This Theorem is from one paper published in IEEE Trans Automatic Control, the details of theorem are as follows:
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Then, I using the following configurations to test this theorem
alpha = -0.07, and beta is -0.28, delta is equal to 0.5.
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If I want to find the miniman vlaue of gamma, i have the following LMI equation
Minimise gamma subject to (10)
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Finally, I develop my own code, please see the following. However, it cannot work well, is there anyone can provide me some advice? Thanks a lot.
%% system definitation
delta = 0.5;A = [delta 0.8 -0.4; -0.5 0.4 0.5; 1.2 1.1 0.8];B = [0 1; 2 -1; 0 1.3];E = [0.1; 0.4; 0.1];C1 = [-1 0 2];D = [0 0];F = 0.3; C2 = [-1 1.2 1; 0 -3 1];H = [0.1; 0.4];alpha = -0.07;beta = -0.28;A_bar = [A E; C1 F];B_bar = [B; D];% K_bar = K;
C_bar = [C2 H];%% define the LMI system
setlmis([]);%% Defining Variables:
gamma = lmivar(1, [1 0]); % gamma(1,1)
[P, n, sP] = lmivar(2, [2 2]); % P(2,2)
[G, n, sG] = lmivar(2, [4 4]); % G(4,4)
[V, n, sV] = lmivar(2, [2 2]); % V(2,2)
[U, n, sU] = lmivar(2, [2 2]); % U(2,2)
[J, n, sJ] = lmivar(2, [4 4]); % J(4,4)
[Xi_11, n, sXi_11] = lmivar(3, [sP, zeros(2, 2); zeros(2, 2), -gamma*gamma*eye(2, 2)]);[Xi_22_bar, n, sXi_22_bar] = lmivar(3, [sP, zeros(2, 2); zeros(2, 2), eye(2, 2)]);[Xi_22, n, sXi_22] = lmivar(3, [-alpha*(sG)-alpha*((sG)')+alpha*alpha*(sXi_22_bar)]);%% Defining LMIs term contents:
% DEFINITION 1-st row
lmiterm([1 1 1 Xi_11], 1, 1); % #1 LMI, the (1, 1) block
% DEFINITION 2-nd row
lmiterm([1 2 1 G], 1, A_bar); % #1 LMI, the (2, 1) block
lmiterm([1 2 1 V], B_bar, C_bar); % #1 LMI, the (2, 1) blocklmiterm([1 2 2 Xi_22], 1, 1); % #1 LMI, the (2, 2) block
lmiterm([1 2 2 J], 1, 1); % #1 LMI, the (2, 2) block% DEFINITION 3-rd row
lmiterm([1 3 1 V], ((B_bar')*B_bar), C_bar); % #1 LMI, the (3, 1) block
lmiterm([1 3 2 0], 0); % #1 LMI, the (3, 2) block
lmiterm([1 3 3 U], beta*((B_bar')*B_bar), -1, 's'); % #1 LMI, the (3, 3) block
% DEFINITION 4-th row
lmiterm([1 4 1 0], 0); % #1 LMI, the (4, 1) block
lmiterm([1 4 2 0], 0); % #1 LMI, the (4, 2) block
lmiterm([1 4 3 G], 1, B_bar); % #1 LMI, the (4, 3) block
lmiterm([1 4 3 U], B_bar, -1); % #1 LMI, the (4, 3) blocklmiterm([1 4 4 J], (1/(beta*beta)), -1); % #1 LMI, the (4, 4) block
lmisys = getlmis;%% check the feasibility
% [tmin, xfeas] = feasp(lmisys);
% V = dec2mat(lmisys, xfeas, V);
% U = dec2mat(lmisys, xfeas, U);
%% Defining vector "c" for C'x in mincx
Num = decnbr(lmisys);c = zeros(Num,1);c(Num)=1;%% Solving LMIs:
[copt,xopt] = mincx(lmisys,c);%% Finding Feedback gain and u:
% display(gopt)
% K = inv(U)*V
Best Answer