Modern Control Engineering · Modern Control Engineering 2nd Edition. Dr. K.P.Mohandas. This book contains information obtained from authentic and highly regarded sources.Reprinted

Modern ControlEngineering

Second Edition

Modern ControlE n g i n e e r i n g

Second Edition

Dr. K. P. Mohandas(Former Dean - PG Studies & Research at N I T Calicut)

Dean Academic & ProfessorDepartment of Electrical & Electronics Engineering

M E S College of Engineering, KuttippuramThrikkanapuram , PO Malappuram

Kerala State INDIA 679 573.

Sanguine Technical Publishers Bangalore.

2016

Modern Control Engineering 2nd Edition.Dr. K.P.Mohandas.This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

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PREFACE TO FIRST EDITION

Since the early days of teaching and learning of Control and Systems Theory in our Uni-versities, the subject has evolved considerably. In the sixties when the author graduatedin Engineering, Control Theory was taught as a part of a course in Applied or IndustrialElectronics whereas now, at least two courses in Control Theory are being offered in most ofthe undergraduate programmes. While several text books from foreign and Indian authorsand publishers are available for the first course in Control, there are very few text booksavailable for the second course in Control Theory which includes ‘something of everything’in Advanced Control Theory. It is expected that this book will fill this gap for such a course.The material discussed in the book has been tested in the class rooms by the author for atleast two decades and it is hoped that the book will be useful for students, teachers andpractising professionals as a ready reference. Every effort has been made to bring out thepractical and conceptual aspects of Control Systems Engineering rather than have a purelymathematical approach as usually given in many books. Personally, the author has taken uplearning Control and Systems theory as a challenge as early as 1968 when he graduated inElectrical Engineering and the book is the sum total of his understanding on the advancedtopics in Control.

The book is divided into ten chapters with the first chapter being a very brief introduc-tion to classical control theory. The second chapter gives the classical design techniquesusing Bode plots and root locus technique. Analysis of discrete time systems is presented inChapter 3 using z-transforms. Chapters 4, 5 and 6 deal with state space modelling, solutionof state equation and design of control systems using state space model with a glimpse onthe design of observers, and state feedback controllers. Chapters 7and 8 deal with nonlinearsystems, the former on phase plane analysis and the latter on describing function method.Even though both these methods were developed long time back, these methods are stilluseful to get some insight into the behaviour of nonlinear systems. Chapter 9 discusses indepth the Lyapunov’s method for stability analysis of systems and Chapter 10 is a briefintroduction to concepts and methods of optimal control. Several worked examples and asummary–‘points to remember’ have been added in each chapter. A set of multiple choice

v

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vi Preface

questions has been added at the end of the book useful for students in the preparation forobjective type tests and answers to additional numerical problems have been provided.MATLAB software package developed by mathworks Inc. is a very useful simulation soft-ware for the study of control systems. An introduction to the software package is given inAppendix.

It is sincerely hoped that the students and teachers in several universities in our coun-try will find this book useful as a text book for the second course in Control SystemsEngineering.

The author wishes to acknowledge the encouragement and facilities provided by FormerDirector in-charge Dr. B.N. Nagaraj, National Institute of Technology, Calicut and thepresent Director Dr. G.R.C. Reddy, National Institute of Technology, Calicut. The largenumber of students whom I have taught and the feedback from them is the inspirationbehind this effort and I am truly grateful to them.

Dr. K. P. Mohandas

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PREFACE TO SECOND EDITION

The author is thankful to the teachers and students in different parts of the country for thewarm welcome offered to the first edition of this book. It is gratifying to note that this bookhas been suggested as text/reference books for under graduate students in many universitiesfor the second course in Control Systems Theory. This meets exactly the purpose for whichthe book has been written. I understand that two reprints have been sold out in quick time.In spite of requests from the publishers for a third reprint, the author wanted to bring arevised edition as the next print, though a bit late.

The revision has followed these lines. Contents of Chapter 9 on Stability analysis andChapter 10 on Introduction to Optimal Control have been thoroughly revised. Popov cri-terion, circle criterion and ideas on input output stability have and been included in Chapter 9.In Chapter 10, new topics on classical optimal control, LQR and LQG problems and a com-parison of the latter two have been included. As a teacher I found that even some of theM Tech students were using this book for preliminary learning of nonlinear systems andoptimal control theory, the author thought that these will be useful additions.

Robust Control systems and preliminary ideas of Optimal Estimation theory have beenrecently included in the B Tech syllabus of few universities. These two chapters have beenadded in this edition. Hope this will help the students of such universities. The optimalestimation theory requires basic knowledge of stochastic processes and for ready referenceimportant results on random variables and stochastic processes have been introduced in thenew Appendix II of the book.

I hope this revision will make the book more useful for UG and PG students of engineer-ing. Now that even on–line editions are available for sale (sometimes even chapter-wise),subsequent revisions can be included on the online versions as early as possible, with a hardcopy revision as soon as a next reprint is required.

If anyone is making a critique of the book, kindly send me/publishers a copy of the sameby e-mail. We shall be very much obliged.

Dr. K. P. Mohandas

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CONTENTS

Chapter 1

Review of Classical Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Transfer Function Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3. Time Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1. Test Inputs for Time Response Analysis . . . . . . . . . . . . . . . . . . 3

1.4. Performance Specifications in Time Domain . . . . . . . . . . . . . . . . . . . . 7

1.5. Time Response of First Order Systems . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6. Time Response of Second Order Systems . . . . . . . . . . . . . . . . . . . . . . 11

1.7. Response of Higher Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8. Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.8.1. Root Locus Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.8.2. Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8.3. Frequency Domain Performance Measures . . . . . . . . . . . . . . . . 22

1.8.4. Determination of Frequency Domain Measures . . . . . . . . . . . . . 25

1.9. Correlation Between Time Domain and Frequency Domain Specifications 27

1.9.1. Peak Resonance, Resonant Frequency and Bandwidth in terms ofζ and ωn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.9.2. Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.10. Stability of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.10.1. Bounded Input Bounded Output Stability . . . . . . . . . . . . . . . . . 29

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1.10.2. Characteristic Equation Roots and Impulse Response . . . . . . . . . 29

1.10.3. Stability from Polar Plot and Bode Plot . . . . . . . . . . . . . . . . . . 29

1.10.4. Determination of Stability—Routh Hurwitz Criterion . . . . . . . . . 32

Chapter 2

Conventional Controllers and Classical Design . . . . . . . . . . . . . . . . . . . . . . 35

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2. Types of Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3. Compensating Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.1. Phase Lag Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.2. Lead Lag Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4. Design of Lead Compensator Using Bode Plot . . . . . . . . . . . . . . . . . . 46

2.5. Design of Lead Compensator Using Root Locus . . . . . . . . . . . . . . . . . 53

2.6. Design of Lag Compensator Using Bode Plot . . . . . . . . . . . . . . . . . . . 58

2.6.1. Lag Compensator Design Using Bode Plot . . . . . . . . . . . . . . . . 58

2.7. Design of Lag Compensator Using Root Locus . . . . . . . . . . . . . . . . . . 64

2.8. System Design Using PI Controllers . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.9. Design of Lead-Lag Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.9.1. Lead-Lag Design by Bode Plot Method . . . . . . . . . . . . . . . . . . 71

2.9.2. Lead-Lag Compensation Using Root Locus . . . . . . . . . . . . . . . 74

2.10. PID Controller Design and Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.10.1. Ziegler-Nichols Rules for Tuning PID Controllers . . . . . . . . . . . 77

2.11. Feedback Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.12. Comparison of Different Types of Controllers . . . . . . . . . . . . . . . . . . . 84

2.13. Additional Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.14. Points to Remember . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.15. Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Chapter 3

Discrete Data Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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3.2. Basic Elements of a Discrete Data System . . . . . . . . . . . . . . . . . . . . . 103

3.3. Advantages of Discrete Data Systems . . . . . . . . . . . . . . . . . . . . . . . . 105

3.4. Different Types of Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . 106

3.4.1. Pulse Amplitude Modulated Systems . . . . . . . . . . . . . . . . . . . . 106

3.4.2. Pulse Width Modulated Systems . . . . . . . . . . . . . . . . . . . . . . . 106

3.4.3. Pulse Frequency Modulated (PFM) Systems . . . . . . . . . . . . . . . 107

3.5. Examples of Discrete Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.5.1. Stepper Motor Control of Hard Disk Drives . . . . . . . . . . . . . . . 109

3.5.2. A Simplified Single-Axis Autopilot Control System . . . . . . . . . . 109

3.5.3. A Digital Computer Controlled Rolling Mill Regulating System . 111

3.5.4. Computer Control of Turbine and Generator . . . . . . . . . . . . . . . 111

3.6. Sampling Process and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.6.1. Analysis of the Output of an Ideal Sampler in Time Domain . . . . 112

3.6.2. Analysis of the Output of a Practical Sampler in Frequency Domain 116

3.7. Sampling Theorem and Significance . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.8. Data Reconstruction and Hold Circuits . . . . . . . . . . . . . . . . . . . . . . . . 122

3.8.1. Zero Order Hold and Its Frequency Response . . . . . . . . . . . . . . 123

3.8.2. First Order Hold and Data Reconstruction . . . . . . . . . . . . . . . . 125

3.9. Mathematical Analysis of Discrete Data Systems . . . . . . . . . . . . . . . . . 128

3.9.1. z-Transforms—Basic Definition . . . . . . . . . . . . . . . . . . . . . . . 128

3.9.2. z-Transform Directly from Laplace Transform . . . . . . . . . . . . . 130

3.9.3. Example for Evaluation of z-Transforms . . . . . . . . . . . . . . . . . 131

3.9.4. Properties of z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.9.5. Inverse z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.10. Solution of Difference Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.11. Pulse Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.12. Block Diagram Reduction for Pulse Transfer Functions . . . . . . . . . . . . 146

3.12.1. Cascaded Elements with Sampler in Between . . . . . . . . . . . . . . 146

3.12.2. Cascaded Elements without Sampler in Between . . . . . . . . . . . . 147

3.12.3. Closed Loop Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . 148

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3.12.4. Zero-Order Hold and G(s) in Cascade . . . . . . . . . . . . . . . . . . . 152

3.13. Time Response Analysis of Discrete Time Systems . . . . . . . . . . . . . . . 154

3.13.1. Time Response of Open Loop Systems . . . . . . . . . . . . . . . . . . 154

3.13.2. Response of Closed Loop Systems . . . . . . . . . . . . . . . . . . . . . 155

3.14. Mapping Properties of Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3.15. Stability of Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3.15.1. Routh-Hurwitz Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3.15.2. Jury’s Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

3.16. Response of Discrete Data Systems Between Sampling Instants . . . . . . . 165




Chapter 4

State Space Analysis of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.1. Limitations of Classical Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.2. Concept of State, State Space and State Variables . . . . . . . . . . . . . . . . . 186

4.3. State Model for Typical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.3.1. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.3.2. Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4.4. State Model of Linear Systems from Differential Equations . . . . . . . . . . 193

4.5. State Variable Diagram and Block Diagram Representation of State Models 196

4.6. State Space Model for Physical Systems . . . . . . . . . . . . . . . . . . . . . . . 198

4.6.1. Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

4.6.2. Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

4.6.3. Electro-Mechanical Systems—DC Motors . . . . . . . . . . . . . . . . 202

4.7. State Space Model from Transfer Functions . . . . . . . . . . . . . . . . . . . . 205

4.7.1. Transfer Functions without Numerator Dynamics . . . . . . . . . . . 205

4.7.2. Transfer Function with Numerator Dynamics . . . . . . . . . . . . . . 207

4.7.3. Series Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

4.7.4. Parallel Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

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4.8. State Model for a Multi-Input Multi-Output System from Block Diagrams 216

4.9. Non-Uniqueness of State Space Model . . . . . . . . . . . . . . . . . . . . . . . . 217

4.10. Transfer Function from State Model . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.11. Canonical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

4.11.1. Phase Variable Form or Controllable Canonical Model . . . . . . . . 220

4.11.2. Observable Canonical Model . . . . . . . . . . . . . . . . . . . . . . . . . 221

4.11.3. Diagonal Canonical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

4.11.4. Jordan Canonical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

4.12. Diagonalisation of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

4.13. State Variable Description of Discrete Time Systems . . . . . . . . . . . . . . 224




Chapter 5

Time Domain Analysis in State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.1. Solution of Time Invariant State Equation . . . . . . . . . . . . . . . . . . . . . . 245

5.1.1. Scalar Differential Equation—Homogeneous Case . . . . . . . . . . . 245

5.1.2. Vector Matrix Differential Equation—Homogeneous Case . . . . . 246

5.1.3. Laplace Transforms Method . . . . . . . . . . . . . . . . . . . . . . . . . . 247

5.2. State Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

5.3. Solution of State Equation with Input . . . . . . . . . . . . . . . . . . . . . . . . . 249

5.4. Laplace Transform Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

5.5. State Transition Matrix from Cayleigh–Hamilton Theorem . . . . . . . . . . 252

5.5.1. Cayleigh–Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 253

5.5.2. Computation of e At—Method 1 . . . . . . . . . . . . . . . . . . . . . . . . 253

5.5.3. Computation of e At—Method 2 . . . . . . . . . . . . . . . . . . . . . . . . 254

5.6. Solution of Linear Time Varying State Equation . . . . . . . . . . . . . . . . . 258

5.7. Solution of State Equation in Canonical Forms . . . . . . . . . . . . . . . . . . 260

5.7.1. Diagonal Canonical Form and State Transition Matrix . . . . . . . . 260

5.7.2. Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

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5.8. Solution of Time Invariant Discrete Time State Equation . . . . . . . . . . . . 262




Chapter 6

Design of State Feedback Controllers and Observers . . . . . . . . . . . . . . . . . . . 271

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

6.2. Controllability of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

6.2.1. Criterion for Controllability for Continuous Systems . . . . . . . . . 272

6.2.2. Controllability of Canonical Forms . . . . . . . . . . . . . . . . . . . . . 277

6.2.3. Output Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

6.2.4. Transformation to Controllable Canonical Form . . . . . . . . . . . . 282

6.2.5. Controllability of Discrete Time Systems . . . . . . . . . . . . . . . . . 283

6.3. Observability of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

6.3.1. Concept of Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

6.3.2. Criterion for Observability of a System . . . . . . . . . . . . . . . . . . 285

6.3.3. Observability Using Canonical Forms . . . . . . . . . . . . . . . . . . . 288

6.3.4. Transformation to Observable Canonical Form . . . . . . . . . . . . . 291

6.3.5. Observability of Discrete Time Systems . . . . . . . . . . . . . . . . . . 293

6.4. Significance of Controllability and Observability . . . . . . . . . . . . . . . . . 295

6.5. Transfer Function and Controllability/Observability . . . . . . . . . . . . . . . 295

6.6. State Feedback Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

6.6.1. Pole Placement for Plants Represented in Phase Variable Form . . 297

6.6.2. Determination of Feedback Gain K Using Ackerman’s Formula . 301

6.7. State Feedback Controller Design with Any State Model . . . . . . . . . . . . 305

6.8. Desired Location of Poles by Performance . . . . . . . . . . . . . . . . . . . . . 309

6.9. Design of Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

6.9.1. State Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

6.9.2. Separation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

6.9.3. Reduced Order Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

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6.10. Discrete Time Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319




Chapter 7

Nonlinear Systems and Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . 341

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

7.2. Characteristics of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 341

7.2.1. Superposition Principle is Not Valid . . . . . . . . . . . . . . . . . . . . 341

7.2.2. Multiple Equilibrium States and Equilibrium Zones . . . . . . . . . . 342

7.2.3. Limit Cycles or Sustained Oscillations . . . . . . . . . . . . . . . . . . . 342

7.2.4. Harmonics and Sub-Harmonic Oscillation in OutputUnder Sinusoidal Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

7.2.5. Jump Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

7.2.6. Frequency Entrainment or Synchronisation . . . . . . . . . . . . . . . . 343

7.3. Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

7.3.1. Linearisation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7.3.2. Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7.3.3. Describing Function Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 344

7.3.4. Lyapunov’s Method for Stability . . . . . . . . . . . . . . . . . . . . . . . 344

7.4. Classification of Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

7.4.1. Inherent or Intentional Nonlinearities . . . . . . . . . . . . . . . . . . . . 345

7.4.2. Static and Dynamic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 345

7.4.3. Functional and Piecewise Linear . . . . . . . . . . . . . . . . . . . . . . . 345

7.4.4. Memory Type (Multi-Valued) or Memory Less or Single-Valued . 345

7.5. Common Physical Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

7.5.1. Saturation or Limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

7.5.2. Dead Zone or Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

7.5.3. Different Types of Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

7.5.4. Different Types of Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

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7.5.5. Different Types of Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

7.5.6. Back Lash in Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

7.6. Linearisation of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 352

7.7. Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7.7.1. Phase Plane Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

7.7.2. Phase Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

7.7.3. Analytical Methods for the Construction of Phase Trajectories . . . 358

7.7.4. Graphical Method of Construction of PhaseTrajectory—Isoclines Method . . . . . . . . . . . . . . . . . . . . . . . . . 361

7.7.5. Delta Method of Construction of Phase Trajectory . . . . . . . . . . . 366

7.7.6. Pell’s Method of Construction of Phase Trajectory . . . . . . . . . . . 369

7.8. Evaluation of Time on Phase Trajectory . . . . . . . . . . . . . . . . . . . . . . . 372

7.9. Analysis and Classification of Singular Points . . . . . . . . . . . . . . . . . . . 374

7.10. Limit Cycles on Phase Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

7.11. Extension to Systems with Piecewise Constant Inputs . . . . . . . . . . . . . . 385




Chapter 8

Describing Function Analysis of Nonlinear Systems . . . . . . . . . . . . . . . . . . . 407

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

8.2. Basic Definition of Describing Function . . . . . . . . . . . . . . . . . . . . . . . 407

8.3. Basis of Describing Function Analysis . . . . . . . . . . . . . . . . . . . . . . . . 409

8.4. Describing Function for Typical Nonlinearities . . . . . . . . . . . . . . . . . . 410

8.4.1. Describing Function for Ideal Relay . . . . . . . . . . . . . . . . . . . . 411

8.4.2. Relay with Dead Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

8.4.3. Simple Dead Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

8.4.4. Saturation or Limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

8.4.5. Relay with Hysteresis and Dead Zone . . . . . . . . . . . . . . . . . . . 417

8.4.6. Friction Controlled Backlash . . . . . . . . . . . . . . . . . . . . . . . . . 421

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8.5. Application of Describing Function . . . . . . . . . . . . . . . . . . . . . . . . . . 422

8.5.1. Closed Loop Stability Using Describing Function . . . . . . . . . . . 422

8.5.2. Stability of the Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 425

8.5.3. Accuracy of Describing Function Analysis . . . . . . . . . . . . . . . . 426

8.5.4. Relative Stability from Describing Function . . . . . . . . . . . . . . . 434

8.5.5. Closed Loop Frequency Response . . . . . . . . . . . . . . . . . . . . . . 434




Chapter 9

Stability of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

9.1. Concept of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

9.2. Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

9.3. Stability in the Small and Stability in the Large . . . . . . . . . . . . . . . . . . 451

9.4. Lyapunov’s Stability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

9.5. Local Linearisation and Stability in the Small . . . . . . . . . . . . . . . . . . . 454

9.6. Stability by the Method of Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . 457

9.6.1. First Method of Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

9.6.2. Concept of Lyapunov’s Stability Theorems . . . . . . . . . . . . . . . . 458

9.6.3. Sign Definiteness of Scalar Functions . . . . . . . . . . . . . . . . . . . 459

9.6.4. Lyapunov’s Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . 463

9.7. Lyapunovs Method for Linear Time Invariant Systems . . . . . . . . . . . . . 467

9.7.1. Stability of Linear Continuous Time Systems . . . . . . . . . . . . . . 467

9.7.2. Stability of Linear Discrete Time Systems . . . . . . . . . . . . . . . . 470

9.8. Stability of Nonlinear Systems by Methodof Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

9.9. Krasovskii’s Theorem on Lyapunov Function . . . . . . . . . . . . . . . . . . . 471

9.9.1. Variable Gradient Method of Generating aLyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

9.10. Application of Lyapunov Function to Estimate Transients . . . . . . . . . . . 478

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9.11. Absolute Stability and Popov’s Criterion . . . . . . . . . . . . . . . . . . . . . . 480

9.11.1. The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

9.11.2. Sector type nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

9.11.3. Absolute stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

9.11.4. Lure’ type system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

9.11.5. Aizerman’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

9.11.6. Kalman’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

9.11.7. Popov’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

9.11.8. Characteristics of the linear part of the plant . . . . . . . . . . . . . . 485

9.11.9. Restrictions on the Nonlinear Element . . . . . . . . . . . . . . . . . . 486

9.11.10. Fundamental Theorem of Popov . . . . . . . . . . . . . . . . . . . . . . 488

9.12. Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

9.12.1. Pole and Zero Transformation . . . . . . . . . . . . . . . . . . . . . . . . 490

9.12.2. Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

9.13. Input Output Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494




Chapter 10

Introduction to Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

10.2. Classical Control and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 512

10.2.1. Integral of the error (IE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

10.2.2. Integral of absolute error (IAE) . . . . . . . . . . . . . . . . . . . . . . 512

10.2.3. Integral of the squared error ( ISE) . . . . . . . . . . . . . . . . . . . . 512

10.2.4. Integral of the time absolute error (ITAE) . . . . . . . . . . . . . . . 512

10.2.5. Integral of time squared error (ITSE) . . . . . . . . . . . . . . . . . . 513

10.3. Formulation of the Optimal Control Problem . . . . . . . . . . . . . . . . . . . 517

10.3.1. Characteristics of the Plant . . . . . . . . . . . . . . . . . . . . . . . . . 518

10.3.2. Requirements on the Plant . . . . . . . . . . . . . . . . . . . . . . . . . 518

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10.4. Typical Optimal Control Performance Measures . . . . . . . . . . . . . . . . . 519

10.4.1. Minimum Time Control Problem . . . . . . . . . . . . . . . . . . . . . 519

10.4.2. Minimum Energy Problem . . . . . . . . . . . . . . . . . . . . . . . . . 519

10.4.3. Minimum Fuel Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

10.4.4. State Regulator Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

10.4.5. Output Regulator Problem . . . . . . . . . . . . . . . . . . . . . . . . . 521

10.4.6. Tracking Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

10.4.7. Choice of Performance Measure . . . . . . . . . . . . . . . . . . . . . 521

10.4.8. Nature of Information about the Plant Suppliedto the Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

10.5. State Regulator Problem - Continuous System . . . . . . . . . . . . . . . . . . 522

10.5.1. Continuous system – Matrix Ricatti equation . . . . . . . . . . . . . 522

10.5.2. General Remarks on the Solution of Regulator Problem . . . . . 527

10.5.3. Properties of LQR Designed System . . . . . . . . . . . . . . . . . . 528

10.6. State Regulator Problem – Discrete Time Case . . . . . . . . . . . . . . . . . . 530

10.7. LQR Steady State Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . 535

10.8. Optimal Control System Design Using Second Method of Lyapunov . . . 536

10.8.1. Parameter Optimisation via Second Method of Lyapunov . . . . 536

10.9. Optimal Observers – Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . 539

10.10. Linear Qudratic Gaussian (LQG) Problem . . . . . . . . . . . . . . . . . . . . . 541

10.10.1. LQG Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

10.10.2. Comparison LQR and LQG designs . . . . . . . . . . . . . . . . . . . 543

10.10.3. Critique on LQG Control . . . . . . . . . . . . . . . . . . . . . . . . . . 544

10.11. Additional Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544


10.13. Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

Chapter 11

Introduction to Robust Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

11.2. Robust Control and System Sensitivity . . . . . . . . . . . . . . . . . . . . . . . 556

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11.3. Sensitivity Of Systems To Parameter Variations . . . . . . . . . . . . . . . . . 558

11.3.1. System sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

11.3.2. Parameter sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

11.4. Analysis of Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

11.5. Stability of Systems with Parameter Uncertainty . . . . . . . . . . . . . . . . 565

11.6. Design of Robust Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 568

11.7. Design of Robust PID Controlled Systems . . . . . . . . . . . . . . . . . . . . 570

11.7.1. Robust Controller Design using ITAE performance index. . . . . 572

11.8. Internal Model Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

11.8.1. Principle of Internal Model Control . . . . . . . . . . . . . . . . . . . 578

11.8.2. Internal Model Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

11.9. Robust Internal Model Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

11.10. Additional Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585


11.12. Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

Chapter 12

Introduction to Optimal Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

12.2. Some Simple Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

12.2.1. Least Squares Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 601

12.2.2. Weighted Least Squares Estimation . . . . . . . . . . . . . . . . . . . 604

12.2.3. Recursive Least Squares Estimation . . . . . . . . . . . . . . . . . . . 605

12.3. Filtering, Prediction and Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . 607

12.3.1. Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

12.3.2. Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

12.3.3. Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

12.4. Optimal Estimation for Discrete Time Systems . . . . . . . . . . . . . . . . . 609

12.4.1. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

12.4.2. The Estimation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

12.4.3. Fundamental Theorem on Estimation . . . . . . . . . . . . . . . . . . 611

Modern Control Engineering

Publisher : Sanguine Publishers ISBN : 9789383506514 Author : Dr. K. P. Mohandas

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Modern Control Engineering · Modern Control Engineering 2nd Edition. Dr. K.P.Mohandas. This book contains information obtained from authentic and highly regarded sources.Reprinted