Time Domain Analysis of Control System

A dynamic control system often contains energy-storage elements such as inductors and capacitors. Their state cannot usually change instantaneously, so the output takes time to move from its initial condition toward a final condition. The resulting changes in voltages, currents or other variables form the transient response.

A transient response can be monotonic or oscillatory, and it can decay, grow or remain sustained. Its form depends on the system model, parameters, initial conditions and input. For a linear time-invariant model, differential equations or an equivalent transfer function describe this behaviour. Solving for the output as a function of time is called time domain analysis of the control system.

Step Function

Consider an ideal voltage source or battery connected to a voltmeter through switch s. While the switch is open, the measured voltage v(t) is zero:

If the switch closes at t = 0, the ideal source applies V volts instantaneously and the voltage remains at V for t ≥ 0:

Combining the two time ranges gives:

Setting V = 1 defines the unit step u(t):

The Laplace transform follows from multiplying u(t) by e^-st and integrating from 0 to infinity.
Fig 6.2.1

Therefore a step of amplitude V has R(s) = V/s, and a unit step has R(s) = 1/s:

Ramp Function

A ramp input is zero before t = 0 and then changes linearly with time. A positive slope rises from the origin, while a negative slope falls.

In this expression, k is the ramp slope. The unit ramp has k = 1.
Fig 6.2.2
The Laplace transform is found by multiplying the ramp by e^-st and integrating from 0 to infinity:

The result is R(s) = k/s², or 1/s² for a unit ramp:

Parabolic Function

A parabolic input is zero for t<0 and varies with the square of time for t > 0. The standard unit parabolic input uses t²/2 so that it is the integral of the unit ramp:

Its Laplace transform is found by multiplying the time function by e^-st and integrating from 0 to infinity.
Fig 6.2.3

The unit parabolic transform is 1/s³:

Impulse Function

The Dirac delta δ(t) models an ideal impulse. Its value is zero away from t = 0. Its idealised duration approaches zero while its peak grows so that the integral stays fixed. A unit impulse has area one. Distribution theory defines the delta as the derivative of the unit step, so its one-sided Laplace transform is 1.
Fig 6.2.4

Time Response of First Order Control Systems

After common pole-zero factors are cancelled, a system is first order when its transfer-function denominator is a first-degree polynomial in s. A standard stable form is G(s) = K/(Ts + 1), with T > 0.

Time Response for Step Function

Apply a unit step to the standard unity-gain first-order system. The output is:

Fig 6.3.2 For T > 0, the response rises exponentially toward one and the tracking error decays toward zero. The response reaches its final value only in the limit as t approaches infinity.

At t = T, the output is:

T is the time constant. At one time constant, the response reaches 1 – e⁻¹ = 63.2% of its final value. At t = 4T:

The response is 1 – e⁻⁴ = 98.17% of its final value. The approximate 2% settling time is 4T. After this time, the response remains within 2% of its final value, while the mathematical transient continues to decay. A smaller positive time constant produces a faster first-order response.

Time Response for Ramp Function

For the unity-gain first-order model, the steady ramp output has the same slope as the input but is delayed by T. A unit ramp therefore has steady tracking error T; a ramp of slope k has error kT. Reducing T reduces this tracking error.

Time Response for Impulse Function

The unit step is the derivative of the unit ramp, and the unit impulse is the derivative of the unit step. For a linear time-invariant system with zero initial conditions, the corresponding outputs have the same derivative relationships: the step response is the derivative of the ramp response, and the impulse response is the derivative of the step response.