ajit k mandal

This page intentionally left blank

Copyright 2006 New Age International (P) Ltd., Publishers Published 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 retrieval system, 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, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com

Dedicated to: my the memory of my parents

K. L. Mandaland

Rohini Mandalappreciation in grateful reverence and appreciation


PrefaceControl engineering is a very important subject to warrant its inclusion as a core course in the engineering program of studies in universities throughout the world. The subject is multidisciplinary in nature since it deals with dynamic systems drawn from the disciplines of electrical, electronics, chemical, mechanical, aerospace and instrumentation engineering. The common binding thread among all these divergent disciplines is a mathematical model in the form of differential or difference equations or linguistic models. Once a model is prepared to describe the dynamics of the system, there is little to distinguish one from the other and the analysis depends solely on characteristics like linearity or nonlinearity, stationary or time varying, statistical or deterministic nature of the system. The subject has a strong mathematical foundation and mathematics being a universal language; it can deal with the subject of interdisciplinary nature in a unified manner. Even though the subject has strong mathematical foundation, emphasis throughout the text is not on mathematical rigour or formal derivation (unless they contribute to understanding the concept), but instead, on the methods of application associated with the analysis and design of feedback system. The text is written from the engineers point of view to explain the basic concepts involved in feedback control theory. The material in the text has been organized for gradual and sequential development of control theory starting with a statement of the task of a control engineer at the very outset. The book is intended for an introductory undergraduate course in control systems for engineering students. The numerous problems and examples have been drawn from the disciplines of electrical, electronics, chemical, mechanical, and aerospace engineering. This will help students of one discipline with the opportunity to see beyond their own field of study and thereby broaden their perceptual horizon. This will enable them to appreciate the applicability of control system theory to many facets of life like the biological, economic, and ecological control systems. This text presents a comprehensive analysis and design of continuous-time control systems and includes more than introductory material for discrete systems with adequate guidelines to extend the results derived in connection with continuous-time systems. The prerequisite for the reader is some elementary knowledge of differential equations, vector-matrix analysis and mechanics. Numerous solved problems are provided throughout the book. Each chapter is followed by 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 been derived in the Appendix at the end of the book. Most of the materials including solved and unsolved problems presented in the book have been class-tested in senior undergraduates and first year graduate level courses in the vii

viii 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 and MATLAB 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 analysis of control systems. Some representative MATLAB scripts used for solving problems are included at the end of each chapter whenever thought relevant. However, the student is encouraged to compute simple answers by hand in order to judge that the computers output is sound and not garbage. Most of the graphical figures were generated using MATLAB and some representative scripts for those are also included in the book. We hope that this text will give the students a broader understanding of control system design and analysis and prepare them for an 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 of control 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 to perform. We have reviewed, in the second chapter, the common mathematical tools used for analysis and design of linear control systems. This is followed by the procedure for handling the block diagrams and signal flow graphs containing the transfer functions of various components constituting the overall system. In chapter 3, the concept of state variable representation along with the solution of state equations is discussed. The concept of controllability and observability are also introduced in this chapter along with the derivation of transfer function from state variable representation. The specifications for transient state and steady state response of linear systems have been discussed in chapter 4 along with the Bode technique for frequency domain response of linear control systems. In chapter 5, the concept of stability has been introduced and Routh-Hurwitz technique along with the Direct method of Lyapunov have been presented. Frequency domain stability test by Nyquist criteria has been presented in chapter 6. The root locus technique for continuous system has been discussed in chapter 7 and its extension to discrete cases has been included. The design of compensators has been taken up in chapter 8. In chapter 9, we present the concept of pole assignment design along with the state estimation. In chapter 10, we consider the representation of digital control system and its solution. In chapter 11, we present introductory material for optimal problem and present the solution of linear regulator problem. Chapter12 introduces the concepts of fuzzy set and fuzzy logic needed to understand Fuzzy Logic Control Systems presented in chapter 13. The reader must be familiar with the basic tools available for analyzing systems that incorporate unwanted nonlinear components or deliberately introduced (relay) to improve system performance. Chapter 14 has been included to deal with nonlinear components and their analysis using MATLAB and SIMULINK through user defined s-functions. Finally, Chapter 15 is concerned with the implementation of digital controllers on finite bit computer, which will bring out the problems associated with digital controllers. We have used MATLAB and SIMULINK tools for getting the solution of system dynamics and for rapid verification of controller designs. Some notes for using MATLAB script M-files and function M-files are included at the end of the book. The author is deeply indebted to a number of individuals who assisted in the preparation of the manuscript, although it is difficult to name everyone in this Preface. I would like to thank Saptarshi for his support and enthusiasm in seeing the text completed, Maya for the many hours she spent reviewing and editing the text and proof reading. I would like to thank many

ix people who have provided valuable support for this book project : Ms Basabi Banerjee for her effort in writing equations in MSword in the initial draft of the manuscript, Mr. U. Nath for typing 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 who have solved many problems used in the book, with special appreciation to Ms Sumitra Mukhopadhyay, who provided feedback and offered helpful comments when reading a draft version of the manuscript. A. K. Mandal


ContentsPreface viiAND THE

1

CONTROL SYSTEMS1.0 1.1 1.2 1.3 1.4 1.5 1.6

TASK OF

A

CONTROL ENGINEER

120

1.7 1.8 1.9

Introduction to Control Engineering 1 The Concept of Feedback and Closed Loop Control 2 Open-Loop Versus Closed-Loop Systems 2 Feedforward Control 7 Feedback Control in Nature 9 A Glimpse of the Areas where Feedback Control Systems have been Employed by Man 10 Classification of Systems 10 1.6.1 Linear System 11 1.6.2 Time-Invariant System 11 Task of Control Engineers 13 Alternative Ways to Accomplish a Control Task 14 A Closer Look to the Control Task 15 1.9.1 Mathematical Modeling 16 1.9.2 Performance Objectives and Design Constraints 17 1.9.3 Controller Design 19 1.9.4 Performance Evaluation 19

2

MATHEMATICAL PRELIMINARIES2.0 2.1 2.2 2.3 2.4 2.5 The Laplace Transform 21 Complex Variables And Complex Functions 21 2.1.1 Complex Function 21 Laplace Transformation 22 2.2.1 Laplace Transform and Its Existence 23 Laplace Transform of Common Functions 23 2.3.1 Laplace Table 26 Properties of Laplace Transform 27 Inverse Laplace Transformation 31 2.5.1 Partial-Fraction Expansion Method 32 2.5.2 Partial-Fraction Expansion when F(s) has only Distinct Poles 2.5.3 Partial-Fraction Expansion of F(s) with Repeated Poles 34 xi

2164

32

xii 2.6 2.7

CONTENTS

Concept of Transfer Function 35 Block Diagrams 36 2.7.1 Block Diagram Reduction 39 2.8 Signal Flow Graph Representation 42 2.8.1 Signal Flow Graphs 42 2.8.2 Properties of Signal Flow Graphs 43 2.8.3 Signal Flow Graph Algebra 43 2.8.4 Representation of Linear Systems by Signal Flow Graph 44 2.8.5 Masons Gain Formula 45 2.9 Vectors and Matrices 48 2.9.1 Minors, Cofactors and Adjoint of a Matrix 49 2.10 Inversion of a Nonsingular Matrix 51 2.11 Eigen Values and Eigen Vectors 52 2.12 Similarity Transformation 53 2.12.1 Diagonalization of Matrices 53 2.12.2 Jordan Blocks 54 2.13 Minimal Polynomial Function and Computation of Matrix Function Using Sylvesters Interpolation 55 MATLAB Scripts 57 Review Exercise 58 Problems 60

3

STATE VARIABLE REPRESENTATION EQUATIONS3.1 3.2 3.3 3.4

AND

SOLUTION

OF

STATE

6588

Introduction 65 System Representation in State-variable Form 66 Concepts of Controllability and Observability 69 Transfer Function from State-variable Representation 73 3.4.1 Computation of Resolvent Matrix from Signal Flow Graph 3.5 State Variable Representation from Transfer Function 77 3.6 Solution of State Equation and State Transition Matrix 81 3.6.1 Properties of the State Transition Matrix 82 Review Exercise 83 Problems 85

75

4

ANALYSIS4.1 4.2

OF

LINEAR SYSTEMS

89130

4.3

Time-Domain Performance of Control Systems 89 Typical Test Inputs 89 4.2.1 The Step-Function Input 89 4.2.2 The Ramp-Function Input 90 4.2.3 The Impulse-Function Input 90 4.2.4 The Parabolic-Function Input 90 Transient State and Steady State Response of Analog Control System

91

CONTENTS

xiii

4.4

Performance Specification of Linear Systems in Time-Domain 92 4.4.1 Transient Response Specifications 92 4.5 Transient Response of a Prototype Second-order System 93 4.5.1 Locus of Roots for the Second Order Prototype System 94 4.5.1.1 Constant wn Locus 94 4.5.1.2 Constant Damping Ratio Line 94 4.5.1.3 Constant Settling Time 94 4.5.2 Transient Response with Constant wn and Variable 95 4.5.2.1 Step Input Response 95 4.6 Impulse Response of a Transfer Function 100 4.7 The Steady-State Error 101 4.7.1 Steady-State Error Caused by Nonlinear Elements 102 4.8 Steady-State Error of Linear Control Systems 102 4.8.1 The Type of Control Systems 103 4.8.2 Steady-State Error of a System with a Step-Function Input 104 4.8.3 Steady-State Error of A System with Ramp-Function Input 105 4.8.4 Steady-State Error of A System with Parabolic-Function Input 106 4.9 Performance Indexes 107 4.9.1 Integral of Squared Error (ISE) 108 4.9.2 Integral of Time Multiplied Squared Error (ITSE) Criteria 108 4.9.3 Integral of Absolute Error (IAE) Criteria 108 4.9.4 Integral of Time Multiplied Absolute Error (ITAE) 109 4.9.5 Quadratic Performance Index 110 4.10 Frequency Domain Response 110 4.10.1 Frequency Response of Closed-Loop Systems 111 4.10.2 Frequency-Domain Specifications 112 4.11 Frequency Domain Parameters of Prototype Second-Order System 112 4.11.1 Peak Resonance and Resonant Frequency 112 4.11.2 Bandwidth 114 4.12 Bode Diagrams 115 4.12.1 Bode Plot 115 4.12.2 Principal Factors of Transfer Function 116 4.13 Procedure for Manual Plotting of Bode Diagram 121 4.14 Minimum Phase and Non-Minimum Phase Systems 122 MATLAB Scripts 123 Review Exercise 125 Problems 126

5

THE STABLILITY5.1 5.2

OF

LINEAR CONTROL SYSTEMS

131158

5.3

The Concept of Stability 131 The Routh-Hurwitz Stability Criterion 134 5.2.1 Relative Stability Analysis 139 5.2.2 Control System Analysis Using Rouths Stability Criterion Stability by the Direct Method of Lyapunov 140

139

xiv 5.3.1 Introduction to the Direct Method of Lyapunov 140 5.3.2 System Representation 141 5.4 Stability by the Direct Method of Lyapunov 141 5.4.1 Definitions of Stability 143 5.4.2 Lyapunov Stability Theorems 144 5.5 Generation of Lyapunov Functions for Autonomous Systems 147 5.5.1 Generation of Lyapunov Functions for Linear Systems 147 5.6 Estimation of Settling Time Using Lyapunov Functions 150 MATLAB Scripts 153 Review Exercise 154 Problems 155

CONTENTS

6

FREQUENCY DOMAIN STABILITY ANALYSIS CRITERION6.1

AND

NYQUIST

159191

Introduction 159 6.1.1 Poles and Zeros of Open Loop and Closed Loop Systems 159 6.1.2 Mapping Contour and the Principle of the Argument 160 6.2 The Nyquist Criterion 165 6.2.1 The Nyquist Path 166 6.2.2 The Nyquist Plot Using a Part of Nyquist Path 175 6.3 Nyquist Plot of Transfer Function with Time Delay 176 6.4 Relative Stability: Gain Margin and Phase Margin 177 6.4.1 Analytical Expression for Phase Margin and Gain Margin of a Second Order Prototype 182 6.5 Gain-Phase Plot 183 6.5.1 Constant Amplitude (M) and Constant Phase (N) Circle 183 6.6 Nichols Plot 186 6.6.1 Linear System Response Using Graphical User Interface in MATLAB 188 MATLAB Scripts 188 Review Exercise 189 Problems 190

7

ROOT LOCUS TECHNIQUE7.1 7.2 7.3 Correlation of System-Roots with Transient Response 192 The Root Locus DiagramA Time Domain Design Tool 192 Root Locus Technique 193 7.3.1 Properties of Root Loci 194 7.4 Step by Step Procedure to Draw the Root Locus Diagram 201 7.5 Root Locus Design Using Graphical Interface in MATLAB 211 7.6 Root Locus Technique for Discrete Systems 212 7.7 Sensitivity of the Root Locus 213 MATLAB Scripts 213 Review Exercise 214 Problems 217

192217

CONTENTS

xvOF

8

DESIGN8.1 8.2

COMPENSATORS

218254

Introduction 218 Approaches to System Design 218 8.2.1 Structure of the Compensated System 219 8.2.2 Cascade Compensation Networks 220 8.2.3 Design Concept for Lag or Lead Compensator in Frequency-Domain 224 8.2.4 Design Steps for Lag Compensator 226 8.2.5 Design Steps for Lead Compensator 226 8.2.6 Design Examples 226 8.3 Design of Compensator by Root Locus Technique 238 8.3.1 Design of Phase-lead Compensator Using Root Locus Procedure 238 8.3.2 Design of Phase-lag Compensator Using Root Locus Procedure 240 8.4 PID Controller 241 8.4.1 Ziegler-Nichols Rules for Tuning PID Controllers 242 8.4.2 First Method 242 8.4.3 Second Method 243 8.5 Design of Compensators for Discrete Systems 246 8.5.1 Design Steps for Lag Compensator 248 8.5.2 Design Steps for Lead Compensator 248 MATLAB Scripts 249 Review Exercise 252 Problems 253

9

STATE FEEDBACK DESIGN9.1

255275

Pole Assignment Design and State Estimation 255 9.1.1 Ackermans Formula 256 9.1.2 Guidelines for Placement of Closed Loop System Poles 258 9.1.3 Linear Quadratic Regulator Problem 258 9.2 State Estimation 259 9.2.1 Sources of Error in State Estimation 260 9.2.2 Computation of the Observer Parameters 261 9.3 Equivalent Frequency-Domain Compensator 264 9.4 Combined Plant and Observer Dynamics of the Closed Loop System 265 9.5 Incorporation of a Reference Input 266 9.6 Reduced-Order Observer 267 9.7 Some Guidelines for Selecting Closed Loop Poles in Pole Assignment Design 270 MATLAB Scripts 271 Review Exercise 272 Problems 275

10 SAMPLED DATA CONTROL SYSTEM10.0 Why We are Interested in Sampled Data Control System? 276

276332

xvi 10.1 10.2 10.3 10.4 Advantage of Digital Control 276 Disadvantages 277 Representation of Sampled Process 278 The Z-Transform 279 10.4.1 The Residue Method 280 10.4.2 Some Useful Theorems 282 10.5 Inverse Z-Transforms 286 10.5.1 Partial Fraction Method 286 10.5.2 Residue Method 286 10.6 Block Diagram Algebra for Discrete Data System 287 10.7 Limitations of the Z-Transformation Method 292 10.8 Frequency Domain Analysis of Sampling Process 292 10.9 Data Reconstruction 297 10.9.1 Zero Order Hold 299 10.10 First Order Hold 302 10.11 Discrete State Equation 305 10.12 State Equations of Systems with Digital Components 308 10.13 The Solution of Discrete State Equations 308 10.13.1 The Recursive Method 308 10.14 Stability of Discrete Linear Systems 311 10.14.1 Jurys Stability Test 313 10.15 Steady State Error for Discrete System 316 10.16 State Feedback Design for Discrete Systems 321 10.16.1 Predictor Estimator 321 10.16.2 Current Estimator 322 10.16.3 Reduced-order Estimator for Discrete Systems 325 10.17 Provision for Reference Input 326 MATLAB Scripts 327 Review Exercise 329 Problems 331

CONTENTS

11 OPTIMAL CONTROL11.1 11.2 11.3 11.4

333370

Introduction 333 Optimal Control Problem 333 Performance Index 336 Calculus of Variations 336 11.4.1 Functions and Functionals 337 A. Closeness of Functions 338 B. Increment of a Functional 339 C. The Variation of a Functional 339 11.4.2 The Fundamental Theorem of the Calculus of Variations 342 11.4.3 Extrema of Functionals of a Single Function 343 11.4.3.1 Variational Problems and the Euler Equation 343 11.4.3.2 Extrema of Functionals of n Functions 346 11.4.3.3 Variable End Point Problems 347

CONTENTS

xvii

11.4.4 Optimal Control Problem 352 11.4.5 Pontryagins Minimum Principle 354 11.5 The LQ Problem 357 11.5.1 The Hamilton-Jacobi Approach 358 11.5.2 The Matrix Riccati Equation 359 11.5.3 Finite Control Horizon 360 11.5.4 Linear Regulator Design (Infinite-time Problem) 362 11.6 Optimal Controller for Discrete System 363 11.6.1 Linear Digital Regulator Design (Infinite-time Problem) 365 MATLAB Scripts 367 Review Exercise 367 Problems 369

12 FUZZY LOGIC12.1 12.2 12.3

FOR

CONTROL SYSTEM

371418

12.4

12.5 12.6

12.7 12.8 12.9

The Concept of Fuzzy Logic and Relevance of Fuzzy Control 371 Industrial and Commercial Use of Fuzzy Logic-based Systems 373 Fuzzy Modeling and Control 373 12.3.1 Advantages of Fuzzy Controller 374 12.3.2 When to Use Fuzzy Control 375 12.3.3 Potential Areas of Fuzzy Control 375 12.3.4 Summary of Some Benefits of Fuzzy Logic and Fuzzy Logic Based Control System 376 12.3.5 When Not to Use Fuzzy Logic 377 Fuzzy Sets and Membership 377 12.4.1 Introduction to Sets 377 12.4.2 Classical Sets 378 12.4.3 Fuzzy Sets 379 Basic Definitions of Fuzzy Sets and a Few Terminologies 379 12.5.1 Commonly Used Fuzzy Set Terminologies 381 Set-Theoretic Operations 384 12.6.1 Classical Operators on Fuzzy Sets 384 12.6.2 Generalized Fuzzy Operators 386 12.6.2.1 Fuzzy Complement 386 12.6.2.2 Fuzzy Union and Intersection 387 12.6.2.3 Fuzzy Intersection: The T-Norm 387 12.6.2.4 Fuzzy Union: The T-Conorm (or S-Norm) 388 MF Formulation and Parameterization 388 12.7.1 MFs of One Dimension 389 From Numerical Variables to Linguistic Variables 391 12.8.1 Term Sets of Linguistic Variables 393 Classical Relations and Fuzzy Relations 394 12.9.1 Cartesian Product 394 12.9.2 Crisp Relations 394 12.9.3 Fuzzy Relations 395 12.9.4 Operation on Fuzzy Relations 396

xviii 12.10 Extension Principle 402 12.11 Logical Arguments and Propositions 403 12.11.1 Logical Arguments 403 12.11.2 Modus Ponens 407 12.11.3 Modus Tollens 407 12.11.4 Hypothetical Syllogism 407 12.12 Interpretations of Fuzzy If-then Rules 407 12.12.1 Fuzzy Relation Equations 409 12.13 Basic Principles of Approximate Reasoning 410 12.13.1 Generalized Modus Ponens 410 12.13.2 Generalized Modus Tollens 410 12.13.4 Generalized Hypothetical Syllogism 411 12.14 Representation of a Set of Rules 411 12.14.1 Approximate Reasoning with Multiple Conditional Rules MATLAB Scripts 416 Problems 417

CONTENTS

413

13 FUZZY LOGIC BASED CONTROLLER13.1

419452

The Structure of Fuzzy Logic-based Controller 419 13.1.1 Knowledge Base 420 13.1.2 Rule Base 421 13.1.2.1 Choice of Sate Variables and Controller Variables 421 13.1.3 Contents of Antecedent and Consequent of Rules 422 13.1.4 Derivation of Production Rules 422 13.1.5 Membership Assignment 423 13.1.6 Cardinality of a Term Set 423 13.1.7 Completeness of Rules 423 13.1.8 Consistency of Rules 424 13.2 Inference Engine 424 13.2.1 Special Cases of Fuzzy Singleton 426 13.3 Reasoning Types 427 13.4 Fuzzification Module 428 13.4.1 Fuzzifier and Fuzzy Singleton 428 13.5 Defuzzification Module 429 13.5.1 Defuzzifier 429 13.5.2 Center of Area (or Center of Gravity) Defuzzifier 430 13.5.3 Center Average Defuzzifier (or Weighted Average Method) 431 13.6 Design Consideration of Simple Fuzzy Controllers 432 13.7 Design Parameters of General Fuzzy Controllers 433 13.8 Examples of Fuzzy Control System Design: Inverted Pendulum 434 13.9 Design of Fuzzy Logic Controller on Simulink and MATLAB Environment 441 13.9.1 Iterative Design Procedure of a PID Controller in MATLAB Environment 441 13.9.2 Simulation of System Dynamics in Simulink for PID Controller Design 444

CONTENTS

xix 13.9.3 Simulation of System Dynamics in Simulink for Fuzzy Logic Controller Design 446

Problems 449

14 NONLINEAR SYSTEMS: DESCRIBING FUNCTION ANALYSIS14.1

AND

PHASE-PLANE 453492

Introduction 453 14.1.1 Some Phenomena Peculiar to Nonlinear Systems 454 14.2 Approaches for Analysis of Nonlinear Systems: Linearization 457 14.3 Describing Function Method 458 14.4 Procedure for Computation of Describing Function 459 14.5 Describing Function of Some Typical Nonlinear Devices 460 14.5.1 Describing Function of an Amplifying Device with Dead Zone and Saturation 460 14.5.2 Describing Function of a Device with Saturation but without any Dead Zone 463 14.5.3 Describing Function of a Relay with Dead Zone 464 14.5.4 Describing Function of a Relay with Dead Zone and Hysteresis 464 14.5.5 Describing Function of a Relay with Pure Hysteresis 466 14.5.6 Describing Function of Backlash 466 14.6 Stability Analysis of an Autonomous Closed Loop System by Describing Function 468 14.7 Graphical Analysis of Nonlinear Systems by Phase-Plane Methods 471 14.8 Phase-Plane Construction by the Isocline Method 472 14.9 Pells Method of Phase-Trajectory Construction 474 14.10 The Delta Method of Phase-Trajectory Construction 476 14.11 Construction of Phase Trajectories for System with Forcing Functions 477 14.12 Singular Points 477 14.13 The Aizerman and Kalman Conjectures 481 14.13.1 Popovs Stability Criterion 482 14.13.2 The Generalized Circle Criteria 482 14.13.3 Simplified Circle Criteria 483 14.13.4 Finding Sectors for Typical Nonlinearities 484 14.13.5 S-function SIMULINK Solution of Nonlinear Equations 485 MATLAB Scripts 489 Problems 492

15 IMPLEMENTATION COMPUTER15.1 15.2

OF

DIGITAL CONTROLLERS

ON

FINITE BIT

493521

Introduction 493 Implementation of Controller Algorithm 15.2.1 Realization of Transfer Function

493 493

xx

CONTENTS

15.2.2 Series or Direct Form 1 494 15.2.3 Direct Form 2 (Canonical) 495 15.2.4 Cascade Realization 496 15.2.5 Parallel Realization 497 15.3 Effects of Finite Bit Size on Digital Controller Implementation 500 15.3.1 Sign Magnitude Number System (SMNS) 500 15.3.1.1 Truncation Quantizer 500 15.3.1.2 Round-off Quantizer 500 15.3.1.3 Mean and Variance 502 15.3.1.4 Dynamic Range of SMNS 503 15.3.1.5 Overflow 503 15.3.2 Twos Complement Number System 504 15.3.2.1 Truncation Operation 504 15.3.2.2 Round-off Quantizer in Twos CNS 505 15.3.2.3 Mean and Variance 505 15.3.2.4 Dynamic Range for Twos CNS 506 15.3.2.5 Overflow 506 15.4 Propagation of Quantization Noise Through the Control System 507 15.5 Very High Sampling Frequency Increases Noise 507 15.6 Propagation of ADC Errors and Multiplication Errors through the Controller 508 15.6.1 Propagated Multiplication Noise in Parallel Realization 508 15.6.2 Propagated Multiplication Noise in Direct Form Realization 510 15.7 Coefficient Errors and Their Influence on Controller Dynamics 511 15.7.1 Sensitivity of Variation of Coefficients of a Second Order Controller 511 15.8 Word Length in A/D Converters, Memory, Arithmetic Unit and D/A Converters 512 15.9 Quantization gives Rise to Nonlinear Behavior in Controller 515 15.10 Avoiding the Overflow 517 15.10.1 Pole Zero Pairing 517 15.10.2 Amplitude Scaling for Avoiding Overflow 518 15.10.3 Design Guidelines 518 MATLAB Scripts 519 Problems 520 Appendex A 522 Appendex B 579 Appendex C 585 Notes on MATLAB Use 589 Bibliography 595 Index 601

CHAPTER

1Control Systems and the Task of a Control Engineer1.0 INTRODUCTION TO CONTROL ENGINEERINGThe subject of control engineering is interdisciplinary in nature. It embraces all the disciplines of engineering including Electronics, Computer Science, Electrical Engineering, Mechanical Engineering, Instrumentation Engineering, and Chemical Engineering or any amalgamation of these. If we are interested to control the position of a mechanical load automatically, we may use an electrical motor to drive the load and a gearbox to connect the load to the motor shaft and an electronic amplifier to amplify the control signal. So we have to draw upon our working experience of electronic amplifier, the electrical motor along with the knowledge of mechanical engineering as to how the motor can be connected with the help of a gearbox including the selection of the gear ratio. If we are interested to regulate the DC output voltage of a rectifier to be used for a computer system, then the entire control system consists purely of electrical and electronics components. There will be no moving parts and consequently the response of such systems to any deviations from the set value will be very fast compared to the response of an electromechanical system like a motor. There are situations where we have to control the position of mechanical load that demands a very fast response, as in the case of aircraft control system. We shall recommend a hydraulic motor in place of an electrical motor for 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 electronic engineers. Its use is widespread only because of the tremendous development in the discipline of electronics engineering over the last 30 to 40 years. Amplifiers may be built utilizing the properties of fluids resulting in hydraulic and pneumatic amplifiers. In petrochemical industry pneumatic amplifiers are a common choice while hydraulic amplifiers are widely used in aircraft control systems and steel rolling mills where very large torque are needed to control the position of a mechanical load. But whenever the measurement of any physical parameter of our interest is involved it should be converted to electrical signal at the first opportunity for subsequent amplification and processing by electronic devices. The unprecedented development of electronic devices in the form of integrated circuits and computers over the last few decades coupled with the tremendous progress made in signal processing techniques has made it extremely profitable to convert information about any physical parameter of our interest to electrical form for necessary 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 have suitable transducers to measure the variables and use them as suitable feedback signal for 1

2

INTRODUCTION TO CONTROL ENGINEERING

proper control. Therefore, the knowledge of various transducers is essential to appreciate the intricacies of a practical control system. In this text, we shall endeavor to introduce the basic principles of control systems starting from building mathematical models of elementary control systems and gradually working out the control strategy in a practical control system. We have drawn examples from various disciplines of engineering so as to put students from various disciplines of engineering on a familiar footing. We have assumed an elementary knowledge of differential calculus and the working 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 shall explain these concepts by considering a schematic representation of a system, as shown in Fig. 1.1 that 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 feedback first.

1.1

THE CONCEPT OF FEEDBACK AND CLOSED LOOP CONTROL

The concept of feedback is the single most important idea that governs the life of man in modern societies. In its present state of sophistication, human life would have been miserable without machines and most of the machines used by man could not be made to function with reasonable reliability and accuracy without the utilization of feedback. Most of the machines meant for domestic and industrial applications, for entertainment, health-services and military science, incorporate the concept of feedback. This concept is not exploited solely by man, it is also prevalent in nature and man has learnt it, like many other things, from nature. Our very life, 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 measure a variable of interest and correct it automatically without the intervention of a human whenever the variable is outside acceptable limits. Systems that can perform such self-correcting action are called feedback systems or closed-loop systems whereas non self-correcting type is referred to as open loop system. When the variable that is being monitored and corrected is an objects physical positions and the system involves mechanical movement is assigned a special name: a servo system.

1.2

OPEN-LOOP VERSUS CLOSED-LOOP SYSTEMS

Let 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 by controlling the incoming flow rate as in Fig. 1.1(a). Liquid enters the tank at the top and flows out via the exit pipe at the bottom. One way to attempt to maintain the proper level in the tank is to employ a human operator to adjust the manual valve so that the rate of liquid flow into the tank exactly balances the rate of liquid flow out of the tank when the liquid is at the desired level. It might require a bit of trial and error for the correct valve setting, but eventually an intelligent operator can set the proper valve opening. If the operator stands and watches the system for a while and observes that the liquid level stays constant, s/he may conclude that the proper valve opening has been set to maintain the correct level.

CONTROL SYSTEMS AND THE TASK OF A CONTROL ENGINEERManual valve Supply pipe

3Chapter 1Desired level Outgoing liquid Exit pipe

Incoming liquid

(a)B

A

Pivot point

Supply pipe

Float

Control valve Exit pipe

Incoming liquid

Outgoing liquid (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-correcting

In 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 upstream side of the manual valve might increase for some reason. This would increase the input flow rate with no corresponding increase in output flow rate. The liquid level would start to rise and the tank would soon overflow. Of course, there would be some increase in output flow rate because of the increased pressure at the bottom of the tank when the level rises, but it would be a chance in a million that this would exactly balance the new input flow rate. An increase in supply pressure is just one example of a disturbing force that would upset the liquid level in the 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 rates or a change in a system restriction downstream of the exit pipe would also change the output flow rate. Now consider the setup in Fig. 1.1(b). If the liquid level falls a little too low, the float moves down, thereby opening the tapered valve to increase the inflow of liquid. If the liquid level rises a little too high, the float moves up, and the tapered valve closes a little to reduce the inflow of liquid. By proper construction and sizing of the valve and the mechanical linkage between float and valve, it would be possible to control the liquid level very close to the desired set point. In this system the operating conditions may change causing the liquid level to deviate from the desired point in either direction but the system will tend to restore it to the set value.

4


Our discussion to this point has been with respect to the specific problem of controlling the liquid level in a tank. However, in general, many different industrial control systems have certain things in common. Irrespective of the exact nature of any control system, there are certain relationships between the controlling mechanisms and the controlled variable that are similar. We try to illustrate these cause-effect relationships by drawing block diagrams of our industrial systems. Because of the similarity among different systems, we are able to devise generalized block diagrams that apply to all systems. Such a generalized block diagram of an open loop system is shown in Fig. 1.2(a). The block diagram is basically a cause and effect indicator, but it shows rather clearly that for a given setting the value of the controlled variable cannot be reliably known in presence of disturbances. Disturbances that happen to the process make their effects felt in the output of the processthe controlled variable. Because the block diagram of Fig. 1.2(a) does not show any lines coming back around to make a circular path, or to close the loop, such a system is called an open-loop system. All open-loop systems are characterized by its inability to compare the actual value of the controlled variable to the desired value and to take action based on that comparison. 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 in Fig. 1.2(b). It is found from the diagram that the setting and the value of the controlled variable are compared to each other in a comparator. The output of the comparator represents the difference between the two values. The difference signal, called actuating signal, then feeds into the controller allowing the controller to affect the process.Disturbances

Set point

Controller (a)

Process

Controlled variable

Disturbances Comparator + Set point (b) Controller Process Controlled variable

Fig. 1.2 Block diagrams that show the cause-effect relationships between the different parts of the system (a) for an open-loop system (b) for a closed-loop system

The 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 a closed-loop system. All closed-loop systems are characterized by the ability to compare the actual value of the controlled variable to its desired value and automatically take action based on that comparison. The comparator performs the mathematical operation of summation of two 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 location of the float in the tank. That is, the human operator selects the level that s/he desires by locating the float at a certain height above the bottom of the tank. This setting could be altered

CONTROL SYSTEMS AND THE TASK OF A CONTROL ENGINEER

5Chapter 1

by changing the length of rod A that connects the float to horizontal member B of the linkage in Fig. 1.1(b). The comparator in the block diagram is the float itself together with the linkages A and B in our example. The float is constantly monitoring the actual liquid level, because it moves up or down according to that level. It is also comparing with the setting, which is the desired liquid level, as explained above. If the liquid level and setting are not in agreement, the float sends out a signal that depends on the magnitude and the polarity of the difference between them. That is, if the level is too low, the float causes horizontal member B in Fig. 1.1(b) to be rotated counterclockwise; the amount of counterclockwise displacement of B depends on how low the liquid is. If the liquid level is too high, the float causes member B to be displaced clockwise. Again, the amount of displacement depends on the difference between the setting and the controlled variable; in this case the difference means how much higher the liquid is than the desired level. Thus the float in the mechanical drawing corresponds to the comparator block in the block diagram of Fig. 1.2(b). The controller in the block diagram is the tapered valve in the actual mechanical drawing. In our particular example, there is a fairly clear correspondence between the physical parts of the actual system and the blocks in the block diagram. In some systems, the correspondence is not so clear-cut. It may be difficult or impossible to say exactly which physical parts comprise which blocks. One physical part may perform the function of two different blocks, or it may perform the function of one block and a portion of the function of another block. Because of the difficulty in stating an exact correspondence between the two system representations, 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 of the controlled variable being fed back and compared to the setting, the system is called a closed-loop system. As stated before, such systems have the ability to automatically take action to correct any difference between actual value and desired value, no matter why the difference occurred. Based on this discussion, we can now formally define the concept of feedback control as follows: Definition 1 The feedback control is an operation, which, in the presence of disturbing forces, tends to reduce the difference between the actual state of a system and an arbitrarily varied 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 disturbing forces may be less prominent. A control system in which the desired state (consequently the set point) is constant, it is referred to as a regulator (example: a regulated power supply) and it is called a tracking system if the set point is continuously varying and the output is required to track the set point (example: RADAR antenna tracking an aircraft position). Figure 1.3 shows another industrial process control system for controlling the temperature of a pre-heated process fluid in a jacketed kettle. The temperature of the process fluid 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 is amplified by an electronic DC amplifier Afb to produce a voltage Vb. The battery and the potentiometer provide a reference (set point) voltage Vr, which is calibrated in appropriate temperature scale. The input voltage Ve to the amplifier Ae is the difference of the reference voltage and the feedback voltage Vb. The output voltage Vo of this amplifier is connected to the

6


solenoid coil that produces a force, fs, proportional to the current through the coil ia. The solenoid pulls the valve plug and the valve plug travels a distance x. The steam flow rate q through the valve is directly proportional with the valve-opening x. The temperature of the process fluid will be proportional to the valve opening.Process flow F at temp q1

Stirrer

q Steam Boiler Valve M F, q2 x Temperature set point Rq ia Amplifier Ae M, B, K R Vr Ve V0 L Solenoid E Vb + Amplifier Afb Thermo-couple Trap Condensed water qs

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 decrease since the steam flow has not increased. The action of the control system is governed in such a way as to increase the steam flow rate to the jacketed kettle, until temperature of the process fluid is equal to the desired value set by the reference potentiometer. Let us see the consequences to the increased flow rate F of the process fluid. As the temperature in the kettle falls below the set point, the thermocouple output voltage and its amplified value Vb will decrease in magnitude. Since Ve = Vr Vb and Vr is fixed in this case by the set point potentiometer, a decrease 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 the process fluid. If the flow rate of the process fluid decreases and the temperature inside the kettle increases, Vb increases and Ve decreases. The output of the second amplifier Vo and the travel x of the valve plug decreases with a consequent reduction of the valve opening. The steam flow-rate q, therefore, decreases and results in a decrease of the kettle temperature until it equals to the set temperature.


7Chapter 1

The open loop mode of temperature control for the process fluid is possible by removing the sensing thermocouple along with the associated amplifier Afb from the system. The point E of the amplifier Ae is returned to the ground and the potentiometer is set to the desired temperature. Since in this case Ve is equal to a constant value Vr set by the potentiometer, the output amplifier voltage Vo and travel x of the valve plug is fixed making the opening of the valve fixed. The steam flow rate q is also fixed and the temperature of the kettle will be fixed if the process fluid flow-rate F is maintained to the constant value. This control arrangement is adequate if the flow rate of the process fluid is maintained to constant value. In actual practice, however, the flow rate deviates from a constant value and the temperature fluctuates with the fluctuation in F. The temperature will, in general, differ from the set point and an experienced human operator will have to change the set point interactively. When the disturbances to the system are absent, the final temperature of the kettle will be determined by the experienced skill of the operator and accuracy of calibration of the temperature-setting potentiometer R . But in presence of disturbances, which are always there, no amount of skill and experience of the operator will be adequate and the process product is likely to be adversely affected. The closed loop control system in Fig. 1.3 may be represented by block diagrams as shown in Fig. 1.4. The desired value of the system state is converted to the reference input by the transducers known as reference input elements.Manipulated variable Disturbance Ref. Input Desired value (Set point) Ref input element Indirectly controlled system Indirectly controlled output

+

Actuating signal

Control elements

Controlled system

Primary F.B.

FB elements Controller of the system Controlled variable

Fig. 1.4 Block diagram of a closed loop system

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

1.3

FEEDFORWARD CONTROL

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

8


of steam input. But the effect of this disturbance cannot be immediately sensed in the output temperature change due to the large time constant in the thermal process. Besides, the effect of corrective action in the form of change of steam input will be felt at the processor output at a time when the temperature might have deviated from the set point by a large value. The disturbance in the flow rate depends on the level of the fluid in the tower (Fig. 1.5) and can be easily measured by a flow meter. If we generate a corrective signal by an open loop controller block with a very small time constant and use it as input to the temperature controller (Gf(s) in Fig. 1.5(b)), the transient response of the process temperature might be controlled within a tighter limit. This type of control is known as feedforward control. The motivation behind the feedforward 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 other controller block with fast response. This strategy of using a faster control path will provide a better transient response in the system output. As soon as a change in the flow rate in the input fluid occurs, corrective action will be taken simultaneously, by adjusting the steam input to the heat exchanger. This can be done by feeding both the signal from the flow meter and the signal 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 the forward path from the disturbance input to the process output. Feedforward control can minimize the transient error, but since feedforward control is open loop control, there are limitations of its functional accuracy. Feedforward control will not cancel the effects of unmeasurable disturbances under normal operating condition. It is, therefore, necessary that a feedforward control system include a feedback loop as shown in Fig. 1.5. The feedback control compensates for any imperfections in the functioning of the open loop feedforward control and takes care of any unmeasurable disturbances.Temperature controller Tower Valve Steam Level controller Thermo-couple

Stirrer

Heat exchanger Valve

To condenser Flow meter (a)

Fig. 1.5 (Contd.)


9Chapter 1Disturbance (Change in flow)

Flow meter G1(s) Gn(s) Controller + Set point Gc(s) Heat exchanger G(s) Temperature Thermocouple (b)

+

+

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

1.4

FEEDBACK CONTROL IN NATURE

The examples of feedback control are abundant in nature and it plays a very important role for controlling the activities of animals including man. The life of human being itself is sustained by the action of feedback control in several forms. To cite an example, let us consider the maintenance of body temperature of warm-blooded animals to a constant value. The body temperature of human being is maintained around 38.5C in spite of the wide variation of ambient temperature from minus 20C to plus 50C. The feedback control system is absent in the cold-blooded animals like lizards, so their body 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 a balance in the process of generation and dissipation of heat. The primary source of heat in the case of living creatures is the metabolic activity supplemented by muscle activity and the heat is dissipated from the body by radiation, conduction and convection. When the ambient temperature suddenly rises, the skin senses this increase and sets in a series of operation for dissipating body-heat. Thermal radiation takes place directly from the body surface to the surrounding. The body fluid and the blood flow take part in the convection process. The blood vessels are constricted and the flow is diverted towards the outer surface of the body. The metabolic activity decreases, decreasing the heat generation. The respiration rate increases, so that more heat can be dissipated to the mass of air coming in contact with the lung surface, Blood volume is increased with body fluid drawn into circulation resulting in further cooling. The perspiration rate increases taking latent heat from the body surface thereby decreasing the body temperature. In a similar way when the outside temperature falls, extra heat is chemically generated by increased rate of metabolism. The heat generation is also supplemented by shivering and chattering of the teeth. Other examples of feedback control in human body include: Hydrogen-ion concentration in blood, concentration of sugar, fat, calcium, protein and salt in blood. Some of these variables should be closely controlled and if these were not so convulsions or coma and death would

10


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 importance with the increased sophistication of modern life and soon the role of feedback control in shaping the future of man will be as critical as is found in nature.

1.5

A GLIMPSE OF THE AREAS WHERE FEEDBACK CONTROL SYSTEMS HAVE BEEN EMPLOYED BY MAN

The following list gives a glimpse of the areas of human activities where the feedback control system is extensively used. Domestic applications: Regulated voltage and frequency of electric power, thermostat control of refrigerators and 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 and radio 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, engine cowl flaps, instrument-landing system. In sea going vessels: Automatic steering devices, radar control, Hull control, boiler pressure and turbine speed control, voltage control of its generators. In automobiles: Thermostatic cooling system, steering mechanisms, the gasoline gauge, and collision avoidance, idle speed control, antiskid braking in the latest models and other instruments have feedback loops. Scientific applications: Measuring instruments, analog computers, electron microscope, cyclotron, x-ray machine, 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 such as contour followers, the regulation of quantity, flow, liquid level, chemical concentration, light intensity, colour, electric voltage and current, recording or controlling almost any measurable quantity with suitable transducers. Military applications: Positioning of guns from 30 caliber machine guns in aircraft to mighty 16-inch guns abroad battle ships, search lights, rockets, torpedoes, surface to air missiles, ground to air or air to air missiles, gun computers, and bombsights, and guided missiles.

1.6

CLASSIFICATION OF SYSTEMS

For convenience of description and mathematical analysis, systems are classified into different categories. They are classified according to the nature of inputs, number of inputs, number of outputs and some inherent characteristic of the system. Fig. 1.6 shows the block diagram


11Chapter 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.u(t) System 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 energy and is at rest, prior to the instant at which the input u(t) is applied. The cause and effect relationship 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 and may include operations like differentiation and integration or may be given in probabilistic language. A system is deterministic if the output y(t) is unique for a given input u(t). For probabilistic or non-deterministic system, the output y(t) is not unique but probabilistic, with a probability of 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 future inputs. That is, the output y(to) at any instant to is solely determined by the characteristics of the 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 effect relationship.

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 principle of homogeneity L(k1u1(t)) = k1L(u1(t)) and the principle of additivity L[k1u1(t) + k2u2(t)] = k1L[u1(t)] + k2L[u2(t)] 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 is multiplied by k1. Also the response follows the principle of superposition, that is, y(t) is given by 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 good for a certain range of inputs u1 and u2, the system is linear in that range of inputs and nonlinear beyond that range. (1.2b) (1.2a)

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 the output for a fixed system will be L u(t ) = y(t ) (1.3)

12


A system, which is not time-invariant, is a time-varying one. A few examples of the above definitions are considered below: Example 1.1 A differentiator is characterized by y(t) = Here, the operator L = and

d u(t) dt

d d d . Therefore, y1(t) = [k u (t)] = k1 u (t) dt dt 1 1 dt 1 d d d [k1u1(t) + k2u2(t)] = k1 u1(t) + k2 u (t) dt dt dt 2 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 k1u12(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 It is linear since, y1(t) = t and

d u(t) dt

d d d [k u (t)] = tk1 [u (t)] = k1t [u (t)] dt 1 1 dt 1 dt 1 d d d t [k u (t) + k2u2(t)] = k1t u (t) + k2t u (t) dt 1 1 dt 1 dt 2 du(t ) d u(t ) (t ) d (t ) dt

It is realizable but time varying since t

Example 1.4 A system is characterized by the relationship d y(t) = u(t) u(t) dt The system is nonlinear since k1u1(t) and [k1u1(t) + k2u2(t)]

d d k u (t) k1u1(t) u (t) dt 1 1 dt 1

d d d [k u (t) + k2u2(t)] k1u1(t) u (t) + k2u2(t) u (t) dt 1 1 dt 1 dt 2 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 signal is continuous if it is a function of a continuous independent variable t [see Fig. 1.7(a)]. The above definition of continuous signal is broader than the mathematical definition of continuous function. For example the signal f(t) in Fig. 1.7(a) is continuous in the interval t1 < t < t2, but it is not a continuous function of time in the same interval. A signal is discrete if it is a function of 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 continuous function of time and it will be referred to as a discrete system if the signal at any point of the system is discrete function of time.


13Chapter 1t (b)

f(t)

f(t)

t1 (a)

t2

t

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

1.7

TASK OF CONTROL ENGINEERS

In order to put the control engineers in proper perspective, we endeavor to present, at the very outset, the control-engineering problem together with the task to be performed by a control engineer. (1) Objective of the Control System A 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 and sub systems, based on the laws of the physical world, to perform the task. The quality of performance is normally specified in terms of some mathematical specifications, that the control system is required to satisfy. The specifications may be the accuracy of controlled variable in the steady state (the behavior of the system as time goes to infinity following a change in set point or disturbance) or it is concerned with the transient response- the way the control variable is reaching the steady state following some disturbance or change in set value. (2) Control Problem Since 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 Modeling Since some mathematical problems are to be solved, a mathematical model of the dynamics of the control system components and subsystems are to be formulated. The differential equation and the state variable representation are very popular mathematical models for describing the dynamics of a control system. The transfer function model is applicable if the system is linear. If the description of the system behavior is linguistic then fuzzy logic and fuzzy model of the system will be needed [1-3]. (4) System Analysis Once 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 System If 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 modified by incorporating some additional feedback or by incorporating compensator to modify the system

14


behavior. The lag-lead compensator and state feedback design method are widely used for improving the system performance. (6) Optimal Control Among 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 optimal control, where among all the possible solutions, the solution that optimizes the performance specification (like minimization of energy consumption, maximization of production rate or minimization of time and minimization of waste of material) should be chosen. Consider, for instance, the problem of landing of a space vehicle on the surface of the moon from earth. Since the consumption of fuel is an important consideration for space journey, of the innumerable trajectories from the earth to the surface of the moon, the one that will consume minimum fuel will be chosen and the controller will be programmed for that trajectory. In Habers process of manufacturing Ammonia, the yield per unit time is dependent on the temperature of the reaction chamber. Control laws may be designed to maintain temperature of the reaction chamber such that the yield is maximum. (7) Adaptive Control In some, control systems the process parameters change with time or environmental conditions. The controller designed by assuming fixed system parameters fails to produce acceptable performance. The controller for such systems should adapt to the changes in the process such that the performance of the control system is not degraded in spite of the changes in the process. This gives rise to the problem of designing adaptive controller for a system. For example, the transfer functions of an aircraft changes with its level of flight and velocity, so that the effectiveness of the pilots control stick will change with the flight conditions and the gain of the controller is to be adapted with the flight conditions [4]. In the space vehicle the fuel is an integral part of the mass of the vehicle, so the mass of the vehicle will change with time. An Adaptive controller that adapts itself to the changes in mass is expected to perform better. The design of adaptive controller, therefore, becomes an important issue in control engineering problem. (8) Fuzzy Control When 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 rules characterizing 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 a number 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 current through a heater coil. The average power supplied to the heater coil can be manipulated by varying the firing angle of a Triac or Silicon Controlled Rectifier (SCR with a suitable triggering circuit). (b) The form of energy available is steam: The temperature of the fluid inside the kettle will be proportional to the volume of steam coming into contact with the process fluid in the kettle. By connecting a solenoid-operated valve in the pipe through which steam flows, the volume of steam flow per unit time may be manipulated by opening and closing of the valve.


15Chapter 1

(c) The form of energy available is fuel oil: The temperature of an oil-fired furnace may be controlled by controlling the fuel to air ratio as well as the flow rate of fuel oil. By sensing the temperature of the chemical bath one can adjust the flow rate of fuel and its ratio to air by means of a solenoid operated valve to control the temperature of the furnace directly and the temperature of the fluid inside the kettle indirectly. (d) The solar energy as the power source: The solar energy, concentrated by a set of focusing lens after collecting it from reflectors, may be used to control the temperature of the fluid inside the kettle. The amount of solar energy may be regulated by controlling the position of 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 jacketed kettle may be accomplished in a number of alternative ways. Depending on the form of energy available, any of the above methods may be adopted for a given situation. The choice of a particular method depends on many other factors like technical feasibility and economic viability. The important point that is to be emphasized here is that the control objective may be realized by using a number of alternative methods. In each method, the components and subsystems should be assembled, in a meaningful way, by utilizing the knowledge of the physical world. (ii) Room temperature control: Let us consider the problem of controlling room temperature using a room air conditioner as another example. The compressor of the room air conditioner may be switched off if the room temperature is less than the set temperature and switched on if the room temperature is higher than the set value. The compressor is kept on until the room temperature is equal to the set temperature. A thermostat switch senses the temperature in the room and when the temperature of the room goes below the set point, the power to the compressor is again switched off. The variation of ambient temperature outside the room and the escape of cool air due to opening of the door are the external disturbances to the control system and the change in the number of occupants in the room is the load disturbance. The temperature of the room could have been controlled by a central air conditioning system, where the volume of cool air entering the room could have been controlled by opening and closing of a solenoid operated valve.

1.9

A CLOSER LOOK TO THE CONTROL TASK

The presence of reference input element is always implied in a system and with the directly controlled output variable taken as the system output, the block diagram of Fig. 1.4 is simplifiedd(t)

+ r(t)

e(t)

Control elements

u(t)

Process or plant

y(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 be controlled. Its inputs are u(t), its outputs are y(t), and reference input is r(t). In the process fluid control problem, u(t) is the steam input to the jacketed kettle, y(t) is the temperature of

16


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

1.9.1 Mathematical ModelingAfter the control engineer has interconnected the components, subsystems and actuators in a meaningful way to perform the control job, often one of the next tasks that the designer undertakes is the development of a mathematical model of the process to be controlled, in order to gain understanding of the problem. There are only a few ways to actually generate the model. We can use the laws of the physical world to write down a model (e.g., F = mf ). Another way is to perform system identification via the use of real plant data to produce a model of the system [5]. Sometimes a combined approach is used where we use physical laws to write down a general differential equation that we believe represents the plant behavior, and then we 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 is developed to be as accurate as possible so that it can be used in simulation-based evaluation of control systems. It must be understood, therefore, that there is never a perfect mathematical model for the plant. The mathematical model is abstraction and hence cannot perfectly represent all possible dynamics of any physical process (e.g., certain noise characteristics or failure conditions). Usually, control engineers keep in mind that they only need to use a model that is accurate enough to be able to design a controller that works. However, they often need a very accurate 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 or the inclusion of only certain forms of non-linearities) that will capture the essential plant behavior. Indeed, it is quite an art and science to produce good low-order models that satisfy these constraints. Linear models such as the one in Equation (1.4) has been used extensively in 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)In

(1.4)

the classic optimization problem of traveling sales representative, it is required to minimize the total distance traveled by considering various routes between a series of cities on a particular trip. For a small number of cities, the problem is a trivial exercise in enumerating all the possibilities and choosing the 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 the possible routes. However, an optimal solution with an accuracy within 1 percent of the exact solution will require two days of CPU time on a supercomputer which is about 40 times faster than a personal computer for finding the optimum path (i.e., minimum travel time) between 100,00 nodes in a travel network. If the same problem is taken and the precision requirement is increased by a modest amount to value of 0.75 percent, the computing time approaches seven months! Now suppose we can live with an accuracy of 3.5 percent and increase the nodes in the network to 1000,000; the computing time for this problem is only slightly more than three hours [6]. This remarkable reduction in cost (translating time 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 on the situation, but for the vast majority of problems we encounter daily, the answer is a resounding yes.


17Chapter 1

In this case u is the m-dimensional input; x is the n-dimensional state; y is the p-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 appropriate for use with frequency domain design techniques (e.g., Bode plot and Nyquist plots), the rootlocus method, state-space methods, and so on. Sometimes it is assumed that the parameters of the 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 nonlinear models 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 of their arguments. One form of the nonlinear model that has received significant attention is (1.6) x(t) = f(x(t)) + g(x(t)) u(t) 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 the nonlinear models above is the case where f and g are not completely known and subsequent research focuses on robust control of nonlinear systems. Discrete time versions of the above models are also used when a digital computer is used as a controller and stochastic effects are often taken into account via the addition of a random input or other stochastic effects. Stability is the single most important characteristic of a system and the engineer should pay attention to it at the very early stage of the design (e.g., to see if certain variables remained bounded). The engineer should know if the plant is controllable [9] (otherwise the control inputs will not be able to properly affect the plant) and observable (to see if the chosen sensors will allow the controller to observe the critical plant behavior so that it can be compensated for) or if it is non-minimum phase. These properties would have a fundamental impact on our ability to design effective controllers for the system. In addition, the engineer will try to make a general assessment of how the plant dynamics change over time, and what random effects are present. This analysis of the behavior of the plant gives the control engineer a fundamental understanding of the plant dynamics that will be very useful when the time comes to synthesize the 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 achieve your 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 information and 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 simple cases an open-loop controller may provide adequate performance. Assuming that feedback control is used the closed-loop specifications (or performance objectives) can involve the following issues: (i) Disturbance rejection properties: (with reference to the process fluid temperature control problem in Fig. 1.3, the control system should be able to minimize the variations in the process flow rate F). Basically, the need for minimization of the effect of

18


disturbance make the feedback control superior to open-loop control; for many systems it is simply impossible to meet the specifications without feedback (e.g., for the temperature control problem, if you had no measurement of process fluid, how well could you relate the temperature to the users set-point ?). (ii) Insensitivity to plant parameter variations: (e.g., for the process fluid control problem that the control system will be able to compensate for changes in the level of the 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 flow rate disturbances and change in the ambient conditions, the fluid temperature will be 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 for the actual temperature to get close to the desired temperature when there is a step change in the set-point). (v) Overshoot: (when there is a step change in the set point in Fig. 1.3, how much the temperature 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 the temperature 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 often of 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., solar power control system)? (iv) Reliability: Will the controller always perform properly? What is its meantime between failures? (v) Maintainability: Will it be easy to perform maintenance and routine field adjustments to the controller? (vi) Adaptability: Can the same design be adapted to other similar applications so that the cost of later designs can be reduced ? In other words, will it be easy to modify the process fluid temperature controller to fit on different processes so that the development can be just once? (vii) Understandability: Will the right people be able to understand the approach to control? For example, will the people that implement it or test it be able to fully understand it? (viii) Politics: Is your boss biased against your approach? Can you sale your approach to your colleagues? Is your approach too novel (solar power control!) and does it thereby depart too much from standard company practice? The above issues, in addition to meeting technical specifications, must also be taken into consideration and these can often force the control engineer to make decisions that can significantly affect how, for example, the ultimate process fluid controller is designed. It is important then that the engineer has these issues in mind at early stages of the design process.


19Chapter 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 operations today are PID controllers (or some variation of it like a P or PI). This approach is often viewed as simple, reliable, and easy to understand. Often, like fuzzy controllers, 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 design, 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 to check if the designed controller really meets the closed-loop specifications. This can be particularly important in safety-critical applications such as a nuclear power plant control or in aircraft control. However, in some consumer applications such as the control of washing machine or an electric shaver, it may not be as important in the sense that failures will not imply the loss of life (just the possible embarrassment of the company and cost of warranty expenses), so some rigorous evaluation matters can sometimes be ignored. Basically, there are three general ways to verify that a control system is operating properly (1) mathematical analysis based on the use of formal models, (2) simulation based analysis that most often uses formal models, and (3) experimental investigations on the real system. (a) Reliability of mathematical analysis. In the analysis phase one may examine the stability (asymptotically stable, or bounded-input bounded-output (BIBO) stable) and controllability of the system together with other closed-loop specifications such as disturbance rejection, rise-time, overshoot, settling time, and steady-state errors. However, one should not forget the limitations of mathematical analysis. Firstly, the accuracy of the analysis is no better than that of the mathematical model used in the analysis, which is never a perfect representation of the actual plant, so the conclusions that are arrived at from the analysis are in a sense only as accurate as the model itself. And, secondly, there is a need for the development of analysis techniques for even more sophisticated nonlinear systems since existing theory is somewhat lacking for the analysis of complex nonlinear (e.g., fuzzy) control systems, a large number of inputs and outputs, and stochastic effects. In spite of these limitations, the

20


mathematical analysis does not become a useless exercise in all the cases. Sometimes it helps to uncover fundamental problems with a control design. (b) Simulation of the designed system. In the next phase of analysis, the controller and the actual plant is simulated on analog or digital computer. This can be done by using the laws of the physical world to develop a mathematical model and perhaps real data can be used to specify some of the parameter of the model obtained via system identification or direct parameter measurement. The simulation model can often be made quite accurate, and the effects of implementation considerations such as finite word-length restrictions in digital computer 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 realtime simulation of complex systems or in certain laboratory settings. Simulation (digital, analog or hybrid) too has its limitations. First, as with the mathematical analysis, the model that is developed will never be identical with the actual plant. Besides, some properties simply cannot be fully verified through simulation studies. For instance, it is impossible to verify the asymptotic stability of an ordinary differential equation through simulations since a simulation can only run for a finite amount of time and only a finite 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 before investing more time for the implementation of the controller apart from enhancing the engineers confidence about the closed loop behavior of the designed system. (c) Experimental studies. In the final stage of analysis, the controller is implemented and integrated with the real plant and tested under various conditions. Obviously, implementations require significant resources (e.g., time, hardware), and for some plants implementation would not even be thought of before extensive mathematical and simulationbased studies have been completed. The experimental evaluation throws some light on some other issues involved in control system design such as cost of implementation, reliability, and perhaps maintainability. The limitations of experimental evaluations are, first, problems with the repeatability of experiments, and second, variations in physical components, which make the verification only approximate for other plants that are manufactured at other times. Experimental studies, also, will go a long way toward enhancing the engineers confidence after seeing one real system in operation. There are two basic reasons to choose one or all three of the above approaches of performance evaluation. Firstly, the engineer wants to verify that the designed controller will perform properly. Secondly, if the closed-loop system does not perform properly, then the analysis is expected to reveal a way for undertaking a redesign of the controller to meet the specifications.

CHAPTER

2Mathematical PreliminariesIn this chapter we shall discuss briefly some of the mathematical tools used extensively for the analysis and design of control systems.

2.0

THE LAPLACE TRANSFORM

The Laplace transform method is a very useful mathematical tool [10-11] for solving linear differential equations. By use of Laplace transforms, operations like differentiation and integration can be replaced by algebraic operations such that, a linear differential equation can be transformed into an algebraic equation in a complex variable s. The solution of the differential equation may be found by inverse Laplace transform operation simply by solving the algebraic equation involving the complex variable s. Analysis and design of a linear system are also carried out in the s-domain without actually solving the differential equations describing the dynamics of the system. Graphical techniques like Bode plot, Root Locus and Nyquist Diagram employ the Laplace transform method for evaluation of the system performance. Besides, the transient response as well as the steady-state response of a dynamic system can be obtained by the Laplace transform method.

2.1

COMPLEX VARIABLES AND COMPLEX FUNCTIONS

A complex number system is a two dimensional number system having a real part and an imaginary part. In the case of Laplace transformation, we use the notation s to represent the complex number and written as s = + j, where is the real part and is the imaginary part. If the real part and/or imaginary part are variables, the complex number s is called a complex variable.

2.1.1 Complex FunctionAny function of a complex variable will also be a complex function having real and imaginary parts. A complex function G(s) may be expressed in terms of its real and imaginary components as : G(s) = A + jB2 2 where A and B themselves are real quantities. The magnitude of G(s) is || G(s) || = A + B , and the angle of G(s) is tan1 (B/A). The angle is considered positive when measured in the anticlockwise direction from the positive real axis. Associated with each complex function is a

complex conjugate function. The complex conjugate of G(s) is given by G (s) = A jB. If a complex function G(s) together with its derivatives exist in a region, it is said to be analytic in that region. 21

22 The function G1(s) given by : G1(s) =


1 d 1 1 with its derivative = s+2 ds s + 2 (s + 2) 2

is found to exist everywhere, except at s = 2, so it is an analytic

ajit k mandal

Documents