A possible approach to model the above equations is to add an extra state to your system that represents the constant term. This state will have a derivative of 0 since it is constant.
The code snippet below illustrates this approach.
% State-space matrices (sample values)
A = [2 2 3; 1 2 1; 3 4 5];
B = [3; 4; 7];
C = [7 8 9];
D = 9;
E = [10; 11; 12];
% Number of states, inputs, and outputs
n = size(A, 1);
m = size(B, 2);
p = size(C, 1);
% Augment the A matrix with an extra column for E
A_new = [A, E; zeros(1, n), 0];
% Augment the B matrix with an extra row of zeros
B_new = [B; zeros(1, m)];
% Augment the C matrix with an extra column for effect of E on the output
C_new = [C, zeros(p, 1)];
% D matrix remains the same
D_new = D;
% Create the state-space model
sys = ss(A_new, B_new, C_new, D_new);
Output:
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