How to use the generated LQG controller

I have a finite-horizon discrete-time system given by

to be regulated around a setpoint r

I provided the matrices A, B, C, D = 0, QXU, QVW, QI and created a controller KLQG. For the purposes of this problem, y and r are scalars (1x1). dim(u) = dim(x) = 50. v and w are white Gaussian noise.

I believe the controller takes [y;r] as input and provides u as an output. However, I'm unsure on how to call this function and how to implement this in the overall system. Also noticed that the A,B,C,D matrices inside KLQG are different from the ones I supplied. I'm following this Mathworks page.

sys = ss(A,B,C,D,1)                             %D = 0, discrete-time
KLQG = lqg(sys,QXU,QWV,QI);                     %Create LQG

for n = 1:99                                    %100 time steps, assume x is initialized, i.e. x(:,1) is given
    y = C*X(:,n) + wnoise();                    %wnoise() generates white gaussian noise of length dim(y)
    u(:,n) = f(r,y);                            %Tried KLQG([r;y]) but doesn't work, what is f?
    X(:,n+1) = A*X(:,n) + B*u(:,n) + vnoise();  %vnoise() generates white gaussian noise of length dim(x)
end

y(:,100) = C*X(:,100) + wnoise();               %Hopefully this is close to setpoint r

J = 0;

for n = 1:100                                   %Compute cost 
    J = J + [x(:,n)' u(:,n)']*QXU*[x(:,n); u(:,n)] + xi(:,n)'*QI*xi(:,n);       
end

Ideally since this is a discrete time problem, I'd like my cost function to be a little different with the final state being taken into account like the problem setup here but we can ignore that for now.

TLDR; How to compute xi and u?

Answer 1

[test]

te this is my first app on this app and first comment.

[RV_S]

Lol