Problem with the solution of an LMI having Hermitian in it.

[Abdur Rehman]

Hi,

I am facing an issue with writing an LMI in Matlab with Yalmip toolbox. 

Following is the LMI (Please see the Page 3 in attached paper):

As it can be observed, there are three decision variables

1. Q is greater than zero.

2. U is a diagonal matrix and is greater than zero.

3. L, there is no condition on L, so I assumed it as a 'full' matrix in sdpvar.

4. there is a scaler gamma, which is also greater than zero.

After solving this, my problem become infeasible. I think there is something wrong with my LMI writing. 

• Can I have any suggestion how to put the hermitian condition around the LMI matrix.
  
• Is there any fault in my LMI Code below:
  

This is the code: Q=sdpvar(6,6,'full'); U=sdpvar(2,2,'diagonal'); L=sdpvar(2,6,'full'); gamma=sdpvar(1); F1=[A*Q+B*L B*U zeros(6,2) zeros(6,2);

-L -U eye(2,2) zeros(2,2);

zeros(2,6) zeros(2,2) (-gamma/2)*eye(2,2) zeros(2,2);

C*Q+D*L D*U zeros(2,2) (-gamma/2)*eye(2,2)];

Constraints=[U>=0.000000001*eye(2,2),Q>=0.00000000001*eye(nx,ny),gamma>=0.0000000001,F1<=0.00000000001*eye(12,12)]; options = sdpsettings('savesolveroutput',1,'verbose',1,'solver','sdpt3'); sol = optimize(Constraints,gamma,options);

Thank you

Kind Regards

[Johan Löfberg]

He(X) <= 0 is short-hand for X + X' <= 0, hence you want Fi + Fi' <= ...

(lower bounding gamma is redundant as it has to be positive according to (4). Same with U)

[Abdur Rehman]

I think I have to set the gamma as symbolic decision variable(using sdpvar), if I set it the objective. is that so? 

( I have checked it, but my LMI is still not providing the desired results.(Not all the eigenvalues are positive).

Attached is my code

[Johan Löfberg]

Surely Q is supposed to be symmetric and not fully parameterized? The constraint Q >= eps*I surely indicates you want a symmetric psd matrix (as it is nom you are saying Q has no structure, and all elements are non-negative except the diagonal which should be larger than 1e-120)

https://yalmip.github.io/tutorial/basics/

however, using numbers like 10^-120 makes absolutely no sense. Numerical solvers work in a precision of, say, 10^-8 or something like that.

[Johan Löfberg]

I have no idea what the question about gamma is about

[Abdur Rehman]

Thank You sir.

How should I specify gamma. Should I fix it at some constant value, or use sdpvar for defining it as a decision variable. 

P.S. I want to get L and Q matrices after solving this LMI. 

[Johan Löfberg]

If it is constant, you create it as such, and if it is a decision variable, as it appears to be as you want to minimize it, you define it as an sdpvar. Nothing different compared to any other variable