How to solve an optimization problem with stability constraints using 'fmincon'?

Hi,

Let us say that a plant model of the following form is given:

The output of the model (y) is its position (x) and velocity (v).

Using this output, I am going to construct a PI controller (u) of the following form:

Then, the task is to find the optimal gains () and () such that the following quantity is maximized:

Without any constraints, this problem can be solve easily using the MATLAB command 'fminsearch' or 'fminunc.'

But I'd like to add a stability constraint such that the closed-loop system with PI control is always stable.

Mathematically, it is realized by constraining the eigenvalues of the closed-loop system in the left-hand plane.

But I'm wondering how to add this constraint using MATLAB commands.

Any comments will be greatly appreciated.

Thank you.

Answer 1

[Alan Weiss]

Perhaps this example will give you some ideas. It is written as a multiobjective problem, but you can adapt the code easily to single-objective minimization.

Alan Weiss

MATLAB mathematical toolbox documentation

[Stephan]

Alan could you please explain the reason for the warning the OP gets, when trying to use eig? I could not find something in the documentation so far.

[Alan Weiss]

I have no idea. The link I gave was to an optimization example using eig, so I know that it can work, and the warning the OP referred to must be for something else.

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 2

[Matt J]

Since it isn't clear that such an eigenvalue constraint is smooth, I would use ga instead of fmincon. This is closer to what you were doing with fminsearch, but with ga you can specify a nonlinear constraint function like the following,

function [c,ceq]=nonlcon(x)

 ceq=[];
 c=max(real(eig(x)));
end