# Control Systems Time Response Analysis - Control Systems

## What is time response analysis?

Let’s analyze the response of the control systems across the time domain and the frequency domain. We will discuss about frequency response analysis of control systems in coming chapters. Now discuss about the time response analysis of control systems.

## What is Time Response?

Time response of control system for an input will be different and changes with time, so it is called the time response of the control system. The time response consists of two parts.
• Transient response
Following diagram explains about the response of control system in time domain:
While both the transient and the steady states are mentioned in the above diagram. The responses corresponding to these states are known as transient and steady state responses. $c\left(t\right)={c}_{tr}\left(t\right)+{c}_{ss}\left(t\right)$
Mathematically, we can write the time response c(t) as
Where,
• ctr(t)is the transient response
• css(t)is the steady state response

## Transient Response

Once you apply the input to the control system, then output takes particular time to reach steady state. So, the output will be in transient state till it goes to a steady state. So that the response of the control system whiles the phase of transient state is known as transient response.
Here the transient response will be zero for large values of ‘t’. At the same time this value of ‘t’ is infinity and practically, it is five times constant.

Mathematically, we can write it as $\underset{t\to \mathrm{\infty }}{lim}{c}_{tr}\left(t\right)=0$

The part of the time response will be the same after the transient response value is zero value for the more larger values of ‘t’ is called as steady state response. Then the transient response will be zero when it I sin steady state also.

### Example

Let’s find the transient and steady state terms of the time response of the control system $c\left(t\right)=10+5{e}^{-t}$

Then the secondwill be zero as t denotes infinity. So, this is called as the transient term. The first term 10 will be even as t approaches infinity; this is called as the steady state term.

## Standard Test Signals

The standard test signals are known as impulse, step, ramp and parabolic. These signals are mainly used for the performance of the control systems using time response of the output.

## Unit Impulse Signal

A unit impulse signal,δ(t)is defined as $\delta \left(t\right)=0$ for $t\ne 0$ and ${\int }_{{0}^{-}}^{{0}^{+}}\delta \left(t\right)dt=1$

The following figure shows unit impulse signal.

So, the unit impulse signal will remain same when ‘t’ is equal to zero. The area of this signal for the small interval of time ‘t’ is equal to zero is one. Here the value of unit impulse signal will be zero for all other values of ‘t’.

## Unit Step Signal

A unit step signal, u(t) is defined as $u\left(t\right)=1;t\ge 0$ $=0;t<0$
u(t)=1;t0
=0;t<0
Following figure shows unit step signal.

Here the unit step signal remain same for all positive values of ‘t’ including zero. While at interval its value can be one. The value of the unit step signal is zero for all negative values of ‘t’.

## Unit Ramp Signal

A unit ramp signal, r(t) is defined as $r\left(t\right)=t;t\ge 0$ $=0;t<0$
r(t)=t;t0
=0;t<0
We can write unit ramp signal,r(t)in terms of unit step signal, $u\left(t\right)$ as $r\left(t\right)=tu\left(t\right)$
r(t)=tu(t)
Following figure shows unit ramp signal.
The unit ramp signal remain positive for all positive values of ‘t’ including zero. The value increases linearly with pertain to ‘t’ during this interval. The value of unit ramp signal is zero for all negative values of ‘t’.

## Unit Parabolic Signal

A unit parabolic signal, p(t) is defined as, $p\left(t\right)=\frac{{t}^{2}}{2};t\ge 0$ $=0;t<0$

Let’s write this parabolic signal,p(t)in terms of the unit step signal,u(t)as,

$p\left(t\right)=\frac{{t}^{2}}{2}u\left(t\right)$

Below mentioned figure shows the unit parabolic signal.
The unit parabolic signal remain same for all the positive values of ‘t’ including zero. Its value increases non-linearly with respect to ‘t’ at the time of interval. The value of the unit parabolic signal is zero for all the negative values of ‘t’.