LMI工具箱介绍——俞立

LMILMILMILMIL(x) L x L x L0(1)0 1 1 N NL0 , , , x1, , x NL L1 Nx[x1 , , ] R NxTN11L, , ) ( , , (X1 X n R X X n1 )L( ) R( ) X, ,1 Xn1LyapunovA T X XA0(2)XA1222 1x x1 22 X X 2x x2 3X x1 x x X2 3X1 0 0 1x1 x x2 30 0 1 01A 22 2 03 0 0x1 x x0(3)2 32 034 0 42 1 Lyapunov2 3 32 A n 3n(n1) 2LMI Lyapunov2 X 3HA X T XA X CTBNTCX I D N0BT DTIA X X T R n n RB C D NNA XT XA XCTBL(X, ) CX I DBT DTIX 1 1 L(X, ) L(X, )L(X, ) XL(X, ) B DXA, IA.2LMIN L(X, , X)N M R(X, , X)MT T1 K 1 KX1 , , N MXKL( ) R( )X0X00X1 2X, ,1XKlmisys lmisys LMIlmidem1 4 4 6G(s) C(s I A) 1 B(4)/ Dd10 0 00 d0 01D(5)0 0 d d2 30 0 d4 d 55 D sup DG( j)D 1 1 X R6 6 S D T D R4 4A T X XABTXC SCTXBS0(6)X0(7)S I(8)lmivar lmiterm 6 ~ 8 setlmis([])X=lmivar(1,[6 1])S=lmivar(1,[2 0;2 1])% 1st LMIlmiterm([1 1 1 X],1,A,’s’)lmiterm([1 1 1 S],C’,C)lmiterm([1 1 2 X],1,B)lmiterm([1 2 2 S],-1,1)% 2nd LMIlmiterm([-2 1 1 X],1,1)% 3rd LMIlmiterm([-3 1 1 S],1,1)lmiterm([3 1 1 0],1)lmisys=getlmislmivar X S lmitermgetlmis lmisys lmisyssetlmis getlmissetlmis getlmissetlmis([])lmisosetlmis(lmiso)lmisys=getlmislmisyslmivarlmivarX=lmivar(type,struct)XXtype XstructXType =1 D10 D 2DrDjstruct r 2i(m , n ) mDin 1 0 1 n 1Din 0Dn1DiiType =2 struct= (m , n )Type =3 X0 xnxnnstruct X0,X (i , j ) 0struct (i , j )n , X (i , j ) xnn ,X (i , j ) xn2 X 1 X2X3X3 3 1X 2 42X30 0 5 511 2I 20 0I2 22 2lmivarsetlmis([]) X1=lmivar(1,[3 1]) X2=lmivar(2,[2 4]) X3=lmivar(1,[5 1;1 0;2 0])lmiterm12 3 A T X C T SCPXQ X P Q3LMIlmitermA X T XAB XTC SCTXBSlmiterm([1 1 1 X],1,A,’s’)lmiterm([1 1 1 S],C’,C)lmiterm([1 1 2 X],1,B)lmiterm([1 2 2 S],-1,1)A T X XA C T SC XB S 11 m m-m m2 3 [1 1 2 1]1 2 2 30 k k X-k k 1X X T X 1 S 2k klmiterm 2 3 PXQ PX T Q4 's'12 lmiterm 's' lmitermA T X XA3 I1 3 S Ilmiterm([-3 1 1 S],1,1)lmiterm([3 1 1 0],1)lmivarnewlmi lmiterm1setlmis([])X=lmivar(1,[6 1])S=lmivar(1,[2 0;2 1])BRL=newlmilmiterm([BRL 1 1 X],1,A,’s’)lmiterm([BRL 1 1 S],C’,C)lmiterm([BRL 1 2 X],1,B)lmiterm([BRL 2 2 S],-1,1)Xpos=newlmilmiterm([-Xpos 1 1 X],1,1)Slmi=newlmilmiterm([-Slmi 1 1 S],1,1)lmiterm([Slmi 1 1 0],1)lmisys=getlmisX S X S BRL Xpos Slmi 1 2 3 -Xpos 2 -X Xlmieditlmieditlmiedit1S R Glmivar struct2 MATLABA X TB XT XA X BI[A’*X+X*A X*B;B’*X -1]<0X Xlmivar/lmiterm view commandsdescribe…lmivar/lmitermMATLABMATLAB saveloadlmivar/lmiterm readdescribe the matrix variables describe the LMIs …lmivar/lmiterm writecreateMATLAB mylmi MATLAB mylmi mylmilmiterm0 1 lmiedit lmiterm lmieditX(A*X+B)’*C+C’*(A*X+B)(A+B)’*X+X’*(A+B)lmiterm lmiterm 1newlmi lmivarA.1 lmiedit 1A.1 lmieditA.3LMI lmiinfo lminbr matnbrlmiinfolmiinfolmiinfolmiinfo(lmisys)lmisys getlmislminbr matnbrmatnbr(lmisys)A.4LMI xX, ,1 Xkx RNX, ,1XkA(x) B(x)feaspmin cTxxs.t. A(x) B(x)mincxminxs.t. C(x) D(x) (9)0 B(x) (10)A(x) B(x) (11) gevpfeaspfeasp[tmin,xfeas]=feasp(lmisys,options,target)feaspmin ts.t. A(x) B(x) t Ilmisystmin feasptmin<0 lmisys lmisys feaspxfeas dec2mat lmisysfeasp target tmin tmin<targettarget=0 feaspoptions 5options(1)options(2)100options(3) options(3)= R0Nx2 R2ii 1xfeas R R109xoptions(4)J J t1%10options(5) options(5)=1 options(5)=0options(i)3P I PA1 P PA012T1A2 P PA013T2A3 P PA14T31 20.8 1.5A 1A, A,23131.32.71.4 0.7 0.92.0feaspsetlmis([]) P=lmivar(1,[2 1]) lmiterm([1 1 1 P],1,A1,’s’) % LMI #1 lmiterm([2 1 1 P],1,A2,’s’) % LMI #2 lmiterm([3 1 1 P],1,A3,’s’) % LMI #3 lmiterm([-4 1 1 P],1,1) % LMI #4: P lmiterm([4 1 1 0],1) % LMI #4: Ilmis=getlmisfeasp[tmin,xfeas]=feasp(lmis)tmin 3.1363 lmis dec2matPP=dec2mat(lmis,xfeas,P)P270.8 126.4 126.4 155.1PFrobenius 10 tmin 1[tmin,xfeas]=feasp(lmis,[0,0,10,0,0],-1)tmin1.1745 P( ) maxP 9.6912mincxmincx[copt,xopt]=mincx(lmisys,c,options,xinit,target)lmisysc xdefcxcmincxc T x coptxoptdec2matxoptmincx lmisys c xinit xopt X, , X mat2dec1 kxinit xinittarget x c T x targetoptions 5options(1) copt 10 2options(2) 100options(3) feaspoptions(4) JJ c T x5options(5) options(5)=1 options(5)=0options(i)mincx4minTrace (X)Xs.t. A T X XA XBB T X Q0X1 2 1 1 1 1 0A 3 2 1 , B0 , Q 1 3 121 2 1 1 0 12 36SchurminTrace (X)XA X XA Q XBTs.t. 0B X ITTrace(X) Xmincxmincx1setlmis([])X=lmivar(1,[3 1]) % Xlmiterm([1 1 1 X],1,A,’s’)lmiterm([1 1 1 0],Q)lmiterm([1 2 2 0],-1)lmiterm([1 2 1 X],B’,1)LMIs=getlmis2 Trace(X) c T x x Xc X X Ic=mat2dec(LMIs,eye(3))defcx3 mincx xopt copt=c’*xoptoptions=[1e-5,0,0,0,0][copt,xopt]=mincx(LMIs,c,options)1e-5 coptmincxSolver for linear objective minimization under LMI constraintsIterations : Best objective value so far12 -8.5114763 -13.063640*** new lower bound: -34.0239784 -15.768450*** new lower bound: -25.0056045 -17.123012*** new lower bound: -21.3067816 -17.882558*** new lower bound: -19.8194717 -18.339853*** new lower bound: -19.1894178 -18.552558*** new lower bound: -18.9196689 -18.646811*** new lower bound: -18.80370810 -18.687324*** new lower bound: -18.75390311 -18.705715*** new lower bound: -18.73257412 -18.712175*** new lower bound: -18.72349113 -18.714880*** new lower bound: -18.71962414 -18.716094*** new lower bound: -18.71798615 -18.716509*** new lower bound: -18.71729716 -18.716695*** new lower bound: -18.716873Result: feasible solution of required accuracybest objective value: -18.716695guaranteed relative accuracy: 9.50e-006f-radius saturation: 0.000% of R = 1.00e+009c T xx 4 mincx xoptXopt=dec2mat(LMIs,xopt,X)6.3542 5.8895 2.20465.88956.2855 2.2201Xopt2.2046 2.2201 6.0771gevpgevp[lopt,xopt]=gevp(lmisys,nlfc,options,linit,xinit,target)gevploptxxopt dec2matlmisys 1 6 ~ 88nlfc( ,)linit=xxxinit=gevpx( ,)( ,)xtarget( , x ) target options5 options(1) lopt10 2options(2) 100options(3) feaspoptions(4)JJ5options(5) options(5)=1 options(5)=0options(i)gevpA (x )B (x )A (x )B (x )0 B (x )0 B (x )B (x )B (x ) 01B x , ( )1gevpYA (x )Y 0 0 0,YB 1(x ), B x( )A (x )B (x ), B (x ) 0gevp5 3x (t ) A (t ),i i x(1, 2, 3),A V (x ) x T Px, (i 1, 2, 3)3Lyapunov i d V (x ) 3d tmins.t.I P A PT 1PA 1 PA PT2PA2PA PT 3PA3Pgevpsetlmis([]); P=lmivar(1,[2 1]) lmiterm([1 1 1 0],1) % P>I: Ilmiterm([-1 1 1 P],1,1) % P>I: P lmiterm([2 1 1 P],1,A1,’s’) % LFC #1 (lhs) lmiterm([-2 1 1 P],1,1) % LFC #1 (rhs) lmiterm([3 1 1 P],1,A2,’s’) % LFC #2 (lhs) lmiterm([-3 1 1 P],1,1) % LFC #2 (rhs) lmiterm([4 1 1 P],1,A3,’s’) % LFC #3 (lhs) lmiterm([-4 1 1 P],1,1) % LFC #3 (rhs)lmis=getlmisgevp[alpha,popt]=gevp(lmis,3)alpha = -0.122 0.122P5.58 8.358.35 18.64x mat2dec dec2matX1 X X2 3X1 X2 X3 mat2decxdec=mat2dec(lmisys,X1,X2,X3)lmisys lmisysmincx gevpmat2dec xinit X1 X X2 3xdec dec2mat k2X2=dec2mat(lmisys,xdec,2)dec2mat 2lmivarmatnbr decnbr decinfoA.5LMIevallmi showlmi4 mincx xoptevlmi=evallmi(LMIs,xopt)[lhs,rhs]=showlmi(evlmi,1)xopt 1 eig(lhs-rhs)ans=-2.0387e-04-3.9333e-05-1.8917e-07-4.6680e+01lhs-rhs xopt 1A.6LMI dellmi delmvar setmvarDellmidellmi1 lmisys Xnewsys=dellmi(lmisys,2)2newsys1 2S I3 2newlmi 1 BRL Xpos Slmi Slminewsys=dellmi(lmisys,Xpos)2 S IdelmvardelmvarA T X XA BW W TB T I0X X R4 4 W R2 4Tsetlmis([])X=lmivar(1,[4 1]) % XW=lmivar(2,[2 4]) % Slmiterm([1 1 1 X],1,A,’s’)lmiterm([1 1 1 W],B,1,’s’)lmiterm([1 1 1 0],1)lmisys=getlmisWnewsys=delmvar(lmisys,W)newsys LyapunovA T X XA I0setmvarsetmvarsetmvar1 G 1 G 1D S D T D S IS2I Snewsys=setmvar(lmisys,S,2)2 S3 S 22I newsysA X T XAB XT2C CTXB2IX02I I newsys=dellmi(newsys,3)newsys =dellmi(newsys,Slmi)Slmi newlmi 3A.7lmivar 1 230 xn x n nX XX X n X p n xx, ,1 n pxn1, , x X lmivar n p[X,n,sX]=lmivar(type,struct)n sX Xx, , 1 xnlmivar6X X1X2X X1 X23 3 2 2X1 X1 X2setlmis([])[X1,n,sX1]=lmivar(2,[2 3]) [X2,n,sX2]=lmivar(2,[3 2])sX1 sX2 X1 X2>>sX1sX1=1 2 34 5 6>>sX2sX2=7 89 1011 12sX2(1, 1)=7 X(1, 1) 722 Type3 X X1 X2[X,n,sX]=lmivar(3,[sX1,zeros(2);zeros(3),sX2])X>>sXsX=1 2 3 0 04 5 6 0 00 0 0 7 80 0 0 9 100 0 0 11 127 3 3 X 3 3 Toeplizy 1 y2y3Y y2 y1y2y 3 y2y1Y 3 3 Y X Y n 1 n 2 n 3 n XXsetlmis([])[X,n]=lmivar(1,[3 1])[X, n] 2 n=6 YY=lmivar(3,n+[1 2 3;2 1 2;3 2 1])Y=lmivar(3,toeplitz(n+[1 2 3]))toeplitz MATLAB decinfo X Ylmis=getlmisdecinfo(lmis,X)ans=1 2 42 3 54 5 6decinfo(lmis,Y)ans=7 8 98 7 89 8 78X Y ZX xy, Yzt, Ztxx y z tX Y 1setlmis([])[X,n,sX]=lmivar(1,[1 0;1 0])[Y,n,sY]=lmivar(1,[1 0;1 0])lmivar 3 X Y(x, , , )1x x x2 3 4 (x, y, z, t)sX=1 00 2sY=3 00 4lmivar 3 Z[Z,n,sZ]=lmivar(3,[0 -sX(1,1);-sY(2,2) 0])sX(1, 1) x sY(2, 2)1x4Zx4x1txHermitian matrix L(x) L(x) 0Re(L(x))Im(L(x))I m(L(x))Re(L(x))XX X1 j X2X1 X2AA A j1 A2A 1 A 2X1 X2X1 X2 X2X1XX1 j X2 A1A2A2A1A A1 j A2M H XM X, X X H IM1M2T X1X2M1M2X1X2 M2M1X2X1M2M1X2X1X1X2X2X1IM 1 j MM2X X1 j X2Mi XiM C55M1=real(M), M2=imag(M)bigM=[M1 M2;-M2 M1]setlmis([])% declare bigX=[X1 X2;-X2 X1] with X1=X1’ and X2+X2’=0:[X1,n1,sX1]=lmivar(1,[5 1])[X2,n2,sX2]=lmivar(3,skewdec(5,n1))bigX=lmivar(3,[sX1 sX2;-sX2 sX1])% describe the real counterpart of the complex LMI system:lmiterm([1 1 1 0],1)lmiterm([-1 1 1 bigX],1,1)lmiterm([2 1 1 bigX],bigM’,bigM)lmiterm([-2 1 1 bigX],1,1)lmis=getlmisbigX= X1X2X2X11 X sX11n1 X12 skewdec X skewdec(5,n1)2n1+1, n1+2, ... 5 53 3 X X sX1, sX2 bigX1 2mincx c T xmincx c T x xTrace(X) X u T Xu udefcx cTrace(X) x PxT0 0X P x lmisysX Pxx0=[1 1]setlmis([])X=lmivar(1,[3 0])P=lmivar(1,[2 1])::lmisys=getlmisc T x Trace(X) x Px cT0 0n=decnbr(lmisys)c=zeros(n,1)for j=1:n,[Xj,Pj]=defcx(lmisys,j,X,P)c(j)=trace(Xj)+x0’*Pj*x0end1 2 cfor1 x 1 x X P defcx defcxjlmisys j X PXj Pj2 X=Xj P=Pj c jc=3121n=decnbr(LMI system)c=zeros(n,1)for j=1:n,[matrix values]=defcx(LMI system,j,matrix identifiers)c(j)=objective(matrix values)endA.8 系统模型描述Ex(t) y(t) Ax(t)Cx(t)Bu(t)Du(t)(15)A, B, C, D, E E x(t), u(t), y(t)15E15 E15mx fx kx uy x151 0 0m(t)k1f(t)1u(t) y(t) 1 0 (t)(t) [x(t) x(t)]T15 MATLAB SYSTEMn A j(E I) B0C D00 InfnLMI ltisys ltiss SYSTEM SYSTEMsys=ltisys(-1,1,1,0)x x u, y xSYSTEM SYSTEM sys A, B, C, D [A,B,C,D]=ltiss(sys)ltitf SYSTEM G(s) n(s)d(s)sys=ltisys(‘tf’,n,d)G(s) n d n(s) d(s)sinfo(sys) SYSTEM sysspol(sys) ssubSYSTEM G G 12 3ssub(G,1,2:3)G G SYSTEM sinv G(s) H(s) G 1 (s)G(s) D sbalancA, B, CLMISYSTEMSYSTEMg1 g2 G1(s) G2 (s) SYSTEM saddG1 s( )( ) G2 s SYSTEM smult(g1,g2)SYSTEM 10 G( ) ( ) 2s G s1sdiag(g1,g2) G(s) diag{G1(s), G2 (s)} SYSTEMSr+yG1(s)G2(s)1sloop r y SYSTEMslft w1w2z1z2SYSTEMw1u G1(s) z1 yz2G2(s) w22u R2 , y R3slft(g1,g2,2,3)H。

Matlab中的LMI工具箱的研究与使用Matlab中的LMI工具箱的研究与使用摘要：由于内点算法可以利用计算机求解高阶矩阵不等式，使得线性矩阵不等式（LMI）在控制理论得以应用，因而线性矩阵不等式在控制系统分析、设计中扮演着越来越重的角色。

在毕业设计中，我学习了LMI理论，Matlab中LMI工具箱的使用，研究了常见的控制问题与LMI关系以及其表达式，并研究了基于LMI方法的鲁棒控制器设计问题，推导了如何将鲁棒控制器设计问题转化为LMI形式，给出了通过求解LMI方程构造控制率的算法。

LMI是一种较新的方法，可以在控制系统的许多领域得以应用，如鲁棒控制、非线性控制预测控制等有着巨大的潜力。

关键词：线性矩阵不等式；LMI工具箱；控制系统The research and use of the Matlba LMI toolboxStudent majoring in Automation SUN Peng-kunTutorAbstract：Due to the interior point algorithm can use computer to solve high order matrix inequality,It makes the linear matrix inequality (LMI) in the control theory to application and linear matrix inequality (LMI) played more and more heavy role in the control system analysis and design. At the graduation design, I studied the LMI theory, in the use of Matlab LMI toolbox, and studies the problem of common control with LMI relations and its expression,and I studied the problem of the robust contoeller design that based on lmi method, how totransform the robust controller design problem into the LMI form, and given by solving the LMI equation of tectonic control algorithm. LMI is a new method, it can be used in many control system fields, such as robust control, nonlinear control predictive control, and it has great potential.Key words: linear matrix inequality; LMI toolbox; control system;robust control引言线性矩阵不等式（LMI）工具箱是求解一般线性矩阵不等式问题的一个高性能软件包。

MATLAB（Matrix Laboratory）是一款广泛应用于科学计算和工程领域的专业软件，其功能强大、灵活性高，并且具有丰富的工具箱支持。

LMI（Linear Matrix Inequality）工具箱是MATLAB中的一种工具箱，用于解决线性矩阵不等式相关的问题。

本文将介绍LMI工具箱的基本使用方法，并结合具体实例进行详细讲解。

一、LMI工具箱的安装1.确保已经安装了MATLAB软件，并且软件版本是R2015b及以上版本。

只有在这些版本中，LMI工具箱才会被自动安装。

2.在MATLAB的命令行中输入“ver”，可以查看当前安装的工具箱列表，确认LMI工具箱是否已经成功安装。

二、LMI工具箱的基本功能1. LMI工具箱主要用于解决线性矩阵不等式问题，例如矩阵的稳定性分析、最优控制问题等。

2. LMI工具箱提供了一系列的函数和工具，能够方便地构建和求解线性矩阵不等式问题，同时也包括了一些经典的稳定性分析方法和控制器设计方法。

三、LMI工具箱的基本使用方法1. 定义变量：在使用LMI工具箱时，首先需要定义相关的变量。

可以使用“sdpvar”函数来定义实数变量，使用“sdpvar”函数和“size”函数可以定义矩阵变量。

2. 构建约束：在定义变量之后，需要构建线性矩阵不等式的约束条件。

可以使用“sdpvar”变量的线性组合来构建约束条件，使用“>=”来表示大于等于关系。

3. 求解问题：构建好约束条件之后，即可使用“optimize”函数来求解线性矩阵不等式问题。

在求解问题时，可以指定优化的目标函数和一些额外的约束条件。

四、LMI工具箱的实例应用下面我们通过一个具体的实例来演示LMI工具箱的使用方法。

假设有一个线性时不变系统，其状态方程可以表示为：$\dot{x} = Ax + Bu$其中，A和B分别为系统的状态矩阵和输入矩阵。

我们希望设计一个状态反馈控制器K，使得系统在闭环下能够保持稳定。

LMI⼯具箱介绍——俞⽴LMI ⼯具箱介绍线性矩阵不等式（LMI ）⼯具箱是求解⼀般线性矩阵不等式问题的⼀个⾼性能软件包。

由于其⾯向结构的线性矩阵不等式表⽰⽅式，使得各种线性矩阵不等式能够以⾃然块矩阵的形式加以描述。

⼀个线性矩阵不等式问题⼀旦确定，就可以通过调⽤适当的线性矩阵不等式求解器来对这个问题进⾏数值求解。

LMI ⼯具箱提供了确定、处理和数值求解线性矩阵不等式的⼀些⼯具，它们主要⽤于：z 以⾃然块矩阵形式来直接描述线性矩阵不等式； z 获取关于现有的线性矩阵不等式系统的信息； z 修改现有的线性矩阵不等式系统； z 求解三个⼀般的线性矩阵不等式问题； z 验证结果。

本附录将详细介绍LMI ⼯具箱提供的⽤于解决以上各个问题的相关函数和命令。

A.1 线性矩阵不等式及相关术语⼀个线性矩阵不等式就是具有以下⼀般形式的⼀个矩阵不等式：0<+++=N N x x L L L x L "110)( (1)其中：是给定的对称常数矩阵，是未知变量，称为决策变量，N L L L ,,,10"N x x ,,1"∈=T 1],,[N x x "x N R 是由决策变量构成的向量，称为决策向量。

尽管表达式（1）是线性矩阵不等式的⼀个⼀般表⽰式，但在⼤多数实际应⽤中，线性矩阵不等式常常不是以⼀般表⽰式（1）的形式出现，⽽是具有以下形式：),,(),,(11n n X X R X X L ""<其中的和是矩阵变量的仿射函数，通过适当的代数运算，上式可以写成线性矩阵不等式的⼀般表⽰式（1）的形式。

例如，在系统稳定性问题中经常遇到的Lyapunov 矩阵不等式)(?L )(?R n X X ,,1"0<+XA X A T (2)也是⼀个线性矩阵不等式，其中的是⼀个矩阵变量。

我们以⼀个⼆阶矩阵为例，将矩阵不等式（2）写成⼀般表⽰式（1）的形式。

摘要本文分析了两课程异同点，并给出了应对方法。

提出合理选取教材和分配学时、弱化理论证明等降低理论难度；介绍人物故事、开展双语教学、强化实验等提高学习兴趣；采用比较教学新方法应对教学内容差异。

文末指出探索的新方向。

关键词现代控制理论自动控制原理教学实践比较教学法Exploration on Teaching Based on Analysis of Similarities and Differences between"Automatic Control Theory"and "Modern Control Theory"Courses//Wan Xiongbo,YangFangAbstract This paper analyzes the similarities and differences be-tween the above two courses and gives the corresponding meth-ods.To reduce theoretical difficulties,the methods such as se-lecting appropriate textbooks,suitable distribution of teachinghours,and simplifying the complexity of theory proof,are expect-ed;To improve students'interest in learning,the methods includ-ing introduction of scientists biographies,bilingual education,and strengthening the experiments,are proposed;Compari-son-based new teaching method is utilized to deal with the dif-ferences between the contents of two courses.Finally,new direc-tions for future exploration are discussed.Key words modern control theory;automatic control theory;teaching practice;comparison-based teaching method “自动控制原理”是研究控制系统的一般规律，并为系统的分析和综合提供基本理论和方法的专业基础核心课程[1,2]。

L M IL M IL M IL M I0NN x x L L L x L 110)( (1)NL L L ,,,10Nx x ,,1T1],,[N x x x N R11),,(),,(11n n X X R X X L1Lyapunov)(L )(R nX X ,,10XA X A T (2)21X221A22X3221x x x x X X321x x x X100001100001321x x x XA20400043300222321x x x (3) 21Lyapunov23323A n2)1(n n L M ILyapunov23XH0NID B D ICX BXC XA X A N TTTT TNDCBAT XXnnR RNI D B D I CX B XC XA X A X L TT TT ),(X 11),(X L ),(X L),(X L X),(X L BDIXA ,A.2L M IM X X R M NX X L N ),,(),,(1T 1T K KKX X ,,1N M)(L )(R 0XXX1KX X ,,12l m i s y s l m i s y sL M Il m i de m1446B A IC G 1)()(s s (4)/D543211000000000d d d d d d D(5)5D1)(s up 1DDG jX66R DD ST 44R0S XB XB SC C XA XA T T T (6) 0X (7) I S(8)l m i var l m it er m 6~8setlmis([])X=lmivar(1,[6 1]) S=lmivar(1,[2 0;2 1])% 1st LMIlmiterm([1 1 1 X],1,A,’s’) lmiterm([1 1 1 S],C’,C) lmiterm([1 1 2 X],1,B) lmiterm([1 2 2 S],-1,1) % 2nd LMI lmiterm([-2 1 1 X],1,1)% 3rd LMI lmiterm([-3 1 1 S],1,1) lmiterm([3 1 1 0],1) lmisys=getlmisl m i varX l m it er mge tl m i sl m i s ys l m i s y sSs et l m i sget l m i s s e tl m i s ge tl m i ssetlmis([])l m i s osetlmis(lmiso) lmisys=getlmisl m i s y sl m i varl m i varX=lmivar(type,struct)X Xt ype Xs t ruc tX Type =1rD D D 0000021s t ruc tjD 2r in 10),(n m miD 11n i D 0n iD 1niD Type =2s t ruc t = ),(n m Type =3X0nx s t ruc tnx nXs t ruc tnn x j i n x j i n j i j i ),(,),(,0),(,0),(X X X 221X X3X1X 332X 42 2213I X 0000055122I 22l m i varsetlmis([])X1=lmivar(1,[3 1]) X2=lmivar(2,[2 4]) X3=lmivar(1,[5 1;1 0;2 0])l m i term1 23X A T S CC T P X QXPQ 3L M Il m it er m0SXB XB S CC XA X A TT Tlmiterm([1 1 1 X],1,A,’s’) lmiterm([1 1 1 S],C’,C) lmiterm([1 1 2 X],1,B) lmiterm([1 2 2 S],-1,1)XB S CC XA XA T T S11mm -m m23[1 1 2 1]1223k k-k 1kX k X T k X X 12S l m it er m234's 'P X Q QP X T1 2l m it er m's 'l m it er mXA X A T3I13I S lmiterm([-3 1 1 S],1,1) lmiterm([3 1 1 0],1)l m i varnew l m il m it er m1setlmis([]) X=lmivar(1,[6 1]) S=lmivar(1,[2 0;2 1])BRL=newlmilmiterm([BRL 1 1 X],1,A,’s’) lmiterm([BRL 1 1 S],C’,C) lmiterm([BRL 1 2 X],1,B) lmiterm([BRL 2 2 S],-1,1) Xpos=newlmi lmiterm([-Xpos 1 1 X],1,1)Slmi=newlmi lmiterm([-Slmi 1 1 S],1,1) lmiterm([Slmi 1 1 0],1) lmisys=getlmisX SX BR L Xpo s Sl m i123-Xpo s2-XSXl m i e di tl m i ed itlmiedit1SRGl m i vars t ruc t2M ATLA B0IXB XBXA X A T T[A’*X+X*A X*B;B’*X -1]<0X Xl m i var /l m it er m v i ew co mm and sde s cr i be…l m i var /l m it er mM ATLA BM ATLA B s avel oadl m i var /l m it er m readde s cr i be t he m a t r i x var i ab l e s de s cr i be t he L M I s …l m i var /l m it er m wr it e crea t eM ATLA B m y l m i M ATLA B m y l m i m y l m il m it er m01l m i ed itl m it er m l m i ed it X (A*X+B)’*C+C’*(A*X+B)(A+B)’*X+X’*(A+B)l m it er m l m it er m1new l m i l m i varA.1l m i ed it1A.1l m i ed itA.3L M I l m ii nfo l m i nbr m a t nbrl m iin fol m ii nfol m ii nfolmiinfo(lmisys)l m i s y s ge tl m i sl m inb r mat nb rmatnbr(lmisys)A.4L M I xkX X ,,1NR x kX X ,,1)()(x B x Afea s px c xT m i ns .t . )()(x B x Am i ncxxm i ns .t . )()(x D x C (9))(x B 0(10) )()(x B x A (11)gevpfea s pfea s p[tmin,xfeas]=feasp(lmisys,options,target)fea s pt m i ns .t . I x B x A t )()( l m i s y st m i nfea s pt m i n <0l m i s y sl m i s y s fea s pxfea sdec2m a tl m i s y s fea s pt arge t t m i n t m i n <t arge tt arge t =0fea s pop ti on s5op ti on s (1)op ti on s (2)100 op ti on s (3)op ti on s (3)=R212R x N i ixfea sR910Rxop ti on s (4)J1%10Jt op ti on s (5)op ti on s (5)=1op ti on s (5)=0op ti on s (i )3IPP01T 1P A PA 1202T 2P A PA 13 03T 3P A P A 14.27.09.04.1,7.23.15.18.0,3121321A A A fea s psetlmis([]) P=lmivar(1,[2 1])lmiterm([1 1 1 P],1,A1,’s’) % LMI #1 lmiterm([2 1 1 P],1,A2,’s’) % LMI #2 lmiterm([3 1 1 P],1,A3,’s’) % LMI #3 lmiterm([-4 1 1 P],1,1) % LMI #4: P lmiterm([4 1 1 0],1)% LMI #4: Ilmis=getlmisfea s p[tmin,xfeas]=feasp(lmis)t m i n l m i sdec2m a t1363.3PP=dec2mat(lmis,xfeas,P)1.1554.1264.1268.270PPF roben i u s10t m i n1[tmin,xfeas]=feasp(lmis,[0,0,10,0,0],-1)t m i n1745.1P6912.9)(m axPm in cxm i ncx[copt,xopt]=mincx(lmisys,c,options,xinit,target)l m i s y s cxdefcxcm i ncxcop txop tdec2m a txop tx c Tm i ncx l m i s y sc x i n it xop tm a t 2decx i n itx i n itkX X ,,1t arge t t arge txxc T op ti on s 5op ti on s (1)cop t210op ti on s (2)100op ti on s (3)fea s pop ti on s (4)5JJx c T op ti on s (5)op ti on s (5)=1op ti on s (5)=0op ti on s (i)m i ncx4)(Trace m i n X Xs .t .0Q X XBB XA XA T T X36121231011,101,121123121Q B AS chur)(Trace m i n X Xs .t .0IXB XB Q XA XA TT )(Trace X Xm i ncxm i ncx1setlmis([])X=lmivar(1,[3 1]) % Xlmiterm([1 1 1 X],1,A,’s’)lmiterm([1 1 1 0],Q)lmiterm([1 2 2 0],-1)lmiterm([1 2 1 X],B’,1)LMIs=getlmisTrace X x c T x X2)(c X I Xc=mat2dec(LMIs,eye(3))defcx3m i ncx xop t cop t=c’*xop toptions=[1e-5,0,0,0,0][copt,xopt]=mincx(LMIs,c,options)1e-5cop tm i ncxSolver for linear objective minimization under LMI constraints Iterations : Best objective value so far12 -8.5114763 -13.063640*** new lower bound: -34.0239784 -15.768450*** new lower bound: -25.0056045 -17.123012*** new lower bound: -21.3067816 -17.882558*** new lower bound: -19.8194717 -18.339853*** new lower bound: -19.189417 8 -18.552558 *** new lower bound: -18.919668 9 -18.646811 *** new lower bound: -18.803708 10 -18.687324 *** new lower bound: -18.753903 11 -18.705715 *** new lower bound: -18.732574 12 -18.712175 *** new lower bound: -18.723491 13 -18.714880 *** new lower bound: -18.719624 14 -18.716094 *** new lower bound: -18.717986 15 -18.716509 *** new lower bound: -18.717297 16 -18.716695 *** new lower bound: -18.716873Result: feasible solution of required accuracy best objective value: -18.716695 guaranteed relative accuracy: 9.50e-006 f-radius saturation: 0.000% of R = 1.00e+009x c T x4m i ncxxop tXopt=dec2mat(LMIs,xopt,X)0771.62201.22046.22201.22855.65.88952046.28895.56.3542op tX gev pgevp[lopt,xopt]=gevp(lmisys,nlfc,options,linit,xinit,target)gevpl op tx xop t dec2m a tl m i s y s 16~88n l fcli n it =),(00x 0x i n it =gevpt arge tx ),(00x ),(00x ),(x t arge top ti on s5op ti on s (1)l op t210op ti on s (2)100op ti on s (3)fea s pop ti on s (4)JJ5op ti on s (5)op ti on s (5)=1op ti on s (5)=0op ti on s (i )gevp)()(x B x A)()(x B x A)(x B 0)(x B 00000)(,)()(11x B x B x BgevpY0000)(),(,)(1x B x B Y Y x A0)(),()(x B x B x Agevp53),3,2,1(),()(itt i x Ax3Lyapunov3)3,2,1(,i i A Px xx T)(VtVd)(d xm i ns.t. P IPP AP A1T1PP AP A2T2PP AP A3T3gevpsetlmis([]);P=lmivar(1,[2 1])lmiterm([1 1 1 0],1) % P>I: Ilmiterm([-1 1 1 P],1,1) % P>I: Plmiterm([2 1 1 P],1,A1,’s’) % LFC #1 (lhs)lmiterm([-2 1 1 P],1,1) % LFC #1 (rhs)lmiterm([3 1 1 P],1,A2,’s’) % LFC #2 (lhs)lmiterm([-3 1 1 P],1,1) % LFC #2 (rhs)lmiterm([4 1 1 P],1,A3,’s’) % LFC #3 (lhs)lmiterm([-4 1 1 P],1,1) % LFC #3 (rhs)lmis=getlmisgevp[alpha,popt]=gevp(lmis,3)a l pha =-0.1220.12264.1835.835.858.5Px m a t 2dec dec2m a tX1X2X3m a t 2dec321X X X xdec=mat2dec(lmisys,X1,X2,X3)l m i s y sl m i s y sm i ncx gevpm a t 2decx i n it321X X X xdecdec2m a tk2X2=dec2mat(lmisys,xdec,2)dec2m a t2l m i varm a t nbrdecnbr dec i nfoA.5L M I eva ll m is how l m i4m i ncxxop tevlmi=evallmi(LMIs,xopt) [lhs,rhs]=showlmi(evlmi,1)xop t1eig(lhs-rhs)ans=-2.0387e-04-3.9333e-05-1.8917e-07-4.6680e+01l h s-rh s xop t1A.6L M I de ll m i de l m var s e t m varDe ll m ide ll m i1l m i s y s Xnewsys=dellmi(lmisys,2)2new s y s12I S32new l m i1BR L Xpo s Sl m i Sl m inewsys=dellmi(lmisys,Xpos)2I Sd e l mvarde l m var0I B W B W XA X A T T TT X X44R42R Wsetlmis([]) X=lmivar(1,[4 1]) % X W=lmivar(2,[2 4]) % Slmiterm([1 1 1 X],1,A,’s’) lmiterm([1 1 1 W],B,1,’s’) lmiterm([1 1 1 0],1) lmisys=getlmisWnewsys=delmvar(lmisys,W)new s y sLyapunov0I XA X A Ts e t m vars etmvars e t m var11G1G D DD ST IS I S2Snewsys=setmvar(lmisys,S,2)2S32S I 2new s y s0IXB XB CC XA XA 22TT T0X I I 2newsys=dellmi(newsys,3) newsys =dellmi(newsys,Slmi) Sl m i new l m i 3 A.7l m i var 123 0n x n x n X X X X n X p p n n x x ,,1p n n x x ,,1X l m i var[X,n,sX]=lmivar(type,struct) n s X X nx x ,,1l m i var 621X X X00 X1X 2X 3223X 1 1X 2Xsetlmis([])[X1,n,sX1]=lmivar(2,[2 3]) [X2,n,sX2]=lmivar(2,[3 2])s X1s X2 1X 2X>>sX1sX1=1 2 3 4 5 6>>sX2sX2= 7 89 10 11 12 s X2(1, 1)=7(1, 1)7 2X 2Type 3X 1X 2X [X,n,sX]=lmivar(3,[sX1,zeros(2);zeros(3),sX2]) X>>sXsX= 1 2 3 0 0 4 5 6 0 0 0 0 0 7 8 0 0 0 9 10 0 0 0 11 12 733X 33Toep li z123212321y y y y y y y y y Y Y 33Y X Y 321n n n n XXsetlmis([]) [X,n]=lmivar(1,[3 1]) [X, n]2n =6Y Y=lmivar(3,n+[1 2 3;2 1 2;3 2 1])Y=lmivar(3,toeplitz(n+[1 2 3])) t oep lit zM ATLA B dec i nfo X Ylmis=getlmis decinfo(lmis,X) ans= 1 2 4 2 3 5 4 5 6decinfo(lmis,Y)ans= 7 8 9 8 7 8 9 8 78Y X Z00,00,00t x t z y x Z Y X tz y x X Y1 setlmis([])[X,n,sX]=lmivar(1,[1 0;1 0]) [Y,n,sY]=lmivar(1,[1 0;1 0])l m i var 3X Y),,,(4321x x x x ),,,(t z y x sX=1 0 0 2sY=3 0 04 l m i var 3Z[Z,n,sZ]=lmivar(3,[0 -sX(1,1);-sY(2,2) 0]) s X(1, 1)s Y(2, 2) 1x 4x 000041t x x x ZHer m iti an m a t r i x)(x L 0)(x L 0))(R e())(I m ())(I m ())(R e(x L x L x L x L X21X X X j 1X 2X A21A A A j 1A 2A 21X X 1221X X X X X21X X j 1221A A A A 21A A A j I X X X X M M H H , 122112211221T 1221X X X X M M M M X X X X M M M M I X X X X 1221 21M M M j 21X X X j i i X M 55C M M1=real(M), M2=imag(M) bigM=[M1 M2;-M2 M1] setlmis([])% declare bigX=[X1 X2;-X2 X1] with X1=X1’ and X2+X2’=0:[X1,n1,sX1]=lmivar(1,[5 1])[X2,n2,sX2]=lmivar(3,skewdec(5,n1)) bigX=lmivar(3,[sX1 sX2;-sX2 sX1])% describe the real counterpart of the complex LMI system: lmiterm([1 1 1 0],1) lmiterm([-1 1 1 bigX],1,1)lmiterm([2 1 1 bigX],bigM’,bigM)lmiterm([-2 1 1 bigX],1,1)lmis=getlmis b i gX =1221X X X X 1s X1n1 1X 1X 2s kewdec s kewdec(5,n1)n1+1, n1+2, ... 2X 5533s X1, s X2b i gX 1X 2Xm in cxx c Tm i ncxx c T x )(Trace X XX uu T u defcx c0T 0)(Trace Px x X X P l m i s y s 0x P X x 0 x0=[1 1] setlmis([])X=lmivar(1,[3 0]) P=lmivar(1,[2 1]) : :lmisys=getlmis 0T 0T )(Trace Px x X x c c n=decnbr(lmisys) c=zeros(n,1) for j=1:n,[Xj,Pj]=defcx(lmisys,j,X,P) c(j)=trace(Xj)+x0’*Pj*x0end 12for c 11j x x X P defcx defcx l m i s y s j X P X j Pj 2X =X j P =Pj j cc=3 1 2 1n=decnbr(LMI system) c=zeros(n,1) for j=1:n, [matrix values]=defcx(LMI system,j, matrix identifiers) c(j)=objective(matrix values) end A.8 系统模型描述)()()()()()(t t t t t t D u C x y B u A x x E (15) E D C B A ,,,,E 15)(),(),(t t t y u x E 15E 15 x y u kx x f x m 15 )(01)()(10)(10)(001t t y t u t f k t mT ])()([)(t x t x t 15M ATLA B S Y S TE MInf 00)(0n j D CB I E An L M I lti s y s lti ss S Y S TE M S Y S TE Msys=ltisys(-1,1,1,0)x y u x x , SYSTEM SYSTEM sys DC B A ,,, [A,B,C,D]=ltiss(sys) ltitf SYSTEM )()()(s d s n s G sys=ltisys(‘tf’,n,d) n d )(s G )(s n )(s d sinfo(sys)SYSTEM sys spol(sys)ssub SYSTEM G 123 G ssub(G,1,2:3) G G SYSTEM sinv )(s G )()(1s s G H )(s G D sbalancC B A ,,L M I SYSTEM SYSTEMg1g2SYSTEM sadd )(1s G )(2s G)(1s G )(2s G SYSTEM smult(g1,g2)SYSTEM 10 )()(12s s G G sdiag(g1,g2)SYSTEMS )}(),({d i ag )(21s s s G G G G 1(s )G 2(s )y r + 1 sloop r y SYSTEM slft SYSTEM 21w w 21z z G 1(s )G 2(s )w 1u z 1y w 2z 2 232,R R y u slft(g1,g2,2,3) H。