% LMI-LAB DEMO: EXAMPLE 2.3.1 IN THE USER'S MANUAL % Author: P. Gahinet % Copyright 1995-2004 The MathWorks, Inc. % $Revision: 1.1.6.1 $ echo off load lmidem; clc disp(' '); disp(' '); disp(' LMI CONTROL TOOLBOX '); disp(' ********************* '); disp(' '); disp(' '); disp(' DEMO OF LMI-LAB '); disp(' '); disp(' '); disp(' Specification and manipulation of LMI systems '); disp(' '); disp(' Example 2.3.1 of the Tutorial Section'); disp(' '); disp(' '); disp(' '); disp(' '); echo on pause % Strike any key to continue... clc % % -1 % Scaled RMS gain of the transfer function G(s) = C (sI-A) B % % -1 % We want to minimize the H-infinity norm of D G(s) D over % a set of scaling matrices D with some given structure. This % problem arises in Mu theory (robust stability analysis) % % The system of LMIs is: % % ( A'X + XA + C'SC XB ) % ( ) < 0 % ( B'X -S ) % % X > 0 % % S > I % where % X is symmetric % S = D'*D is symmetric block diagonal with prescribed structure % % ( s11 0 0 0 ) % S = ( 0 s11 0 0 ) % ( 0 0 s22 s23 ) % ( 0 0 s23 s33 ) A pause % Strike any key to continue... clc % % ( A'X + XA + C'SC XB ) % ( ) < 0 % ( B'X -S ) % % X > 0 % % S > I % % % ( s11 0 0 0 ) % X is symmetric, S = ( 0 s11 0 0 ) % ( 0 0 s22 s23 ) % ( 0 0 s23 s33 ) % To specify this LMI system with LMIVAR and LMITERM, % (1) initialize the LMI system to void setlmis([]); % (2) define the 2 matrix variables X,S X=lmivar(1,[6 1]); S=lmivar(1,[2 0;2 1]); pause % Strike any key to continue... clc % % ( A'X + XA + C'SC XB ) % ( ) < 0 % ( B'X -S ) % % X > 0 % % S > I % (3) specify the terms appearing in each LMI. For % convenience, you can give a name to each LMI % with NEWLMI (optional) % 1st LMI BRL=newlmi; lmiterm([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); % 2nd LMI Xpos=newlmi; lmiterm([-Xpos 1 1 X],1,1); % 3rd LMI Slmi=newlmi; lmiterm([-Slmi 1 1 S],1,1); lmiterm([Slmi 1 1 0],1); % (4) get the internal description of this LMI system % with GETLMIS lmisys=getlmis; % Done! A full description of this LMI system is now % stored in the MATLAB variable LMISYS pause % Strike any key to continue... clc % % You can also specify this system with the LMI editor: % % >> lmiedit % % echo off lmiedit echo on % Here you specify the variables in the upper half of the % window and type the LMIs as MATLAB expressions in the % lower half ... % % To see how this should look like, click "LOAD" and % load the string called "demolmi". pause % Strike any key to continue... % % You can % * save this description in a MATLAB string of your % choice ("SAVE") % * generate the internal representation "lmisys" of % this LMI system by clicking on "CREATE" % * visualize the LMIVAR and LMITERM commands that % create "lmisys" (click on "VIEW COMMANDS") % * write in a file this series of commands (click on % "WRITE") % % % Click on "CLOSE" to exit LMIEDIT pause % Strike any key to continue... clc % % You can retrieve various information about the LMI system % you just defined % number of LMIs: lminbr(lmisys) % number of matrix variables: matnbr(lmisys) pause % Strike any key to continue... % variables and terms in each LMI (type q to exit lmiinfo): lmiinfo(lmisys) pause % Strike any key to continue... clc % % We now call FEASP to solve our system of LMIs % % ( A'X + XA + C'SC XB ) % ( ) < 0 % ( B'X -S ) % % X > 0 % % S > I pause % Strike any key to continue... [tmin,xfeas]=feasp(lmisys); % tmin=-1.839011 < 0 : the problem is feasible! % -1 % -> there exists a scaling D such that || D G(s) D ||oo < 1 % % % The output XFEAS is a feasible value of the vector of decision % variables (the free entries of X and S). pause % Strike any key to continue... clc % % Use DEC2MAT to get the corresponding values of the matrix % variables X and S: Xf=dec2mat(lmisys,xfeas,X); Sf=dec2mat(lmisys,xfeas,S); eig(Xf) eig(Sf) % the constraints X > 0 and S > I are satisfied! pause % Strike any key to continue... clc % % To verify that the first LMI is satisfied, % % (1) evaluate the LMI system for the computed decision % vector XFEAS: evlmi = evallmi(lmisys,xfeas); % (2) get the values of the left and right-hand sides of % the first LMI with SHOWLMI: [lhs1,rhs1]=showlmi(evlmi,1); eig(lhs1-rhs1) % the first LMI is indeed satisfied. pause % Strike any key to continue... clc % % Finally, let us check that the H-infinity norm of G(s) % was not less than one from the start. To do this, we % can remove the scaling D by setting S = 2*I and solve % the resulting feasibility problem: % % Find X such that % % ( A'X + XA + C'C XB ) % ( ) < 0 % ( B'X -I ) % % X > 0 % This new LMI system is derived from the previous one % by setting S = 2*I with SETMVAR: newsys=setmvar(lmisys,S,2); % Now call FEASP to solve the modified LMI problem: pause % Strike any key to continue... [tmin,xfeas]=feasp(newsys); % Infeasible! The H-infinity norm of G(s) was larger than one pause % Strike any key to continue... echo off close all return