system with time delay

I would like to know if there is a command in Matlab to extract matrices of system with time delay, I have a nominal system :

x_dot = A(t)*x(t)+B(t)*u(t)

I would like to simulate the system with time delay t_x and t_u as :

x_dot = A(t)*x(t)+B(t)*u(t) + A_d*x(t-t_x)+B_d(t)*u(t-t_u)

then extract the matrices A_d(t), B_d(t) to use them to design a controller.

your help is appreciated.

Answer 1

[thoughtGarden]

This answer has been updated to better match the OP.

doc ss %search for "InternalDelay" and "OutputDelay"

% Define Delays
t_x = 0.005;
t_u = 0.001;

% Fill in your system matrix here
A = [1,0; 0, 1];

B = [0; 0.5];

C = [1, 0];

D = 0;

% Create the no delay system
sys =ss(A,B,C,D,'InputDelay',t_x,'OutputDelay',t_u);

[Chokri Sendi]

Would you please elaborate more on : sys = sys *pade(delayTime,1)

also is the delayTime = 0.01; represent t_x the delay in x or t_u the delay in u.

Thank you

[thoughtGarden]

I just updated the answer showing more clearly the input and output delay. You might also be interested in https://www.mathworks.com/help/control/ug/time-delays-in-linear-systems.html

[thoughtGarden]

If you found this answer to be correct, please accept it. Otherwise, please add aditional comments so that it may be improved.

[Chokri Sendi]

Thank you for your reply, I still dont see how I can use

" sys =ss(A,B,C,D,'InputDelay',t_x,'OutputDelay',t_u),

to write the system as:

x_dot = A(t)*x(t)+B(t)*u(t) + A_d*x(t-t_x)+B_d(t)*u(t-t_u)

in other word since A and B are known how I can find A_d and B_d from "sys" that I just created .

Thanks