Hi @gilad shaul
Minimizing the norm(P) does not work because there are infinite K that can make A'*P + P*A - K'*R*K + Q exactly 0. If you want to solve using genetic algorithm, ga(), then you should directly minimize the Finite-time Performance Index J (a scalar) instead:
Else, if you want to find the elements in the positive-definite matrix 
that minimizes J, then you should use gamultiobj() instead (because there are multiple fitness functions). In 2nd-order systems, there are three fitness functions involving 
, 
, 
because the positive-definite matrix 
looks like this:
that minimizes J, then you should use gamultiobj() instead (because there are multiple fitness functions). In 2nd-order systems, there are three fitness functions involving , , because the positive-definite matrix looks like this:
.% ------- Computation of K via LQR (solving Algebraic Riccati Matrix Equation) -------
% State Space
A = [0 1; 0 -1];
B = [0; 1];
Q = [1 0; 0 1];
R = 1;
% LQR
K = lqr(A, B, Q, R)
% [K, P, E] = lqr(A, B, Q, R)
% Riccati Equation
% A'*P + P*A - (P*B)*inv(R)*(B'*P) + Q
% QQ = - (P*B)*inv(R)*(B'*P) + Q;
% QQ = - (P*B)*K + Q;
% QQ = - K'*R*K + Q;
% PP = lyap(A', QQ) % returns the same positive-definite matrix P
% ------- Computation of K via GA (minimizing Finite-time Performance Index J) -------
fitfun = @costfun;
PopSize = 100;
MaxGens = 200;
nvars = 2;
A = -eye(nvars);
b = zeros(nvars, 1);
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = [];
options = optimoptions('ga', 'PopulationSize', PopSize, 'MaxGenerations', MaxGens);
[KK, fval, exitflag, output] = ga(fitfun, nvars, A, b, Aeq, beq, lb, ub, nonlcon, options)
function J = costfun(param)
% Parameters
A = [0 1; 0 -1];
B = [0; 1];
Q = [1 0; 0 1];
R = 1;
K = [param(1) param(2)];
% State-Space
odefcn = @(t, x) (A - B*K)*x;
tspan = [0 20]; % Finite-time
x0 = [1 0];
[t, x] = ode45(odefcn, tspan, x0);
% Performance Index
u = - K(1)*x(:,1) - K(2)*x(:,2);
itgrd = Q(1,1)*x(:,1).^2 + Q(2,2)*x(:,2).^2 + R*u.^2; % integrand
J = trapz(t, itgrd);
end
