Inverse Laplace Transform in Matlab Programming

The Laplace Transform converts linear time-invariant (LTI) differential equations into algebraic equations in the complex frequency domain ((s)-domain). Once you solve for a system transfer function in the (s)-domain, you must convert the expression back to the time domain ((t)-domain) using the Inverse Laplace Transform to obtain physical time-domain responses.

In MATLAB, the Symbolic Math Toolbox provides the ilaplace() function to compute exact analytical inverse Laplace transforms.

What Is the Inverse Laplace Transform?

The Inverse Laplace Transform converts a frequency-domain transfer function (F(s)) into its corresponding time-domain sequence (f(t)):

This conversion allows control engineers to inspect system transient responses, stability behavior, and time-domain step or impulse signals.

MATLAB Function Syntax: ilaplace()

f = ilaplace(F) f = ilaplace(F, s, t)

• F: Symbolic transfer function in the Laplace domain.
• s: Laplace variable (frequency domain).
• t: Time variable (time domain).

Step-by-Step Examples

Example 1: Second-Order System Response

Find the inverse Laplace transform of:

syms s t F = 1 / (s^2 + 4*s + 3);
f = ilaplace(F, s, t)

Output:

f = (1/2)*exp(-t) - (1/2)*exp(-3*t)

The resulting time-domain expression consists of two decaying exponential terms characteristic of a stable second-order system response.

Example 2: Generalized Symbolic Constants

Compute the inverse Laplace transform containing a symbolic parameter a:

syms s a t F = a / (s^2 + a^2);
f = ilaplace(F, s, t)

Output:

f = sin(a*t)

Example 3: Control System Step Response

Find the time-domain response of the open-loop transfer function:

syms s t G = 5 / (s * (s + 2));
g_t = ilaplace(G, s, t)

Output:

g_t = (5/2) - (5/2)*exp(-2*t)

Plotting Time-Domain Responses in MATLAB

Visualize the resulting continuous analytical function using fplot():

figure;
f
plot(g_t, [0, 5], 'LineWidth', 1.5, 'Color', 'b');
title('System Step Response g(t)');
xlabel('Time t (seconds)');
ylabel('Amplitude g(t)');
grid on;