Optimization problem related to output feedback

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Kamal Bera
Kamal Bera 2017-4-19
I have the following optimization problem related to output feedback (in control theory)
Minimize the Cost J=0.5*trace(P*X) % X=eye(16)
with below three equations
Ac'*P+P*Ac+C'*K'*R*K*C+Q=0 ---(1) % can be solved for P as, P = lyap(Ac',C1); with C1 = C'*K'*R*K*C + Q;
Ac*S+S*Ac'+X=0 ---(2) % can be solved for S as, S = lyap(Ac,X);
gradient, dJ/dK=R*K*C*S*C'-B'*P*S*C' ---(3)
Ac=A-B*K*C; A, B, C, Q, R are input matrices.
Now I am writing the description given in Book ( Optimal Control by Lewis and Syrmos,2nd Ed, page-366) to solve this problem: “A second approach for computing K is to use a gradient-based routine found in MATLAB (Optimization Toolbox). This routine would use all of the design equations (i.e. Eqs 1, 2, 3). For a given value of K, it would solve the two Lyapunov equations (Eqs 1, 2).Third design equation gives the gradient of J with respect to K, which would be used by the routine to update the value of K”. I am attaching all the input matrices (open ABCQR.mat, you will get 5 matrices) along with starting value of K. Please help me to solve this problem.

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