线性矩阵不等式5

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xAxBu设计状态反馈控制律uKx使得闭环系统xABKx渐进稳定。其中1211321,01211AB0TABKPPABK左乘右乘1P1,XPWKX，并记，得0TTTXAAXBWWB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0TTTXAAXBWWB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=getlmis[tmin,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0TTTXAAXBWWB编程实现0TTTXAAXBWWB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+W’B’编程实现0TTTXAAXBWWB％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=getlmis[tmin,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表示LMI存在可行解xfeas％得到变量X和W的可行解可行性问题A(x)B(x)可描述为：mintA(x)-B(x)tI％求解控制增益K0.5269-0.41410.3643-0.41410.4693-0.13800.3643-0.13800.7871-1.2136-0.4220-0.4013XW0.5269-0.41410.46930.3643-0.13800.7871-1.2136-0.4220-0.4013xfeastmin=－0.013057-17.0905-14.55314.8481K例2(定理3.3)针对闭环系统(3-3)和给定的一个常数γ0,如果对所有满足FTFI的实矩阵F，若存在对称正定实矩阵V0,实矩阵W以及标量ε0,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||∞γ,且闭环系统渐进稳定。TTT2TTTTTT1212000AVVABWWBDDMMVCVEWECVIEVEWITTT2TTTTTT1212000AVVABWWBDDMMVCVEWECVIEVEWIclc;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=getlmis[tmin,xfeas]=feasp(lmisys)VV=dec2mat(lmisys,xfeas,V)WW=dec2mat(lmisys,xfeas,W)K=WW*inv(VV)0.4923-0.1625-0.7478-0.16250.12810.0255-0.74780.02551.9338-0.33680.0472-1.10820.7203VWtmin=－0.004612-33.3111-39.3104-12.9358K特征值问题（EVP）例3考虑优化问题minTrace()..0XTTXstAXXAXBBXQ其中X是对称正定阵。根据Schur补，本例中的优化问题等价于1211110321,0,1312121101236ABQminTrace()..0XTTXAXXAQXBstBXI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)minTrace()..0XTTXAXXAQXBstBXIlmiterm([121X],B’,1)lmiterm([1220],-1)lmis=getlmisc=mat2dec(lmis,eye(3))options=[1e-5,0,0,0,0][copt,xopt]=mincx(lmis,c,options)Xopt=dec2mat(lmis,xopt,X)-6.3542-5.8895-6.28552.2046-5.88952.2201-2.20462.6.07712201optX10-5.88-6.3542-6.285951,02.204602.522011-6.0771cxoptminTrace()minTXXcx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min..stGxImin..0TcxstFxmin..0TTcxstGxcxTcxTFxGxcx特征值问题（优化问题）例4(式5.4.31)考虑优化问题212121100*****0*******000**000*0000TXIAXBVBXHHEXEVICXDVICXDVI,,,,minTrace()XVNN其中对称正定阵X和N,矩阵V，标量α,β是变量..st220TNBBX121222TzwTTraceBXB220TNBBX1220TNBXBminTraceN122TBXBN121222minminTzwTTraceBXB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=getlmis[tmin,xfeas]=feasp(lmisys)c=mat2dec(lmisys,0,0,0,0,eye(1))options=[1e-5,0,0,0,0][copt,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9.3.54**,,,,,,min1+XMNWzhd其中对称正定阵X、M和N,矩阵W，标量α,β,ε是变量..st0TIUUX0TIUUM**,,,,,,min1+XMNWzhdc=mat2dec(lmisys,(1+h1)*eye(1),d1*eye(1),0,0,0,0,0)options=[1e-5,0,0,0,0][copt,xopt]=mincx(lmisys,c,options)为mincx确定目标函数cTx考虑优化问题00min()TTraceXxPx其中X、P是对称矩阵变量.x0=[11];setlmis([])X=lmivar(1,[31])P=lmivar(1,[21])….lmisys=getlmisn=decnbr(lmisys)c=zeros(n,1)forj=1:n[Xj,Pj]=defcx(lmisys,j,X,P)c(j)=trace(Xj)+x0’*Pj*x0end[copt,xopt]=mincx(lmisys,c,options)**,,,,,,min1+XMNWzhdn=decnbr(lmisys)c=zeros(n,1)forj=1:n,[alphaj,betaj]=defcx(lmisys,j,alpha,beta)c(j)=(1+h1)*alphaj+d1