LMI - Feedback Stabilization of Time-Delay System

Dear community:

I hope you can help me with the following. I am a beginner in the use of Linear Matrix Inequalities in Control Applications. As guide, i am using the book "LMIs in Control Systems: Analysis, Design and Applications" from Guang-Ren Duan and Hai-Hua yu. In order to solve the example 7.16 of the book; I am trying to program the LMI of the Theorem 7.19 in MATLAB, however i got compilation errors, could anybody help me please?

My MATLAB code is as follows:

% Definition of the Matrices
% Clear the workspace
clc
clear all

% Specify parameter matrices
A = [-1     0         1;
      0     2        -1;
      2      0       -3]
Ad = [  1     0      1;
       2     1      1;
       0     0     -1]
 B = [1  1;
      1  2;
      0  1]

d = 0.1;

% LMI problem definition
%% Definition of the Decision Variables
setlmis([]);
X = lmivar(1,[3 1]);
W = lmivar(2,[2,3]);
Beta = lmivar(1,[1 1]);

% Definitions of the LMI

% LMI #1

% LMI #1 LMI(1,1)
lmiterm([1 1 1 X],1,(A + Ad)');     % #1 LMI(1,1)    / X*(A + Ad)^T
lmiterm([1 1 1 X],(A + Ad),1);      % #1 LMI(1,1)    / + (A + Ad)*X
lmiterm([1 1 1 W],B,1);             % #1 LMI(1,1)    / + BW
lmiterm([1 1 1 -W],1,B');           % #1 LMI(1,1)    / + W^T*B^T
lmiterm([1 1 1 0],d,(Ad*Ad'));      % #1 LMI(1,1)    / + d*Ad*Ad^T

% LMI #1 LMI(2,1)
lmiterm([1 2 1 X],d*A,1);           % #1 LMI(2,1)    / d*A*X
lmiterm([1 2 1 W],(d*B),1);         % #1 LMI(2,1)    / d*B*W

% LMI #1 LMI(2,2)
lmiterm([1 2 2 Beta],-1*d,1);       % #1 LMI(2,1)    / d*A*X

% LMI #1 LMI(3,1)
lmiterm([1 3 1 X],(d*Ad),1);        % #1 LMI(2,1)    / d*Ad*X

% LMI #1 LMI(3,2)
lmiterm([1 3 2 0],0,1);             % #1 LMI(2,1)    / 0

% LMI #1 LMI(3,3)
lmiterm([1 3 3 0],-1*d,1);         % #1 LMI(2,1)    / -d
lmiterm([1 3 3 Beta],d,1);         % #1 LMI(2,1)    / d*Beta

% LMI #2: 0 < X
lmiterm([-2 1 1 X],1,1); % #2 LMI, right-hand side / X
lmiterm([2 1 1 0],0);    % #2 LMI, left-hand side  / 0

% LMI #3: 0 < Beta
lmiterm([-3 1 1 Beta],1,1); % #3 LMI, right-hand side / Beta
lmiterm([3 1 1 0],0);       % #3 LMI, left-hand side  / 0

% LMI #3: -1 < Beta
lmiterm([-4 1 1 Beta],-1,1); % #4 LMI, right-hand side / Beta
lmiterm([4 1 1 0],1);        % #4 LMI, left-hand side  / 0

stabilz2 = getlmis;
%LMI Solution

[tmin,xfeas] = feasp(stabilz2);
Xvalue = dec2mat(stabilz2,xfeas,X)
Wvalue = dec2mat(stabilz2,xfeas,W)
Betavalue = dec2mat(stabilz2,xfeas,Beta)

k = Wvalue*inv(Xvalue)

Answer 1

[Azeddine Elmajidi]

Hi,

I think there was some errors in your lmiterm, i prefer to use the gui editor "lmiedit" it helps to avoid errors, with using it i can produce the same results that you have in the book.

%Definition of the Matrices

% Clear the workspace

clc

clear all

% Specify parameter matrices

A = [-1 0 1;

0 2 -1;

2 0 -3]

Ad = [ 1 0 1;

2 1 1;

0 0 -1]

B = [1 1;

1 2;

0 1]

d = 0.1;

% LMI problem definition

%% Definition of the Decision Variables

setlmis([]);

X = lmivar(1,[3 1]);

W = lmivar(2,[2,3]);

Beta = lmivar(1,[1 1]);

% Definitions of the LMI

% LMI #1

% LMI #1 LMI(1,1)

lmiterm([1 1 1 X],1,A','s'); % LMI #1: X*A'+A*X

lmiterm([1 1 1 X],1,Ad','s'); % LMI #1: X*Ad'+Ad*X

lmiterm([1 1 1 W],B,1,'s'); % LMI #1: B*W+W'*B'

lmiterm([1 1 1 0],d*Ad*Ad'); % LMI #1: d*Ad*Ad'

lmiterm([1 2 1 X],d*A,1); % LMI #1: d*A*X

lmiterm([1 2 1 W],d*B,1); % LMI #1: d*B*W

lmiterm([1 2 2 Beta],.5*d,-eye(3),'s'); % LMI #1: -d*Beta*eye(3) (NON SYMMETRIC?)

lmiterm([1 3 1 X],d*Ad,1); % LMI #1: d*Ad*X

lmiterm([1 3 3 Beta],.5*d,eye(3),'s'); % LMI #1: d*Beta*eye(3) (NON SYMMETRIC?)

lmiterm([1 3 3 0],-d*eye(3)); % LMI #1: -d*eye(3)

lmiterm([-2 1 1 X],1,1); % LMI #2: X

lmiterm([-3 1 1 Beta],1,1); % LMI #3: Beta

lmiterm([4 1 1 Beta],1,1); % LMI #4: Beta

lmiterm([-4 1 1 0],1); % LMI #4: 1

stabilz2=getlmis;

%LMI Solution

[tmin,xfeas] = feasp(stabilz2);

Xvalue = dec2mat(stabilz2,xfeas,X)

Wvalue = dec2mat(stabilz2,xfeas,W)

Betavalue = dec2mat(stabilz2,xfeas,Beta)

k = Wvalue*inv(Xvalue)

[Rodrigo Zenon Guzman Iturra]

Thank you for your help

[Farshid R]

Hi @Azeddine Elmajidi

Could you help me with the problem?

https://www.mathworks.com/matlabcentral/answers/1837598-time-varying-parameter-in-lmi