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 $T\frac{dy}{dt}+y=x$, the solution for a step input $x=H$ at time $t=0$, with $y(0)=0$: $$y=H(1-e^{-\frac{t}{T}})$$ 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

$$u(t)=\{\begin{array}{ll}0& t<0\\ H& t\ge 0\end{array}$$

• 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

Impulse

$$u(t)=\{\begin{array}{ll}0& t\ne 0\\ \infty & t=0\end{array}$$

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

Ramp

$$u(t)=\{\begin{array}{ll}0& t<0\\ kt& t\ge 0\end{array}$$

• An input that starts increasing at a constant rate, starting at $t=0$.

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 $T=10$, the response looks something like this:

The time constant $T$ of a system determines how long the system takes to respond to step input. After 1 time constant, the system is at about $1-\frac{1}{e}$ (63) % of its final value.