Introduction to Control Engineering Modeling, Analysis and Design by Ajit K. Mandal

This pageintentionally left

blank

Copyright 2006 New Age International (P) Ltd., PublishersPublished by New Age International (P) Ltd., Publishers

All rights reserved.No part of this ebook may be reproduced in any form, by photostat, microfilm,xerography, or any other means, or incorporated into any information retrievalsystem, electronic or mechanical, without the written permission of the publisher.All inquiries should be emailed to [email protected]

ISBN : 978-81-224-2414-0

PUBLISHING FOR ONE WORLD

NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS4835/24, Ansari Road, Daryaganj, New Delhi - 110002Visit us at www.newagepublishers.com

Dedicated to:Dedicated to:Dedicated to:Dedicated to:Dedicated to:

the memory of mthe memory of mthe memory of mthe memory of mthe memory of my parentsy parentsy parentsy parentsy parents

K. L. MandalK. L. MandalK. L. MandalK. L. MandalK. L. Mandalandandandandand

Rohini MandalRohini MandalRohini MandalRohini MandalRohini Mandalin grateful reverence and apin grateful reverence and apin grateful reverence and apin grateful reverence and apin grateful reverence and appreciationpreciationpreciationpreciationpreciation


blank

vii

PrefaceControl engineering is a very important subject to warrant its inclusion as a core course in theengineering program of studies in universities throughout the world. The subject ismultidisciplinary in nature since it deals with dynamic systems drawn from the disciplines ofelectrical, electronics, chemical, mechanical, aerospace and instrumentation engineering. Thecommon binding thread among all these divergent disciplines is a mathematical model in theform of differential or difference equations or linguistic models. Once a model is prepared todescribe the dynamics of the system, there is little to distinguish one from the other and theanalysis depends solely on characteristics like linearity or nonlinearity, stationary or timevarying, statistical or deterministic nature of the system. The subject has a strong mathematicalfoundation and mathematics being a universal language; it can deal with the subject ofinterdisciplinary nature in a unified manner.

Even though the subject has strong mathematical foundation, emphasis throughout thetext is not on mathematical rigour or formal derivation (unless they contribute to understand-ing the concept), but instead, on the methods of application associated with the analysis anddesign of feedback system. The text is written from the engineers point of view to explain thebasic concepts involved in feedback control theory. The material in the text has been organizedfor gradual and sequential development of control theory starting with a statement of the taskof a control engineer at the very outset.

The book is intended for an introductory undergraduate course in control systems forengineering students. The numerous problems and examples have been drawn from the disci-plines of electrical, electronics, chemical, mechanical, and aerospace engineering. This will helpstudents of one discipline with the opportunity to see beyond their own field of study and therebybroaden their perceptual horizon. This will enable them to appreciate the applicability of con-trol system theory to many facets of life like the biological, economic, and ecological controlsystems.

This text presents a comprehensive analysis and design of continuous-time control sys-tems and includes more than introductory material for discrete systems with adequate guide-lines to extend the results derived in connection with continuous-time systems. The prerequi-site for the reader is some elementary knowledge of differential equations, vector-matrix analy-sis and mechanics.

Numerous solved problems are provided throughout the book. Each chapter is followedby review problems with adequate hints to test the readers ability to apply the theory involved.Transfer function and state variable models of typical components and subsystems have beenderived in the Appendix at the end of the book.

Most of the materials including solved and unsolved problems presented in the bookhave been class-tested in senior undergraduates and first year graduate level courses in the

field of control systems at the Electronics and Telecommunication Engineering Department,Jadavpur University.

The use of computer-aided design (CAD) tool is universal for practicing engineers andMATLAB is the most widely used CAD software package in universities through out the world.MATLAB scripts are provided so that students can learn to use it for calculations and analysisof control systems. Some representative MATLAB scripts used for solving problems are in-cluded at the end of each chapter whenever thought relevant. However, the student is encour-aged to compute simple answers by hand in order to judge that the computers output is soundand not garbage. Most of the graphical figures were generated using MATLAB and some rep-resentative scripts for those are also included in the book. We hope that this text will give thestudents a broader understanding of control system design and analysis and prepare them foran advanced course in control engineering.

In writing the book, attempt has been made to make most of the chapters self-contained.In the introductory chapter, we endeavored to present a glimpse of the typical applications ofcontrol systems that are very commonly used in industrial, domestic and military appliances.This is followed by an outline of the task that a control-engineering student is supposed toperform. We have reviewed, in the second chapter, the common mathematical tools used foranalysis and design of linear control systems. This is followed by the procedure for handling theblock diagrams and signal flow graphs containing the transfer functions of various componentsconstituting the overall system. In chapter 3, the concept of state variable representation alongwith the solution of state equations is discussed. The concept of controllability and observabil-ity are also introduced in this chapter along with the derivation of transfer function from statevariable representation. The specifications for transient state and steady state response of lin-ear systems have been discussed in chapter 4 along with the Bode technique for frequencydomain response of linear control systems. In chapter 5, the concept of stability has been intro-duced and Routh-Hurwitz technique along with the Direct method of Lyapunov have beenpresented. Frequency domain stability test by Nyquist criteria has been presented in chapter6. The root locus technique for continuous system has been discussed in chapter 7 and its exten-sion to discrete cases has been included. The design of compensators has been taken up inchapter 8. In chapter 9, we present the concept of pole assignment design along with the stateestimation. In chapter 10, we consider the representation of digital control system and itssolution. In chapter 11, we present introductory material for optimal problem and present thesolution of linear regulator problem. Chapter12 introduces the concepts of fuzzy set and fuzzylogic needed to understand Fuzzy Logic Control Systems presented in chapter 13.

The reader must be familiar with the basic tools available for analyzing systems thatincorporate unwanted nonlinear components or deliberately introduced (relay) to improve sys-tem performance. Chapter 14 has been included to deal with nonlinear components and theiranalysis using MATLAB and SIMULINK through user defined s-functions. Finally, Chapter 15is concerned with the implementation of digital controllers on finite bit computer, which willbring out the problems associated with digital controllers. We have used MATLAB andSIMULINK tools for getting the solution of system dynamics and for rapid verification of con-troller designs. Some notes for using MATLAB script M-files and function M-files are includedat the end of the book.

The author is deeply indebted to a number of individuals who assisted in the preparationof the manuscript, although it is difficult to name everyone in this Preface. I would like to thankSaptarshi for his support and enthusiasm in seeing the text completed, Maya for the manyhours she spent reviewing and editing the text and proof reading. I would like to thank many

viii

people who have provided valuable support for this book project : Ms Basabi Banerjee for hereffort in writing equations in MSword in the initial draft of the manuscript, Mr. U. Nath fortyping major part of the manuscript and Mr. S. Seal for drawing some figures.

The author would like to express his appreciation to the former graduate students whohave solved many problems used in the book, with special appreciation to Ms SumitraMukhopadhyay, who provided feedback and offered helpful comments when reading a draftversion of the manuscript.

A. K. Mandal

ix


blank

xi

ContentsPreface vii

1.0 Introduction to Control Engineering 11.1 The Concept of Feedback and Closed Loop Control 21.2 Open-Loop Versus Closed-Loop Systems 21.3 Feedforward Control 71.4 Feedback Control in Nature 91.5 A Glimpse of the Areas where Feedback Control Systems have been

Employed by Man 101.6 Classification of Systems 10

1.6.1 Linear System 111.6.2 Time-Invariant System 11

1.7 Task of Control Engineers 131.8 Alternative Ways to Accomplish a Control Task 141.9 A Closer Look to the Control Task 15

1.9.1 Mathematical Modeling 161.9.2 Performance Objectives and Design Constraints 171.9.3 Controller Design 191.9.4 Performance Evaluation 19

2.0 The Laplace Transform 212.1 Complex Variables And Complex Functions 21

2.1.1 Complex Function 212.2 Laplace Transformation 22

2.2.1 Laplace Transform and Its Existence 232.3 Laplace Transform of Common Functions 23

2.3.1 Laplace Table 262.4 Properties of Laplace Transform 272.5 Inverse Laplace Transformation 31

2.5.1 Partial-Fraction Expansion Method 322.5.2 Partial-Fraction Expansion when F(s) has only Distinct Poles 322.5.3 Partial-Fraction Expansion of F(s) with Repeated Poles 34

xii CONTENTS

2.6 Concept of Transfer Function 352.7 Block Diagrams 36

2.7.1 Block Diagram Reduction 392.8 Signal Flow Graph Representation 42

2.8.1 Signal Flow Graphs 422.8.2 Properties of Signal Flow Graphs 432.8.3 Signal Flow Graph Algebra 432.8.4 Representation of Linear Systems by Signal Flow Graph 442.8.5 Masons Gain Formula 45

2.9 Vectors and Matrices 482.9.1 Minors, Cofactors and Adjoint of a Matrix 49

2.10 Inversion of a Nonsingular Matrix 512.11 Eigen Values and Eigen Vectors 522.12 Similarity Transformation 53

2.12.1 Diagonalization of Matrices 532.12.2 Jordan Blocks 54

2.13 Minimal Polynomial Function and Computation of Matrix Function UsingSylvesters Interpolation 55

MATLAB Scripts 57Review Exercise 58Problems 60

3.1 Introduction 653.2 System Representation in State-variable Form 663.3 Concepts of Controllability and Observability 693.4 Transfer Function from State-variable Representation 73

3.4.1 Computation of Resolvent Matrix from Signal Flow Graph 753.5 State Variable Representation from Transfer Function 773.6 Solution of State Equation and State Transition Matrix 81

3.6.1 Properties of the State Transition Matrix 82Review Exercise 83Problems 85

!

4.1 Time-Domain Performance of Control Systems 894.2 Typical Test Inputs 89

4.2.1 The Step-Function Input 894.2.2 The Ramp-Function Input 904.2.3 The Impulse-Function Input 904.2.4 The Parabolic-Function Input 90

4.3 Transient State and Steady State Response of Analog Control System 91

CONTENTS xiii

4.4 Performance Specification of Linear Systems in Time-Domain 924.4.1 Transient Response Specifications 92

4.5 Transient Response of a Prototype Second-order System 934.5.1 Locus of Roots for the Second Order Prototype System 94

4.5.1.1 Constant wn Locus 944.5.1.2 Constant Damping Ratio Line 944.5.1.3 Constant Settling Time 94

4.5.2 Transient Response with Constant wn and Variable 954.5.2.1 Step Input Response 95

4.6 Impulse Response of a Transfer Function 1004.7 The Steady-State Error 101

4.7.1 Steady-State Error Caused by Nonlinear Elements 1024.8 Steady-State Error of Linear Control Systems 102

4.8.1 The Type of Control Systems 1034.8.2 Steady-State Error of a System with a Step-Function Input 1044.8.3 Steady-State Error of A System with Ramp-Function Input 1054.8.4 Steady-State Error of A System with Parabolic-Function Input 106

4.9 Performance Indexes 1074.9.1 Integral of Squared Error (ISE) 1084.9.2 Integral of Time Multiplied Squared Error (ITSE) Criteria 1084.9.3 Integral of Absolute Error (IAE) Criteria 1084.9.4 Integral of Time Multiplied Absolute Error (ITAE) 1094.9.5 Quadratic Performance Index 110

4.10 Frequency Domain Response 1104.10.1 Frequency Response of Closed-Loop Systems 1114.10.2 Frequency-Domain Specifications 112

4.11 Frequency Domain Parameters of Prototype Second-Order System 1124.11.1 Peak Resonance and Resonant Frequency 1124.11.2 Bandwidth 114

4.12 Bode Diagrams 1154.12.1 Bode Plot 1154.12.2 Principal Factors of Transfer Function 116

4.13 Procedure for Manual Plotting of Bode Diagram 1214.14 Minimum Phase and Non-Minimum Phase Systems 122MATLAB Scripts 123Review Exercise 125Problems 126

5.1 The Concept of Stability 1315.2 The Routh-Hurwitz Stability Criterion 134

5.2.1 Relative Stability Analysis 1395.2.2 Control System Analysis Using Rouths Stability Criterion 139

5.3 Stability by the Direct Method of Lyapunov 140

xiv CONTENTS

5.3.1 Introduction to the Direct Method of Lyapunov 1405.3.2 System Representation 141

5.4 Stability by the Direct Method of Lyapunov 1415.4.1 Definitions of Stability 1435.4.2 Lyapunov Stability Theorems 144

5.5 Generation of Lyapunov Functions for Autonomous Systems 1475.5.1 Generation of Lyapunov Functions for Linear Systems 147

5.6 Estimation of Settling Time Using Lyapunov Functions 150MATLAB Scripts 153Review Exercise 154Problems 155

!!

6.1 Introduction 1596.1.1 Poles and Zeros of Open Loop and Closed Loop Systems 1596.1.2 Mapping Contour and the Principle of the Argument 160

6.2 The Nyquist Criterion 1656.2.1 The Nyquist Path 1666.2.2 The Nyquist Plot Using a Part of Nyquist Path 175

6.3 Nyquist Plot of Transfer Function with Time Delay 1766.4 Relative Stability: Gain Margin and Phase Margin 177

6.4.1 Analytical Expression for Phase Margin and Gain Marginof a Second Order Prototype 182

6.5 Gain-Phase Plot 1836.5.1 Constant Amplitude (M) and Constant Phase (N) Circle 183

6.6 Nichols Plot 1866.6.1 Linear System Response Using Graphical User Interface in

MATLAB 188MATLAB Scripts 188Review Exercise 189Problems 190

" !"

7.1 Correlation of System-Roots with Transient Response 1927.2 The Root Locus DiagramA Time Domain Design Tool 1927.3 Root Locus Technique 193

7.3.1 Properties of Root Loci 1947.4 Step by Step Procedure to Draw the Root Locus Diagram 2017.5 Root Locus Design Using Graphical Interface in MATLAB 2117.6 Root Locus Technique for Discrete Systems 2127.7 Sensitivity of the Root Locus 213MATLAB Scripts 213Review Exercise 214Problems 217

CONTENTS xv

8.1 Introduction 2188.2 Approaches to System Design 218

8.2.1 Structure of the Compensated System 2198.2.2 Cascade Compensation Networks 2208.2.3 Design Concept for Lag or Lead Compensator in Frequency-Domain

2248.2.4 Design Steps for Lag Compensator 2268.2.5 Design Steps for Lead Compensator 2268.2.6 Design Examples 226

8.3 Design of Compensator by Root Locus Technique 2388.3.1 Design of Phase-lead Compensator Using Root Locus Procedure 2388.3.2 Design of Phase-lag Compensator Using Root Locus Procedure 240

8.4 PID Controller 2418.4.1 Ziegler-Nichols Rules for Tuning PID Controllers 2428.4.2 First Method 2428.4.3 Second Method 243

8.5 Design of Compensators for Discrete Systems 2468.5.1 Design Steps for Lag Compensator 2488.5.2 Design Steps for Lead Compensator 248


! "

9.1 Pole Assignment Design and State Estimation 2559.1.1 Ackermans Formula 2569.1.2 Guidelines for Placement of Closed Loop System Poles 2589.1.3 Linear Quadratic Regulator Problem 258

9.2 State Estimation 2599.2.1 Sources of Error in State Estimation 2609.2.2 Computation of the Observer Parameters 261

9.3 Equivalent Frequency-Domain Compensator 2649.4 Combined Plant and Observer Dynamics of the Closed Loop System 2659.5 Incorporation of a Reference Input 2669.6 Reduced-Order Observer 2679.7 Some Guidelines for Selecting Closed Loop Poles in Pole Assignment

Design 270MATLAB Scripts 271Review Exercise 272Problems 275

"

10.0 Why We are Interested in Sampled Data Control System? 276

xvi CONTENTS

10.1 Advantage of Digital Control 27610.2 Disadvantages 27710.3 Representation of Sampled Process 27810.4 The Z-Transform 279

10.4.1 The Residue Method 28010.4.2 Some Useful Theorems 282

10.5 Inverse Z-Transforms 28610.5.1 Partial Fraction Method 28610.5.2 Residue Method 286

10.6 Block Diagram Algebra for Discrete Data System 28710.7 Limitations of the Z-Transformation Method 29210.8 Frequency Domain Analysis of Sampling Process 29210.9 Data Reconstruction 297

10.9.1 Zero Order Hold 29910.10 First Order Hold 30210.11 Discrete State Equation 30510.12 State Equations of Systems with Digital Components 30810.13 The Solution of Discrete State Equations 308

10.13.1 The Recursive Method 30810.14 Stability of Discrete Linear Systems 311

10.14.1 Jurys Stability Test 31310.15 Steady State Error for Discrete System 31610.16 State Feedback Design for Discrete Systems 321

10.16.1 Predictor Estimator 32110.16.2 Current Estimator 32210.16.3 Reduced-order Estimator for Discrete Systems 325

10.17 Provision for Reference Input 326MATLAB Scripts 327Review Exercise 329Problems 331

"

11.1 Introduction 33311.2 Optimal Control Problem 33311.3 Performance Index 33611.4 Calculus of Variations 336

11.4.1 Functions and Functionals 337A. Closeness of Functions 338B. Increment of a Functional 339C. The Variation of a Functional 339

11.4.2 The Fundamental Theorem of the Calculus of Variations 34211.4.3 Extrema of Functionals of a Single Function 343

11.4.3.1 Variational Problems and the Euler Equation 34311.4.3.2 Extrema of Functionals of n Functions 34611.4.3.3 Variable End Point Problems 347

CONTENTS xvii

11.4.4 Optimal Control Problem 35211.4.5 Pontryagins Minimum Principle 354

11.5 The LQ Problem 35711.5.1 The Hamilton-Jacobi Approach 35811.5.2 The Matrix Riccati Equation 35911.5.3 Finite Control Horizon 36011.5.4 Linear Regulator Design (Infinite-time Problem) 362

11.6 Optimal Controller for Discrete System 36311.6.1 Linear Digital Regulator Design (Infinite-time Problem) 365


## "

12.1 The Concept of Fuzzy Logic and Relevance of Fuzzy Control 37112.2 Industrial and Commercial Use of Fuzzy Logic-based Systems 37312.3 Fuzzy Modeling and Control 373

12.3.1 Advantages of Fuzzy Controller 37412.3.2 When to Use Fuzzy Control 37512.3.3 Potential Areas of Fuzzy Control 37512.3.4 Summary of Some Benefits of Fuzzy Logic and Fuzzy Logic Based

Control System 37612.3.5 When Not to Use Fuzzy Logic 377

12.4 Fuzzy Sets and Membership 37712.4.1 Introduction to Sets 37712.4.2 Classical Sets 37812.4.3 Fuzzy Sets 379

12.5 Basic Definitions of Fuzzy Sets and a Few Terminologies 37912.5.1 Commonly Used Fuzzy Set Terminologies 381

12.6 Set-Theoretic Operations 38412.6.1 Classical Operators on Fuzzy Sets 38412.6.2 Generalized Fuzzy Operators 386

12.6.2.1 Fuzzy Complement 38612.6.2.2 Fuzzy Union and Intersection 38712.6.2.3 Fuzzy Intersection: The T-Norm 38712.6.2.4 Fuzzy Union: The T-Conorm (or S-Norm) 388

12.7 MF Formulation and Parameterization 38812.7.1 MFs of One Dimension 389

12.8 From Numerical Variables to Linguistic Variables 39112.8.1 Term Sets of Linguistic Variables 393

12.9 Classical Relations and Fuzzy Relations 39412.9.1 Cartesian Product 39412.9.2 Crisp Relations 39412.9.3 Fuzzy Relations 39512.9.4 Operation on Fuzzy Relations 396

xviii CONTENTS

12.10 Extension Principle 40212.11 Logical Arguments and Propositions 403

12.11.1 Logical Arguments 40312.11.2 Modus Ponens 40712.11.3 Modus Tollens 40712.11.4 Hypothetical Syllogism 407

12.12 Interpretations of Fuzzy If-then Rules 40712.12.1 Fuzzy Relation Equations 409

12.13 Basic Principles of Approximate Reasoning 41012.13.1 Generalized Modus Ponens 41012.13.2 Generalized Modus Tollens 41012.13.4 Generalized Hypothetical Syllogism 411

12.14 Representation of a Set of Rules 41112.14.1 Approximate Reasoning with Multiple Conditional Rules 413

MATLAB Scripts 416Problems 417

## !

13.1 The Structure of Fuzzy Logic-based Controller 41913.1.1 Knowledge Base 42013.1.2 Rule Base 421

13.1.2.1 Choice of Sate Variables and Controller Variables 42113.1.3 Contents of Antecedent and Consequent of Rules 42213.1.4 Derivation of Production Rules 42213.1.5 Membership Assignment 42313.1.6 Cardinality of a Term Set 42313.1.7 Completeness of Rules 42313.1.8 Consistency of Rules 424

13.2 Inference Engine 42413.2.1 Special Cases of Fuzzy Singleton 426

13.3 Reasoning Types 42713.4 Fuzzification Module 428

13.4.1 Fuzzifier and Fuzzy Singleton 42813.5 Defuzzification Module 429

13.5.1 Defuzzifier 42913.5.2 Center of Area (or Center of Gravity) Defuzzifier 43013.5.3 Center Average Defuzzifier (or Weighted Average Method) 431

13.6 Design Consideration of Simple Fuzzy Controllers 43213.7 Design Parameters of General Fuzzy Controllers 43313.8 Examples of Fuzzy Control System Design: Inverted Pendulum 43413.9 Design of Fuzzy Logic Controller on Simulink and MATLAB Environment 441

13.9.1 Iterative Design Procedure of a PID Controller in MATLABEnvironment 441

13.9.2 Simulation of System Dynamics in Simulink for PID ControllerDesign 444

13.9.3 Simulation of System Dynamics in Simulink for Fuzzy Logic ControllerDesign 446

Problems 449

$%

!

14.1 Introduction 45314.1.1 Some Phenomena Peculiar to Nonlinear Systems 454

14.2 Approaches for Analysis of Nonlinear Systems: Linearization 45714.3 Describing Function Method 45814.4 Procedure for Computation of Describing Function 45914.5 Describing Function of Some Typical Nonlinear Devices 460

14.5.1 Describing Function of an Amplifying Device with Dead Zone andSaturation 460

14.5.2 Describing Function of a Device with Saturation but without anyDead Zone 463

14.5.3 Describing Function of a Relay with Dead Zone 46414.5.4 Describing Function of a Relay with Dead Zone and Hysteresis 46414.5.5 Describing Function of a Relay with Pure Hysteresis 46614.5.6 Describing Function of Backlash 466

14.6 Stability Analysis of an Autonomous Closed Loop System by DescribingFunction 468

14.7 Graphical Analysis of Nonlinear Systems by Phase-Plane Methods 47114.8 Phase-Plane Construction by the Isocline Method 47214.9 Pells Method of Phase-Trajectory Construction 47414.10 The Delta Method of Phase-Trajectory Construction 47614.11 Construction of Phase Trajectories for System with Forcing Functions 47714.12 Singular Points 47714.13 The Aizerman and Kalman Conjectures 481

14.13.1 Popovs Stability Criterion 48214.13.2 The Generalized Circle Criteria 48214.13.3 Simplified Circle Criteria 48314.13.4 Finding Sectors for Typical Nonlinearities 48414.13.5 S-function SIMULINK Solution of Nonlinear Equations 485

MATLAB Scripts 489Problems 492

!

15.1 Introduction 49315.2 Implementation of Controller Algorithm 493

15.2.1 Realization of Transfer Function 493

CONTENTS xix

15.2.2 Series or Direct Form 1 49415.2.3 Direct Form 2 (Canonical) 49515.2.4 Cascade Realization 49615.2.5 Parallel Realization 497

15.3 Effects of Finite Bit Size on Digital Controller Implementation 50015.3.1 Sign Magnitude Number System (SMNS) 500

15.3.1.1 Truncation Quantizer 50015.3.1.2 Round-off Quantizer 50015.3.1.3 Mean and Variance 50215.3.1.4 Dynamic Range of SMNS 50315.3.1.5 Overflow 503

15.3.2 Twos Complement Number System 50415.3.2.1 Truncation Operation 50415.3.2.2 Round-off Quantizer in Twos CNS 50515.3.2.3 Mean and Variance 50515.3.2.4 Dynamic Range for Twos CNS 50615.3.2.5 Overflow 506

15.4 Propagation of Quantization Noise Through the Control System 50715.5 Very High Sampling Frequency Increases Noise 50715.6 Propagation of ADC Errors and Multiplication Errors through the

Controller 50815.6.1 Propagated Multiplication Noise in Parallel Realization 50815.6.2 Propagated Multiplication Noise in Direct Form Realization 510

15.7 Coefficient Errors and Their Influence on Controller Dynamics 51115.7.1 Sensitivity of Variation of Coefficients of a Second Order

Controller 51115.8 Word Length in A/D Converters, Memory, Arithmetic Unit and D/A

Converters 51215.9 Quantization gives Rise to Nonlinear Behavior in Controller 51515.10 Avoiding the Overflow 517

15.10.1 Pole Zero Pairing 51715.10.2 Amplitude Scaling for Avoiding Overflow 51815.10.3 Design Guidelines 518

MATLAB Scripts 519Problems 520Appendex A 522Appendex B 579Appendex C 585Notes on MATLAB Use 589Bibliography 595Index 601

xx CONTENTS

1.0 INTRODUCTION TO CONTROL ENGINEERINGThe subject of control engineering is interdisciplinary in nature. It embraces all the disciplinesof engineering including Electronics, Computer Science, Electrical Engineering, MechanicalEngineering, Instrumentation Engineering, and Chemical Engineering or any amalgamationof these. If we are interested to control the position of a mechanical load automatically, wemay use an electrical motor to drive the load and a gearbox to connect the load to the motorshaft and an electronic amplifier to amplify the control signal. So we have to draw upon ourworking experience of electronic amplifier, the electrical motor along with the knowledge ofmechanical engineering as to how the motor can be connected with the help of a gearboxincluding the selection of the gear ratio. If we are interested to regulate the DC output voltageof a rectifier to be used for a computer system, then the entire control system consists purely ofelectrical and electronics components. There will be no moving parts and consequently theresponse of such systems to any deviations from the set value will be very fast compared to theresponse of an electromechanical system like a motor. There are situations where we have tocontrol the position of mechanical load that demands a very fast response, as in the case ofaircraft control system. We shall recommend a hydraulic motor in place of an electrical motorfor fast response, since the hydraulic motor has a bandwidth of the order of 70 radians/sec.

It is to be pointed out that the use of amplifiers is not the exclusive preserve of electronicengineers. Its use is widespread only because of the tremendous development in the disciplineof electronics engineering over the last 30 to 40 years. Amplifiers may be built utilizing theproperties of fluids resulting in hydraulic and pneumatic amplifiers. In petrochemical industrypneumatic amplifiers are a common choice while hydraulic amplifiers are widely used in aircraftcontrol systems and steel rolling mills where very large torque are needed to control the positionof a mechanical load. But whenever the measurement of any physical parameter of our interestis involved it should be converted to electrical signal at the first opportunity for subsequentamplification and processing by electronic devices. The unprecedented development of electronicdevices in the form of integrated circuits and computers over the last few decades coupled withthe tremendous progress made in signal processing techniques has made it extremely profitableto convert information about any physical parameter of our interest to electrical form fornecessary preprocessing.

Since we are interested to control a physical parameter of our interest like temperature,pressure, voltage, frequency, position, velocity, concentration, flow, pressure, we must havesuitable transducers to measure the variables and use them as suitable feedback signal for

1

CHAPTER

1Control Systems and the Task

of a Control Engineer

2 INTRODUCTION TO CONTROL ENGINEERING

proper control. Therefore, the knowledge of various transducers is essential to appreciate theintricacies of a practical control system.

In this text, we shall endeavor to introduce the basic principles of control systems startingfrom building mathematical models of elementary control systems and gradually working outthe control strategy in a practical control system. We have drawn examples from variousdisciplines of engineering so as to put students from various disciplines of engineering on afamiliar footing. We have assumed an elementary knowledge of differential calculus and theworking knowledge for the solution of differential equations and elementary algebra.

The Control systems may be used in open loop or in close loop configuration. We shallexplain these concepts by considering a schematic representation of a system, as shown in Fig. 1.1that maintains the liquid level in a tank by controlling the incoming flow rate of fluid. But,before we explain its operation we shall highlight the importance of the concept of feedbackfirst.

1.1 THE CONCEPT OF FEEDBACK AND CLOSED LOOP CONTROLThe concept of feedback is the single most important idea that governs the life of man inmodern societies. In its present state of sophistication, human life would have been miserablewithout machines and most of the machines used by man could not be made to function withreasonable reliability and accuracy without the utilization of feedback. Most of the machinesmeant for domestic and industrial applications, for entertainment, health-services and militaryscience, incorporate the concept of feedback. This concept is not exploited solely by man, it isalso prevalent in nature and man has learnt it, like many other things, from nature. Our verylife, for instance, is dependent on the utilization of feedback by nature.

Control systems may be classified as self-correcting type and non self-correcting type.The term self-correcting, as used here, refers to the ability of a system to monitor or measurea variable of interest and correct it automatically without the intervention of a human wheneverthe variable is outside acceptable limits. Systems that can perform such self-correcting actionare called feedback systems or closed-loop systems whereas non self-correcting type is referredto as open loop system.

When the variable that is being monitored and corrected is an objects physical positionsand the system involves mechanical movement is assigned a special name: a servo system.

1.2 OPEN-LOOP VERSUS CLOSED-LOOP SYSTEMSLet us illustrate the essential difference between an open-loop system and a closed-loop system.Consider a simple system for maintaining the liquid level in a tank to a constant value bycontrolling the incoming flow rate as in Fig. 1.1(a). Liquid enters the tank at the top and flowsout via the exit pipe at the bottom.

One way to attempt to maintain the proper level in the tank is to employ a humanoperator to adjust the manual valve so that the rate of liquid flow into the tank exactly balancesthe rate of liquid flow out of the tank when the liquid is at the desired level. It might require abit of trial and error for the correct valve setting, but eventually an intelligent operator can setthe proper valve opening. If the operator stands and watches the system for a while and observesthat the liquid level stays constant, s/he may conclude that the proper valve opening has beenset to maintain the correct level.

CONTROL SYSTEMS AND THE TASK OF A CONTROL ENGINEER 3

Chap

ter 1

Supplypipe

Manualvalve

Incomingliquid Desired level

Outgoing liquidExit pipe

(a)

Float

Exit pipe

Outgoing liquid

IncomingliquidControl

valve

Supplypipe

B

APivotpoint

(b)

Fig. 1.1 System for maintaining the proper liquid level in a tank(a) an open-loop system; it has no feedback and is not self-correcting.

(b) A closed-loop system; it has feedback and is self-correctingIn reality, however, there are numerous subtle changes that could occur to upset the

balance s/he has taken trouble to achieve. For example, the supply pressure on the upstreamside of the manual valve might increase for some reason. This would increase the input flowrate with no corresponding increase in output flow rate. The liquid level would start to riseand the tank would soon overflow. Of course, there would be some increase in output flow ratebecause of the increased pressure at the bottom of the tank when the level rises, but it wouldbe a chance in a million that this would exactly balance the new input flow rate. An increase insupply pressure is just one example of a disturbing force that would upset the liquid level inthe tank. There may be other disturbing forces that can upset the constant level. For instance,any temperature change would change the fluid viscosity and thereby changing the flow ratesor a change in a system restriction downstream of the exit pipe would also change the outputflow rate.

Now consider the setup in Fig. 1.1(b). If the liquid level falls a little too low, the floatmoves down, thereby opening the tapered valve to increase the inflow of liquid. If the liquidlevel rises a little too high, the float moves up, and the tapered valve closes a little to reducethe inflow of liquid. By proper construction and sizing of the valve and the mechanical linkagebetween float and valve, it would be possible to control the liquid level very close to the desiredset point. In this system the operating conditions may change causing the liquid level to deviatefrom the desired point in either direction but the system will tend to restore it to the set value.


Our discussion to this point has been with respect to the specific problem of controllingthe liquid level in a tank. However, in general, many different industrial control systems havecertain things in common. Irrespective of the exact nature of any control system, there arecertain relationships between the controlling mechanisms and the controlled variable that aresimilar. We try to illustrate these cause-effect relationships by drawing block diagrams of ourindustrial systems. Because of the similarity among different systems, we are able to devisegeneralized block diagrams that apply to all systems. Such a generalized block diagram of anopen loop system is shown in Fig. 1.2(a).

The block diagram is basically a cause and effect indicator, but it shows rather clearlythat for a given setting the value of the controlled variable cannot be reliably known in presenceof disturbances. Disturbances that happen to the process make their effects felt in the outputof the processthe controlled variable. Because the block diagram of Fig. 1.2(a) does not showany lines coming back around to make a circular path, or to close the loop, such a system iscalled an open-loop system. All open-loop systems are characterized by its inability to comparethe actual value of the controlled variable to the desired value and to take action based on thatcomparison. On the other hand, the system containing the float and tapered valve of Fig. 1.1(b)is capable of this comparison. The block diagram of the system of Fig. 1.1(b) is shown inFig. 1.2(b). It is found from the diagram that the setting and the value of the controlled variableare compared to each other in a comparator. The output of the comparator represents thedifference between the two values. The difference signal, called actuating signal, then feedsinto the controller allowing the controller to affect the process.

Set point Controller

Comparator+

Process

Disturbances

Controlledvariable

(b)

Disturbances

ControllerSet point Process Controlledvariable

(a)

Fig. 1.2 Block diagrams that show the cause-effect relationshipsbetween the different parts of the system

(a) for an open-loop system (b) for a closed-loop systemThe fact that the controlled variable comes back around to be compared with the setting

makes the block diagram look like a closed loop. A system that has this feature is called aclosed-loop system. All closed-loop systems are characterized by the ability to compare theactual value of the controlled variable to its desired value and automatically take action basedon that comparison. The comparator performs the mathematical operation of summation oftwo or more signals and is represented by a circle with appropriate signs.

For our example of liquid level control in Fig. 1.1(b), the setting represents the locationof the float in the tank. That is, the human operator selects the level that s/he desires bylocating the float at a certain height above the bottom of the tank. This setting could be altered


Chap

ter 1

by changing the length of rod A that connects the float to horizontal member B of the linkagein Fig. 1.1(b).

The comparator in the block diagram is the float itself together with the linkages A andB in our example. The float is constantly monitoring the actual liquid level, because it movesup or down according to that level. It is also comparing with the setting, which is the desiredliquid level, as explained above. If the liquid level and setting are not in agreement, the floatsends out a signal that depends on the magnitude and the polarity of the difference betweenthem.

That is, if the level is too low, the float causes horizontal member B in Fig. 1.1(b) to berotated counterclockwise; the amount of counterclockwise displacement of B depends on howlow the liquid is. If the liquid level is too high, the float causes member B to be displacedclockwise. Again, the amount of displacement depends on the difference between the settingand the controlled variable; in this case the difference means how much higher the liquid isthan the desired level.

Thus the float in the mechanical drawing corresponds to the comparator block in theblock diagram of Fig. 1.2(b). The controller in the block diagram is the tapered valve in theactual mechanical drawing.

In our particular example, there is a fairly clear correspondence between the physicalparts of the actual system and the blocks in the block diagram. In some systems, thecorrespondence is not so clear-cut. It may be difficult or impossible to say exactly which physicalparts comprise which blocks. One physical part may perform the function of two differentblocks, or it may perform the function of one block and a portion of the function of anotherblock. Because of the difficulty in stating an exact correspondence between the two systemrepresentations, we will not always attempt it for every system we study.

The main point to be realized here is that when the block diagram shows the value ofthe controlled variable being fed back and compared to the setting, the system is called aclosed-loop system. As stated before, such systems have the ability to automatically take actionto correct any difference between actual value and desired value, no matter why the differenceoccurred.

Based on this discussion, we can now formally define the concept of feedback control asfollows:

Definition 1 The feedback control is an operation, which, in the presence of disturbingforces, tends to reduce the difference between the actual state of a system and an arbitrarilyvaried desired state of the system and which does so on the basis of this difference.

In a particular system, the desired state may be constant or varying and the disturbingforces may be less prominent. A control system in which the desired state (consequently theset point) is constant, it is referred to as a regulator (example: a regulated power supply) andit is called a tracking system if the set point is continuously varying and the output is requiredto track the set point (example: RADAR antenna tracking an aircraft position).

Figure 1.3 shows another industrial process control system for controlling thetemperature of a pre-heated process fluid in a jacketed kettle. The temperature of the processfluid in the kettle is sensed by transducers like a thermocouple immersed in the process fluid.Thermocouple voltage, which is in tens of milli-volts, represents the fluid temperature and isamplified by an electronic DC amplifier Afb to produce a voltage Vb. The battery and thepotentiometer provide a reference (set point) voltage Vr, which is calibrated in appropriatetemperature scale. The input voltage Ve to the amplifier Ae is the difference of the referencevoltage and the feedback voltage Vb. The output voltage Vo of this amplifier is connected to the


solenoid coil that produces a force, fs, proportional to the current through the coil ia. Thesolenoid pulls the valve plug and the valve plug travels a distance x. The steam flow rate qthrough the valve is directly proportional with the valve-opening x. The temperature of theprocess fluid will be proportional to the valve opening.

MF, q2

Condensed water qs

Thermo-couple

Solenoid

R

L

M, B, K

V0

ia

VeVr

Rq

E

+ Vb

x

Stirrer

q

Temperature set point

Amplifier Ae

Steam Boiler

Amplifier Afb

Trap

Process flow F at temp q1

Valve

Fig. 1.3 Feedback control system for temperature control of a process fluid

If the flow rate, F of the process fluid increases, the temperature of the kettle will decreasesince the steam flow has not increased. The action of the control system is governed in such away as to increase the steam flow rate to the jacketed kettle, until temperature of the processfluid is equal to the desired value set by the reference potentiometer. Let us see the consequencesto the increased flow rate F of the process fluid. As the temperature in the kettle falls belowthe set point, the thermocouple output voltage and its amplified value Vb will decrease inmagnitude. Since Ve = Vr Vb and Vr is fixed in this case by the set point potentiometer, adecrease in Vb causes Ve to increase. Consequently the amplifier output voltage Vo increases,thereby increasing the travel x of the valve plug. As a result, the opening of the valve increases,resulting in an increase to the steam flow rate q, which increases the temperature of theprocess fluid.

If the flow rate of the process fluid decreases and the temperature inside the kettleincreases, Vb increases and Ve decreases. The output of the second amplifier Vo and the travelx of the valve plug decreases with a consequent reduction of the valve opening. The steamflow-rate q, therefore, decreases and results in a decrease of the kettle temperature until itequals to the set temperature.


Chap

ter 1

The open loop mode of temperature control for the process fluid is possible by removingthe sensing thermocouple along with the associated amplifier Afb from the system. The point Eof the amplifier Ae is returned to the ground and the potentiometer is set to the desiredtemperature. Since in this case Ve is equal to a constant value Vr set by the potentiometer, theoutput amplifier voltage Vo and travel x of the valve plug is fixed making the opening of thevalve fixed. The steam flow rate q is also fixed and the temperature of the kettle will be fixedif the process fluid flow-rate F is maintained to the constant value. This control arrangementis adequate if the flow rate of the process fluid is maintained to constant value. In actualpractice, however, the flow rate deviates from a constant value and the temperature fluctuateswith the fluctuation in F. The temperature will, in general, differ from the set point and anexperienced human operator will have to change the set point interactively. When thedisturbances to the system are absent, the final temperature of the kettle will be determinedby the experienced skill of the operator and accuracy of calibration of the temperature-settingpotentiometer R . But in presence of disturbances, which are always there, no amount of skilland experience of the operator will be adequate and the process product is likely to be adverselyaffected.

The closed loop control system in Fig. 1.3 may be represented by block diagrams asshown in Fig. 1.4. The desired value of the system state is converted to the reference input bythe transducers known as reference input elements.

Controlledsystem

Indirectlycontrolled

system

Disturbance

Manipulated variable

IndirectlycontrolledoutputControl

elements

Actuatingsignal

FBelements

Refinput

element

Ref. Input

Desiredvalue

(Set point) +

PrimaryF.B.

Controller of the systemControlledvariable

Fig. 1.4 Block diagram of a closed loop system

The controlled variable is converted to the feedback signal by the feedback element, alsoit is converted to the indirectly controlled variable, which is actual output of the entire feedbackcontrol system. The subtraction of feedback signal and reference input to obtain the actuatingsignal is indicated by a small circle with a sign to represent the arithmetic operation. Theparts of the diagram enclosed by the dotted line constitute the controller of the system.

1.3 FEEDFORWARD CONTROLWith reference to the temperature control system in Fig. 1.3, the effect of the disturbance inflow rate F is manifested by the change in temperature of the process fluid for a constant value


of steam input. But the effect of this disturbance cannot be immediately sensed in the outputtemperature change due to the large time constant in the thermal process. Besides, the effectof corrective action in the form of change of steam input will be felt at the processor output ata time when the temperature might have deviated from the set point by a large value. Thedisturbance in the flow rate depends on the level of the fluid in the tower (Fig. 1.5) and can beeasily measured by a flow meter. If we generate a corrective signal by an open loop controllerblock with a very small time constant and use it as input to the temperature controller (Gf(s) inFig. 1.5(b)), the transient response of the process temperature might be controlled within atighter limit. This type of control is known as feedforward control. The motivation behind thefeedforward control is to provide the corrective action for the disturbance, if it is measurable,not by using the delayed feedback signal at the output of the process but by using some othercontroller block with fast response. This strategy of using a faster control path will provide abetter transient response in the system output. As soon as a change in the flow rate in theinput fluid occurs, corrective action will be taken simultaneously, by adjusting the steam inputto the heat exchanger. This can be done by feeding both the signal from the flow meter and thesignal from the thermocouple to the temperature controller.

The feedforward controller block Gf(s) in part (b) of the Fig. 1.5, is found to be in theforward path from the disturbance input to the process output.

Feedforward control can minimize the transient error, but since feedforward control isopen loop control, there are limitations of its functional accuracy. Feedforward control will notcancel the effects of unmeasurable disturbances under normal operating condition. It is,therefore, necessary that a feedforward control system include a feedback loop as shown inFig. 1.5. The feedback control compensates for any imperfections in the functioning of the openloop feedforward control and takes care of any unmeasurable disturbances.

To condenser

Flow meter

Valve

Level controller

SteamValve

Temperature controllerTower

Thermo-couple

Stirrer

Heat exchanger

(a)

Fig. 1.5 (Contd.)


Chap

ter 1

G (s)1

G (s)c G(s)

G (s)n

+

Set point

Thermocouple

+ +

Temperature

ControllerHeat

exchanger

Flow meter

Disturbance(Change in flow)

(b)

Fig. 1.5 (a) Feedforward temperature control system (b) Block diagram

1.4 FEEDBACK CONTROL IN NATUREThe examples of feedback control are abundant in nature and it plays a very important role forcontrolling the activities of animals including man. The life of human being itself is sustainedby the action of feedback control in several forms. To cite an example, let us consider themaintenance of body temperature of warm-blooded animals to a constant value. The bodytemperature of human being is maintained around 38.5C in spite of the wide variation ofambient temperature from minus 20C to plus 50C.

The feedback control system is absent in the cold-blooded animals like lizards, so theirbody temperature varies with the ambient temperature. This severely restricts their activities;they go for hibernation in the winter when their body processes slow up to the extreme end.They become active again in the summer, and the warmer part of each day.

The body temperature of a human being is regulated to a constant value by achieving abalance in the process of generation and dissipation of heat. The primary source of heat in thecase of living creatures is the metabolic activity supplemented by muscle activity and the heatis dissipated from the body by radiation, conduction and convection.

When the ambient temperature suddenly rises, the skin senses this increase and sets ina series of operation for dissipating body-heat. Thermal radiation takes place directly from thebody surface to the surrounding. The body fluid and the blood flow take part in the convectionprocess. The blood vessels are constricted and the flow is diverted towards the outer surface ofthe body. The metabolic activity decreases, decreasing the heat generation. The respirationrate increases, so that more heat can be dissipated to the mass of air coming in contact withthe lung surface, Blood volume is increased with body fluid drawn into circulation resulting infurther cooling. The perspiration rate increases taking latent heat from the body surface therebydecreasing the body temperature. In a similar way when the outside temperature falls, extraheat is chemically generated by increased rate of metabolism. The heat generation is alsosupplemented by shivering and chattering of the teeth.

Other examples of feedback control in human body include: Hydrogen-ion concentrationin blood, concentration of sugar, fat, calcium, protein and salt in blood. Some of these variablesshould be closely controlled and if these were not so convulsions or coma and death would


result. Mans utilization of the feedback control is not as critical as in nature, except probably,in the space and undersea voyages. But utilization of feedback control is gaining importancewith the increased sophistication of modern life and soon the role of feedback control in shapingthe future of man will be as critical as is found in nature.

1.5 A GLIMPSE OF THE AREAS WHERE FEEDBACK CONTROL SYSTEMSHAVE BEEN EMPLOYED BY MAN

The following list gives a glimpse of the areas of human activities where the feedback controlsystem is extensively used.

Domestic applications:

Regulated voltage and frequency of electric power, thermostat control of refrigeratorsand electric iron, temperature and pressure control of hot water supply in cold countries,pressure of fuel gas, automatic volume and frequency control of television, camcorder andradio receivers, automatic focusing of digital cameras.

Transportation:

Speed control of the airplane engines with governors, control of engine pressure,instruments in the pilots cabin contain feedback loops, control of rudder and aileron, enginecowl flaps, instrument-landing system.

In sea going vessels:

Automatic steering devices, radar control, Hull control, boiler pressure and turbine speedcontrol, voltage control of its generators.

In automobiles:

Thermostatic cooling system, steering mechanisms, the gasoline gauge, and collisionavoidance, idle speed control, antiskid braking in the latest models and other instrumentshave feedback loops.

Scientific applications:

Measuring instruments, analog computers, electron microscope, cyclotron, x-raymachine, x-y plotters, space ships, moon-landing systems, remote tracking of satellites.

In industry:

Process regulators, process and oven regulators, steam and air pressure regulators,gasoline and steam engine governors, motor speed regulators, automatic machine tools suchas contour followers, the regulation of quantity, flow, liquid level, chemical concentration,light intensity, colour, electric voltage and current, recording or controlling almost anymeasurable quantity with suitable transducers.

Military applications:

Positioning of guns from 30 caliber machine guns in aircraft to mighty 16-inch gunsabroad battle ships, search lights, rockets, torpedoes, surface to air missiles, ground to air orair to air missiles, gun computers, and bombsights, and guided missiles.

1.6 CLASSIFICATION OF SYSTEMSFor convenience of description and mathematical analysis, systems are classified into differentcategories. They are classified according to the nature of inputs, number of inputs, number ofoutputs and some inherent characteristic of the system. Fig. 1.6 shows the block diagram


Chap

ter 1

representation of a system with only a single input u(t) and a single output y(t) (SISO). Similarly,we might have multi inputs and multi outputs (MIMO) systems, single input multi output(SIMO) and multi input and single output (MISO) systems.

Systemu(t) y(t)

Fig. 1.6 Block diagram representation of a system

In the SISO system in Fig. 1.6, it is assumed that it has no internal sources of energyand is at rest, prior to the instant at which the input u(t) is applied. The cause and effectrelationship may be represented in short as

y(t) = Lu (t) (1.1)where L is an operator that characterizes the system. It may be a function of u, y or time t andmay include operations like differentiation and integration or may be given in probabilisticlanguage.

A system is deterministic if the output y(t) is unique for a given input u(t). For probabilisticor non-deterministic system, the output y(t) is not unique but probabilistic, with a probabilityof occurrence for a given input. If the input to a deterministic system is statistical, like, noise,the output is also statistical in nature.

A system is said to be non-anticipative if the present output is not dependent on futureinputs. That is, the output y(to) at any instant to is solely determined by the characteristics ofthe system and the input u(t) for t > to. In particular, if u(t) = 0 for t > to ; then y(t) = 0.

An anticipatory system cannot be realized since it violates the normal cause and effectrelationship.

1.6.1 Linear SystemLet us assume that the outputs of the systems in Fig. 1.6 are y1(t) and y2(t) respectively

corresponding to the inputs u1(t) and u2(t). Let k1 and k2 represent two arbitrary constants.The system in Fig. 1.6 will be linear if the output response of the system satisfies the principleof homogeneity

L(k1u1(t)) = k1L(u1(t)) (1.2a)and the principle of additivity

L[k1u1(t) + k2u2(t)] = k1L[u1(t)] + k2L[u2(t)] (1.2b)

The principle in (1.2b) is also known as the principle of superposition.

In other words, a system is linear if its response y(t) is multiplied by k1 when its input ismultiplied by k1. Also the response follows the principle of superposition, that is, y(t) is givenby k1y1(t) + k2y2(t) when the input u(t) becomes k1u1(t) + k2u2(t) for all u1, u2, k1 and k2.

If the principle of homogeneity together with the principle of superposition holds goodfor a certain range of inputs u1 and u2, the system is linear in that range of inputs and non-linear beyond that range.

1.6.2 Time-Invariant SystemFor a time-invariant or fixed system, the output is not dependent on the instant at

which the input is applied. If the output at t is y(t) corresponding to an input u(t) then theoutput for a fixed system will be

L u(t ) = y(t ) (1.3)


A system, which is not time-invariant, is a time-varying one. A few examples of theabove definitions are considered below:

Example 1.1 A differentiator is characterized by

y(t) = ddt

u(t)

Here, the operator L = ddt

. Therefore, y1(t) = ddt

[k1u1(t)] = k1 ddt

u1(t)

andddt

[k1u1(t) + k2u2(t)] = k1 ddt

u1(t) + k2 ddt

u2(t)

Hence the system is linear. It is also realizable and time-invariant.

Example 1.2 A system is characterized by y(t) = [u(t)]2.In this case, the operator L is the squarer and since [k1u1(t)]

2 k1[u1(t)]2 and [k1u1(t) +

k2u2(t)]2 k1u1

2(t) + k2u22(t), the system is nonlinear.

It is realizable and time invariant.

Example 1.3 A system is characterized by the relationship

y(t) = t ddt

u(t)

It is linear since, y1(t) = t ddt

[k1u1(t)] = tk1 ddt

[u1(t)] = k1t ddt

[u1(t)]

and t ddt

[k1u1(t) + k2u2(t)] = k1t ddt

u1(t) + k2t ddt

u2(t)

It is realizable but time varying since t ddt

u(t ) (t ) du td t

( )( )

Example 1.4 A system is characterized by the relationship

y(t) = u(t) ddt

u t( )

The system is nonlinear since k1u1(t) ddt

k1u1(t) k1u1(t) ddt

u1(t)

and [k1u1(t) + k2u2(t)] ddt

[k1u1(t) + k2u2(t)] k1u1(t) ddt

u1(t) + k2u2(t) ddt

u2(t)

It is realizable and time-invariant.Systems are also classified based on the nature of signals present at all the different

points of the systems. Accordingly systems may be described as continuous or discrete. A signalis continuous if it is a function of a continuous independent variable t [see Fig. 1.7(a)]. Theabove definition of continuous signal is broader than the mathematical definition of continuousfunction. For example the signal f(t) in Fig. 1.7(a) is continuous in the interval t1 < t < t2, but itis not a continuous function of time in the same interval. A signal is discrete if it is a functionof time at discrete intervals [see Fig. 1.7(b)].

A system is continuous if the signals at all the points of the system are a continuousfunction of time and it will be referred to as a discrete system if the signal at any point of thesystem is discrete function of time.


Chap

ter 1

f(t)

t1 t2 t

(a) (b)t

f(t)

Fig. 1.7 Classification of systems based on nature of signals;(a) continuous signal and (b) discrete signal

1.7 TASK OF CONTROL ENGINEERSIn order to put the control engineers in proper perspective, we endeavor to present, at the veryoutset, the control-engineering problem together with the task to be performed by a controlengineer.

(1) Objective of the Control SystemA control system is required to do a job and is specified to the control engineer by the

user of the control system. The engineer is expected to assemble the various components andsub systems, based on the laws of the physical world, to perform the task. The quality ofperformance is normally specified in terms of some mathematical specifications, that the controlsystem is required to satisfy. The specifications may be the accuracy of controlled variable inthe steady state (the behavior of the system as time goes to infinity following a change in setpoint or disturbance) or it is concerned with the transient response- the way the control variableis reaching the steady state following some disturbance or change in set value.

(2) Control ProblemSince the control system is required to satisfy a performance specification expressed in

mathematical terms, the control engineer needs to solve a mathematical problem.

(3) System ModelingSince some mathematical problems are to be solved, a mathematical model of the

dynamics of the control system components and subsystems are to be formulated. Thedifferential equation and the state variable representation are very popular mathematicalmodels for describing the dynamics of a control system. The transfer function model is applicableif the system is linear. If the description of the system behavior is linguistic then fuzzy logicand fuzzy model of the system will be needed [1-3].

(4) System AnalysisOnce the mathematical model of the basic system is obtained, its analysis is performed

by using the existing mathematical tools available to the control engineer to study its behavior.The analysis may be carried out in the frequency domain or in time domain.

(5) Modification of the Control SystemIf the analysis reveals some sort of shortcoming in meeting one or more of the performance

specifications in the steady state and /or transient state, the basic system needs to be modifiedby incorporating some additional feedback or by incorporating compensator to modify the system


behavior. The lag-lead compensator and state feedback design method are widely used forimproving the system performance.

(6) Optimal ControlAmong a number of alternative design solutions, some control laws are superior to others

so far as the performance specifications are concerned. This leads to the problem of optimalcontrol, where among all the possible solutions, the solution that optimizes the performancespecification (like minimization of energy consumption, maximization of production rate orminimization of time and minimization of waste of material) should be chosen. Consider, forinstance, the problem of landing of a space vehicle on the surface of the moon from earth. Sincethe consumption of fuel is an important consideration for space journey, of the innumerabletrajectories from the earth to the surface of the moon, the one that will consume minimum fuelwill be chosen and the controller will be programmed for that trajectory. In Habers process ofmanufacturing Ammonia, the yield per unit time is dependent on the temperature of the reactionchamber. Control laws may be designed to maintain temperature of the reaction chamber suchthat the yield is maximum.

(7) Adaptive ControlIn some, control systems the process parameters change with time or environmental

conditions. The controller designed by assuming fixed system parameters fails to produceacceptable performance. The controller for such systems should adapt to the changes in theprocess such that the performance of the control system is not degraded in spite of the changesin the process. This gives rise to the problem of designing adaptive controller for a system. Forexample, the transfer functions of an aircraft changes with its level of flight and velocity, sothat the effectiveness of the pilots control stick will change with the flight conditions and thegain of the controller is to be adapted with the flight conditions [4]. In the space vehicle thefuel is an integral part of the mass of the vehicle, so the mass of the vehicle will change withtime. An Adaptive controller that adapts itself to the changes in mass is expected to performbetter. The design of adaptive controller, therefore, becomes an important issue in controlengineering problem.

(8) Fuzzy ControlWhen the system description and performance specification are given in linguistic terms,

the control of the system can better be handled by Fuzzy Logic setting down certain rulescharacterizing the system behavior and common sense logic (vide Chapter 12 and Chapter 13).

1.8 ALTERNATIVE WAYS TO ACCOMPLISH A CONTROL TASK(i) Process temperature control: Let us now consider the problem of controlling the

temperature of a process fluid in a jacketed kettle (Fig. 1.3). The problem can be solved in anumber of alternative ways depending on the form of energy available to the control engineer.

(a) The form of energy available is electricity: If the available source of energy is electricity,the temperature of the fluid inside the kettle may be controlled by manipulating the currentthrough a heater coil. The average power supplied to the heater coil can be manipulated byvarying the firing angle of a Triac or Silicon Controlled Rectifier (SCR with a suitable triggeringcircuit).

(b) The form of energy available is steam: The temperature of the fluid inside the kettlewill be proportional to the volume of steam coming into contact with the process fluid in thekettle. By connecting a solenoid-operated valve in the pipe through which steam flows, thevolume of steam flow per unit time may be manipulated by opening and closing of the valve.


Chap

ter 1

(c) The form of energy available is fuel oil: The temperature of an oil-fired furnace maybe controlled by controlling the fuel to air ratio as well as the flow rate of fuel oil. By sensingthe temperature of the chemical bath one can adjust the flow rate of fuel and its ratio to air bymeans of a solenoid operated valve to control the temperature of the furnace directly and thetemperature of the fluid inside the kettle indirectly.

(d) The solar energy as the power source: The solar energy, concentrated by a set offocusing lens after collecting it from reflectors, may be used to control the temperature of thefluid inside the kettle. The amount of solar energy may be regulated by controlling the positionof the reflectors as the position of the sun in the sky changes.

It is, therefore, apparent that the job of controlling the temperature of a fluid in a jacketedkettle may be accomplished in a number of alternative ways. Depending on the form of energyavailable, any of the above methods may be adopted for a given situation. The choice of aparticular method depends on many other factors like technical feasibility and economicviability. The important point that is to be emphasized here is that the control objective maybe realized by using a number of alternative methods. In each method, the components andsubsystems should be assembled, in a meaningful way, by utilizing the knowledge of the physicalworld.

(ii) Room temperature control: Let us consider the problem of controlling roomtemperature using a room air conditioner as another example. The compressor of the room airconditioner may be switched off if the room temperature is less than the set temperature andswitched on if the room temperature is higher than the set value. The compressor is kept onuntil the room temperature is equal to the set temperature. A thermostat switch senses thetemperature in the room and when the temperature of the room goes below the set point, thepower to the compressor is again switched off. The variation of ambient temperature outsidethe room and the escape of cool air due to opening of the door are the external disturbances tothe control system and the change in the number of occupants in the room is the loaddisturbance. The temperature of the room could have been controlled by a central airconditioning system, where the volume of cool air entering the room could have been controlledby opening and closing of a solenoid operated valve.

1.9 A CLOSER LOOK TO THE CONTROL TASKThe presence of reference input element is always implied in a system and with the directlycontrolled output variable taken as the system output, the block diagram of Fig. 1.4 is simplified

Controlelements

Processor

planty(t)

d(t)

u(t)e(t)r(t)

+

F B elements

Fig. 1.8 Block diagram of a basic control system

and shown as the basic control system in Fig. 1.8. The process (or plant) is the object to becontrolled. Its inputs are u(t), its outputs are y(t), and reference input is r(t). In the processfluid control problem, u(t) is the steam input to the jacketed kettle, y(t) is the temperature of


the process fluid and r(t) is the desired temperature specified by the user. The plant is thejacketed kettle containing the process fluid. The controller is the thermocouple, amplifiers andsolenoid valve (elements inside the dotted line in Fig. 1.4). In this section, we provide an overviewof the steps involved in the design of the controller shown in Fig. 1.8. Basically, these aremodeling, controller design, and performance evaluation.

1.9.1 Mathematical ModelingAfter the control engineer has interconnected the components, subsystems and actuators in ameaningful way to perform the control job, often one of the next tasks that the designerundertakes is the development of a mathematical model of the process to be controlled, inorder to gain understanding of the problem. There are only a few ways to actually generate themodel. We can use the laws of the physical world to write down a model (e.g., F = mf ). Anotherway is to perform system identification via the use of real plant data to produce a model ofthe system [5]. Sometimes a combined approach is used where we use physical laws to writedown a general differential equation that we believe represents the plant behavior, and thenwe perform experiments in the plant to determine certain model parameters or functions.

Often more than one mathematical model is produced. An actual model is one that isdeveloped to be as accurate as possible so that it can be used in simulation-based evaluation ofcontrol systems. It must be understood, therefore, that there is never a perfect mathematicalmodel for the plant. The mathematical model is abstraction and hence cannot perfectly representall possible dynamics of any physical process (e.g., certain noise characteristics or failureconditions).

Usually, control engineers keep in mind that they only need to use a model that isaccurate enough to be able to design a controller that works. However, they often need a veryaccurate model to test the controller in simulation before it is tested in an experimental setting.Lower-order design-models may be developed satisfying certain assumptions (e.g., linearity orthe inclusion of only certain forms of non-linearities) that will capture the essential plantbehavior. Indeed, it is quite an art and science to produce good low-order models that satisfythese constraints. Linear models such as the one in Equation (1.4) has been used extensivelyin the modern control theory for linear systems and is quite mature [vide Chapter 3].

( )x t = A x(t) + Bu(t)

y(t) = Cx(t) + Du(t) (1.4)

In the classic optimization problem of traveling sales representative, it is required to minimize thetotal distance traveled by considering various routes between a series of cities on a particular trip. Fora small number of cities, the problem is a trivial exercise in enumerating all the possibilities and choosingthe shortest route. But for 100 cities there are factorial 100 (or about 10200) possible routes to consider!No computers exist today that can solve this problem through a brute-force enumeration of all thepossible routes. However, an optimal solution with an accuracy within 1 percent of the exact solutionwill require two days of CPU time on a supercomputer which is about 40 times faster than a personalcomputer for finding the optimum path (i.e., minimum travel time) between 100,00 nodes in a travelnetwork. If the same problem is taken and the precision requirement is increased by a modest amountto value of 0.75 percent, the computing time approaches seven months! Now suppose we can live withan accuracy of 3.5 percent and increase the nodes in the network to 1000,000; the computing time forthis problem is only slightly more than three hours [6]. This remarkable reduction in cost (translatingtime to money) is due solely to the acceptance of a lesser degree of precision in the optimum solution.The big question is can humans live with a little less precision? The answer to this question depends onthe situation, but for the vast majority of problems we encounter daily, the answer is a resounding yes.


Chap

ter 1

In this case u is the m-dimensional input; x is the n-dimensional state; y is thep-dimensional output; and A, B, C and D are matrices of appropriate dimension. Such models,or transfer functions (G(s) = C(sI A)1 B + D where s is the Laplace variable), are appropriatefor use with frequency domain design techniques (e.g., Bode plot and Nyquist plots), the root-locus method, state-space methods, and so on. Sometimes it is assumed that the parameters ofthe linear model are constant but unknown, or can be perturbed from their nominal values,then techniques for robust control or adaptive control are developed [7-8].

Much of the current focus in control is on the development of controllers using nonlinearmodels of the plant of the form:

( )x t = f {x(t), u(t)} y(t) = g{x(t), u(t)} (1.5)

where the variables are defined as in the linear model and f and g are nonlinear function oftheir arguments. One form of the nonlinear model that has received significant attention is

( )x t = f(x(t)) + g(x(t)) u(t) (1.6)since it is possible to exploit the structure of this model to construct nonlinear controllers (e.g.,in feedback linearization or nonlinear adaptive control). Of particular interest with both thenonlinear models above is the case where f and g are not completely known and subsequentresearch focuses on robust control of nonlinear systems.

Discrete time versions of the above models are also used when a digital computer isused as a controller and stochastic effects are often taken into account via the addition of arandom input or other stochastic effects.

Stability is the single most important characteristic of a system and the engineer shouldpay attention to it at the very early stage of the design (e.g., to see if certain variables remainedbounded). The engineer should know if the plant is controllable [9] (otherwise the controlinputs will not be able to properly affect the plant) and observable (to see if the chosen sensorswill allow the controller to observe the critical plant behavior so that it can be compensatedfor) or if it is non-minimum phase. These properties would have a fundamental impact on ourability to design effective controllers for the system. In addition, the engineer will try to makea general assessment of how the plant dynamics change over time, and what random effectsare present. This analysis of the behavior of the plant gives the control engineer a fundamentalunderstanding of the plant dynamics that will be very useful when the time comes to synthesizethe controller.

1.9.2 Performance Objectives and Design ConstraintsController design entails constructing a controller to meet the performance specifications.

Often the first issue to address is whether to use open-or closed-loop control. If you can achieveyour objectives with open-loop control (for example, position control using a stepper motor),why turn to feedback control? Often, you need to pay for a sensor for the feedback informationand there should be justification for this cost. Moreover, feedback can destabilize this system.One should not develop a feedback control just because one is used to do it, in some simplecases an open-loop controller may provide adequate performance.

Assuming that feedback control is used the closed-loop specifications (or performanceobjectives) can involve the following issues:

(i) Disturbance rejection properties: (with reference to the process fluid temperaturecontrol problem in Fig. 1.3, the control system should be able to minimize the variationsin the process flow rate F). Basically, the need for minimization of the effect of


disturbance make the feedback control superior to open-loop control; for many systemsit is simply impossible to meet the specifications without feedback (e.g., for thetemperature control problem, if you had no measurement of process fluid, how wellcould you relate the temperature to the users set-point ?).

(ii) Insensitivity to plant parameter variations: (e.g., for the process fluid controlproblem that the control system will be able to compensate for changes in the level ofthe process fluid in the kettle, or its thermal constant).

(iii) Stability: (e.g., in the control system of Fig. 1.3, to guarantee that in absence of flowrate disturbances and change in the ambient conditions, the fluid temperature willbe equal to the desired set point).

(iv) Rise-time: (e.g., in the control system of Fig. 1.3, a measure of how long it takes forthe actual temperature to get close to the desired temperature when there is a stepchange in the set-point).

(v) Overshoot: (when there is a step change in the set point in Fig. 1.3, how much thetemperature will increase above the set point).

(vi) Settling time: (e.g., in the control system of Fig. 1.3, how much time it takes for thetemperature to reach to within a pre-assigned percent (2 or 5% ) of the set point).

(vii) Steady-state error: (in absence of any disturbance in the control system of Fig. 1.3,whether the error between the set-point and actual temperature will become zero).

Apart from these technical issues, there are other issues to be considered that are oftenof equal or greater importance. These include:

(i) Cost: How much money and time will be needed to implement the controller?(ii) Computational complexity: When a digital computer is used as a controller, how

much processor power and memory will it take to implement the controller?(iii) Manufacturability: Does your controller has any extraordinary requirements with

regard to manufacturing the hardware that is required to implement it (e.g., solarpower control system)?

(iv) Reliability: Will the controller always perform properly? What is its meantimebetween failures?

(v) Maintainability: Will it be easy to perform maintenance and routine field adjustmentsto the controller?

(vi) Adaptability: Can the same design be adapted to other similar applications so thatthe cost of later designs can be reduced ? In other words, will it be easy to modify theprocess fluid temperature controller to fit on different processes so that the develop-ment can be just once?

(vii) Understandability: Will the right people be able to understand the approachto control? For example, will the people that implement it or test it be able to fullyunderstand it?

(viii) Politics: Is your boss biased against your approach? Can you sale your approach toyour colleagues? Is your approach too novel (solar power control!) and does it therebydepart too much from standard company practice?

The above issues, in addition to meeting technical specifications, must also be takeninto consideration and these can often force the control engineer to make decisions that cansignificantly affect how, for example, the ultimate process fluid controller is designed. It isimportant then that the engineer has these issues in mind at early stages of the design process.


Chap

ter 1

1.9.3 Controller DesignConventional control has provided numerous methods for realizing controllers for

dynamic systems. Some of these are:(i) Proportional-integral-derivative (PID) control: Over 90 % of the controllers in opera-

tions today are PID controllers (or some variation of it like a P or PI). This approachis often viewed as simple, reliable, and easy to understand. Often, like fuzzy control-lers, heuristics are used to tune PID controllers (e.g., the Zeigler-Nichols tuning rules).

(ii) Classical control: lead-lag compensation, Bode and Nyquist methods, root-locus de-sign, and so on.

(iii) State-space methods: State feedback, observers, and so on.(iv) Optimal control: Linear quadratic regulators, use of Pontryagins minimum principle

or dynamic programming, and so on.(v) Robust control: H2 or H infinity methods, quantitative feedback theory, loop shaping,

and so on.(vi) Nonlinear methods: Feedback linearization, Lyapunov redesign, sliding mode control,

backstepping, and so on.(vii) Adaptive control: Model reference adaptive control, self-tuning regulators, nonlinear

adaptive control, and so on.(viii) Discrete event systems: Petri nets, supervisory control, Infinitesimal perturbation

analysis and so on.If the engineers do not fully understand the plant and just take the mathematical model

as such, it may lead to development of unrealistic control laws.

1.9.4 Performance EvaluationThe next step in the design process is the system analysis and performance evaluation.

The performance evaluation is an essential step before commissioning the control system tocheck if the designed controller really meets the closed-loop specifications. This can beparticularly important in safety-critical applications such as a nuclear power plant control orin aircraft control. However, in some consumer applications such as the control of washingmachine or an electric shaver, it may not be as important in the sense that failures will notimply the loss of life (just the possible embarrassment of the company and cost of warrantyexpenses), so some rigorous evaluation matters can sometimes be ignored. Basically, there arethree general ways to verify that a control system is operating properly (1) mathematicalanalysis based on the use of formal models, (2) simulation based analysis that most often usesformal models, and (3) experimental investigations on the real system.

(a) Reliability of mathematical analysis. In the analysis phase one may examinethe stability (asymptotically stable, or bounded-input bounded-output (BIBO) stable) andcontrollability of the system together with other closed-loop specifications such as disturbancerejection, rise-time, overshoot, settling time, and steady-state errors. However, one should notforget the limitations of mathematical analysis. Firstly, the accuracy of the analysis is nobetter than that of the mathematical model used in the analysis, which is never a perfectrepresentation of the actual plant, so the conclusions that are arrived at from the analysis arein a sense only as accurate as the model itself. And, secondly, there is a need for the developmentof analysis techniques for even more sophisticated nonlinear systems since existing theory issomewhat lacking for the analysis of complex nonlinear (e.g., fuzzy) control systems, a largenumber of inputs and outputs, and stochastic effects. In spite of these limitations, the


mathematical analysis does not become a useless exercise in all the cases. Sometimes it helpsto uncover fundamental problems with a control design.

(b) Simulation of the designed system. In the next phase of analysis, the controllerand the actual plant is simulated on analog or digital computer. This can be done by using thelaws of the physical world to develop a mathematical model and perhaps real data can be usedto specify some of the parameter of the model obtained via system identification or directparameter measurement. The simulation model can often be made quite accurate, and theeffects of implementation considerations such as finite word-length restrictions in digitalcomputer realization can be included. Currently simulations are done on digital computers,but there are occasions where an analog computer is still quite useful, particularly for real-time simulation of complex systems or in certain laboratory settings.

Simulation (digital, analog or hybrid) too has its limitations. First, as with themathematical analysis, the model that is developed will never be identical with the actualplant. Besides, some properties simply cannot be fully verified through simulation studies. Forinstance, it is impossible to verify the asymptotic stability of an ordinary differential equationthrough simulations since a simulation can only run for a finite amount of time and only afinite number of initial conditions can be tested for these finite-length trajectories. But,simulation-based studies can provide valuable insights needed to redesign the controller beforeinvesting more time for the implementation of the controller apart from enhancing the engineersconfidence about the closed loop behavior of the designed system.

(c) Experimental studies. In the final stage of analysis, the controller is implementedand integrated with the real plant and tested under various conditions. Obviously,implementations require significant resources (e.g., time, hardware), and for some plantsimplementation would not even be thought of before extensive mathematical and simulation-based studies have been completed. The experimental evaluation throws some light on someother issues involved in control system design such as cost of implementation, reliability, andperhaps maintainability. The limitations of experimental evaluations are, first, problems withthe repeatability of experiments, and second, variations in physical components, which makethe verification only approximate for other plants that are manufactured at other times.Experimental studies, also, will go a long way toward enhancing the engineers confidenceafter seeing one real system in operation.

There are two basic reasons to choose one or all three of the above approaches ofperformance evaluation. Firstly, the engineer wants to verify that the designed controller willperform properly. Secondly, if the closed-loop system does not perform properly, then theanalysis is expected to reveal a way for undertaking a redesign of the controller to meet thespecifications.

21

In this chapter we shall discuss briefly some of the mathematical tools used extensively for theanalysis and design of control systems.

2.0 THE LAPLACE TRANSFORMThe Laplace transform method is a very useful mathematical tool [10-11] for solving lineardifferential equations. By use of Laplace transforms, operations like differentiation andintegration can be replaced by algebraic operations such that, a linear differential equationcan be transformed into an algebraic equation in a complex variable s. The solution of thedifferential equation may be found by inverse Laplace transform operation simply by solvingthe algebraic equation involving the complex variable s.

Analysis and design of a linear system are also carried out in the s-domain withoutactually solving the differential equations describing the dynamics of the system. Graphicaltechniques li

Documents

Introduction to Control Engineering Modeling, Analysis and Design by Ajit K. Mandal