Linearization Observer-Based State Feedback

[André Teles]

I am trying to linearize my LMI that designs a static gain K and did not find a solution to this problem so far.

The LMI follows

[-l*P    Atil'    0
 Atil  -P^-1    Btil
  0     Btil' (l-1)*I]<=0

Btil and l are known. The variables are P>=0 (I considered P=[Px 0; 0 Pe] to simplify things) and the gain K inside of Atil:

Atil = [A+B*K   -B*K
         0     A-L*C]

L is also known.

I tried a congruence transformation by multiplying both sides by diag(Q,I,I), where Q=P^-1... but then Atil*Q becomes

AtilQ = [A*Qx+B*K*Qx    -B*K*Qe
             0        A*Qe-L*C*Qe]

I tried creating new variables K*Qx=W, and K*Qe=V, but this requires another nonlinear constraint between W and V.

Does someone have an idea how to deal with that?

I would be extremely thankful!