First Order Step Response

Modelling is about predicting the behaviour of a system. Often, need to know

  • What is the output for a given input?
  • Is the system stable?
  • If the input changes quickly, how will the output change?

First Order Systems

First order systems are those with only one energy store, and can be modelled by a first order differential equation.


For the general form of the equation , the solution for a step input at time , with : T is the time constant of the system.

Free and Forced Response

  • Free response:
    • The response of a system to its stored energy when there is no input
    • Zero Input
    • Non-zero initial Conditions
    • Homogenous differential equation
  • Forced response:
    • The response of a system to an input when there is no energy initially in the system
    • Non-zero input
    • Zero initial Conditions
    • Non-homogeneous differential equation
  • Total system response is a linear combination of the two

System Inputs

Different inputs can be used to determine characteristics of the system.

Step Input

  • A sudden increase of a constant amplitude input
  • Can see how quickly the system responds
  • Is there is any delay/oscillation?
  • Is it stable?

Sine Wave

  • Can vary frequency and amplitude
  • Shows frequency response of a system


  • A spike of infinite magnitude at an infinitely small time step


  • An input that starts increasing at a constant rate, starting at .

Step Response

  • The step response of the system is the output when given a step input
    • System must have zero initial conditions
  • Characteristics of a response:
    • Final/resting value
    • Rise time
    • Delay
    • Overshoot
    • Oscillation (frequency & damping factor)
    • Stability

For a system with time constant , the response looks something like this:

The time constant of a system determines how long the system takes to respond to step input. After 1 time constant, the system is at about (63) % of its final value.

Time (s)% of final value