Transfer Functions

  • A transfer function is a representation of the system which maps from input to output
    • Useful for system analysis
    • Carried out in the Laplace Domain

The Laplace Domain

  • Problems can be easier to solve in the Laplace domain, so the equation is Laplace transformed to make it easier to work with
  • Given a problem such as "what is the output given a differential equation in and the step input ?"
    • Express step input in Laplace domain
    • Express differential equation in Laplace domain and find transfer function
    • Find output in Laplace domain
    • Transfer back to time domain to get
FunctionTime domainLaplace domain

The laplace domain is particularly useful in this case, as a differential equation in the time domain becomes an algebraic one in the Laplace domain.

Transfer Function Definition

The transfer function is the ratio of output to input, given zero initial conditions.

For a general first order system of the form

The transfer function in the Laplace domain can be derived as:

Step Input in the Laplace Domain

Step input has a constant value for

For a first order system, the output will therefore be:


Find the transfer function for the system shown:

The system has input-output equation (in standard form):

Taking the Laplace transform of both sides:

Rearranging to obtain the transfer function:

Using Matlab

In matlab the tf function (Matlab docs) can be used to generate a system model using it's transfer function. For example, those code below generates a transfer function , and then plots it's response to a step input of amplitude 1.

G = tf([1],[2 3]);


For the system shown below, where , , , plot the step response and obtain the undamped natural frequency and damping factor .

system = tf([1],[100 40 100]);
step(system, 15); % plot 15 seconds of the response

%function to obtain system parameters
[wn,z] = damp(system)

The script will output wn=1, and z = 0.2. The plotted step response will look like: