This document discusses time domain analysis of control systems. It introduces standard test signals used to analyze dynamic systems, including impulse, step, ramp, and parabolic signals. These signals mimic characteristics of actual inputs like sudden shock, changes, constant velocity, and acceleration. The time response of a system has two components - transient response as it moves from rest to steady state, and steady-state response once settled. Standard signals are used to examine a system's transient response and steady-state response depends on both system dynamics and input type.

1. Feedback Control Systems (FCS)
Lecture-12
Time Domain Analysis of Control systems
Dr. Imtiaz Hussain
email: imtiaz.hussain@faculty.muet.edu.pk
URL :http://imtiazhussainkalwar.weebly.com/

2. Introduction
• In time-domain analysis the response of a dynamic system
to an input is expressed as a function of time.
• It is possible to compute the time response of a system if the
nature of input and the mathematical model of the system
are known.
• Usually, the input signals to control systems are not known
fully ahead of time.
• For example, in a radar tracking system, the position and the
speed of the target to be tracked may vary in a random
fashion.
• It is therefore difficult to express the actual input signals
mathematically by simple equations.

3. Standard Test Signals
• The characteristics of actual input signals are a
sudden shock, a sudden change, a constant
velocity, and constant acceleration.
• The dynamic behavior of a system is therefore
judged and compared under application of
standard test signals – an impulse, a step, a
constant velocity, and constant acceleration.
• Another standard signal of great importance is a
sinusoidal signal.

4. Standard Test Signals
• Impulse signal
– The impulse signal imitate the
sudden shock characteristic of
actual input signal.
(t )
A
0
0
t
t
δ(t)
A
0
0
– If A=1, the impulse signal is
called unit impulse signal.
t

5. Standard Test Signals
• Impulse signal
Source: English Wikipedia. Iain. Original image: [1]

6. Standard Test Signals
• Step signal
– The step signal imitate
the sudden change
characteristic of actual
input signal.
u( t )
A
0
0
t
t
0
– If A=1, the step signal is
called unit step signal
u(t)
A
0
t

7. Standard Test Signals
• Ramp signal
r(t)
– The ramp signal imitate
the constant velocity
characteristic of actual
input signal.
t
0
r(t )
At
0
0
t
t
0
– If A=1, the ramp signal
is called unit ramp
signal
r(t)
ramp signal with slope A
r(t)
unit ramp signal

8. Standard Test Signals
p(t)
• Parabolic signal
– The
parabolic
signal
imitate the constant
acceleration characteristic
of actual input signal.
p(t )
At
2
0
t
0
2
p(t)
t
t
0
0
parabolic signal with slope A
p(t)
– If A=1, the parabolic
signal is called unit
parabolic signal.
Unit parabolic signal

9. Relation between standard Test Signals
• Impulse
• Step
• Ramp
• Parabolic
(t )
u( t )
r(t )
p(t )
A
0
0
t
t
0
t
t
d
dt
0
t
At
At 2
2
0
0
t
0
d
dt
0
t
A
0
0
t
0
d
dt
0

10. Laplace Transform of Test Signals
• Impulse
(t )
A
0
L{ (t )}
• Step
u( t )
0
t
0
t
( s)
A
0
A
0
t
t
0
L{u(t )} U ( s )
A
S

11. Laplace Transform of Test Signals
• Ramp
r(t )
At
0
L{ r(t )}
• Parabolic
p(t )
L{ p(t )}
0
t
0
t
A
R( s )
At 2
2
0
s2
t
t
P( s )
0
0
2A
S3

12. Time Response of Control Systems
• Time response of a dynamic system is response to an input
expressed as a function of time.
System
• The time response of any system has two components
• Transient response
• Steady-state response.

13. Time Response of Control Systems
• When the response of the system is changed form rest or
equilibrium it takes some time to settle down.
• Transient response is the response of a system from rest or
equilibrium to steady state.
-3
6
Step Response
x 10
Step Input
Steady State Response
4
Amplitude
• The response of the
system after the transient
response is called steady
state response.
5
Response
3
Transient Response
2
1
0
0
2
4
6
8
10
Time (sec)
12
14
16
18
20

14. Time Response of Control Systems
• Transient response dependents upon the system poles only and
not on the type of input.
• It is therefore sufficient to analyze the transient response using a
step input.
• The steady-state response depends on system dynamics and the
input quantity.
• It is then examined using different test signals by final value
theorem.

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END OF LECTURE-12