran_vec (), ldc, lscale.fortran_vec (), rscale.fortran_vec (), dwork, info)); if (f77_exception_encountered) error ("ss: prescale: __sl_tg01ad__: exception in SLICOT subroutine TG01AD"); if (info != 0) error ("ss: prescale: __sl_tg01ad__: TG01AD returned info = %d", info); // return values retval(0) = a; retval(1) = e; retval(2) = b; retval(3) = c; retval(4) = lscale; retval(5) = rscale; } return retval; } control/src/sl_tg01fd.cc0000644000076500000240000001044612273411257014432 0ustar lukasstaff/* Copyright (C) 2013 Thomas Vasileiou This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Orthogonal reduction of a descriptor system to a SVD-like coordinate form. Uses SLICOT TG01FD by courtesy of NICONET e.V. Author: Thomas Vasileiou Created: September 2013 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tg01fd, TG01FD) (char& COMPQ, char& COMPZ, char& JOBA, int& L, int& N, int& M, int& P, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* C, int& LDC, double* Q, int& LDQ, double* Z, int& LDZ, int& RANKE, int& RNKA22, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tg01fd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tg01fd__, args, nargout, "-*- texinfo -*-\n\ Slicot TG01FD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 6) { print_usage (); } else { // arguments in char compq; char compz; char joba = 'T'; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); const int qz_flag = args(4).int_value (); double tol = args(5).double_value (); if (qz_flag == 0) { compq = 'N'; compz = 'N'; } else { compq = 'I'; compz = 'I'; } int l = a.rows (); int n = l; int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, l); int lde = max (1, l); int ldb = max (1, l); int ldc = max (1, p); int ldq = max (1, l); int ldz = max (1, n); // arguments out Matrix q(l, l, 0.); Matrix z(n, n, 0.); Matrix empty(0, 0); int ranke, rnka22; // workspace int ldwork = max (1, n+p, min (l,n) + max (3*n-1, m, l)); OCTAVE_LOCAL_BUFFER (int, iwork, n); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info = 0; // SLICOT routine TG01FD F77_XFCN (tg01fd, TG01FD, (compq, compz, joba, l, n, m, p, a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, c.fortran_vec (), ldc, q.fortran_vec (), ldq, z.fortran_vec (), ldz, ranke, rnka22, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("__sl_tg01fd__: exception in SLICOT subroutine TG01FD"); if (info != 0) error ("__sl_tg01fd__: TG01FD returned info = %d", info); // return values retval(0) = a; retval(1) = e; retval(2) = b; retval(3) = c; retval(4) = octave_value (ranke); retval(5) = octave_value (rnka22); if (qz_flag == 0) { retval(6) = empty; retval(7) = empty; } else { retval(6) = q; retval(7) = z; } } return retval; } control/src/sl_tg01hd.cc0000644000076500000240000001010112273411257014420 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Staircase controllability form for descriptor models. Uses SLICOT TG01HD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2010 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tg01hd, TG01HD) (char& JOBCON, char& COMPQ, char& COMPZ, int& N, int& M, int& P, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* C, int& LDC, double* Q, int& LDQ, double* Z, int& LDZ, int& NCONT, int& NIUCON, int& NRBLCK, int* RTAU, double& TOL, int* IWORK, double* DWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tg01hd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tg01hd__, args, nargout, "Slicot TG01HD Release 5.0") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in char jobcon = 'C'; char compq = 'I'; char compz = 'I'; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); double tol = args(4).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int lde = max (1, n); int ldb = max (1, n); int ldc = max (1, p); int ldq = max (1, n); int ldz = max (1, n); // arguments out Matrix q (ldq, n); Matrix z (ldz, n); int ncont; int niucon; int nrblck; OCTAVE_LOCAL_BUFFER (int, rtau, n); // workspace int ldwork = max (n, 2*m); OCTAVE_LOCAL_BUFFER (int, iwork, m); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info; // SLICOT routine TG01HD F77_XFCN (tg01hd, TG01HD, (jobcon, compq, compz, n, m, p, a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, c.fortran_vec (), ldc, q.fortran_vec (), ldq, z.fortran_vec (), ldz, ncont, niucon, nrblck, rtau, tol, iwork, dwork, info)); if (f77_exception_encountered) error ("__sl_tg01hd__: exception in SLICOT subroutine TG01HD"); if (info != 0) error ("__sl_tg01hd__: TG01HD returned info = %d", info); // resize a.resize (n, n); e.resize (n, n); b.resize (n, m); c.resize (p, n); q.resize (n, n); z.resize (n, n); // return values retval(0) = a; retval(1) = e; retval(2) = b; retval(3) = c; retval(4) = q; retval(5) = z; retval(6) = octave_value (ncont); } return retval; } control/src/sl_tg01id.cc0000644000076500000240000001014712273411257014433 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Staircase observability form for descriptor models. Uses SLICOT TG01ID by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2010 Version: 0.1 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tg01id, TG01ID) (char& JOBOBS, char& COMPQ, char& COMPZ, int& N, int& M, int& P, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* C, int& LDC, double* Q, int& LDQ, double* Z, int& LDZ, int& NOBSV, int& NIUOBS, int& NLBLCK, int* CTAU, double& TOL, int* IWORK, double* DWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tg01id__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tg01id__, args, nargout, "Slicot TG01ID Release 5.0") { int nargin = args.length (); octave_value_list retval; if (nargin != 5) { print_usage (); } else { // arguments in char jobobs = 'O'; char compq = 'I'; char compz = 'I'; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); double tol = args(4).double_value (); int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int lde = max (1, n); int ldb = max (1, n); int ldc = max (1, m, p); int ldq = max (1, n); int ldz = max (1, n); b.resize (ldb, max (m, p)); // arguments out Matrix q (ldq, n); Matrix z (ldz, n); int nobsv; int niuobs; int nlblck; OCTAVE_LOCAL_BUFFER (int, ctau, n); // workspace int ldwork = max (n, 2*p); OCTAVE_LOCAL_BUFFER (int, iwork, p); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info; // SLICOT routine TG01ID F77_XFCN (tg01id, TG01ID, (jobobs, compq, compz, n, m, p, a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, c.fortran_vec (), ldc, q.fortran_vec (), ldq, z.fortran_vec (), ldz, nobsv, niuobs, nlblck, ctau, tol, iwork, dwork, info)); if (f77_exception_encountered) error ("__sl_tg01id__: exception in SLICOT subroutine TG01ID"); if (info != 0) error ("__sl_tg01id__: TG01ID returned info = %d", info); // resize a.resize (n, n); e.resize (n, n); b.resize (n, m); c.resize (p, n); q.resize (n, n); z.resize (n, n); // return values retval(0) = a; retval(1) = e; retval(2) = b; retval(3) = c; retval(4) = q; retval(5) = z; retval(6) = octave_value (nobsv); } return retval; } control/src/sl_tg01jd.cc0000644000076500000240000001355312273411257014440 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Minimal realization of descriptor state-space models. Uses SLICOT TG01JD by courtesy of NICONET e.V. Author: Lukas Reichlin Created: September 2010 Version: 0.4 */ #include #include #include "common.h" extern "C" { int F77_FUNC (tg01jd, TG01JD) (char& JOB, char& SYSTYP, char& EQUIL, int& N, int& M, int& P, double* A, int& LDA, double* E, int& LDE, double* B, int& LDB, double* C, int& LDC, int& NR, int* INFRED, double& TOL, int* IWORK, double* DWORK, int& LDWORK, int& INFO); } // PKG_ADD: autoload ("__sl_tg01jd__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tg01jd__, args, nargout, "-*- texinfo -*-\n\ Slicot TG01JD Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 8) { print_usage (); } else { // arguments in char job; char systyp; char equil; Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); double tol = args(4).double_value (); const int scaled = args(5).int_value (); const int ijob = args(6).int_value (); const int isystyp = args(7).int_value (); if (scaled == 0) equil = 'S'; else equil = 'N'; switch (ijob) { case 0: job = 'I'; break; case 1: job = 'C'; break; case 2: job = 'O'; break; default: error ("__sl_tg01jd__: argument job invalid"); } switch (isystyp) { case 0: systyp = 'R'; break; case 1: systyp = 'S'; break; case 2: systyp = 'P'; break; default: error ("__sl_tg01jd__: argument systyp invalid"); } int n = a.rows (); // n: number of states int m = b.columns (); // m: number of inputs int p = c.rows (); // p: number of outputs int lda = max (1, n); int lde = max (1, n); int ldb = max (1, n); int ldc; if (n == 0) ldc = 1; else ldc = max (1, m, p); a.resize (lda, n); e.resize (lde, n); if (job == 'C') b.resize (ldb, m); else b.resize (ldb, max (m, p)); c.resize (ldc, n); // arguments out int nr; int infred[7]; // workspace int liwork = n + max (m, p); int ldwork; // int ldwork = max (n, 2*m, 2*p); // int ldwork = n * (2*n + m + p) + max (n, 2*m, 2*p); if (equil == 'S') ldwork = max (8*n, 2*m, 2*p); else // if EQUIL = 'N' ldwork = max (n, 2*m, 2*p); // FIXME: larger ldwork should give better results, // but it breaks the test that Slicot provides. /* LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX(8*N,2*M,2*P), if EQUIL = 'S'; LDWORK >= MAX(N,2*M,2*P), if EQUIL = 'N'. If LDWORK >= MAX(2*N*N+N*M+N*P)+MAX(N,2*M,2*P) then more accurate results are to be expected by performing only those reductions phases (see METHOD), where effective order reduction occurs. This is achieved by saving the system matrices before each phase and restoring them if no order reduction took place. */ OCTAVE_LOCAL_BUFFER (int, iwork, liwork); OCTAVE_LOCAL_BUFFER (double, dwork, ldwork); // error indicators int info = 0; // SLICOT routine TG01JD F77_XFCN (tg01jd, TG01JD, (job, systyp, equil, n, m, p, a.fortran_vec (), lda, e.fortran_vec (), lde, b.fortran_vec (), ldb, c.fortran_vec (), ldc, nr, infred, tol, iwork, dwork, ldwork, info)); if (f77_exception_encountered) error ("dss: minreal: __sl_tg01jd__: exception in SLICOT subroutine TG01JD"); if (info != 0) error ("dss: minreal: __sl_tg01jd__: TG01JD returned info = %d", info); // resize a.resize (nr, nr); e.resize (nr, nr); b.resize (nr, m); c.resize (p, nr); // return values retval(0) = a; retval(1) = e; retval(2) = b; retval(3) = c; retval(4) = octave_value (nr); } return retval; } control/src/sl_tg04bx.cc0000644000076500000240000000674312273411257014462 0ustar lukasstaff/* Copyright (C) 2009-2014 Lukas F. Reichlin This file is part of LTI Syncope. LTI Syncope is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. LTI Syncope is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with LTI Syncope. If not, see . Gain of descriptor state-space models. Based on SLICOT TB04BX.f. Author: Lukas Reichlin Created: March 2011 Version: 0.1 */ #include #include #include "common.h" #include #include extern "C" { int F77_FUNC (tg04bx, TG04BX) (int& IP, int& IZ, double* A, int& LDA, double* E, double* B, double* C, double* D, double* PR, double* PI, double* ZR, double* ZI, double& GAIN, int* IWORK); } // PKG_ADD: autoload ("__sl_tg04bx__", "__control_slicot_functions__.oct"); DEFUN_DLD (__sl_tg04bx__, args, nargout, "-*- texinfo -*-\n\ Slicot TG04BX Release 5.0\n\ No argument checking.\n\ For internal use only.") { int nargin = args.length (); octave_value_list retval; if (nargin != 9) { print_usage (); } else { // arguments in Matrix a = args(0).matrix_value (); Matrix e = args(1).matrix_value (); Matrix b = args(2).matrix_value (); Matrix c = args(3).matrix_value (); Matrix d = args(4).matrix_value (); ColumnVector pr = args(5).column_vector_value (); ColumnVector pi = args(6).column_vector_value (); ColumnVector zr = args(7).column_vector_value (); ColumnVector zi = args(8).column_vector_value (); int n = a.rows (); // n: number of states int ip = pr.length (); // ip: number of finite poles int iz = zr.length (); // iz: number of zeros // For ss, IP = n is always true. // However, dss models with poles at infinity // (filtered by pole.m) may have IP <= n // Take pr.length == pi.length == ip for granted, // and the same for iz, zr and zi. int lda = max (1, n); // arguments out double gain; // workspace OCTAVE_LOCAL_BUFFER (int, iwork, lda); F77_XFCN (tg04bx, TG04BX, (ip, iz, a.fortran_vec (), lda, e.fortran_vec (), b.fortran_vec (), c.fortran_vec (), d.fortran_vec (), pr.fortran_vec (), pi.fortran_vec (), zr.fortran_vec (), zi.fortran_vec (), gain, iwork)); if (f77_exception_encountered) error ("dss: zero: __sl_tg04bx__: exception in TG04BX"); // return values retval(0) = octave_value (gain); } return retval; } control/src/slicot.tar.gz0000644000076500000240001156520412273411257014775 0ustar lukasstaff}Lslicot.taris9/k~yq-"gbw%kOLܠ%Y2K6)Z\5rĚH$@/G?֟P=Y¿"KbW8L_?7]/Φ_-.'hϚm"0"u_+e~_4jH}q'5`'I!Wпw>A#5OP ֽb&Q4*PQfF1&]deH{ VߋA[[Qjs|TQ!Irn(9Dd[L<%AWOd&,E['>66nEBTu<4ad$ ^;I17E02abA2*ֺ:"okJlFf WWSQ1g_[@UIaEe*pw+F+(EQFj PhvQBlFYs{?ߚͳiB:?aUzo..((uZu-e)2a(oh «KiGaaEᴨ,I(ͭlh.R@I% 3M%ͰS[ CfjStykN=E~\Iv,48]S݌3Օ t$wઽYJj:foYisWZ~~B\.QD>}95j}a8jW֜m??R֜'c$s+R2۳(K#$N#c|)[D|n`7fEP灪4 M0E4ҍFR{%ܣT GxN'8UCtVn@`vd8X/%'gY0W̜h)W]{"*kH;ARVL CuhLH$zpvDB%60vgyj'4݉e~@ɈB90ZS;HHN"PYS2ӑj$K<`ka8ۈ2ŚjEK԰Gńv"6,9q:-SdZfȠ0N"6r{DJ'qǐxjj9J%!pdtNh{lbYl&Ke#pL-"ˇDy{$b:D9&y/CaIPRkpĉɼUC*PzX8US'Ղ<*Qn&C+,tf iQQ0%,jEQX&)I(HfZW|&: *F\MK %#MN 4D"{R' 5 X#qb927JLFIIvL}/*ڧ}:5)uUS;FnMc #TtA1MGFHͭA ZBhM#["#@&m)yT諨.Pșۥ ̷nWI\O-"c@z,raYΏp:2"8u3ޓ̎| e'Z{tjGB_R"Ŀ$q1S% ]ʭלZ3~~ZZ)M0(C-05V>IlgAI?fG#N&iGPP؄%m-Soyfa!M4rgXIdR2+rĻa5#+ QfΜAKHwq\ [Q( `ݢFZY-xhO37\-X[dw1P*hX&F\*v"z2{Xi! :罢0 FJ2]4J\s]R""V6Sv%Xbyh,vl@2;hIlg}f`g,tPv4D` ͻiz/i͑8ب %;/,9%ױdNm$54:Kh 6r UbUZa4bqN%%Ӑ9o9CןQO]1qꞪּ~1f&sOb`ׄQ` R!M0nHFT'@6UTL8u \]YaP J[zA`YecR ׮ТKjߵ Ro'3`#a PmC: xQo=# ajR9tȧƞPQ 1Q0DPL ҏO!KO4bunr! !dVTl e!B.4G@45g-VX&YdNF>ED,wN2C *"7ȉC}ӈpWQcBfd]-T;+‰tKB3qX:/op@ca 5LGamy#aC`SR"w} HUGb} AŽfIG" D!,@E67CFF4C;TϭtdKS0\ܶ=L2$FⲚ>8.z!u)16Z];vRK2L U—d\C4^ vA$\hYVIXWe$%BTN~ArtìCdF1Bn*&OIT\u.@R/N-efFhy.n;-PpADUfA[Ye1r$оoI4ȵNC+a 3 GMMJ+4 eQ3 'G0AVF"AEKlFlm2rܵ-]Vv $IjNc `!5XaYcd6-Th\GXibY`W}?D#3'$T]VeIkV<6jG$mT=t70gXv= Si~Rlq C(B%M 2sW JO \T8-Mv@^JPٕvEj;ҩ( UAfhæuܸ5U͡gjAE5qrT0"ʝ1h" *,:v@cg_J{`QV+UZQE-SDA`S(u*-<_> ~ceVKOf>=Ѳ^x\rfcCJ'B(v>`/vZ>IdKfJ1$(ʐ~GaFX~u,e9-tҫjG~oe[G'Y4|*1+qAqO☍YިYGfH<1Rro3.٥اkU{ףK ^z>='KfKo ]!$eGEt}zGx[؃0MAe>Eyy1ur Kx{(,R>bqG3yi0զZ2-H9Ze>d(aO1Xp.8dEZ'n0Jks&]&q)3B,oZ lE7BVscH,et"TwZn%3ԥZKy.NK )GGĵ% /d8`(=䁪g> mĩy! 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Db n6ǭʫs꼭jV?@S'2]E4}.DK/ko+Ƈz5 |gGGE6C<=HD?e??PV__uMVK^Bx}Uw%! ;qr9]Va. a)|iEB͒wс:Z*ϨbmRm '{qi H1Z K?Ke"|&Wb?b)-œeѤ#@ZjO]?\@jv `+pz v@ TG9XFb9Y 6_k b,ڱJ ͘Μ/\73?ԒBo% Jh%<ZW>NZ}PӈkM1_^LF)yJ5P59*8<Ɛ}h\w[9b+߄<^~0=0ƌ W'ДAq|Aj L ^)@ٔ7B ڤy6. C C PURPOSE C C To compute the gain of a single-input single-output linear system, C given its state-space representation (A,b,c,d), and its poles and C zeros. The matrix A is assumed to be in an upper Hessenberg form. C The gain is computed using the formula C C -1 IP IZ C g = (c*( S0*E - A ) *b + d)*Prod( S0 - Pi )/Prod( S0 - Zi ) , C i=1 i=1 (1) C C where Pi, i = 1 : IP, and Zj, j = 1 : IZ, are the poles and zeros, C respectively, and S0 is a real scalar different from all poles and C zeros. C C ARGUMENTS C C Input/Output Parameters C C IP (input) INTEGER C The number of the system poles. IP >= 0. C C IZ (input) INTEGER C The number of the system zeros. IZ >= 0. C C A (input/output) DOUBLE PRECISION array, dimension (LDA,IP) C On entry, the leading IP-by-IP part of this array must C contain the state dynamics matrix A in an upper Hessenberg C form. The elements below the second diagonal are not C referenced. C On exit, the leading IP-by-IP upper Hessenberg part of C this array contains the LU factorization of the matrix C A - S0*I, as computed by SLICOT Library routine MB02SD. C C LDA INTEGER C The leading dimension of array A. LDA >= max(1,IP). C C B (input/output) DOUBLE PRECISION array, dimension (IP) C On entry, this array must contain the system input C vector b. C On exit, this array contains the solution of the linear C system ( A - S0*I )x = b . C C C (input) DOUBLE PRECISION array, dimension (IP) C This array must contain the system output vector c. C C D (input) DOUBLE PRECISION C The variable must contain the system feedthrough scalar d. C C PR (input) DOUBLE PRECISION array, dimension (IP) C This array must contain the real parts of the system C poles. Pairs of complex conjugate poles must be stored in C consecutive memory locations. C C PI (input) DOUBLE PRECISION array, dimension (IP) C This array must contain the imaginary parts of the system C poles. C C ZR (input) DOUBLE PRECISION array, dimension (IZ) C This array must contain the real parts of the system C zeros. Pairs of complex conjugate zeros must be stored in C consecutive memory locations. C C ZI (input) DOUBLE PRECISION array, dimension (IZ) C This array must contain the imaginary parts of the system C zeros. C C GAIN (output) DOUBLE PRECISION C The gain of the linear system (A,b,c,d), given by (1). C C Workspace C C IWORK INTEGER array, dimension (IP) C On exit, it contains the pivot indices; for 1 <= i <= IP, C row i of the matrix A - S0*I was interchanged with C row IWORK(i). C C METHOD C C The routine implements the method presented in [1]. A suitable C value of S0 is chosen based on the system poles and zeros. C Then, the LU factorization of the upper Hessenberg, nonsingular C matrix A - S0*I is computed and used to solve the linear system C in (1). C C REFERENCES C C [1] Varga, A. and Sima, V. C Numerically Stable Algorithm for Transfer Function Matrix C Evaluation. C Int. J. Control, vol. 33, nr. 6, pp. 1123-1133, 1981. C C NUMERICAL ASPECTS C C The algorithm is numerically stable in practice and requires C O(IP*IP) floating point operations. C C CONTRIBUTORS C C V. Sima, Research Institute for Informatics, Bucharest, May 2002. C Partly based on the BIMASC Library routine GAIN by A. Varga. C C REVISIONS C C 2011-02-08 (Lukas Reichlin) Modifications for Descriptor Systems. C C KEYWORDS C C Eigenvalue, state-space representation, transfer function, zeros. C C ****************************************************************** C C .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, P1, ONEP1 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, $ P1 = 0.1D0, ONEP1 = 1.1D0 ) C .. Scalar Arguments .. DOUBLE PRECISION D, GAIN INTEGER IP, IZ, LDA C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), E(LDA,*), B(*), C(*), PI(*), PR(*), $ ZI(*), ZR(*) INTEGER IWORK(*) C .. Local Scalars .. INTEGER I, J, INFO DOUBLE PRECISION S0, S C .. External Functions .. DOUBLE PRECISION DDOT EXTERNAL DDOT C .. External Subroutines .. EXTERNAL MB02RD, MB02SD C .. Intrinsic Functions .. INTRINSIC ABS, MAX C .. C .. Executable Statements .. C C For efficiency, the input scalar parameters are not checked. C C Quick return if possible. C C IF( IP.EQ.0 ) THEN C GAIN = ZERO C RETURN C END IF C C Compute a suitable value for S0 . C S0 = ZERO C DO 10 I = 1, IP S = ABS( PR(I) ) IF ( PI(I).NE.ZERO ) $ S = S + ABS( PI(I) ) S0 = MAX( S0, S ) 10 CONTINUE C DO 20 I = 1, IZ S = ABS( ZR(I) ) IF ( ZI(I).NE.ZERO ) $ S = S + ABS( ZI(I) ) S0 = MAX( S0, S ) 20 CONTINUE C S0 = TWO*S0 + P1 IF ( S0.LE.ONE ) $ S0 = ONEP1 C C Form A - S0*E . C DO 30 J = 1, LDA DO 25 I = 1, LDA A(I,J) = A(I,J) - S0*E(I,J) 25 CONTINUE 30 CONTINUE C C Compute the LU factorization of the matrix A - S0*E C (guaranteed to be nonsingular). C C CALL MB02SD( IP, A, LDA, IWORK, INFO ) CALL MB02SD( LDA, A, LDA, IWORK, INFO ) C C Solve the linear system (A - S0*E)*x = b . C C CALL MB02RD( 'No Transpose', IP, 1, A, LDA, IWORK, B, IP, INFO ) C CALL MB02RD( 'No Transpose', IP, 1, A, LDA, IWORK, B, LDA, INFO ) CALL MB02RD( 'No Transpose', LDA, 1, A, LDA, IWORK, B, LDA, INFO ) C -1 C Compute c*(S0*E - A) *b + d . C C GAIN = D - DDOT( IP, C, 1, B, 1 ) GAIN = D - DDOT( LDA, C, 1, B, 1 ) C C Multiply by the products in terms of poles and zeros in (1). C I = 1 C C WHILE ( I <= IP ) DO C 40 IF ( I.LE.IP ) THEN IF ( PI(I).EQ.ZERO ) THEN GAIN = GAIN*( S0 - PR(I) ) I = I + 1 ELSE GAIN = GAIN*( S0*( S0 - TWO*PR(I) ) + PR(I)**2 + PI(I)**2 ) I = I + 2 END IF GO TO 40 END IF C C END WHILE 40 C I = 1 C C WHILE ( I <= IZ ) DO C 50 IF ( I.LE.IZ ) THEN IF ( ZI(I).EQ.ZERO ) THEN GAIN = GAIN/( S0 - ZR(I) ) I = I + 1 ELSE GAIN = GAIN/( S0*( S0 - TWO*ZR(I) ) + ZR(I)**2 + ZI(I)**2 ) I = I + 2 END IF GO TO 50 END IF C C END WHILE 50 C RETURN C *** Last line of TG04BX *** END