1. CONTROL SYSTEMS
IMTHIAS AHAMED T.P. MTech, PhD
Professor
Head of the Department
Department of Electrical Engineering
T K M College of Engineering
Kollam -691005
Kerala, India

2. Copyright © 2016 by Phasor Books
All rights reserved
Price :
Published by Phasor Books
No.10 Chinoise
Sree Narayana College Junction
Kollam - 691001, Kerala
Ph : 0474-2715660, Mob : +91 9447452074

3. Dedicated to my mother
Mrs. Sakeena Hashim
who taught me the art of communication

4. PREFACE
here are many books on control systems, some of them are very good books
from the teacher’s point of view, and some of them are very easy from the
student’s point of view. Most of the teachers follow one of these very good
books and universities prescribe these books in the syllabus. However most of
the students find these books very difficult to understand and they follow one
of the easy books. More often than not the so called easy books train only to solve
problems. Most of the students learn time response by by-hearting few formulas
without understanding what time response is. They learn to solve “root-locus
problems”, “Bode plot problems” and “Nyquist Stability problems” without
understanding the fundamentals. This results in frustration. The aim of this book is
to present fundamentals of control systems in an easy way and explain the
meaning of step response, root locus, Bode plot etc.
The so called very good books often intimidate the students. The reason why
students are put off is because of the mathematical contents. This problem is
tackled in three ways.
1. The basic mathematics tools required to understand the fundamentals of
control systems, differential equations and Laplace transforms, are explained
in chapter two of the book.
2. Other necessary mathematical concepts are given in a gradual manner when
and where it is required.
3. Necessary but complicated mathematical concepts are pushed to the end of
the chapter.
Another complaint about control system from students is: it is all theory. In this
book, students will be motivated to learn the theory. Moreover to give a “physical
feel”, many simple experiments like step response and frequency response of RC
network are included through out the text. Studying these experiments has an
added benefit as they are part of practical subjects in the higher semester labs. For
example, in Kerala University, students after studying Control Systems, have a
practical subject titled “Systems and Control Lab”. In other universities also
T

5. students do some control system experiments. Thus correlating theory and
practical paper will help students to get a physical feel for the subject.
Writing this book was a very long exercise. It started way back in 2008. On this long
journey many teachers and students have helped me. I would like to thank all of
them. First and foremost I would like to thank the authorities of T.K.M. College of
Engineering for giving all the support for this project. Many chapters of the book
were used to teach the course control systems in T.K.M. College of Engineering,
Kollam, Dofhar University, Salalah, Sultanate of Oman. I thank the students of both
these institutes for their valuable feedback. I would like to thank my Colleagues
Prof. Asha Ravindranath, Prof. B. Premlet, T. K. M. College of Engineering, Prof. M.
Shahin, Government College of Engineering, Kannur and Prof. Bindhu G. R, College
of Engineering, Trivandrum for giving valuable suggestions.
This book would not have materialized without the help of management and staff
of M/S Phasor Books, Kollam. Special thanks are due to Mr. Anoop George for
digitizing and formatting the manuscript. Anoop, it was very interesting to work
with you. Thank you Mr. Nizam for proving all the logistic support for this project.
For any work of this magnitude one needs a peaceful and happy “space”. Thank
you Shemy and Reha for providing this space and making our house a home.

6. CONTENTS
1 INTRODUCTION 1
1.1 Closed loop control systems 2
1.2 Open loop control systems 3
1.3 Automatic control system 4
1.4 CONTROL problems 6
1.5 Computer controlled systems 8
1.6 Study of control systems 8
Summary 11
2 LAPLACE TRANSFORMS AND SOLUTION OF DIFFERENTIAL
EQUATIONS 13
2.1 Introduction 14
2.2 Laplace transforms 14
2.2.1 Complex variables and functions 14
2.2.2 Definition 15
2.2.3 Laplace transform of simple functions 16
2.3 Properties of laplace transforms 19
2.4 Partial fraction expansion 37
2.4.1 Proper fractions 38
2.4.2 Improper fraction 42
2.5 Inverse laplace transform 45
2.6 Meaning of solution of differential equation 53
2.7 Solution of differential equation 55
Summary 57
3 TRANSFER FUNCTION OF PHYSICAL SYSTEMS 65
3.1 Introduction 66
3.2 Transfer function 67
3.3 Steps for finding transfer function 68

7. 3.4 Transfer function of electrical systems 69
3.5 Transfer function of mechanical systems 81
3.6 Transfer function of eletromechanical system 88
3.7 Analogous systems 95
3.8 Block diagram reduction techniques 103
3.8.1 Blocks in cascade (series) 104
3. 8.2 Blocks in parallel 105
3.8.3 Feedback loop 107
3.8.4 Shifting of branching point 110
3.8.4 Equivalent summing points 114
3.8.5 Shifting of summing point (sp) 114
3.9 Signal flow graphs 122
3.10 Masons gain formula 124
Exercise 132
4 TIME RESPONSE 141
4.1 Introduction 142
4.2 Standard test signals 143
4.2.1 Step signal 143
4.2.2 Ramp signal 144
4.2.3 Parabolic signal 144
4.2.4 Impulse signal 144
4.3 Time response of a typical first order system 146
4.4 Step response 147
4.5 Step response of a rc network: a practical example 152
4.6 Correlation between step responses obtained
Mathematically and experimentally 153
4.7 First order systemwith steady state gain 155
4.8 Impulse response of first order system 157
4.9 Ramp response of first order system 158
4.10 Step response of ii order system 161

8. 4.11 Time domain specifications 172
4.12 Derivation of time domain specifications 173
4.13 Step response of an rlc network: a practical example 179
4.14 Type number of a system 182
4.15 Steady state error 184
4.16 Relation between type number and steady state error 187
Summary 190
Exercise 195
5 CONCEPT OF STABILITY 203
5.1 Introduction 204
5.2 Poles and zeros of a transfer function 204
5.3 The s-plane 207
5.4 Closed loop poles and characteristic equation 209
5.5. Correlation between closed loop poles and step response 212
5.6 Relative stability 221
5.7 Routh stability criterion 222
5.8 Relative stability analysis 232
Summary 235
Exercise 240
6 ROOT LOCUS 243
6.1 Introduction 244
6.2 Root locus of first order system 244
6.3 Root locus of second order system 247
6.4 Root locus of third and higher order system 248
6.4.1 Magnitude and angle condition 248
6.5 Steps for drawing root locus of simple systems 257
6.6 Determination of open loop gain on a specified point on the root locus 263
6.7 root locus of systems with complex open loop poles 277
6.8 Few examples of root locus 282

9. Summary 284
Exercise 285
7 FREQUENCY DOMAIN ANALYSIS 289
7.1 Introduction 290
7.2 Frequency response of a rc network 291
7.3 Sinusoidal transfer function 294
7.4 Bode plot 295
7.4.1 Semilog sheet 296
7.4.2 Bode plot of first order system 297
7.4.3. Bode plot of a typical system 300
7.4.4 Steps for drawing bode plot 313
7.4.5 Errors in asymptotic approximations 328
7.5 Polar graph335
7.6 Log magnitude versus phase plot (nichols plot) 343
7.7 Frequency domain specifications 345
7.7.1 Resonant frequency, r 346
7.7.2 Resonant peak, mr 346
7.7.3 Cut-off frequency, c 347
7.7.4 Bandwidth, b 347
7.7.5 Derivation of expressions for mr, r and b 347
7.7.6 Correlation between frequency domain specification and time
domain specification 350
7.8 Nyquist stability criterion 353
7.8.1 Mapping points 355
7.8.2 Mapping of contours 358
7.8.3 Mapping theorem 359
7.8.4 Application of mapping / theoremfor stability analysis 361
7.8.5 Nyquist stability criterion (for systemwithout open loop poles or
zeros on the imaginary axis) 362
7.8.6 Nyquist stability criterion : general case 368

10. 7.8.7 Nyquist stability criterion for openloop stable systems 370
7.9 Gain margin 380
7.10 Phase margin 384
7.10 All pass systems 387
7.11 Minimum phase systems and non-minimumphase systems 388
7.12 Constant mand n circles 390
7.13 Constant angle loci (n-circle) 393
7.14 Experimental determination of transfer function 396
Summary 401
Exercises 409
8 COMPENSATOR DESIGN 413
8.1 Introduction 414
8.2 Lead compensator 415
8.2.1 Bode plot of a lead compensator 416
8.2.2 Realization of lead compensator 417
8.3 Lag compensator 429
8.3.1 Bode plot of lag compensator 430
8.3.2 Realization of lag compensator 433
8.4 Lag – lead compensator 435
8.4.1 Bode plot of lag-lead compensator 437
8.4.2 Realization of lag-lead compensator 438
8.5 Design of lead compensator using bode plot 444
8.5.1 Steps for designing lead compensator 447
8.6 Design of lag compensator using bode plot 455
8.6.1 Steps for designing a lag compensator 459
8.7 Design of compensators in time domain 464
8.7.1 Correlation between closed loops poles and time
domain specifications 464
8.7.2 Dominant closed loop poles 468
8.7.3 Effect of variation of gain on ess and mos 471

11. 8.7.4 Effect of addition of zero on the root locus 474
8.7.5 Angle condition 475
8.8 Design of lead compensator in time domain 476
8.9 Design of lag compensators in time domain 482
8.10 Proportional controller 483
8.11 Integral controller 485
8.12 Pi- controller 486
8.13 Derivative controller 487
8.14 Pid – controller 488
Summary 489
9 CONTROL SYSTEM COMPONENTS 497
9.1 Introduction 498
9.2 Actuators 499
9.2.1 DC servo motor 499
9.2.2 AC servomotor 504
9.2.3 Stepper motor 509
9.3 Sensors 515
9.3.1 Potentiometers 515
9.3.2 Synchro 516
9.3.3 Tachogenerators 519