Issue on H-infinity norm

[Rodrigo Trentini]

Hello!

First, I need to apologize for my inexperience on LMIs and it solvers.

I'm trying to solve the attached H-infinity norm problem. My code is the following:

--------------------------

clear all

close all

clc

A = [0 0 1 0; 0 0 0 1; -1 1 -0.2 0.2; 0.5 -2.5 0.1 -0.15];

B = [0 0; 0 0; 1 0; 0 0.5];

C = [1 0 0 0; 0 1 0 0];

D = zeros(2);

I = eye(2);

P = sdpvar(length(A));                      

gamma = sdpvar(1);

F = set([A'*P+P*A+C'*C , P*B+C'*D ; B'*P+D'*C , D'*D-gamma^2*I]< 0)...

    +set(P>0);       

sol = solvesdp(F,gamma^2,sdpsettings('solver','sedumi'));

Pf = double(P)

Gamma = double(sqrt(gamma))

------------------------------

It returns:

Warning: Solver not applicable (sedumi)

Pf =

   NaN   NaN   NaN   NaN

   NaN   NaN   NaN   NaN

   NaN   NaN   NaN   NaN

   NaN   NaN   NaN   NaN

Gamma =

   NaN

Could anyone help me understanding the problem? I'm sure that the there is a reachable solution, and I already know it, however I cannot achieve it by YALMIP.

Yet, if I change the constrain to F = set([A'*P+P*A+C'*C , P*B+C'*D ; B'*P+D'*C , D'*D-gamma*I]< 0) +set(P>0), it works almost properly, the result is very near to the correct one, but the constrain is not right...

Does it make sense?

Thanks in advance.

Rodrigo

[Johan Löfberg]

First, SET is obsolete, don't use it
http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Commands.Set

Second, strict inequalties are not really supported
http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Blog.Prepare-your-code

The model is not linear if you have gamma^2 in the constraints, hence, *linear* matrix inequality solvers are not applicable

Hence, you should express the whole thing in the squared gain, i.e., instead of letting gamma represent the Hinf gain, you let it define the squared Hinf gain, and thus have a linear constraint and linear objective.