Least square parameter estimation of MIMO ARX model

I know that the ARX function in Matlab can estimate the parameters of the multi input multi output ARX model.

But I can't find the estimated main program when I open ARX function.

I want to find a program to estimate the parameters of MIMO ARX model with least square method.

I came to ask for help.

Answer 1

[Ivo Houtzager]

Example script to obtain the parameters of a MIMO ARX model (VARX) using least squares.

N = 1000; % number of samples
p = 5; % past window of VARX model
directfeedthrough = 1; % VARX model includes direct feedthrough
r = 2; % number of inputs
l = 3; % number of outputs
u = randn(r,N); % input data
y = randn(l,N); % output data

% concatenate the past data vectors
m = r+l;
z = [u; y];
Z = zeros(p*m,N-p);
for i = 1:p
    Z((i-1)*m+1:i*m,:) = z(:,i:N+i-p-1);
end

% solve VARX problem
Y = y(:,p+1:N);
U = u(:,p+1:N);
if directfeedthrough
    Z = [Z; U];
end  
VARX = Y/Z; % least squares estimate

% convert solution to idpoly object 
A = cell(l,l);
B = cell(l,r);
if directfeedthrough
    VARX0 = [VARX eye(l)];
else
    VARX0 = [VARX zeros(l,r) eye(l)];
end
for i = 1:l
    for j = 1:l
        A{i,j} = fliplr(VARX0(i,r+j:m:m*p+r+j));
    end
    
    for k = 1:r
        B{i,k} = fliplr(VARX0(i,k:m:m*p+k));
    end
end
Ts = 1; % sample time
E = Y - VARX*Z;
NoiseVariance = cov(E');
sys = idpoly(A,B,[],[],[],NoiseVariance,Ts); % MIMO ARX model object

For larger models, the least squares problem can become ill conditioned and would require regularization to get good estimate. If regularization is needed, I recommend to look at source code of the following matlab function.

https://github.com/TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox/blob/master/dordvarx.m