Linear Matrix Inequality Lecture.pdf

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1、EE363Winter2008-09Lecture15LinearmatrixinequalitiesandtheS-procedure•Linearmatrixinequalities•Semidefiniteprogramming•S-procedureforquadraticformsandquadraticfunctions15–1LinearmatrixinequalitiessupposeF0,...,Fnaresymmetricm×mmatricesaninequalityoftheformF(x)=F0+x1F1+···+xnFn≥0niscalled

2、alinearmatrixinequality(LMI)inthevariablex∈Rnm×mhere,F:R→RisanaffinefunctionofthevariablexLinearmatrixinequalitiesandtheS-procedure15–2LMIs:•canrepresentawidevarietyofinequalities•ariseinmanyproblemsincontrol,signalprocessing,communications,statistics,...mostimportantforus:LMIscanbesolve

3、dveryefficientlybynewlydevelopedmethods(EE364)“solved”means:wecanfindxthatsatisfiestheLMI,ordeterminethatnosolutionexistsLinearmatrixinequalitiesandtheS-procedure15–3Examplex1+x2x2+1F(x)=≥0x2+1x301101100F0=,F1=,F2=,F3=10001001LMIF(x)≥0equivalenttox1+x2≥0,x3≥0(x+x)x−(x+1)2=xx+xx−x

4、2−2x−1≥01232132322...asetofnonlinearinequalitiesinxLinearmatrixinequalitiesandtheS-procedure15–4CertifyinginfeasibilityofanLMI•ifA,BaresymmetricPSD,thenTr(AB)≥0:21/21/21/21/21/21/2Tr(AB)=TrABBA=ABF•supposeZ=ZTsatisfiesZ≥0,Tr(F0Z)<0,Tr(FiZ)=0,i=1,...,n•thenifF(x)=F0+x1F1+···+xnFn≥0,0≤T

5、r(ZF(x))=Tr(ZF0)<0acontradiction•ZiscertificatethatprovesLMIF(x)≥0isinfeasibleLinearmatrixinequalitiesandtheS-procedure15–5Example:Lyapunovinequalityn×nsupposeA∈RtheLyapunovinequalityATP+PA+Q≤0isanLMIinvariablePmeaning:PsatisfiestheLyapunovLMIifandonlyifthequadraticformV(z)=zTPzsatisfiesV

6、˙(z)≤−zTQz,forsystemx˙=AxthedimensionofthevariablePisn(n+1)/2(sinceP=PT)here,F(P)=−ATP−PA−QisaffineinP(wedon’tneedspecialLMImethodstosolvetheLyapunovinequality;wecansolveitanalyticallybysolvingtheLyapunovequationATP+PA+Q=0)LinearmatrixinequalitiesandtheS-procedure15–6ExtensionsmultipleLM

7、Is:wecanconsidermultipleLMIsasone,largeLMI,byformingblockdiagonalmatrices:(1)(k)(1)(k)F(x)≥0,...,F(x)≥0⇐⇒diagF(x),...,F(x)≥0example:wecanexpressasetoflinearinequalitiesasanLMIwithdiagonalmatrices:TTTTa1x≤b1,...,akx≤bk⇐⇒diag(b1−a1x,...,bk−akx)≥0linearequalityconstraints:aTx=bisthesa