How to simulate the forced response of a transfer function

I have a transfer function G(s) = (1-s)/(s^2 + 2s+ 1) that I want to simulate while being under a forced input of u(t) = 2*cos(3t), but I am not sure how. I hope someone can help.

Answer 1

[Star Strider]

Use the lsim function —

s = tf('s');
G = (1-s)/(s^2 + 2*s + 1)

G =
 
     -s + 1
  -------------
  s^2 + 2 s + 1
 
Continuous-time transfer function.

t = linspace(0, 1E+1, 1E+3);
u = @(t) 2*cos(3*t);

Gu  = lsim(G,u(t),t);

figure
plot(t, u(t), ':k', 'DisplayName','u(t)')
hold on
plot(t, Gu, '-r', 'DisplayName','G(u(t))')
hold off
grid
legend('Location','best')

.

Answer 2

[Sam Chak]

Hi @Ali Almakhmari

The 'lsim()' command is great when the Control System Toolbox is available. Here is an alternative approach to generate the time response G(s) subject to the forced input u(t) without using the Control System Toolbox. However, it requires the Symbolic Math Toolbox. Both toolboxes are bundled in the MATLAB and Simulink Student Suite.

You can also find examples at the following links:

https://www.mathworks.com/help/symbolic/sym.laplace.html

https://www.mathworks.com/help/symbolic/sym.ilaplace.html

syms F(s) u(t);

% Forced input function
u(t) = 2*cos(3*t)

u(t) = 

U(s) = laplace(u)

U(s) = 

% Plant transfer function
G(s) = (1 - s)/(s^2 + 2*s + 1)

G(s) = 

% Convolution of two functions
F(s) = U*G

F(s) = 

% Applying the inverse Laplace transform
f(t) = ilaplace(F, s, t)

f(t) = 

% Plots
fplot(u, [0 10], ':k'), hold on, grid on
fplot(f, [0 10], 'LineWidth', 1.5), hold off

% Labels
xlabel('Time (seconds)')
ylabel('Amplitude')
legend('u(t)', 'f(t)')
title({'Time response of $G(s)$ due to input $u(t) = 2 \cos(3 t)$'}, 'Interpreter', 'LaTeX', 'FontSize', 12)