MATLAB - Laplace Transform

Quiz

The Laplace Transform is a powerful mathematical technique used to solve differential equations and analyze systems in engineering, physics, and other fields. It's named after the French mathematician Pierre-Simon Laplace.

In simple terms, the Laplace Transform converts a function of time into a function of a complex variable, usually denoted by s. This transformation changes differential equations into algebraic equations, making them easier to solve.

One of the key advantages of the Laplace Transform is its ability to handle signals and systems with discontinuities or complex behavior. It's widely used in control theory, signal processing, communication systems, and many other areas of science and engineering.

In MATLAB, the Laplace Transform is often used in the context of analyzing and solving differential equations, particularly in the field of control systems and signal processing.

MATLAB provides the laplace function, which can be used to compute the Laplace Transform of a given function.

Syntax

F = laplace(f)
F = laplace(f,transVar)
F = laplace(f,var,transVar)

Syntax Explanation

F = laplace(f) − Calculates the Laplace Transform of the function f. By default, the function is assumed to be a function of time (t) and the Laplace Transform is a function of a complex variable (s).

F = laplace(f,transVar) − Calculates the Laplace Transform of the function f using a different variable, transVar, instead of the default variable s.

F = laplace(f,var,transVar) − Calculates the Laplace Transform of the function f, where the function depends on the variable var and the Laplace Transform is with respect to the variable transVar, instead of the default variables t and s.

Working Examples on Laplace using laplace() function

Let us see a few examples to show how you can use the laplace function in MATLAB to compute Laplace Transforms.

Example 1: Compute Laplace transform for function f(t) = e-at

Suppose we have a simple function f(t) = e-at , where a is a constant.We want to compute its Laplace Transform using the laplace function in MATLAB.

syms t s a
f = exp(-a*t);
F = laplace(f);
disp(F);

In the example above we have −

We start by defining the symbolic variables t, s, and a using syms.
Then, we define the function f(t) as exp(-a*t), which represents e-at
Next, we use the laplace function with just the function f as the argument. Since we haven't specified the independent variable or the transformation variable, MATLAB assumes the default values of t for the independent variable and s for the transformation variable.
Finally, we use disp to display the Laplace Transform of the function f.

When the code is executed we get the output as shown below −

>> syms t s a
f = exp(-a*t);
F = laplace(f);
disp(F); 

1/(a + s)
 
>>

Example 2: Compute the Laplace transform of 1/sqrt(x)

The code we have is −

syms x
f = 1/sqrt(x);
F = laplace(f);
disp(F);

In the example we have −

syms x declares the variable x as a symbolic variable, indicating that x can represent any mathematical expression.
f = 1/sqrt(x); defines the function f(x) = 1/sqrt(x) using the symbolic variable x.
F = laplace(f); computes the Laplace transform of f with respect to the default variable t and assigns the result to F.
disp(F); displays the Laplace transform of f(x) as a function of the Laplace variable s.

When the code is executed the output we get is as follows −

>> syms x
f = 1/sqrt(x);
F = laplace(f);
disp(F);

pi^(1/2)/s^(1/2)
 
>>

Example 3: Computing the Laplace Transform of a Sine Function with a Parameter

The code we have is as follows −

syms t s
syms omega real
f = sin(omega*t);
f_subs = subs(f, t, s/omega); % Substitute t with s/omega
F = laplace(f_subs, s);
disp(F);

In the example,

syms t s − This line declares the variables t and s as symbolic, meaning they can represent mathematical expressions.syms omega real: This line declares the variable omega as a symbolic real constant. This is done to ensure that omega is treated as a real number when performing computations.
f = sin(omega*t) − This line defines the function f(t) = sin(omega*t). Here, omega is a parameter that controls the frequency of the sine wave.
f_subs = subs(f, t, s/omega) − This line performs a substitution in the function f. It replaces every occurrence of the variable t with s/omega, effectively transforming the function to f(s/omega).
F = laplace(f_subs, s) − This line computes the Laplace transform of the substituted function f_subs with respect to the variable s. Since s is the default variable for Laplace transforms in MATLAB, we only need to specify s as the second argument to the laplace function.
disp(F) − This line displays the result of the Laplace transform, which is a function of s. The result will be in terms of omega, reflecting the frequency parameter in the original function.

On execution the output is −

>> syms t s
syms omega real
f = sin(omega*t);
f_subs = subs(f, t, s/omega); % Substitute t with s/omega
F = laplace(f_subs, s);
disp(F);
1/(s^2 + 1)
 
>>

Example 4: Computing the Laplace Transform of an Exponential-Cosine Function

The code we have is as follows −

syms t s
f = exp(-2*t)*cos(3*t);
F = laplace(f, t, s/2);
disp(F);

The function used is f(t) = e-2t cos(3t) .We compute its Laplace transform with respect to t and use s/2 as the Laplace variable.

In the example we have −

syms t s − This line declares the variables t and s as symbolic, indicating that they can represent mathematical expressions.
f = exp(-2*t)*cos(3*t) − This line defines the function f(t) = e^{-2t} \cos(3t).
F = laplace(f, t, s/2) − This line computes the Laplace transform of the function f with respect to the variable t and using the Laplace variable s/2 instead of the default variable s.
disp(F) − This line displays the result of the Laplace transform, which is a function of s/2. The result will be in terms of s, reflecting the transformation variable used in the Laplace transform.

On execution in matlab command window the output we get is −

>> syms t s
f = exp(-2*t)*cos(3*t);
F = laplace(f, t, s/2);
disp(F);

(s/2 + 2)/((s/2 + 2)^2 + 9)
 
>>

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