锥补线性化问题和应用线性矩阵不等式解决控制问题

A=[1 0.25;0 1]; B=[0.0063;0.0500]; C=[1 0]; D=0; U=[0 0;0 0]; setlmis([]) P=lmivar(1,[2 1]); Q=lmivar(1,[2 1]); R=lmivar(1,[2 1]); S=lmivar(1,[2 1]); M=lmivar(1,[2 1]); N=lmivar(1,[2 1]); K=lmivar(2,[1 2]); F=lmivar(2,[2 1]); lmiterm([ 1 1 1 S],1,1); lmiterm([ 1 1 1 P],-1,1); lmiterm([ 1 5 1 0],A); lmiterm([ 1 5 1 K],B,1); lmiterm([ 1 6 1 F],-1,C); lmiterm([ 1 2 2 S],-1,1); lmiterm([ 1 6 2 F],1,C); lmiterm([ 1 3 3 R],1,1); lmiterm([ 1 3 3 Q],-1,1); lmiterm([ 1 5 3 K],B,-1); lmiterm([ 1 6 3 F],1,C);

lmiterm([ 1 6 3 0],-A); lmiterm([ 1 4 4 R],-1,1); lmiterm([ 1 5 5 M],-1,1); lmiterm([ 1 6 6 N],-1,1); lmiterm([ -2 1 1 P],1,1); lmiterm([ -2 2 1 0],1); lmiterm([ -2 2 2 M],1,1); lmiterm([ -3 1 1 Q],1,1); lmiterm([ -3 2 1 0],1); lmiterm([ -3 2 2 N],1,1); lmiterm([ -4 1 1 P],1,1); lmiterm([ -5 1 1 Q],1,1); lmiterm([ -6 1 1 R],1,1); lmiterm([ -7 1 1 S],1,1); lmiterm([ -8 1 1 M],1,1); lmiterm([ -9 1 1 N],1,1); lmisys =getlmis ;

[tmin,xfeasp] =feasp(lmisys) PP =dec2mat(lmisys,xfeasp,P) QQ =dec2mat(lmisys,xfeasp,Q) RR =dec2mat(lmisys,xfeasp,R) SS =dec2mat(lmisys,xfeasp,S) MM =dec2mat(lmisys,xfeasp,M) NN =dec2mat(lmisys,xfeasp,N) KK =dec2mat(lmisys,xfeasp,K) FF =dec2mat(lmisys,xfeasp,F) for i =1:100

n =decnbr(lmisys); c =zeros(n ,1); for j =1:n

[ Pj,Qj,Rj,Sj,Mj,Nj,Kj,Fj] =defcx(lmisys,j,P,Q,R,S,M,N,K,F); c(j)=trace(PP*Mj+MM*Pj +QQ*Nj+NN*Qj); end

options =[ 1e-4,0,0,0,0] ;

[ copt, xopt]=mincx(lmisys,c,options) PPP =dec2mat(lmisys,xopt,P); QQQ =dec2mat(lmisys,xopt,Q); RRR =dec2mat(lmisys,xopt,R); SSS =dec2mat(lmisys,xopt,S); MMM =dec2mat(lmisys,xopt,M); NNN =dec2mat(lmisys,xopt,N); KKK =dec2mat(lmisys,xopt,K); FFF =dec2mat(lmisys,xopt,F);

Z =[ SSS-PPP,U,U,U,(A+B*KKK)',-(FFF*C)'; U,-SSS,U,U,U,(FFF*C)';

U,U,RRR-QQQ,U,-(B*KKK)',-(A-FFF*C)'; U,U,U,-RRR,U,U;

A+B*KKK,U,-B*KKK,U,-inv(PPP),U; -FFF*C,FFF*C,-(A -FFF*C),U,U,-inv(QQQ)] ; Y =eig(Z); i2 =0 ;

for i1 =1 :length(Y), if(Y(i1 , 1)<0), i2 =i2 +1 ;

end end

if(i2 ==length(Y)), break ; end PP =PPP QQ =QQQ RR =RRR SS =SSS MM =MMM NN =NNN KK =KKK FF =FFF end if(i==10),

disp(' There is no result' ); end

Solver for linear objective minimization under LMI constraints

Iterations : Best objective value so far

1 49.439545 2 36.063874 3 23.102877 4 14.567845 5 14.567845 6 13.442476

*** new lower bound: 0.296420 7 13.442476