Control Engineering: Time Domain Analysis

CONTROL ENGINEERING
Course Code 18ME71 CIE Marks 40
Teaching Hours / Week (L:T:P) 3:0:0 SEE Marks 60
Credits 03 Exam Hours 03
[AS PER CHOICE BASED CREDIT SYSTEM (CBCS) SCHEME]
SEMESTER – VII
Dr. Mohammed Imran
B. E. IN MECHANICAL ENGINEERING

Course Objectives
 To develop comprehensive knowledge and understanding of
modern control theory, industrial automation, and systems
analysis.
 To model mechanical, hydraulic, pneumatic and electrical
systems.
 To represent system elements by blocks and its reduction
 To represent system elements by blocks and its reduction
techniques.
 To understand transient and steady state response analysis
of a system.
 To carry out frequency response analysis using polar plot,
Bode plot.
 To analyse a system using root locus plots.
 To study different system compensators and characteristics
of linear systems.
Dr. Mohammed Imran

Course outcomes
On completion of the course the student will be able to
CO1: Identify the type of control and control actions.
CO2: Develop the mathematical model of the physical systems.
CO3: Estimate the response and error in response of first and
second order systems subjected standard input signals.
second order systems subjected standard input signals.
CO4: Represent the complex physical system using block diagram
and signal flow graph and obtain transfer function.
CO5: Analyse a linear feedback control system for stability using
Hurwitz criterion, Routh‟s criterion an root Locus technique in
complex domain.
CO6: Analyse the stability of linear feedback control systems in
frequency domain using polar plots, Nyquist and Bode plots.
Dr. Mohammed Imran

Module-2
 Time domain performance of control
systems:
 Typical test signal,
Unit step response and time domain
 Unit step response and time domain
specifications of first order,
 Unit step response and time domain
specifications of second order system.
 Steady state error, error constants.
10 Hours
Dr. Mohammed Imran

Text Books:
 Automatic Control
Systems, Farid G., Kuo
B. C, McGraw Hill
Education, 10th
Edition,2018
Dr. Mohammed Imran
Edition,2018
 Control systems, Manik
D. N, Cengage, 2017

Reference Books:
 Modern control Engineering K. Ogeta Pearson 5th
Edition, 2010
 Control Systems Engineering Norman S Nice Fourth
Edition, 2007
Modern control Systems Richard C Dorf Pearson
Dr. Mohammed Imran
 Modern control Systems Richard C Dorf Pearson
2017
 Control Systems Engineering IjNagrath, M Gopal
New Age International (P) Ltd 2018
 Control Systems Engineering S Palani Tata McGraw
Hill Publishing Co Ltd ISBN-13 9780070671935

MODULE -2
PART-A
PART-A
Dr. Mohammed Imran CONTROL ENGINEERING

Module-2
Time domain performance of control systems:
 The most control systems, time is used as an independent variable, so
control systems are inherently Time-domain system. Therefore, response
analysis of a control system has become an important factor for the
design and analysis of control system.
 Response analysis means study of the system output behavior as a function
of time, when subjected to known input. This is also known as time
response.
response.
 This output behavior with respect to time must be with in specified limits
for obtaining satisfactory performance.
 The time response analysis of control system helps in evaluating the
stability of the system, accuracy of the system.
 Therefore., in most control system, the complete or final evaluation of the
performance of the system is based on the time response analysis and
corresponding results
 Definition : The time response of a control system is the output of the
system as a function time, when subjected to known input
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Parts of Time Response
The time response of a control system is usually consists of two pans :
1. The Transient response and
2. The Steady-State response
(1) Transient response
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Fig.1. Response of the system and steady
state error
Transient response is defined as the part of
the total time response which decays or
diesout or goes to zero over a period of time
or as time becomes very large or after large
interval of time. It is denoted by yt(t).
Mathematically it is given by

(2) Steady-State response
The steady state response is
defined as the part of the total
time response which remains
time response which remains
after the transient response has
diesout. It is denoted by yss
Thus, the total time response
y(t), of a control system can
written as
y(t) = yt(t)+ yss(t)
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state error

Response of the system and steady state error
 In general, all the control systems exhibit transient behaviour to
some extent before a steady state is attained because of energy
stating elements such as, spring mass etc. which are always part
of a control system and cannot he avoided. Due to these energy
stetting elements, the response of the control system cannot follow
sudden changes in the input instantaneously and transients are
observed. Thus, transient response analysis is necessarily
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state error
observed. Thus, transient response analysis is necessarily
important, as it plays significant rule in the dynamic behaviour of
the system and infact, deviation of output response sad input must
be control, before steady state is reached.
 The steady-state response of a control system is also very
important as it indicates final output of the system.
 This points towards the accuracy of the system.
 In general, if the steady state response of the system does not
agree with the desired reference exactly, then, the system is said
to be in steady state error.
 Thus, the difference in the desired output and the actual output of
the system is called Sleady state error and is denoted by e„ All
these definition can be shown in the wave form as shown in fig.1.

1. Typical test signal or Standard Test Inputs
 In evaluating performance of a system, it is necessary to
assume some basic standard test input signals.
 By assuming these standard test signals, not only
mathematical analysis of the system is made easy but also
these input signals response allows to predict the system's
performance to other more complex inputs.
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performance to other more complex inputs.
 So for the ease of the time-domain analysis, following
standard test signals sue used
1. Step Input
2. Ramp Input
3. Parabolic Input
4. Impulse Input

1. Step Input
A Step input is a standard test input whose value
changes from one position to another position
in zero time. It is also known as position input.
Thus Step input represents an instantaneous
change in the input as shown in figure
where A is real constant
If A =1, then it is known as unit Step function and denoted by
The Laplace transform of Step input is
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Mathematically, Step input is represented as

2. Ramp Input
A ramp input is a standard input signal which
changes with time linearly. it is also known as
velocity input. Graphical representation is as
shown in figure
Mathematically., Ramp input is represented as
Where A is a real constant =slope of line
The Laplace transform of Ramp input is
If A= 1 . than the ramp signal is known as unit ramp signal
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Mathematically., Ramp input is represented as

3. Parabolic Input
A Parabolic input is a standard input signal in which
instantaneous value of a parabolic signal varies bas
square of the time from an initial value of zero at time t =
0. Graphically it is as shown in figure. It is also known as
acceleration input.
Mathematically., Parabolic input is represented as
where A is a real constant
The Laplace transform of Parabolic input is
If A= 1 . than the Parabolic signal is known as unit Parabolic signal
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4. Impulse Input
A impulse input is a standard input signal whose value
is zero everywhere except t = 0. It is generally called -
function.
Mathematically., impulse input is represented as
Graphically it is as shown in figure. (If area of the impulse is unit),
If A =1 it is known as unit impulse input.
Note that area of the impulse in nothing but magnitude of impulse
signal.
The Laplace transform of impulse signal is
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In general, the mathematical equation representing
standard test inputs are given is Table -1.
Table -1
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Table -1

1.1 ORDER AND TYPE OF THE CONTROL SYSTEM
1.1.1 ORDER OF THE SYSTEM
Consider a feedback control system as shown in figure
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1.1.2 Types OF SYSTEM
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1.1.2 Types OF SYSTEM : Example
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2. Unit step response and time domain
specifications of first order system.
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3. Unit step response and time domain specifications of second order system
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Formula list for solving problems on Response
of transfer function
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Problems-1 on Response of transfer function
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Problems-1 on Response of transfer function
Cont……
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MODULE -2
PART-B
PART-B
Dr. Mohammed Imran CONTROL ENGINEERING

Module-2
4. Steady state error, error constants.
4.1 Steady state error
 The steady state error e is the difference between the
input or desired value and the output or actual of a
closed loop system for a known input as response tends to
infinity.
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infinity.
 The steady state error is an important aspect of system
behaviour by which system accuracy can be measured.
 Definition : Steady state error is the difference between the
input value (desired value) and the output (actual response)
of the control system.

Module-2
Expression for error constant and steady state error:
Consider a closed loop linear control system as shown in
figure
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figure
Fig. Feedback control system

Module-2
From figure
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Module-2
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Module-2
4.2 Effect of standard test inputs on
Steady state error or error constants.
1. Unit setup input and positional error
1. Unit setup input and positional error
2. Unit ramp Input and velocity error
3. Unit parabolic input and acceleration
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Module-2
4.2 Effect of standard test inputs on Steady state error or error constants.
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Module-2
4.3 Types of Steady state error
1. Steady state error of Type-0 with
unity feed back
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unity feed back
unity feed back

1. Steady state error of Type-0 with unity feed back
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Module-2
Steady state error of Types.
 So steady state errors for different types of system when
subjected to standard test inputs are given Table 2
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Module-2
Problems on Error
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Control Engineering: Time Domain Analysis

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