% QUADPERF Assess quadratic Hinf performance of P-systems % % [PERF,P] = QUADPERF(PDS,G,OPTIONS) computes an upper bound on the % worst-case RMS gain of the affine or polytopic parameter-dependent % systems PDS. % % If OPTION(1)=0, QUADPERF estimates the quadratic H-infinity % performance (an upper bound on the worst-case I/O gain) of PDS when G=0, % or tests whether the performance G >0 can be guaranteed. % % If OPTION(1)=1, QUADPERF maximizes the parameter region where the % quadratic H-infinity performance G >0 can be sustained. The maximal % region is obtained by scaling the prescribed parameter box around its % center. This option is only available for affine PDS. % % Input: % PDS polytopic or parameter-dependent system (see PSYS) % G prescribed performance (default=0) % OPTIONS optional 3-entry vector. % OPTIONS(1) : see above (default=0) % OPTIONS(2)=0 : fast mode (default) % =1 : least conservative mode % OPTIONS(3) : bound on the condition number of P % (default = 1e9) % Output: % PERF if OPTIONS(1)=0, quadratic RMS gain. Otherwise, % percentage of the parameter box PV where the % performance G can be guaranteed (PERF=1 means 100%) % P Lyapunov matrix P % % See also ROBUSTPERF, QUADSTAB, PSYS. % Author: P. Gahinet 10/94 % Copyright 1995-2004 The MathWorks, Inc. % $Revision: 1.1.6.1 $ function [gama,P] = quadperf(pdsys,g,options) if nargin<1, error('usage: [perf,P] = quadperf(pdsys,g,options)'); elseif ~ispsys(pdsys), error('PDS is not a polytopic or parameter-dependent system'); else if nargin<2, g=0; end if nargin<3, options=[0 0 1e9]; elseif length(options)~=3, error('OPTIONS should be a 3-entry vector'); end end if sum(size(g))~=2, error('G must be a scalar'); end condp=options(3); if condp==0, condp=1e9; end [pdtype,nv,ns,ni,no]=psinfo(pdsys); if strcmp(pdtype,'aff'), pv=psinfo(pdsys,'par'); elseif ~strcmp(pdtype,'pol'), error('PDS must be a polytopic or parameter-dependent system'); elseif ni==0 | no==0, error('This system has no input or no output'); end % prepare data if strcmp(pdtype,'aff'), [typ,np]=pvinfo(pv); if ~strcmp(typ,'box'), pdsys=aff2pol(pdsys); pdtype='pol'; else % generate the matrix of vertices (columns) range=pvinfo(pv,'par'); vmat=[]; ls=1; for bnds=range', vmat=[vmat vmat;bnds(1)*ones(1,ls) bnds(2)*ones(1,ls)]; ls=size(vmat,2); end end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Treat various cases %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Q > 0 setlmis([]); [Q,base]=lmivar(1,[ns,1]); % Q lmipos=newlmi; lmiterm([-lmipos 1 1 Q],1,1); suffcond=0; if options(1)==0, % QUAD PERF MINIMIZATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [gam,base]=lmivar(1,[1 0]); % gamma lilab=[]; % labels of li variables if strcmp(pdtype,'aff'), %%%% AFFINE % multi-convexity constraints addc=[]; if pdsys(7,1)>=100, % some pj enters both A(p),E(p) suffcond=1; for j=1:np, [a,b,c,d,e]=ltiss(psinfo(pdsys,'sys',j+1)); if (norm(b,1) | norm(a,1)>0) & norm(e,1)>0, % parameter pj enters both A,E or B,E multc=newlmi; % multi-conv. LMI if options(2), % accurate mode li=lmivar(1,[ns+no 1]); % li lipos=newlmi; % li > 0 lmiterm([lipos 1 1 li],-1,1); lilab=[lilab,li]; else % fast mode -> li=diag(l1*I,l2*I) [li1,nli1]=lmivar(1,[1 0]); [li2,nli2]=lmivar(1,[1 0]); lipos1=newlmi; % li1 > 0 lipos2=newlmi; % li2 > 0 li=lmivar(3,diag([nli1*ones(ns,1);nli2*ones(no,1)])); % li lmiterm([lipos1 1 1 li1],-1,1); lmiterm([lipos2 1 1 li2],-1,1); lilab=[lilab,li]; end lmiterm([-multc 1 1 Q],[a;c],[e;zeros(size(c))]','s'); lmiterm([-multc 1 1 li],1,1); addc=[addc,j]; end end end % vertex constraints for vertx=vmat, vertc=newlmi; % vertex LMI [a,b,c,d,e]=ltiss(psinfo(pdsys,'eval',vertx)); lmiterm([vertc 1 1 Q],[a;c],[e;zeros(size(c))]','s'); lmiterm([vertc 1 1 gam],-1,mdiag(zeros(ns),eye(no))); lmiterm([vertc 1 2 0],[b;d]); lmiterm([vertc 2 2 gam],-1,1); % add li terms if ~isempty(addc), ind=find(vertx(addc)~=0); else ind=[]; end % ind = nonzero parameter values for j=addc(ind), % j -> j-th parameter lmiterm([vertc 1 1 lilab(j)],vertx(j),vertx(j)); end end else %%%%%%% POLYTOPIC if any(pdsys(7,1)==[1 10 0]), % Aj or Ej constant % vertex constraints for i=1:nv, vertc=newlmi; % vertex LMI [a,b,c,d,e]=ltiss(psinfo(pdsys,'sys',i)); if max(max(abs(e-eye(ns))))==0, e=1; end lmiterm([vertc 1 1 Q],a,e','s'); lmiterm([vertc 1 2 Q],e,c'); lmiterm([vertc 1 3 0],b); lmiterm([vertc 2 2 gam],-1,1); lmiterm([vertc 2 3 0],d); lmiterm([vertc 3 3 gam],-1,1); end else suffcond=1; for i=1:nv, [a,b,c,d,e]=ltiss(psinfo(pdsys,'sys',i)); if max(max(abs(e-eye(ns))))==0, e=1; end eval(['e' num2str(i) '=e;']); end struct=zeros(nv); for i=1:nv, eval(['ei=e' num2str(i) ';']); [ai,bi,ci,di,e]=ltiss(psinfo(pdsys,'sys',i)); for j=1:i, vertc=newlmi; % vertex LMI tij=lmivar(1,[1 0]); % tij base=base+1; struct(i,j)=base; struct(j,i)=base; eval(['ej=e' num2str(j) ';']); [aj,bj,cj,dj,e]=ltiss(psinfo(pdsys,'sys',j)); lmiterm([vertc 1 1 Q],ai,ej','s'); lmiterm([vertc 1 1 Q],aj,ei','s'); lmiterm([vertc 1 2 Q],ej,ci','s'); lmiterm([vertc 1 2 Q],ei,cj','s'); lmiterm([vertc 1 3 0],bi+bj); lmiterm([vertc 2 2 gam],-2,1); lmiterm([vertc 2 3 0],di+dj); lmiterm([vertc 3 3 gam],-2,1); lmiterm([-vertc 1 1 tij],1,2); end,end tcons=newlmi; tmat=lmivar(3,struct); lmiterm([tcons 1 1 tmat],1,1); % [tij]<0 end end lmis=getlmis; % solve mincx problem options=[0 200 condp 0 0]; cobj=zeros(1,decnbr(lmis)); cobj(decinfo(lmis,gam))=1; [copt,xopt]=mincx(lmis,cobj,options,[],g); if isempty(copt), disp(' This system is not quadratically stable'); gama=Inf; P=[]; else Q=dec2mat(lmis,xopt,Q); P=inv(Q); gama=copt; if suffcond, disp(sprintf(' Upper bound on the quadratic RMS gain: %7.3e ',gama)); else disp(sprintf(' Quadratic RMS gain: %7.3e ',gama)); end end else % STABILITY BOX MAXIMIZATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if g <= 0, error('G must be positive'); end if strcmp(pdtype,'pol'), %%%% POLYTOPIC error('Option "maximal stability region" not available for polytopic PDS') elseif strcmp(pdtype,'aff'), if rem(pdsys(7,1),10)==1, error('Not tractable when the E matrix depends on p'); end % compute center range=pvinfo(pv,'par'); pmed=(range(:,1)+range(:,2))/2; [a0,b0,c0,d0,e0]=ltiss(psinfo(pdsys,'eval',pmed)); M0=[e0;zeros(size(c0))]'; CT0=mdiag(zeros(ns),-g*eye(no)); % negativity at center lcent=newlmi; lmiterm([lcent 1 1 Q],[a0;c0],M0,'s'); lmiterm([lcent 1 1 0],CT0); lmiterm([lcent 1 2 0],[b0;d0]); lmiterm([lcent 2 2 0],-g); % vertex conditions for i=1:2^np, vertx=vmat(:,i); vertc=newlmi; % vertex LMI [a,b,c,d,e]=ltiss(psinfo(pdsys,'eval',vertx)); a=a-a0; b=b-b0; c=c-c0; d=d-d0; lmiterm([vertc 1 1 Q],[a;c],M0,'s'); lmiterm([vertc 1 2 0],[b;d]); lmiterm([-vertc 1 1 Q],-[a0;c0],M0,'s'); lmiterm([-vertc 1 1 0],-CT0); lmiterm([-vertc 1 2 0],-[b0;d0]); lmiterm([-vertc 2 2 0],g); end nlfc=2^np; end lmis=getlmis; % solve gevp problem options=[1e-2 150 condp 0 0]; [gama,xopt]=gevp(lmis,nlfc,options); if ~isempty(xopt), gama=1/gama; Q=dec2mat(lmis,xopt,Q); P=inv(Q); disp(sprintf('\n\n')); disp([' Quadratic stability established on ' ... num2str(100*gama) '% of the']); disp(' prescribed parameter box'); else P=[]; gama=0; disp(' Optimization failed'); end end