AdvancedControl Theory

ThirdEdition

for BE, BTech, ME, MTech Courses

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9 789389 396294

ThirdEdition

• Different types of fonts for text, proof and solved problems for better understanding

• Exercise at the end of each chapter for self-practice• Bode plot, polar plot and root locus are presented in exact graph sheets with proper

scale

• Follows a cohesive approach to portray the basics

is designed for universities that teaches advance course in control systems as an elective in BE/BTech (EEE | E&I | ICE) and various branches of ME/MTech. It presents the topics in an easy to understand manner with thorough explanations and detailed illustrations, to make students understand the basic underlying concepts.

• Solutions to university questions for better scoring

Salient Features

• Step-by-step details to solved problems

This book is organized into 5 chapters and appendices. The conventional and modern design concepts of continuous and discrete time control systems are presented in a very easiest and elaborative manner. The analysis and design of nonlinear control systems are included with clear explanations. Throughout the book, carefully chosen examples are presented so that the reader will have a clear understanding of the concepts discussed.

• Clear explanations of concepts with appropriate illustrations

A Nagoor Kani BE, MS is a multifaceted personality with an efficient technical expertise and management skills. He obtained his BE in EEE from Thiagarajar College of Engineering, Madurai, and MS (Electronics and Control) through Distance Learning Program from BITS, Pilani. He started his career as a self-employed industrialist (1986–1989) and then changed his career to teaching in 1989. He worked as Lecturer in Dr MGR Engineering College (1989–1990) and as Assistant Professor in Satyabhama Engineering College (1990–1997). He started his own coaching center for BE students named Institute of Electrical Engineering which was renamed as RBA Tutorials in 2005. The well-known companies currently run by him are RBA Engineering (manufacturing laboratory equipment and microprocessor trainer kits), RBA Innovations (involved in developing projects for engineering students and industries), RBA Tutorials (conducting coaching classes for engineering and GATE students) and RBA Publications (publishing of engineering books). He is an eminent writer and has written thirteen engineering books.

ISBN: 978-93-89396-29-4

Advanced Control Theory

for BE, BTech, ME, MTech Courses

Third Edition

CBS Publishers & Distributors Pvt LtdNew Delhi • Bengaluru • Chennai • Kochi • Kolkata • Mumbai

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A Nagoor Kani

Advanced Control Theory

for BE, BTech, ME, MTech Courses

Third Edition

ISBN: 978-93-89396-29-4

© All rights reserved by CBS Publishers & Distributors Pvt Ltd

Third Edition: 2020Second Edition: 1999First Edition: 1998

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DisclaimerScience and technology are constantly changing fields. New research and experience broaden the scope of information and knowledge. The author has tried his best in giving information available to him while preparing the material for this book. Although all efforts have been made to ensure optimum accuracy of the material, yet it is quite possible some errors might have been left uncorrected. The publisher, the printer and the author will not be held responsible for any inadvertent errors or inaccuracies.

to

my beloved sister Ms A Yuhanitha Beevi

PREFACE

Advanced control theory primarily deals with modern concepts of control systems analysis and design. This text is designed for an advance study of control systems as an elective course for undergraduate and postgraduate engineering students and also for research scholars.

This book is organized with 5 chapters. The design of linear, nonlinear and discrete time control systems by conventional methods and the modern state space methods are presented in a very easiest and elaborative manner. Throughout the book, carefully chosen examples are presented so that the reader will have a clear understanding of the concepts discussed.

Chapter 1 presents the detailed conventional methods of linear control systems design using controllers and compensators in both frequency and time domain in order to meet the desired performance specifications. The design using lag, lead and lag-lead compensators, PI, PD and PID controllers are presented with appropriate examples.

Chapter 2 focuses on analysis and design of nonlinear control systems. The development of describing functions of various nonlinearities like dead-zone, saturation, hysteresis and backlash are presented with clear steps.

Chapter 3 elaborates on sampled data or discrete time control systems. The fundamental concepts of modeling and analysis of sampled data control systems are presented in this chapter. An introduction to Z-transform is included in this chapter, which is an important prerequirement for modeling and analysis of sampled data control systems. The stability of sampled data control systems is also presented with clear examples.

Chapter 4 deals with state space analysis of continuous time and discrete time control systems. State space representation and solution of state equations are given with explanation and examples.

Chapter 5 is concerned with controllability and observability of state models of control systems. The design of control systems using state feedback is also presented with illustrated design examples.

Several concepts and procedures are illustrated through simple examples rather than generalized formulation. Each chapter provides the foundation and practical implications of their own topic with more number of solved problems for better understanding.

Special care has been taken in presenting the concept, choosing examples and solved problems. I hope that the teaching and student community will welcome the book. The readers can feel free to convey their suggestions or criticism for further improvement of the book.

A Nagoor Kani

ACKNOWLEDGEMENTS

I express my heartfelt thanks to my wife Mrs C Gnanaparanjothi Nagoor Kani, and my sons N Bharath Raj alias Chandrakani Allaudeen and N Vikram Raj for their support, encouragement and cooperation they have extended to me throughout my career. I thank Ms T A Benazir, Manager, RBA Group, and all my office staff for their cooperation in carrying out my day-to-day activities.

I sincerely acknowledge the contributions of our technical editors, Ms C Mohana Priya and Ms S Saranya and Ms P Kanimozhi, for editing, proof-reading and type-setting of the manuscript and preparing the layout of the book.

My sincere thanks to all reviewers for their valuable suggestions and comments which helped me to explore the subject to a greater depth.

I am also grateful to Mr Satish K Jain, CMD, CBS Publishers & Distributors, for his keen interest in publishing this work in CBS banner. My sincere thanks to all team members of CBS Publishers & Distributors, for their concern and care in publishing this work.

Finally, a special note of appreciation is due to my sisters, brothers, relatives, friends, students and the entire teaching community, for their overwhelming support and encouragement to my writing.

A Nagoor Kani

CONTENTS

Preface vii

CHAPTER 1: Linear System Design 1.1–1.18

1.1 Introduction to design using compensators 1.1 1.2 Lag compensator 1.4 1.3 Lead compensator 1.29 1.4 Lag–Lead compensator 1.54 1.5 PI, PD and PID controllers 1.71 1.6 Feedback compensation 1.92 1.7 Short-answer questions 1.107 1.8 Exercises 1.115

CHAPTER 2: Nonlinear Systems 2.1–2.76

2.1 Introduction to nonlinear systems 2.1 2.2 Describing function 2.8 2.3 Describing function of dead-zone and saturation non-linearity 2.9 2.4 Describing function of saturation nonlinearity 2.13 2.5 Describing function of dead-zone nonlinearity 2.16 2.6 Describing function of relay with dead-zone and hysteresis 2.18 2.7 Describing function of backlash nonlinearity 2.22 2.8 Describing function analysis of nonlinear systems 2.26 2.9 Review of polar plot and Nichols plot 2.29 2.10 Phase plane and phase trajectories 2.45 2.11 Short-answer questions 2.64 2.12 Exercises 2.73

CHAPTER 3: Sampled Data Control Systems 3.1–3.78

3.1 Introduction 3.1 3.2 Sampling process 3.2 3.3 Analysis of sampling process in frequency domain 3.3 3.4 Reconstruciton of sampled signals using hold circuits 3.6 3.5 Discrete sequence (Discrete time signal) 3.6 3.6 Z-Transform 3.10 3.7 Linear discrete time systems 3.36 3.8 Transfer function of LDS system (Pulse Transfer Function) 3.38 3.9 Analysis of sampler and zero-order hold 3.43 3.10 Analysis of systems with impulse sampling 3.44 3.11 Analysis of sampled data control systems using Z-transform 3.49 3.12 The z- and s-domain relationship 3.58 3.13 Stability analysis of sampled data control systems 3.59 3.14 Short-answer questions 3.68 3.15 Exercises 3.76

xii Advanced Control Theory

CHAPTER 4: State Space Analysis 4.1–4.82

4.1 Introduction 4.1 4.2 State space formulation 4.1 4.3 State model of linear system 4.2 4.4 State diagram 4.4 4.5 State space representation using physical variables 4.6 4.6 State space representation using phase variables 4.23 4.7 State space representation using canonical variables 4.34 4.8 Solution of state equations 4.42 4.9 State space representation of discrete time systems 4.54 4.10 Short-answer questions 4.69 4.11 Exercises 4.79

CHAPTER 5: Analysis and Design of Control Systems in State Space 5.1–5.80

5.1 Definitions involving matrices 5.1 5.2 Eigenvalues and eigenvectors 5.2 5.3 Similarity transformation 5.3 5.4 Cayley–Hamilton theorem 5.5 5.5 Tranformation of state model 5.14 5.6 Concept of controllability and observability 5.24 5.7 Controllable phase variable form of state model 5.34 5.8 Control system design via pole placement by state feedback 5.41 5.9 Observable phase variables form of state model 5.56 5.10 State observers 5.60 5.11 Short-answer questions 5.68 5.12 Exercises 5.77

APPENDICES A.1–A.26

Appendix I: Bode Polt A.1–A.13Appendix II: Root Locus A.14–A.26

Index 1.1 – 1.4