线性矩阵不等式5.ppt

,MATLAB的LMI工具箱介绍,主要内容,可行性问题(LMIP)特征值问题(EVP)广义特征值问题(GEVP),可行性问题,例1考虑系统,设计状态反馈控制律,使得闭环系统,渐进稳定。,其中,左乘右乘,，并记,，得,若要闭环系统渐进稳定，需满足如下矩阵不等式：,LMI,编程实现,clcclearallA=-1-21;321;1-2-1;B=1;0;1;setlmis()X=lmivar(1,31)W=lmivar(2,13)lmiterm(111X,A,1,s)lmiterm(111W,B,1,s),lmiterm(-211X,1,1)lmisys=getlmistmin,xfeas=feasp(lmisys)XX=dec2mat(lmisys,xfeas,X)WW=dec2mat(lmisys,xfeas,W)K=WW*inv(XX),clcclearallA=-1-21;321;1-2-1;B=1;0;1;,清除命令窗口,清除工作空间,定义矩阵A,定义矩阵B,编程实现,编程实现,setlmis()X=lmivar(1,31)W=lmivar(2,13),进入线性矩阵不等式编程环境,定义线性矩阵不等式系,统的矩阵变量X，W,与getlmis相配对,矩阵类型,矩阵维数结构,lmiterm(111X,A,1,s)lmiterm(111W,B,1,s)lmiterm(-211X,1,1),AX+XA,BW+WB,编程实现,X0,属于哪个矩阵不等式,为正在左,为负在右,所描述的项所在块的位置,若是常数则为0,若是变量则指出说明,lmisys=getlmistmin,xfeas=feasp(lmisys)XX=dec2mat(lmisys,xfeas,X)WW=dec2mat(lmisys,xfeas,W)K=WW*inv(XX),LMI编程结束,并取名为lmisys,tmin0,实矩阵W以及标量0,使得如下线性矩阵不等式成立,则u(t)=WV-1x(t)为闭环系统的H鲁棒控制律，即|Gyw|,且闭环系统渐进稳定。,clc;clearallA=-1-21;321;1-2-1;B=1;0;1;C=100;D=1;1;1;M=0.2*eye(3);E1=100;020;001;E2=1;0;1;gama=3;setlmis()V=lmivar(1,31)W=lmivar(2,13)eps=lmivar(1,11)lmiterm(111V,A,1,s)lmiterm(111W,B,1,s)lmiterm(111eps,gama(-2)*D*D,1)lmiterm(1110,M*M),lmiterm(121V,C,1)lmiterm(122eps,-1,1)lmiterm(131V,E1,1)lmiterm(131W,E2,1)lmiterm(1330,-1)lmiterm(-211V,1,1)lmiterm(-311eps,1,1)lmisys=getlmistmin,xfeas=feasp(lmisys)VV=dec2mat(lmisys,xfeas,V)WW=dec2mat(lmisys,xfeas,W)K=WW*inv(VV),tmin=0.004612,特征值问题（EVP）,例3考虑优化问题,其中X是对称正定阵。,根据Schur补，本例中的优化问题等价于,clc;clearallA=-1-21;321;1-2-1;B=1;0;1;Q=1-10;-1-3-12;0-12-36;setlmis()X=lmivar(1,31)lmiterm(111X,1,A,s)lmiterm(1110,Q),lmiterm(121X,B,1)lmiterm(1220,-1)lmis=getlmisc=mat2dec(lmis,eye(3)options=1e-5,0,0,0,0copt,xopt=mincx(lmis,c,options)Xopt=dec2mat(lmis,xopt,X),特征值问题的由来,特征值问题(EVP)：求矩阵G(x)的特征值的最小化问题,特征值问题（优化问题）,例4(式5.4.31)考虑优化问题,其中对称正定阵X和N,矩阵V，标量,是变量,clcclearA=-0.2500;-0.50.52;-0.75-1-1.5;B1=001;B2=00-1;C0=111;C1=-110;D0=1;D1=0.5;H=0.25-0.500.75;E1=00.501.00;E2=0;gama=10;setlmis()alpha=lmivar(1,11)beta=lmivar(1,11),X=lmivar(1,31)V=lmivar(2,13)N=lmivar(1,11)lmiterm(111X,-1,1)lmiterm(131X,A,1)lmiterm(131V,B1,1)lmiterm(132alpha,1,B2)lmiterm(141X,E1,1)lmiterm(141V,E2,1)lmiterm(151X,C1,1)lmiterm(151V,D1,1)lmiterm(161X,C0,1),lmiterm(161V,D0,1)lmiterm(122alpha,-1,gama*gama*1)lmiterm(133X,-1,1)lmiterm(133beta,1,H*H)lmiterm(144beta,-1,1)lmiterm(155alpha,-1,1)lmiterm(1660,-1)lmiterm(211N,-1,1)lmiterm(2210,B2)lmiterm(222X,-1,1)lmisys=getlmistmin,xfeas=feasp(lmisys),c=mat2dec(lmisys,0,0,0,0,eye(1)options=1e-5,0,0,0,0copt,xopt=mincx(lmisys,c,options)XOPT=dec2mat(lmisys,xopt,X)VOPT=dec2mat(lmisys,xopt,V)alphaopt=dec2mat(lmisys,xopt,alpha)betaopt=dec2mat(lmisys,xopt,beta)NOPT=dec2mat(lmisys,xopt,N)K=VOPT*inv(XOPT)J=trace(B2*inv(XOPT)*B2),特征值问题（优化问题）,例5(例9.3.4)考虑优化问题,其中对称正定阵X、M和N,矩阵W，标量,是变量,c=mat2dec(lmisys,(1+h1)*eye(1),d1*eye(1),0,0,0,0,0)options=1e-5,0,0,0,0copt,xopt=mincx(lmisys,c,options),为mincx确定目标函数cTx,考虑优化问题,其中X、P是对称矩阵变量.,x0=11;setlmis()X=lmivar(1,31)P=lmivar(1,21).lmisys=getlmis,n=decnbr(lmisys)c=zeros(n,1)forj=1:nXj,Pj=defcx(lmisys,j,X,P)c(j)=trace(Xj)+x0*Pj*x0endcopt,xopt=mincx(lmisys,c,options),n=decnbr(lmisys)c=zeros(n,1)forj=1:n,alphaj,betaj=defcx(lmisys,j,alpha,beta)c(j)=(1+h1)*alphaj+d1*betajendoptions=1e-5,0,0,0,0copt,xopt=mincx(lmisys,c,options),广义特征值问题（GEVP),广义特征值问题(GEVP):求矩阵A,B的广义特征值的最小化问题,广义特征值问题,例6考虑优化问题,其中对称正定阵P是变量,setlmis()P=lmivar(1,21);lmiterm(1110,1)lmiterm(-111P,1,1)lmiterm(211P,1,A1,s)lmiterm(-211P,1,1)lmiterm(311P,1,A2,s),lmiterm(-311P,1,1)lmiterm(411P,1,A3,s)lmiterm(-411P,1,1)lmis=getlmistmin,xfeas=feasp(lmis)alpha,popt=gevp(lmis,3)pp=dec2mat(lmis,popt,P),针对广义特征值的最小化问题，在调用求解器gevp时，须遵循以下规则：确定包含的线性矩阵不等式：A(x)B(x)（注意没有）总是把A(x)B(