I want to include a 3x3 metzler matrix (non-negative off-diagonal elements) with the elements W(1,3), W(2,1), W(3,1) and W(3,2) to be zero.

The theorem is provided in the figure:

I want to implement Theorem 1 here. The solution is feasible but constraint (10) in Theorem 1 as provided in the image is violated.

The structure of the W matrix should be Metzler matrix with the additional constraint that the off-diagonal elements of W should be the zero as that of the off-diagonal elements of the matrix A22 (system matrix) which are exactly zero.

The Matlab code is provided below:

%%%%%%%%%%% **MATLAB CVX CODE** %%%%%%%%%%%%%%%%%

% Design of positive observer x_hat_dot = G x_hat + L y, where G=Metzler;L>>0

%for system dynamics x_dot = A x, y = Cx

clear all

clc

% Parameters

p20=0.0155;p30=1.288e-5;p40=0.0365;p50=0.0105;GP0=1.792;

% Maximum parametric perturbations

p2=p20+0.3*p20;p3=p30+0.3*p30;p4=p40+0.3*p40;p5=p50+0.3*p50;

% Minimum parametric perturbations

p22=p20-0.3*p20;p33=p30-0.3*p30;p44=p40-0.3*p40;p55=p50-0.3*p50;

% Enter system and output matrices

A220=[-p20 p30 0;0 -p40 p40;0 0 -p50]; % System matrix nominal

A22=[-p2 p3 0;0 -p4 p4;0 0 -p5]; % System matrix max

A22m=[-p22 p33 0;0 -p44 p44;0 0 -p55]; % Syatem matrix min

C=[1 0 0]; % Output matric

% LMI for observer design

cvx_begin sdp

variable P(3,3) diagonal

variable Q(3,3) diagonal

variable V(3)

variable W(3,3)

% LMI constraints

Ass=[Q*A22m-V*C-W];

Ass>=0;

P>=0;

Q>=0;

1e-8 <= V([1 2 3]) <= 1e-6;

1e-8 <= W([2 3 6]) <= 1e-5;

[P*A22+A22’*P A22’*Q-C’ V’-W’;QA22-V*C-W W+W’] <= 0;

cvx_end

P

Q

W

V

G = inv(Q)*W

L = inv(Q)*V