Question about the state space model SS

Hi everyone

I have a question about the state space model. I have linearized my equations with Taylor at first order around a stationary point. If i consider my stationary point different from zero, I obtain the following model: x_dot = Ax+Bu+E and y = Cx+Du where E is the matrix that contains only known terms related to linearization constant. So my question is if there's a way to pass from this two equations to the state space model, cause I always used sys = ss(A,B,C,D), but this time I have also the matrix E.

Answer 1

[Abhinav Aravindan]

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:

Please find below similar queries to yours and relevant documentation for reference:

https://www.mathworks.com/matlabcentral/answers/13506

https://www.mathworks.com/matlabcentral/answers/723093

https://www.mathworks.com/matlabcentral/answers/60305

https://www.mathworks.com/help/releases/R2020b/control/ref/ss.html

[Sam Chak]

Hi @Abhinav Aravindan, it seems that you placed the E constants into the state matrix A. What will be the initial value of augmented state ?