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:100220411(2006)0520593207LMIH(,210003):Takagi2Sugeno(T2S),H.(LMI),H.,H.,,.,,.3:;T2S;H;;:TP13:ALMI2basedHControlofUncertainNonlinearSystemsFANChun2xia(CollegeofAutomation,NanjingUniversityofPostsandTelecommunications,Nanjing210003,China)Abstract:BasedonTakagi2Sugeno(T2S)fuzzymodelofnonlinearsystems,Hcontrolisstudiedforaclassofnonlinearsystemswithstructureuncertainties.ThesystematicdesignprocedureoffuzzyHoutputfeedbackcontrol2lersisachievedvialinearmatrixinequality(LMI)approach.InteractionsamongthesubsystemsareconsideredandsufficientconditionforexistenceofthefuzzyHoutputfeedbackcontrollerisobtained.Andthesufficientconditionisexpressedwithonlyamatrixinequality,whichissimpleinformandrelaxesthesufficientconditionfortheexistenceofthecontroller.Finally,Lorentzchaoticsystemistakenasanexampletodemonstratethevalidityoftheproposedcontroller.Keywords:uncertainty;T2Sfuzzymodel;Hcontrol;linearmatrixinequality(LMI);chaoticsystem1(Introduction),,,.,,,.Takagi2Sugeno(T2S)[1],T2SRn[2,3].,T2S[46].,T2S,,[7][8].T2SH.,,H,,,,.,H,.,,,H.2T2S(T2Sfu2zzymodelbasedproblemformulation)T2S355200610InformationandControlVol.35,No.5Oct.,20063:2005-08-31:(NY205052)x=ri=1hi((t))[(Ai+Ai(t))x(t)+(B1i+Bi)u(t)+B2iw]z=ri=1hi((t))(Cix(t)+Diu(t))(1),xRn,uRm,wRl,zRp,AiRnn,B1iRnm,B2iRnl,CRpn,DRpm,AiB1i,Ai=EiFi(t)H2i,B1i=EiFi(t)H1i,EiRnn,H1iRnn,H2iRnn,Fi(t)RnnF(t)TiF(t)iI,I(2)ri=1hi((t))=1,hi((t))0,i=1,2,,ri((t))=kj=1Mij((t))(3)hi((t))=i((t))ri=1i((t))(4)Mij((t)).u,(1),d,0d,z2dw2,2L2,(1)H.3H(DesignofHcontrol2ler)3.1,.1[9]:DE,S,FTFRS+DFE+ETFTDT0(5):0S+[-1ETD]R00I-1EDT0(6).H,,H=H11H1rwHr1Hrr,Hii=H2i+H1iKiHij=H2j+H1jKi(7)M=M11M1rwMr1Mrr,Mii=H2i+H1iMiMij=H2j+H1jMi(8)Ep=PE1wPEr(9)F=F1wFr(10)2:P-120T11P+P11T1rP+P1rwTr1P+Pr1TrrP+Prr+(HTEp)-1IIHETP-I(11),0,ii=Ai+B1iKi,2ij=Ai+B1iKj+Aj+B1jKi,KiRmn,HEp(7)(9),u(t)=ri=1hi((t))Kix(t)(12)(1)w(t)0,.:V(x(t))=x(t)TPx(t)(13)(13),(12)V(x(t))=xT(t)Px(t)+x(t)TPx(t)=ri=1hi((t))[(Ai+EiFi(t)H2i)x(t)+(B1i+EiFi(t)H1i)u(t)]TPx+xT(t)Pri=1hi((t))[(Ai+EiFi(t)H2i)x(t)+(B1i+EiFi(t)H1i)u(t)]=ri=1h2i((t))xT(t)[(Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Ki]TP+P[Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Ki]x(t)49535+2rijhi((t))hj((t))xT(t)[Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Kj2+Aj+EjFj(t)H2j+(B1j+EjFj(t)H1j)Ki2]TP+P[Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Kj2+Aj+EjFj(t)H2j+(B1j+EjFj(t)H1j)Ki2]x(t)=h1((t))x(t)hr((t))x(t)TT11P+P11T1rP+P1rwTr1P+Pr1TrrP+Prr(14)+HTFTETp+EpFHh1((t))x(t)hr((t))x(t),F(10).1,(11)T11P+P11T1rP+P1rwTr1P+Pr1TrrP+Prr+HTFTETp+EpFH-I(15),V(x(t))-xT(t)x(t)(16)w(t)0,(1).1:Mj(j=1,2,,r),-0.50Q,111rwr1rr+(MTE)-1IIMET0(17),M(8),E,(E1,,Er).ii=QATi+AiQ+MTiBT1i+B1iMiij=Q(Ai+Aj)T+(Ai+Aj)Q+MTiBT1j+B1jMi+MTjBT1i+B1iMj2i,j=1,2,,r(1)(12),w(t)0,.:P=Q-1,Mi=KiQ-1.(17)diag(Q-1,,Q-1),Q,Q-1wQ-1111rwr1rr+(MTE)-1IIMETQ-1wQ-1=T11P+P11T1rP+P1rwTr1P+Pr1TrrP+Prr+(HTEp)-1IIHETp0(18)2,1.:(11)(1)r,(11)(17)(1).[8],TiiP+Pii0TijP+Pij0.3.2H3:0,P,KiRmn-0.505955:LMIH111rwr1rr+(HTEp)-1IIHETp-I(19),Ep(9),ii=(Ai+B1iKi)TP+P(Ai+B1iKi)+(Ci+DKi)T(Ci+DKi)+12PB2iBT2iPij=12[(Ai+B1iKj+Aj+B1jKi)TP+P(Ai+B1iKj+Aj+B1jKi)+2(Ci+DKi)T(Cj+DKj)+12PB2iBT2jP]i,j=1,2,,r(1)(12),,H.:ii=Ai+B1iKi,2ij=Ai+B1iKj+Aj+B1jKi.,w(t)0,(12),(1).0-I111rwr1rr+(HTEp)-1IIHETp=T11P+P11T1rP+P1rwTr1P+Pr1TrrP+Prr+12PB21PB2rPB21PB2rT+CT1+KT1DCTr+KTrDCT1+KT1DCTr+KTrDT+(HTEp)-1IIHETpT11P+P11T1rP+P1rwTr1P+Pr1TrrP+Prr+(HTEp)-1IIHETp(20)(20),1,(12),w(t)0,(1).,H.V(x(t))=x(t)TPx(t)(21)(21),(1)z,1,V(x(t))=xT(t)Px(t)+x(t)TPx(t)=ri=1hi((t))[(Ai+EiFi(t)H2i)x(t)+(B1i+EiFi(t)H1i)u(t)]+B2iwTPx+xT(t)Pri=1hi((t))[(Ai+EiFi(t)H2i)x(t)+(B1i+EiFi(t)H1i)u(t)]+B2iw=ri=1h2i((t))xT(t){[Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Ki]TP+P[Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Ki]+(Ci+DKi)T(Ci+DKi)+12PB2iBT2iP}x(t)+2rijhi((t))hj((t))xT(t){[Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Kj269535+Aj+EjFj(t)H2j+(B1j+EjFj(t)H1j)Ki2]TP+P[Ai+EiFi(t)H2i+(B1i+EiFi(t)H1i)Kj+Aj+EjFj(t)H2j2(22)+(B1j+EjFj(t)H1j)Ki2+(Ci+DKi)T(Ci+DKi)+12PB2iBT2jP]}x(t)-ri=1rj=1hi((t))hj((t))x(t)T[(Ci+DKi)T(Cj+DKj)+12PB2iBT2jP]x(t)+ri=1hi((t))(wTBT2ix(t)+x(t)TB2iw)-[w-1ri=1hi((t))BT2ix(t)]T(w-1ri=1hi((t))BT2ix(t))-x(t)Tx(t)-zTz+2wTwx(0)=0,V(0)=0.(22),z222w22-V()-x(t)22-2w-12ri=1hi((t))BT2iPx(t)222w22(23)3.2:0,Mi,Q-120,F11F1rwFr1Frr+-1MTM+EETQCT1+MT1DTB21QCTr+MTrDTB2rC1Q+DM1CrQ+DMrBT21BT2r-I00-2I0(24),M(8),E,(E1,,Er).Fii=AiQ+QATi+MTiBT1i+B1iMiFij=12(AiQ+QATi+MTiBT1j+B1jMi+AjQ+QATj+MTiBT1j+B1jMi)i,j=1,2,,r(1)(12),,H.:Shur,(24)F11F1rwFr1Frr+-1MTM+EET+12B21B2rB21B2rT+QCT1+MT1DTQCTr+MTrDTQCT1+MT1DTQCTr+MTrDTT0(25)(25)diag((Q-1)T,,(Q-1)T)diag((Q-1),,(Q-1)),2(12),(1)2.,(1)(12),(1)H,B1iRnm(1)H.3:0,P,B1iRnm-0.50(19),(1)(12),,H.:,.SchurB1iRnm.4(Simulation).7955:LMIHx1=a(x2-x1)+w1+ux2=bx1-x2-x1x3+w2x3=x1x2-cx3+w3z=x1(26),a=10,b=28,c=8/3,w=(w1,w2,w3)T=(0,0,0)T,u=0,(26);z.x=(x1,x2,x3)T,(26)T2SIFx1isM1,x2isandx3isthenx=(A1+A1)x+(B11+B11)u+B21wz=x1IFx1isM2,x2isandx3isthenx=(A2+A2)x+(B12+B12)u+B22wz=x1,B11=B12=(0,0,0)T,w=(8e-0.5t,0,0)T,[A1,A2]=[F1H11,F2H12],M1:f1=-x1(t)+d2d2-d1,M2:f2=x1(t)-d1d2-d1,d1=-20,d2=30,A1=-1010028-1-d10d1-8/3B11=100B21=100A2=-1010028-1d20-d2-8/3B12=100B22=100,30%,F1=F2=0.30000.30000.3H11=H12=-101002800