system state feedback in LMI with

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1、State-feedbackControlofSystemswithMultiplicativeNoiseviaLinearMatrixInequalitiesLaurentElGhaouiEcoleNationaleSuperieuredeTechniquesAvancees32,Bd.Victor,75015Paris,Francee-mail:elghaoui@ensta.frAbstractWeconsiderLTIsystemsperturbedbyparametricuncertainties,modele

2、daswhitenoisedisturbances.Weshowhowtomaximize,viastate-feedbackcontrol,thesmallestnormofthenoiseintensityvectorproducinginstabilityinthemeansquaresense,usingconvexoptimizationoverlinearmatrixinequalities.Wealsoshowhowtomaximizeperformancerobustness,whereperformance

3、ismeasuredbyexpectedoutputenergy,witheitherboundedinitialconditionsandzeroinputs(classicalLQGcost),orzeroinitialconditionsanddeterministicinputsofboundedenergy(ageneralizationoftheHnorm).1Keywords:linearmatrixinequalities,systemswithmultiplicativenoise,stochasticro

4、bust-ness,Lgain.21IntroductionInthispaper,weconsidertheIt^odierentialequationdx(t)=Ax(t)dt+Bu(t)dt+Bw(t)dt+uwLX(Ax(t)+Bu(t)+Bw(t))dp(t);iu;iw;iii=1(1)z(t)dt=Cx(t)dt+Du(t)dt+Dw(t)dt+zzuzwLX(Cx(t)+Du(t)+Dw(t))dp(t);z;izu;izw;iii=1nnnnuwzwherex2Risthestate,u2Ristheco

5、ntrolinput,w2Ristheexogeneousinput,z2Ristheoutput,andthevectorprocessp=(p;:::;p)isastandardWienerprocess.Thatis,p(t),1Lii=1;:::;L,areBrownianmotionssatisfying22E(dp(t))=0andE(dp(t))=dtfori=1;:::;L;iiiE(dp(t)dp(t))=0for1j6=kL:jkThispaperappearedinSystemsandContr

6、olLett.,February1995.1Thescalars0,i=1;:::;Lre ectthenoiseintensitylevelofthedisturbances.Weshallidenoteby=(;:::;)thenoiseintensityvectorofthesystem(for=0,wehaveanLTI1Lsystem,whichwillbetermedominal").Sometimes,weshallassumethatisunknown-but-bounded":2,w

7、here=fj0;i=1;:::;Lg,where>0isagivenscalar.i(Theanalysisextendstomoregeneralconvexsets.)Likewise,weshallassumethattheinitialnconditionx(0)iseitherknown,oronlyknowntobelongtoagivenpolytopeorellipsoidofR,inwhichcaseweshallsimplysaythatx(0)isunknown-but-boun

8、ded.Inthispaper,weshowhowtocomputestatic,state-feedbacklawsoftheformu=Kx,withKaconstantmatrix,soastoachievevariouspropertiesfortheclosed-loopsyst