Hello everyone,

I am trying to solve the following cvx optimization problem:

clear all

clc

A=[0 1;-2 3];B=[0;1];Qc=[100 0;0 1];Rc=10;

h=0.05;tav=0.08;

Q2=[[4.9915 0.1284;0.1284 0.0629] [1.1616e-04;2.3852e-04] [0.0019;0.0013] zeros(2,1);…

[1.1616e-04;2.3852e-04]’ .2000 6.7902e-06 0;…

[0.0019;0.0013]’ 6.7902e-06 .2000 0;zeros(1,2) 0 0 0];

fi2=[expm(A*h) [2.0405e-04;.0206] [.0011;.0333];zeros(1,2) 0 0;zeros(1,2) 1 0];

sigma=0.01;

landa2=[0;1;0;0];

%% cvx

cvx_begin

cvx_solver sedumi

variable S(4,4) symmetric

variable w2(1,4)

variable alfa2

minimize(trace_inv(S))

subject to

[S zeros(4,4) (fi2*S+landa2*w2).’ S.’ (sqrtm(Q2)*[S;w2]).’;…

zeros(4,4) 2*S-alfa2*eye(4,4) (landa2*w2).’ zeros(4,4) (sqrtm(Q2)*[zeros(4,4);w2]).’;…

fi2*S+landa2*w2 landa2*w2 S zeros(4,4) zeros(4,5);…
S zeros(4,4) zeros(4,4) eye(4,4) alfa2/(sigma^2) zeros(4,5);…
sqrtm(Q2)[S;w2] sqrtm(Q2)*[zeros(4,4);w2] zeros(5,4) zeros(5,4) eye(5,5)]>= 0

alfa2>=0;

S>=0;

cvx_end

K2=w2*(S^-1)

status is solved, but in fact after solving ,when I check eigenvalue of the matrix:

[S zeros(4,4) (fi2*S+landa2*w2).’ S.’ (sqrtm(Q2)*[S;w2]).’;…

zeros(4,4) 2*S-alfa2*eye(4,4) (landa2*w2).’ zeros(4,4) (sqrtm(Q2)*[zeros(4,4);w2]).’;…

fi2*S+landa2*w2 landa2*w2 S zeros(4,4) zeros(4,5);…
S zeros(4,4) zeros(4,4) eye(4,4) alfa2/(sigma^2) zeros(4,5);…
sqrtm(Q2)[S;w2] sqrtm(Q2)*[zeros(4,4);w2] zeros(5,4) zeros(5,4) eye(5,5)]>= 0

values are negative!!! ( this matrix should be PD)

I have been at this issue for a while.

Help and comments are much appreciated.