CONTROL SYSTEMS

IMTHIAS AHAMED T.P. MTech, PhDProfessor

Head of the DepartmentDepartment of Electrical Engineering

T K M College of EngineeringKollam -691005

Kerala, India

Copyright © 2016 by Phasor BooksAll rights reserved

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Published by Phasor BooksNo.10 ChinoiseSree Narayana College JunctionKollam - 691001, KeralaPh : 0474-2715660, Mob : +91 9447452074

Dedicated to my mother Mrs. Sakeena Hashim

who taught me the art of communication

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.

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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 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.

CONTENTS

1 INTRODUCTION 11.1 Closed loop control systems 21.2 Open loop control systems 31.3 Automatic control system 4

1.4 CONTROL problems 61.5 Computer controlled systems 81.6 Study of control systems 8Summary 11

2 LAPLACE TRANSFORMS AND SOLUTION OF DIFFERENTIAL EQUATIONS 132.1 Introduction 142.2 Laplace transforms 14

2.2.1 Complex variables and functions 142.2.2 Definition 152.2.3 Laplace transform of simple functions 16

2.3 Properties of laplace transforms 192.4 Partial fraction expansion 37

2.4.1 Proper fractions 382.4.2 Improper fraction 42

2.5 Inverse laplace transform 452.6 Meaning of solution of differential equation 532.7 Solution of differential equation 55Summary 57

3 TRANSFER FUNCTION OF PHYSICAL SYSTEMS 653.1 Introduction 663.2 Transfer function 673.3 Steps for finding transfer function 68

3.4 Transfer function of electrical systems 693.5 Transfer function of mechanical systems 81 3.6 Transfer function of eletromechanical system 883.7 Analogous systems 953.8 Block diagram reduction techniques 103

3.8.1 Blocks in cascade (series) 1043. 8.2 Blocks in parallel 1053.8.3 Feedback loop 1073.8.4 Shifting of branching point 1103.8.4 Equivalent summing points 1143.8.5 Shifting of summing point (sp) 114

3.9 Signal flow graphs 1223.10 Masons gain formula 124Exercise 132

4 TIME RESPONSE 1414.1 Introduction 1424.2 Standard test signals 143

4.2.1 Step signal 1434.2.2 Ramp signal 1444.2.3 Parabolic signal 1444.2.4 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 system with steady state gain 1554.8 Impulse response of first order system 1574.9 Ramp response of first order system 1584.10 Step response of ii order system 161

4.11 Time domain specifications 1724.12 Derivation of time domain specifications 1734.13 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 187Summary 190Exercise 195

5 CONCEPT OF STABILITY 2035.1 Introduction 2045.2 Poles and zeros of a transfer function 2045.3 The s-plane 2075.4 Closed loop poles and characteristic equation 2095.5. Correlation between closed loop poles and step response 212 5.6 Relative stability 2215.7 Routh stability criterion 222 5.8 Relative stability analysis 232Summary 235Exercise 240

6 ROOT LOCUS 2436.1 Introduction 2446.2 Root locus of first order system 2446.3 Root locus of second order system 2476.4 Root locus of third and higher order system 248

6.4.1 Magnitude and angle condition 2486.5 Steps for drawing root locus of simple systems 2576.6 Determination of open loop gain on a specified point on the root locus 2636.7 root locus of systems with complex open loop poles 2776.8 Few examples of root locus 282

Summary 284Exercise 285

7 FREQUENCY DOMAIN ANALYSIS 2897.1 Introduction 2907.2 Frequency response of a rc network 2917.3 Sinusoidal transfer function 2947.4 Bode plot 295

7.4.1 Semilog sheet 2967.4.2 Bode plot of first order system 2977.4.3. Bode plot of a typical system 3007.4.4 Steps for drawing bode plot 3137.4.5 Errors in asymptotic approximations 328

7.5 Polar graph 3357.6 Log magnitude versus phase plot (nichols plot) 3437.7 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 3477.7.6 Correlation between frequency domain specification and time

domain specification 3507.8 Nyquist stability criterion 353

7.8.1 Mapping points 3557.8.2 Mapping of contours 3587.8.3 Mapping theorem 3597.8.4 Application of mapping / theorem for stability analysis 3617.8.5 Nyquist stability criterion (for system without open loop poles or

zeros on the imaginary axis) 3627.8.6 Nyquist stability criterion : general case 368

7.8.7 Nyquist stability criterion for openloop stable systems 3707.9 Gain margin 3807.10 Phase margin 3847.10 All pass systems 3877.11 Minimum phase systems and non-minimum phase systems 3887.12 Constant m and n circles 3907.13 Constant angle loci (n-circle) 3937.14 Experimental determination of transfer function 396Summary 401Exercises 409

8 COMPENSATOR DESIGN 4138.1 Introduction 4148.2 Lead compensator 415

8.2.1 Bode plot of a lead compensator 4168.2.2 Realization of lead compensator 417

8.3 Lag compensator 4298.3.1 Bode plot of lag compensator 4308.3.2 Realization of lag compensator 433

8.4 Lag – lead compensator 4358.4.1 Bode plot of lag-lead compensator 4378.4.2 Realization of lag-lead compensator 438

8.5 Design of lead compensator using bode plot 4448.5.1 Steps for designing lead compensator 447

8.6 Design of lag compensator using bode plot 4558.6.1 Steps for designing a lag compensator 459

8.7 Design of compensators in time domain 4648.7.1 Correlation between closed loops poles and time

domain specifications 4648.7.2 Dominant closed loop poles 4688.7.3 Effect of variation of gain on ess and mos 471

8.7.4 Effect of addition of zero on the root locus 4748.7.5 Angle condition 475

8.8 Design of lead compensator in time domain 4768.9 Design of lag compensators in time domain 4828.10 Proportional controller 4838.11 Integral controller 4858.12 Pi- controller 4868.13 Derivative controller 487 8.14 Pid – controller 488Summary 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