What are the different methods to obtain response of a state space model?

Please explain (or redirect to sources) with codes on methods to obtain ss response like from commands lsim or methods like runge kutta method and more. Examples would help.

Answer 1

[Hari]

Hi Ram,

I understand that you are looking for different methods to obtain the response of a state-space model in MATLAB, including using built-in functions like "lsim" and numerical methods such as the Runge-Kutta method.

I assume you have a state-space representation of your system and are interested in both analytical and numerical methods for response analysis.

In order to obtain the response of a state-space model, you can follow the below steps:

Define the State-Space Model:

Use the "ss" function to define your state-space model with matrices A, B, C, and D.

A = [0 1; -2 -3];
B = [0; 1];
C = [1 0];
D = 0;
sys = ss(A, B, C, D);

Simulate the Response with "lsim":

Use "lsim" to simulate the system's response to a custom input signal over a specified time period.

t = 0:0.01:10; % Time vector
   u = sin(t);    % Input signal
   [y, t, x] = lsim(sys, u, t);
   plot(t, y), title('Response using lsim'), xlabel('Time'), ylabel('Output');

Implement the Runge-Kutta Method:

Numerically solve the state equations using the Runge-Kutta method for a given input signal.

   dt = 0.01; % Time step
   x = [0; 0]; % Initial state
   for i = 1:length(t)-1
       k1 = Ax(:, i) + Bu(i);
       k2 = A(x(:, i) + 0.5dtk1) + Bu(i);
       x(:, i+1) = x(:, i) + dt*k2;
   end
   y_rk = C*x;
   plot(t, y_rk), title('Response using Runge-Kutta'), xlabel('Time'), ylabel('Output');

Use Other Built-in Functions:

Explore other functions like "step" for step response and "impulse" for impulse response.

   step(sys); % Step response

Compare and Analyze:

Compare the results from different methods to ensure consistency and accuracy of the responses.

   figure;
   plot(t, y, 'b', t, y_rk, 'r--');
   legend('lsim', 'Runge-Kutta');

Refer to the documentation of "lsim" function to know more about the properties supported: https://www.mathworks.com/help/control/ref/lsim.html

Hope this helps!