Common Lyapunov function - linear time-varying system

1

Hello, I’m new with CVX.
Itry to find positive matrix P, to use a common Lyapunov function for a family of linear time-varying system.
A matix is \begin{pmatrix} -1 & \alpha \\\alpha & -2 \end{pmatrix}, where |\alpha(t)|\leq1.
Here is my script:

cvx_begin
variable P(2, 2) symmetric
variable alpha_var
A = [-1, alpha_var; alpha_var, -2];
variable lambda
minimize(lambda)
subject to
abs(alpha_var) <= 1
A’P + PA + lambda*P <= 0;
P >= 0; % ensure P is positive definite
cvx_end

I go the following error:
Error using *
Disciplined convex programming error:
Only scalar quadratic forms can be specified in CVX

Thanks!

2

A and P both contain variables, and are multiplied. hence the error message.

if you are to use CVX, you will need a reformulation.

Also, if you are to use P >= 0 for a PSD constraint, you will need to use sdp mode. Alternative ways are addressed in Semidefinite programming mode — CVX Users' Guide .

1 Like
3

Idefined P to be semidrfinite in order to ensure P is positive definite:

variable P(2, 2) semidefinite

What “reformulation” are you recommend?
Thanks again

4

That’s for you to determine. Perhaps you need to more carefully re-read the link.