How to solve a LMI with inverse matrix

[Francisco Ronay López Estrada]

Dear Pr Löfberg

It is possible to solve LMIs with inverse matrix like this who result for apply bounded real lemma for discrete systems. Small example attached. 

before hand, thanks for your help

Ronay

[Johan Löfberg]

No. (the paper is odd as they don't seem to comment on the fact that the problem they derive is a nonconvex SDP and thus essentially intractable). They either perform the trick I descibe below, or they hide a lot of information about how they actually manage to solve the problem

This looks like it might be possible to transform to a linear exression in X (or the inverse of X), by suitable congruence transformation (diagonal with blocks (I,I,X,I) looks like the trick), variable changes and/or Schur complements (the standard road towards LMIs from nonconvex SDPs)

[Francisco Ronay López Estrada]

Thanks for your help, I did what  you suggest and works, thanks a lot. 

[Muhammad Mohsin]

Respected Sir,
 how to solve these type of LMI's please share ideas.
Francisco Ronay López Estrada sir, 
                                                        How to solve your problem and attach file if possible.

[Johan Löfberg]

You cannot. Inverse is not linear, hence no LMI. You have to find a way to eliminate the inverse, or re-parameterize the problem. For instance, in the pdf above it is easily converted to a problem linear in X by congruence transformation with a matrix with diagonal blocks (I,I,X,I).

[Muhammad Mohsin]

 Respected Johan Löfberg,
                                                  Image of LMI is attached which i want to solve. plz give some detail answer.
 In this LMI Q and R matrices are constant and N and M are variables.

[Johan Löfberg]

I don't see any way of lnearizing this. However, are you sure about this form. It looks a bit odd as it is cubic in M if you perform a Schur complement over the M^-1 block. Having cubic stuff in SDP constraints arising from stability arguments is something I rarely see

[Muhammad Mohsin]

there is no derivation of this LMI or theorem, but this is probably correct.
variables are shown in this image.

[Johan Löfberg]

What paper are you trying to implement?

[Muhammad Mohsin]

 ''Coordinated control of AFS and DYC for vehicle handling and stability based on optimal guaranteed cost theory'' by  Xiujian Yang  , Zengcai Wang  & Weili Peng

[Johan Löfberg]

To be honest, I don't trust that reference. No derivation, and they do not in any way reflect on the fact that the problem is nonlinear in epsilon, and later simply reference to it as an LMI (which it isn't). I wouldn't be surprised if the inverse on Mtilde should be removed. The notation with Ntilde is absolutely weird. They define Ntilde using inverse of epsilon, and then they invert Ntilde (which means they just as well could have defined Ntilde with epsilon directly)

Go back to basics and derive these things on your own. The reference in the paper is L Yu. Very similar stuff (with no inverse on Mtilde as I suspected) can be found in e.g. http://www.ijcas.org/admin/paper/files/IJCAS_v3_n3_pp.396-402.pdf

[Johan Löfberg]

In http://www.ijcas.org/admin/paper/files/IJCAS_v3_n3_pp.396-402.pdf, Mtilde is not inverted, but nonlinearities in epsilon are still there due to Ntilde. However, they hint in the text that they don't actually optimize over epsilon ("constant epsilon"...), so I guess they just fix it to some value and try to solve the problem

[Muhammad Mohsin]

I see this paper and implement.
 Thank you  for guidance Johan Löfberg

[Muhammad Mohsin]

ok, i will do. Thanks