Problem with quadratic forms constraint

Nonconvex
1

Hi,

I am writing the following code using CVX,

clc
clear
close all
cvx_setup

A = [-1 0
     0 -2];
B1 = eye(2);
B2 = eye(2);
C = eye(2);
g2 = 3; g3 = -1;

cvx_begin
    variable k(2, 2)
    variable W(2, 2) semidefinite
    minimize (trace_inv(W));
    subject to
        (A - B1*k)*W+W*(A - B1*k)' == - B2*B2';
        trace(C'*W*C) <= g2;
        max(eig(A - B1*k)) <= g3;
cvx_end
k

but I am getting the following error,

Only scalar quadratic forms can be specified in CVX

Besides, I am going to define W variable as a positive definite matrix.

It’s appreciated if anyone helps.

Thanks
Mohammad

2

You are multiplying 2 matrix variables in the first constraint. That is not allowed in CVX. I will presume that the first constraint is non-convex unless you show otherwise.

Additionally, you need to change the last constraint to
lambda_max(A - B1*k) <= g3;
CVX will enforce that the argument of lambda_max is symmetric.

3

Thanks for your reply.

Is it possible to use “gram” command in CVX (to compute the controllability grammian)?

4

No. gram command cannot be used in CVX (on CVX variables or expressions). Controllability grammians are not my area, so I can’t help with model formulations. I leave it to you to determine whether you can formulate an appropriate LMI, which will be acceptable to CVX.