Solving "Lyapunov-Equation" with LMIs

[PhilippZ]

Dear Mr. Johan Löfberg!

In advance, many thanks for your great work on YALMIP and, hopefully, some help for my little problem here.

In general, what I am trying to do is to solve some LMIs. These LMIs (the constraints) are based on the so called W-Problem, which is a slight 

alteration of the Lyapunov equation (makes it convex) for a continous state-space system of the form

x' = Ax + Bu

y = Cx  

So shortly, I am trying to find a controller gain matrix K, where the system matrix A is extended by some integral error states (PI controller).

Furthermore, the output is restricted to these error states and some additional states ( no state feedback! ). 

In the end, the controller gain matrix K should have some defined structure like K = [a 0 b 0;0 c 0 d] and is dependend on the decision variables of the LMI: K = N*inv(M). 

By randomly generating some K matrices of that form K = [a 0 b 0;0 c 0 d], I was able to get some, that would stabilize my closed-loop system A-BKC < 0. 

Hence, the LMIs should be solvable - by definition of the W-Problem. However, as it seems the program runs into some numerical problems which I am not fully 

capable to understand. I tried however two things

One: I changed some entries of the system matrix A to change the condition number of it. It then helps to make the problem "more" solvable, as the constraints seem to be fulfilled. Nevertheless, I would like to use the system matrix as it is.

Two: I tried to restrict the number of decision variables to make it maybe easier for the solver to solve the LMI. In the case of where the controller gain matrix is then restricted to K = [a 0 b 0;0 c 0 d], the solver was not able to fulfill all constraints. Althoug I found some of that form by randomly generating it.

Could you have a look at my code and tell me where the problem lies?

There is the main script (LMI_W_Problem.m) and two functions to check the constraints (strict and nonstrict) of the W-Problem.

Again, thanks in advance!

[Johan Löfberg]

Note sure what you are saying here. I don't think it is known in which tractability class optimization over output feedback is, i.e., whether it is solvable in polynomial time or not. It is thus definitely not possible to solve using LMIs, and any LMI method would be conservative, and thus you could have problems where output feedback is possible but the LMI for the approximation would be infeasible

Have you tried another solver, such as mosek