Computing matrices for Model Predictive Control - matlab

Question

In model predictive control, an optimization problem is solved at every time instant and it is very common to write down the matrices in a compact form. Without going into the details of the optimization problem, suppose I have matrices and . I need to compute the matrices and defined as
Note that N_p is called "prediction horizon" and it is not the order of matrix A. How can I compute these matrices in a fast and efficient way? In Matlab, I have done the following, but maybe there is a more efficient way to compute these matrices:
A_cal = zeros(length(A)*Np, length(A)); %calligraphic A matrix
B_cal = zeros(size(B,1)*Np, size(Bd,2)*Np); %calligraphic B matrix
temp = eye(size(A));
for j = 1:Np
A_cal(1+(j-1)*length(A):j*length(A),:)= temp;
if j > 1
%The current row is obtained as shift of the previous row, and only the block in the first column is computed
B_cal(1+(j-1)*size(B,1):j*size(B,1),:) = circshift(B_cal(1+(j-2)*size(B,1):(j-1)*size(B,1),:),size(B,2),2);
B_cal(1+(j-1)*size(B,1):j*size(B,1),1:size(B,2)) = temp_prev*B_cent;
end
temp_prev = temp; %this variable contains A^(j-1)
temp = temp * A_cent; %use temp variable to speed up the matrix power computation
end

Answer 1

I assume you already solved your problem, but here is the code from my exercises sheet, creating your matrices.
S_x being the first matrix.
function S_x = compute_Sx(A,N)
S_x = eye(size(A));
for i=1:N
S_x = [S_x;A^i];
end
end
function S_u = compute_Su(A,B,N)
S_u = zeros(size(A,1)*N,size(B,2)*N);
for i=1:N
S_u = S_u + kron(diag(ones(N-i+1,1),-i+1),A^(i-1)*B);
end
S_u = [zeros(size(A,1),size(B,2)*N);S_u];
end
Best Regards