% finding the closed loop poles for the optimal infinte horizion 
% control law for the system of the form:
% X(k+1) = AX(k) + Bu(k)
% while minimizing a quadratic performance of the
% form:
% J = 1/2*sum(X(k)'QX(k) + Ru(k)^2)
% using Riccati analytic solution   
% Ref.: Optimal Control by F. Leuis & V. Syrmous, 1995

% set the problem parameters

A = [1 .5;0 1];
B = [0.125;.5];
R = 1;
syms alfa betta
Q = [alfa 0;0 betta];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

H = [A^-1 A^-1*B*R^-1*B'; Q*A^-1 A'+Q*A^-1*B*R^-1*B'];

[W,D] = eig(H);

W21 = W(3:4,1:2);
W11 = W(1:2,1:2);

S_inf = W21*inv(W11);
K_inf = R^-1*B'*S_inf;

closed_loop_poles = eig(A-B*K_inf)