LMI.rar_Inequality_LMI_Linear matrix_linear inequality

clc clear all close all %% system page 85 A{1}=[-1.9 -1;2 1]; A{2}=[-0.08 0.8;0.58 0.69]; B{1}=[0.26;0.92]; B{2}=[-1;0.21]; % A{1}=[0 1;-0.59 2.18]; % A{2}=[0 1;-0.36 2.26]; % B{1}=[0;-0.603]; % B{2}=[0;-1.12]; W=eye(1);R=eye(1); C=[1 0]; %% LMI setlmis([]) X=lmivar(1,[2 1]) ; for i=1:2 M{i}=lmivar(2,[1 2]) ; end % n=1; % for i=1:2 % newlmi % lmiterm([-n 1 1 X] , 1 , 1) ; % lmiterm([-n 2 1 X] , A{i} , 1) ; % lmiterm([-n 2 1 M{i}] , -B{i} , 1) ; % lmiterm([-n 2 2 X], 1,1) ; % n=n+1; % end % n=1; % for i=1:2 % for j=1:2 % if j>i % newlmi % lmiterm([-n 1 1 X] , 1 , 1) ; % lmiterm([-n 2 1 X], .5.*A{i} ,1) % lmiterm([-n 2 1 X], .5.*A{j} , 1); % lmiterm([-n 2 1 M{i}], -.5.*B{j},1) ; % lmiterm([-n 2 1 M{j}],-.5.*B{i},1) ; % lmiterm([-n 2 2 X],1,1); % end % end % n=n+1; % end n=1; for i=1:2 for j=1:2 newlmi lmiterm([-n 1 1 X],1,1) lmiterm([-n 2 1 X],A{i} , 1) lmiterm([-n 2 1 M{j}],-B{i} ,1) lmiterm([-n 2 2 X],1 ,1) n=n+1; end end lmisys=getlmis; [tmin,xfeas]=feasp(lmisys) clc M1=dec2mat(lmisys,xfeas,M{1}) M2=dec2mat(lmisys,xfeas,M{2}) X=dec2mat(lmisys,xfeas,X) F1=M1*inv(X) F2=M2*inv(X) P=inv(X) % xfeas=dec2mat(lmisys,xfeas,X) % for i=1:2 % mfeas=dec2mat(lmisys,xfeas,M{i}) % end %% x1(1)=1;x2(1)=-.5; % A1=[0 1;-0.59 2.18]; % A2=[0 1;-0.36 2.26]; % B1=[0;-0.603]; % B2=[0;-1.12]; A1=[-1.9 -1;2 1]; A2=[-0.08 0.8;0.58 0.69]; B1=[0.26;0.92]; B2=[-1;0.21]; G1=A1-B1*F1;G2=A2-B2*F2; C1=[1 0];C2=[1 0];W=eye(1);R=eye(1);J=0; my_fis=newfis('TS','sugeno'); my_fis=addvar(my_fis,'input','x2',[0 1]); my_fis=addmf(my_fis,'input',1,'M1','trapmf',[0 0 0.3 0.6]); my_fis=addmf(my_fis,'input',1,'M2','trapmf',[0.4 0.7 1 1]); my_fis=addvar(my_fis,'output','A11',[-10 10]); my_fis=addmf(my_fis,'output',1,'a11','constant',A1(1,1)); my_fis=addmf(my_fis,'output',1,'a11','constant',A2(1,1)); my_fis=addvar(my_fis,'output','A12',[-10 10]); my_fis=addmf(my_fis,'output',2,'a12','constant',A1(1,2)); my_fis=addmf(my_fis,'output',2,'a12','constant',A2(1,2)); my_fis=addvar(my_fis,'output','A21',[-10 10]); my_fis=addmf(my_fis,'output',3,'a21','constant',A1(2,1)); my_fis=addmf(my_fis,'output',3,'a21','constant',A2(2,1)); my_fis=addvar(my_fis,'output','A22',[-10 10]); my_fis=addmf(my_fis,'output',4,'a22','constant',A1(2,2)); my_fis=addmf(my_fis,'output',4,'a22','constant',A2(2,2)); my_fis=addvar(my_fis,'output','B1',[-10 10]); my_fis=addmf(my_fis,'output',5,'b1','constant',B1(1)); my_fis=addmf(my_fis,'output',5,'b1','constant',B2(1)); my_fis=addvar(my_fis,'output','B2',[-10 10]); my_fis=addmf(my_fis,'output',6,'b2','constant',B1(2)); my_fis=addmf(my_fis,'output',6,'b2','constant',B2(2)); my_fis=addvar(my_fis,'output','C1',[-10 10]); my_fis=addmf(my_fis,'output',7,'c1','constant',C1(1)); my_fis=addmf(my_fis,'output',7,'c1','constant',C2(1)); my_fis=addvar(my_fis,'output','C2',[-10 10]); my_fis=addmf(my_fis,'output',8,'c2','constant',C1(2)); my_fis=addmf(my_fis,'output',8,'c2','constant',C2(2)); my_fis=addvar(my_fis,'output','F1',[-10 10]); my_fis=addmf(my_fis,'output',9,'f1','constant',F1(1)); my_fis=addmf(my_fis,'output',9,'f1','constant',F2(1)); my_fis=addvar(my_fis,'output','F2',[-10 10]); my_fis=addmf(my_fis,'output',10,'f2','constant',F1(2)); my_fis=addmf(my_fis,'output',10,'f2','constant',F2(2)); Rulelist = [1 1 1 1 1 1 1 1 1 1 1 1 1;2 2 2 2 2 2 2 2 2 2 2 1 1]; my_fis = addrule(my_fis, Rulelist); V(1)=[x1(1);x2(1)]'*P*[x1(1);x2(1)]; n=50;x=[x1;x2]; for i=2:n G=evalfis(x1(i-1),my_fis); A=[G(1) G(2); G(3) G(4)]; B=[G(5);G(6)]; C=[G(7) G(8)]; F=[G(9) G(10)]; x(:,i)=(A+B*F)*x(:,i-1); x1(i)=x(1,i); x2(i)=x(2,i); u(i)=F*x(:,i); V(i)=[x1(i);x2(i)]'*P*[x1(i);x2(i)]; J=J+x(:,i)'*C'*W*C*x(:,i)+x(:,i)'*F'*R*F*x(:,i); end m=0:n-1; figure subplot(2,1,1) plot(m,x1,'marker','.') grid on xlabel('sample time') ylabel('x_1') subplot(2,1,2) plot(m,x2,'marker','.') grid on xlabel('sample time') ylabel('x_2') figure subplot(2,1,1) plot(m,V,'marker','.') ylabel('Lyapunov Function') grid on xlabel('sample time') subplot(2,1,2) plot(m,u,'marker','.') ylabel('control signal') grid on xlabel('sample time')