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Elsevier Academic Press
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Library of Congress Cataloging-in-Publication Data
Ellis, George (George H.)
Control system design guide: a practical guide/George Ellis.Ð3rd ed.
p. cm.
ISBN 0-12-237461-4 (hardcover : alk. paper)
1. Automatic control. 2. System design. I. Title.
TJ213.E5625 2003
629.8
H
3Ðdc22 2003023742
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN: 0-12-237461-4
For all information on all Academic Press publications
visit our website at www.academicpress.com
Printed in the United States of America
040506070809 987654321
//SYS21/F:/PAGINATION/ELSEVIER US/CSDG/3B2/FINALS_03-01-04/PRELIMS.3D ± 5 ± [1±24/24] 12.1.2004 7:20PM
To my loving wife, LeeAnn, and to Gretchen and Brandon, who both make us proud.
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//SYS21/F:/PAGINATION/ELSEVIER US/CSDG/3B2/FINALS_03-01-04/PRELIMS.3D ± 7 ± [1±24/24] 12.1.2004 7:20PM
Contents
Preface xxi
Section I Applied Principles of Controls 1
Important Safety Guidelines for Readers 3
Chapter 1 Introduction to Controls 5
1.1 Visual ModelQ Simulation Environment 6
1.1.1 Installation of Visual ModelQ 6
1.1.2 Errata 6
1.2 The Control System 7
1.2.1 The Controller 7
1.2.2 The Machine 8
1.3 The Controls Engineer 8
Chapter 2The Frequency Domain 11
2.1 The Laplace Transform 11
2.2 Transfer Functions 12
2.2.1 What Is s?12
2.2.1.1 DC Gain 13
2.2.2 Linearity, Time Invariance, and Transfer
Functions 13
2.3 Examples of Transfer Functions 14
2.3.1 Transfer Functions of Controller Elements 15
2.3.1.1 Integration and Differentiation 15
2.3.1.2 Filters 15
2.3.1.3 Compensators 15
2.3.1.4 Delays 15
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2.3.2 Transfer Functions of Power Conversion 16
2.3.3 Transfer Functions of Physical Elements 16
2.3.4 Transfer Functions of Feedback 18
2.4 Block Diagrams 18
2.4.1 Combining Blocks 18
2.4.1.1 Simplifying a Feedback Loop 19
2.4.2 Mason's Signal Flow Graphs 20
2.4.2.1 Step-by-Step Procedure 20
2.5 Phase and Gain 22
2.5.1 Phase and Gain from Transfer Functions 23
2.5.2 Bode Plots: Phase and Gain versus Frequency 24
2.6 Measuring Performance 25
2.6.1 Command Response 25
2.6.2 Stability 27
2.6.3 Time Domain versus Frequency Domain 28
2.7 Questions 29
Chapter 3 Tuning a Control System 31
3.1 Closing Loops 31
3.1.1 The Source of Instability 32
3.2 A Detailed Review of the Model 34
3.2.1 Integrator 34
3.2.2 Power Converter 36
3.2.3 PI Control Law 37
3.2.4 Feedback Filter 38
3.3 The Open-Loop Method 39
3.4 Margins of Stability 40
3.4.1 Quantifying GM and PM 40
3.4.2 Experiment 3A: Understanding the
Open-Loop Method 41
3.4.3 Open Loop, Closed Loop, and the
Step Response 43
3.5 A Zone-Based Tuning Procedure 45
3.5.1 Zone One: Proportional 46
3.5.2 Zone Two: Integral 47
3.6 Variation in Plant Gain 48
3.6.1 Accommodating Changing Gain 50
3.7 Multiple (Cascaded) Loops 50
3.8 Saturation and Synchronization 51
3.8.1 Avoid Saturation When Tuning 54
3.9 Questions 54
Chapter 4 Delay in Digital Controllers 57
4.1 How Sampling Works 57
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CONTENTS
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4.2 Sources of Delay in Digital Systems 58
4.2.1 Sample-and-Hold Delay 58
4.2.2 Calculation Delay 60
4.2.3 Velocity Estimation Delay 60
4.2.4 The Sum of the Delays 61
4.3 Experiment 4A: Understanding Delay in Digital Control 61
4.3.1 Tuning the Controller 62
4.4 Selecting the Sample Time 64
4.4.1 Aggressive Assumptions for General Systems 65
4.4.2 Aggressive Assumptions for Position-Based
Motion Systems 65
4.4.3 Moderate and Conservative Assumptions 66
4.5 Questions 67
Chapter 5 The z-Domain 69
5.1 Introduction to the z-Domain 69
5.1.1 De®nition of z 69
5.1.2 z-Domain Transfer Functions 70
5.1.3 Bilinear Transform 71
5.2 z Phasors 71
5.3 Aliasing 73
5.4 Experiment 5A: Aliasing 74
5.4.1 Bode Plots and Block Diagrams in z 76
5.4.2 DC Gain 76
5.5 From Transfer Function to Algorithm 76
5.6 Functions for Digital Systems 78
5.6.1 Digital Integrals and Derivatives 78
5.6.1.1 Simple Integration 78
5.6.1.2 Alternative Methods of Integration 80
5.6.2 Digital Derivatives 81
5.6.2.1 Inverse Trapezoidal Differentiation 82
5.6.2.2 Experiment 5B: Inverse Trapezoidal
Differentiation 84
5.6.3 Sample-and-Hold 85
5.6.4 DAC/ADC: Converting to and from Analog 86
5.7 Reducing the Calculation Delay 87
5.8 Selecting a Processor 88
5.8.1 Fixed- and Floating-Point Math 88
5.8.2 Overrunning the Sample Time 89
5.8.3 Other Algorithms 90
5.8.4 Ease of Programming 90
5.8.5 The Processor's Future 90
5.8.6 Making the Selection 90
CONTENTS
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5.9 Quantization 91
5.9.1 Limit Cycles and Dither 91
5.9.2 Offset and Limit Cycles 93
5.10 Questions 94
Chapter 6 Six Types of Controllers 97
6.1 Tuning in This Chapter 98
6.2 Using the Proportional Gain 98
6.2.1 P Control 99
6.2.1.1 How to Tune a Proportional Controller 100
6.3 Using the Integral Gain 102
6.3.1 PI Control 103
6.3.1.1 How to Tune a PI Controller 103
6.3.1.2 Analog PI Control 104
6.3.2 PI
Control 107
6.3.2.1 Comparing PI
and PDFF 108
6.3.2.2 How to Tune a PI
Controller 108
6.4 Using the Differential Gain 111
6.4.1 PID Control 112
6.4.1.1 How to Tune a PID Controller 112
6.4.1.2 Noise and the Differential Gain 115
6.4.1.3 The Ziegler±Nichols Method 115
6.4.1.4 Popular Terminology for PID Control 117
6.4.1.5 Analog Alternative to PID: Lead-Lag 117
6.5 PID Control 118
6.5.1 How to Tune a PID
Controller 119
6.6 PD Control 121
6.6.1 How to Tune a PD Controller 121
6.7 Choosing the Controller 124
6.8 Experiments 6A±6F 124
6.9 Questions 125
Chapter 7 Disturbance Response 127
7.1 Disturbances 128
7.1.1 Disturbance Response of a Power Supply 130
7.2 Disturbance Response of a Velocity Controller 134
7.2.1 Time Domain 136
7.2.1.1 Proportional Controller 137
7.2.2 Frequency Domain 137
7.3 Disturbance Decoupling 140
7.3.1 Applications for Disturbance Decoupling 141
7.3.1.1 Power Supplies 141
7.3.1.2 Multizone Temperature Controller 142
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CONTENTS
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7.3.1.3 Web Handling 143
7.3.2 Experiment 7B: Disturbance Decoupling 145
7.4 Questions 149
Chapter 8 Feed-Forward 151
8.1 Plant-Based Feed-Forward 151
8.1.1 Experiment 8A: Plant-Based Feed-Forward 152
8.2 Feed-Forward and the Power Converter 154
8.2.1 Experiment 8B: Power Converter Compensation 156
8.2.2 Increasing the Bandwidth vs. Feed-Forward
Compensation 159
8.3 Delaying the Command Signal 160
8.3.1 Experiment 8C: Command-Path Delay 161
8.3.2 Experiment 8D: Power Converter Compensation
and Command Path Delay 162
8.3.3 Tuning and Clamping with Feed-Forward 164
8.4 Variation in Plant and Power Converter Operation 165
8.4.1 Variation of the Plant Gain 166
8.4.2 Variation of the Power Converter Operation 167
8.5 Feed-Forward for the Double-Integrating Plant 167
8.6 Questions 168
Chapter 9 Filters in Control Systems 171
9.1 Filters in Control Systems 171
9.1.1 Filters in the Controller 172
9.1.1.1 Using Low-Pass Filters to Reduce Noise
and Resonance 172
9.1.1.2 Using Low-Pass Filters to Reduce Aliasing 173
9.1.1.3 Using Notch Filters for Noise and Resonance 174
9.1.2 Filters in the Power Converter 175
9.1.3 Filters in the Feedback 175
9.2 Filter Passband 175
9.2.1 Low-Pass Filters 176
9.2.1.1 First-Order Low-Pass Filters 176
9.2.1.2 Second-Order Low-Pass Filters 176
9.2.1.3 A Simple Model for a Closed Loop System 178
9.2.1.4 Higher-Order Low-Pass Filters 178
9.2.1.5 Butterworth Low-Pass Filters 178
9.2.2 Notch 180
9.2.3 Experiment 9A: Analog Filters 182
9.2.4 Bi-Quad Filters 182
9.3 Implementation of Filters 183
9.3.1 Passive Analog Filters 184
9.3.2 Active Analog Filters 184
CONTENTS
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9.3.3 Switched Capacitor Filters 184
9.3.4 IIR Digital Filters 185
9.3.4.1 First-Order Low-Pass IIR Filter 185
9.3.4.2 Second-Order IIR Filter 186
9.3.4.3 Experiment 9C: Digital Filters 186
9.3.4.4 Higher-Order Digital Filters 187
9.3.5 FIR Digital Filters 187
9.4 Questions 188
Chapter 10 Introduction to Observers in Control Systems 191
10.1 Overview of Observers 191
10.1.1 Observer Terminology 192
10.1.2 Building the Luenberger Observer 193
10.1.2.1 Two Ways to Avoid G
s
(S) T1 194
10.1.2.2 Simulating the Plant and Sensor in Real
Time 195
10.1.2.3 Adding the Observer Compensator 196
10.2 Experiments 10A±10C: Enhancing Stability with an Observer 196
10.2.1 Experiment 10D: Elimination of Phase Lag 200
10.3 Filter Form of the Luenberger Observer 201
10.3.1 Low-Pass and High-Pass Filtering 203
10.3.2 Block Diagram of the Filter Form 204
10.3.3 Comparing the Loop and Filter Forms 204
10.4 Designing a Luenberger Observer 205
10.4.1 Designing the Sensor Estimator 206
10.4.1.1 Sensor Scaling Gain 206
10.4.2 Sensor Filtering 207
10.4.3 Designing the Plant Estimator 207
10.4.3.1 Plant Scaling Gain (K) 208
10.4.3.2 Order of Integration 209
10.4.3.3 Filtering Effects 209
10.4.3.4 Experiment 10E: Determining the Gain
Experimentally 209
10.4.4 Designing the Observer Compensator 211
10.5 Introduction to Tuning an Observer Compensator 211
10.5.1 Step 1: Temporarily Con®gure the Observer for
Tuning 213
10.5.2 Step 2: Adjust the Observer Compensator for
Stability 214
10.5.2.1 Modifying the Tuning Process for
Noncon®gurable Observers 214
10.5.2.2 Tuning the Observer Compensator
Analytically 215
10.5.2.3 Frequency Response of Experiment 10G 215
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CONTENTS
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10.5.3 Step 3: Restore the Observer to the Normal
Luenberger Con®guration 217
10.6 Questions 217
Section II Modeling 219
Chapter 11 Introduction to Modeling 221
11.1 What Is a Model? 221
11.2 Frequency-Domain Modeling 222
11.2.1 How the Frequency Domain Works 222
11.3 Time-Domain Modeling 224
11.3.1 State Variables 224
11.3.1.1 Reducing Multiple-Order Equations 224
11.3.1.2 Matrix Equations 225
11.3.1.3 Time-Based Simulation 226
11.3.2 The Modeling Environment 226
11.3.2.1 The Differential Equation Solver 226
11.3.2.2 Advanced Differential Equation Solvers 228
11.3.2.3 Selecting ÁT 228
11.3.3 The Model 229
11.3.3.1 Initial Conditions 229
11.3.3.2 Writing the Modeling Equations 230
11.3.3.3 Modeling an RC Circuit 230
11.3.3.4 Modeling a Two-Pole Low-Pass Filter 231
11.3.3.5 Modeling an Analog PI Controller 232
11.3.3.6 Modeling a Digital PI Controller 234
11.3.3.7 Adding Calculation Delay 236
11.3.3.8 Adding Saturation 236
11.3.4 Frequency Information from Time-Domain
Models 237
11.4 Questions 238
Chapter 12Nonlinear Behavior and Time Variation 239
12.1 LTI Versus non-LTI 239
12.2 Non-LTI Behavior 240
12.2.1 Slow Variation 240
12.2.2 Fast Variation 241
12.3 Dealing with Nonlinear Behavior 242
12.3.1 Modify the Plant 242
12.3.2 Tuning for Worst Case 243
12.3.3 Gain Scheduling 243
12.4 Ten Examples of Nonlinear Behavior 245
12.4.1 Plant Saturation 245
CONTENTS
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xiii
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