I want to implement this form for MPC but the matrix changes over the horizon and I cannot figure out how to implement this form;

I want to implement this form for MPC but the matrix changes over the horizon and I cannot figure out how to implement this form. Does someone know a good way to implement this form?

N = 10;

Al = zeros((N*12)+1);

for i = 0:N

Al(2+(12*i):13+(12*i),1:4) = (A^(i) * B);

end

Answer 1

[Pratyush Swain]

Hi Emiel,

It seems that you are trying to construct a block lower-triangular matrix that is typically used in Model Predictive Control (MPC) to represent the system dynamics over a prediction horizon.This matrix is often referred to as the "dynamics matrix" or "predictive model matrix".

Here is a corrected version of the code snippet and an example implementation that should achieve the desired form for the MPC matrix:

% Define the number of states (n) and inputs (m)
n = 4; % Number of states
m = 2; % Number of inputs

% Defining dummy state transition matrix A (n x n)
A = rand(n, n);

% Defining dummy input matrix B (n x m)
B = rand(n, m);

% Prediction horizon
N = 10; 

% Initialize the matrix Al with zeros
Al = zeros(N*n, N*m);

% Fill in the matrix Al
for i = 1:N
    for j = 1:i
        Al((i-1)*n+1:i*n, (j-1)*m+1:j*m) = A^(i-j) * B;
    end
end

% Display the resulting matrix Al
disp(Al);

"A" is a 4x4 matrix with random values, representing the state transition dynamics of the system and "B" is a 4x2 matrix with random values, representing control input matrix.

"Al" is the resulting MPC matrix that maps the sequence of future control inputs to future states over the prediction horizon "N".

For more information on use of matrices in MATLAB, please refer to https://www.mathworks.com/help/matlab/learn_matlab/matrices-and-arrays.html

Hope this helps.