MEM Chapter 4b. LMI Lab commands

Transcription

1 1 MEM8-7 Chapter 4b LMI Lab commands setlms lmvar lmterm getlms lmedt lmnbr matnbr lmnfo feasp dec2mat evallm showlm setmvar mat2dec mncx dellm delmvar gevp

2 2 Intalzng the LMI System he descrpton of an LMI system should begn wth setlms and end wth getlms. he functon setlms ntalzes the LMI system descrpton. When specfyng a new system, type setlms([]) whch resets the nternal varbales used for creatng LMIs so that you can create a new system of LMIs. o add on to an exstng LMI system wth nternal representaton LMI SYS, type setlms(lmi SYS) whch uses an exstng LMI system LMISYS as the startng pont for creatng a new system of LMIs. Subsequent commands wll add to the LMI system LMISYS to construct a new set of LMIs. Example: %Create A'*X +X*A < setlms([]) X = lmvar(1,[2,1]) lmterm([1 1 1 X],A',1,'s') lmsys = getlms; % A'*X +X*A

3 3 Specfyng the LMI Varables he matrx varables are declared one at a tme wth lmvar whch creates a new matrx-valued varable n LMI system. X=lmvar(YPE,SRUC) adds a new matrx varable X to the LMI system currently specfed (see setlms). YPE 1 symmetrc block dagonal structure: D1 D2 X.. Dr where D s square and ether zero, a full symmetrc matrx, or a scalar matrx D d I, d R. D s determned by ts correspondng SRUC [ j, k ], where j s the sze of D and k specfes the structure: k D s a scalar matrx, k 1 D s a full symmetrc matrx, k 1 D s a zero matrx. YPE 2 rectangular structure: SRUC=[ mn, ] f X s a m n martrx. YPE 3 general structures: hs thrd type s used to descrbe more sophstcated structures and/or correlatons between the matrx varables. he prncple s as follows: each entry of X s specfed ndependently as ether, x n, or xn where x n denotes the n-th decson varable n the problem. For detals on how to use ype 3, see Structured Matrx Varables on page 4-33 as the lmvar entry n the reference pages of the Robust Control Users Gude manual.

4 4 Examples: % Create a 2x2 full symmetrc matrx % varbale X=lmvar(1,[2 1]) %Create a 2x3 matrx varable Y Y=lmvar(2,[2 3]) %Create a 3x3 dagonal matrx Z Z=lmvar(1,[3 ]);

5 5 Specfyng Indvdual LMIs After declarng the matrx varables wth lmvar, we are left wth specfyng the term content of each LMI. Recall that LMI terms fall nto three categores: he constant terms,.e., fxed matrces lke I n the left-hand sde of the LMI S > I he varable terms,.e., terms nvolvng a matrx varable. For nstance, A X and C SC n AX XACSC XB B X S Varable terms are of the form PXQ where X s a varable and P, Q are gven matrces called the left and rght coeffcents, respectvely. he outer factors

6 6 Ex. LMI terms are specfed one at a tme wth lmterm. For nstance, the LMI AX XACSC XB B X S s descrbed by lmterm([ ],1,A,'s'); lmterm([ ],C',C); lmterm([ ],1,B); lmterm([ ],-1,1); LMIERM(ERMID,A,B,FLAG) ERMID 4-entry vector specfyng the term locaton and nature Whch LMI? ERMID(1) = +n -> left-hand sde of the n-th LMI ERMID(1) = -n -> rght-hand sde of the n-th LMI Whch block? For outer factors, set ERMID(2:3) = [ ]. Otherwse, set ERMID(2:3) = [ j] f the term belongs to the (,j) block of the LMI What type of term? ERMID(4) = -> constant term ERMID(4) = X -> varable term A*X*B ERMID(4) = -X -> varable term A*X'*B where X s the varable dentfer returned by LMIVAR

7 7 A B FLAG value of the outer factor, constant term, or left coeffcent n varable terms A*X*B or A*X'*B rght coeffcent n varable terms A*X*B or A*X'*B quck way of specfyng the expresson A*X*B+B'*X'*A' n a DIAGONAL block. Set FLAG='s' to specfy t wth only one LMIERM command Ex. % 3rd LMI S I Slm=newlm; lmterm([-slm 1 1 S],1,1); lmterm([slm 1 1 ],1); lmterm([ ],1,1); lmterm([3 1 1 ],1); % 2nd LMI X Xpos=newlm; lmterm([-xpos 1 1 X],1,1); stance, the LMI

8 8 he hree Generc LMI Problems (1) Fndng a soluton x to the LMI system A( x) (4.4a) s called the feasblty problem. (2) Mnmzng a convex objectve under LMI constrants s also a convex problem. In partcular, the lnear objectve mnmzaton problem Mnmze cxsubject to Ax ( ) (4.4b) plays an mportant role n LMI-based desgn. (3) he generalzed egenvalue mnmzaton problem A( x) B( x) Mnmze subject to Bx ( ) Cx ( ) s quas-convex and can be solved by smlar technques. (4.4c)

9 9 LMI Solvers for the hree Generc LMI Problems LMI solvers are provded for the followng three generc optmzaton problems (here x denotes the vector of decson varables,.e., of the free entres of the matrx varables X 1,..., X K ): (1) Feasblty Problem N Fndng x R (or equvalently matrces X 1,..., X K wth prescrbed structure) that satsfes the LMI system A( x) B( x) (4.5a) he correspondng solver s called feasp. (2) he Problem of Mnmzng a Lnear Objectve Under LMI Constrants N Mnmze c x over x R subject to A( x) B( x) (4.5b) he correspondng solver s called mncx. (3) Generalzed Egenvalue Mnmzaton Problem N Mnmze over x R subject to Cx ( ) Dx ( ), B( x), A( x) B( x) (4.5c) he correspondng solver s called gevp. he three LMI solvers feasp, mncx, and gevp take as nput the nternal representaton LMISYS of an LMI system and return a feasble or optmzng value x* of the decson varables. he correspondng values of the matrxvarables X 1,..., X K are derved from x* wth the functon dec2mat.

10 When usng gevp, you should follow these three rules to ensure proper specfcaton of thenproblem: Specfy the LMIs nvolvng λ as A(x) < B(x) (wthout the λ) Specfy them last n the LMI system. gevp systematcally assumes that the last L LMIs are lnear-fractonal f L s the number of LMIs nvolvng λ. Add the constrant < B(x) or any other constrant that enforces t. hs postvty constrant s requred for well-posedness of the problem and s not automatcally added by gevp (see the reference pages for detals). 1

11 11 Modfyng a System of LMIs Once specfed, a system of LMIs can be modfed n several ways wth the functons dellm, delmvar, and setmvar. Deletng an LMI Ex. In Example 8.1 he system of LMIs was: AX XACSC XB, X, S I B X S Suppose we want to remove the postvty constrant on X. hs s done by NEWSYS = dellm(lmisys,2) where the second argument specfes deleton of the second LMI. he resultng system of two LMIs s returned n NEWSYS. he LMI dentfers (ntal rankng of the LMI n the LMI system) are not altered by deletons. As a result, the last LMI S I remans known as the thrd LMI even though t now ranks second n the modfed system. o avod confuson, t s safer to refer to LMIs va the dentfers returned by newlm. If BRL, Xpos, and Slm are the dentfers attached to the three LMIs as n Example 8.1, Slm keeps pontng to S > I even after deletng the second LMI by NEWSYS = dellm(lmisys,xpos) Deletng a Matrx Varable

12 12 Another way of modfyng an LMI system s to delete a matrx varable, that s, to remove all varable terms nvolvng ths matrx varable. hs operaton s performed by delmvar. Ex. Consder the LMI A X XABW W B I 44 where X X R 2 4 and W R. hs LMI s defned by setlms([]) X = lmvar(1,[4 1]) % X W = lmvar(2,[2 4]) % W lmterm([1 1 1 X],1,A,'s') lmterm([1 1 1 W],B,1,'s') lmterm([1 1 1 ],1) LMISYS = getlms o delete the varable W, type the command NEWSYS = delmvar(lmisys,w) he resultng NEWSYS now descrbes the Lyapunov nequalty A X XAI Note that deletng a matrx varable s equvalent to settng t to the zero matrx of the same dmensons wth setmvar.

13 13 Settng a Matrx Varable he functon setmvar s used to set a matrx varable to some gven value. Ex. In Example 8.1 AX XACSC XB, X, S I B X S o set S to 2I, enter NEWSYS = setmvar(lmisys,s,2) he second argument s the varable dentfer S, and the thrd argument s the value to whch S should be set. Here the value 2 s shorthand for 2I. he resultng system NEWSYS reads A X XA 2C C XB, X, 2I I B X 2I Note that the last LMI s now free of varable and trvally satsfed. It could, therefore, be deleted by NEWSYS = dellm(newsys,3) or NEWSYS = dellm(newsys,slm) f Slm s the dentfer returned by newlm.

14 14 Problems wth lnear equalty constrants Ex.5 k k P, A PPA, rp 1, where Px ( ) P( x) R s the varable. Assume k=2, then p p P p11 p12 p12 1 p and hence, F( x) F x F x F where F P A P PA f and only f P xp 1 1x2P2,.e., P and,,1, 2 ( APPA) x( APPA) x( APPA) e., ( APPA) x( APPA) x( APPA).e., A P x P x P P x P x P A.e., A PPA

15 15 Structured Sngular Values M: a gven n n matrx he uncertanty set s descrbed by block dag(,...,, I,..., I ) : ( mc, ) 1 f 1 c1 q cq mm R, R M wth f q. 1 1 n m c A smlarty scalng set D s defned as block dag( d1i m,...,, 1 fim D1,..., D ) : f q D = * cc dr, D D R he structure sngular value ( M ) s defned as ( M ) 1 mn ( ) : det( I M) ( mc, ) Note that ( M ) = f no It can be shown that 1 ( M ) nf ( DMD ) D D ( mc, ) satsfes det( I M ).

16 16 heorem Structured robust stablty 1 nf DM ( s) D 1 DD 1 D M j D sup nf ( ) ( ) ( ) 1 D( ) D M j sup ( ) 1 H Scalng G Let mnmzed. 1 Gs () CsI ( A) B. Fnd D D such that 1 DG() s D s

17 17 herorem he followng statements are equvalent: () Gs () s stable and there exsts D D such that 1 DG() s D. () here exst X and S S D D D such that Proof: 2 AX XA CSC XB ( XB) S () s equvalent to the exstence of X and S D DD such that (See Gahnet, Proceedngs of the 3 th Conference on Decson and Control) 2 A X XA C SC XBS 1 ( XB) hen the equvalence wth () s obtaned by a standard Shur complement argument.