Control Systems with Actuator Saturation: Analysis and Design

396 Pages·2001·12.38 MB·English

by  Tingshu Hu

Control Engineering
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Control Systems with Actuator Saturation
T. Hu, Z. Lin
Control Systems with
Actuator Saturation
Analysis and Design
Tingshu Hu
Zongli Lin
With 67 Figures
Springer Science+Business Media, LLC
Tingshu Hu
Zongli Lin
Department of Electrical Engineering
University of Virginia
Charlottesville, V A 22903
USA
Library of Congress Cataloging-in-Publication Data
Hu, Tingshu.
Control systems with actuator saturation : analysis and design / Tingshu Hu, Zongli Lin.
p. cm.-{Control engineering)
Includes bibliographical references and indeX.
ISBN 978-1-4612-6661-7 ISBN 978-1-4612-0205-9 (eBook)
DOI 10.1007/978-1-4612-0205-9
1. Automatic control. 2. Actuators. 1. Lin, Zongli, 1964- II. Title. III. Control
engineering (Birkhăuser)
TJ213.H73 2001
629.8-dc21 2001025374
CIP
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
Origina1ly published by Birkhăuser Boston in 2001
Softcover reprint of the hardcover 1s t edition 2001
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ISBN 978-1-4612-6661-7 SPIN 10833455
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Typeset by the authors in TeX.
9 8 765 432 l
To Jianping and Sylvia
T.H.
To Jian, Tony, and Vivian
Z.L.
Contents
Preface xiii
1 Introduction 1
1.1 Linear Systems with Actuator Saturation 1
1.2 Notation, Acronyms, and Terminology .. 3
2 Null Controllability - Continuous-Time Systems 11
2.1 Introduction .................... . 11
2.2 Preliminaries and Definitions .......... . 12
2.3 General Description of Null Controllable Region 15
2.4 Systems with Only Real Eigenvalues .. . 21
2.5 Systems with Complex Eigenvalues ... . 27
2.6 Some Remarks on the Description of C(T) 33
2.7 Asymptotically Null Controllable Region. 34
2.8 Conclusions................. 35
3 Null Controllability - Discrete-Time Systems 37
3.1 Introduction .................. . 37
3.2 Preliminaries and Definitions . . . . . . . . . 38
3.3 General Description of Null Controllable Region 41
3.4 Systems with Only Real Eigenvalues 44
3.5 Systems with Complex Eigenvalues . . . . 48
3.6 An Example. . . . . . . . . . . . . . . . . 50
3.7 Asymptotically Null Controllable Region . 51
3.8 Conclusions........ 53
Vll
viii Contents
4 Stabilization on Null Controllable Region
Continuous-Time Systems 55
4.1 Introduction ................. . . 55
4.2 Domain of Attraction - Planar System under
Saturated Linear Feedback ......... . 57
4.3 Semi-Global Stabilization - Planar Systems 67
4.4 Semi-Global Stabilization- Higher Order Systems 74
4.5 Conclusions.............. . . . .. 83
5 Stabilization on Null Controllable Region-
Discrete-Time Systems 85
5.1 Introduction ................ . 85
5.2 Global Stabilization at Set of Equilibria-
Planar Systems . . . . . . . . . . . . . . . 86
5.3 Global Stabilization - Planar Systems. . . 99
5.4 Semi-Global Stabilization - Planar Systems 105
5.5 Semi-Global Stabilization - Higher Order Systems. 108
5.6 Conclusions . ...... . . . ..... .... .. . 111
6 Practical Stabilization on Null Controllable Region 113
6.1 Introduction....... . ....... 113
6.2 Problem Statement and Main Results ... . . . .. . 114
6.2.1 Problem Statement. . . . . . . . . . . . . . . . 114
6.2.2 Main Results: Semi-Global Practical Stabilization 114
6.3 Proof of Main Results . . . . . . . . . . . . . . . . . 1.1 5
6.3.1 Properties of the Trajectories of Second Order
Linear Systems . . . . .. ... 115
6.3.2 Properties of the Domain of Attraction 119
6.3.3 Proof of Theorem 6.2.1: Second Order Systems 127
6.3.4 Proof of Theorem 6.2.1: Higher Order Systems 141
6.4 An Example. . . 144
6.5 Conclusions.. 147
6.A Proof of Lemma 6.3.1 149
6.B Proof of Lemma 6.3.2 153
7 Estimation of the Domain of Attraction under
Saturated Linear Feedback 157
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 157
Contents ix
7.2 A Measure of Set Size .............. . 159
7.3 Some Facts about Convex Hulls ......... . 160
7.4 Continuous-Time Systems under State Feedback 163
7.4.1 A Set Invariance Condition Based on
Circle Criterion . . . . . . . . . . . . . . . 164
7.4.2 An Improved Condition for Set Invariance . 165
7.4.3 The Necessary and Sufficient Condition
Single Input Systems. . . . . . . . . . . 167
7.4.4 Estimation of the Domain of Attraction 169
7.5 Discrete-Time Systems under State Feedback . 173
7.5.1 Condition for Set Invariance ...... . 173
7.5.2 The Necessary and Sufficient Condition
Single Input Systems . . . . . . . . . . . 177
7.5.3 Estimation of the Domain of Attraction 179
7.6 Extension to Output Feedback 180
7.7 Conclusions ............... . 181
8 On Enlarging the Domain of Attraction 183
8.1 Introduction ....... . 183
8.2 Continuous-Time Systems 183
8.3 Discrete-Time Systems 185
8.4 Conclusions ....... . 191
9 Semi-Global Stabilization with Guaranteed
Regional Performance 195
9.1 Introduction ................. . 195
9.2 Expansion of the Domain of Attraction 197
9.3 Semi-Globalization - Discrete-Time Systems 199
9.4 Semi-Globalization - Continuous-Time Systems 205
9.5 An Example. 207
9.6 Conclusions ............. . 208
10 Disturbance Rejection with Stability 211
10.1 Introduction ........ . 211
10.2 Continuous-Time Systems . . . . . . 213
10.2.1 Problem Statement ..... . 213
10.2.2 Condition for Set Invariance . 214
x Contents
10.2.3 Disturbance Rejection with Guaranteed Domain
of Attraction . . 216
10.2.4 An Example 219
10.3 Discrete-Time Systems . 221
10.3.1 Problem Statement. 221
10.3.2 Condition for Set Invariance . 223
10.3.3 Disturbance Rejection with Guaranteed Domain
of Attraction 225
10.4 Conclusions . . . . . . . . . . . . . . . 228
11 On Maximizing the Convergence Rate 229
11.1 Introduction .............. . 229
11.2 Continuous-Time Systems. . . . . . . . . . . . . . . 233
11.2.1 Maximal Convergence Control and Maximal
Invariant Ellipsoid . . . . . . . 233
11.2.2 Saturated High Gain Feedback . . . . . . . . 242
11.2.3 Overall Convergence Rate. . . . . . . . . . . 247
11.2.4 Maximal Convergence Control in the Presence
of Disturbances . 255
11.3 Discrete-Time Systems . 258
11.4 Conclusions . . . . . . . 264
12 Output Regulation - Continuous-Time Systems 265
12.1 Introduction. . . . . . . . . . . . . . . . . . . . . 265
12.2 Preliminaries and Problem Statement ...... 267
12.2.1 Review of Linear Output Regulation Theory 267
12.2.2 Output Regulation in the Presence of
Actuator Saturation 270
12.3 The Regulatable Region . . 271
12.4 State Feedback Controllers 279
12.5 Error Feedback Controllers 290
12.6 An Example. 297
12.7 Conclusions . . . . . . . . . 301
13 Output Regulation - Discrete-Time Systems 305
13.1 Introduction. . . . . . . . . . . . . . . . . . . 305
13.2 Preliminaries and Problem Statement .... 306
13.2.1 Review of Linear Output Regulation Theory 306
Contents Xl
13.2.2 Output Regulation in the Presence of
Actuator Saturation 307
13.3 The Regulatable Region .. 309
13.4 State Feedback Controllers 315
13.5 Error Feedback Controllers 324
13.6 Conclusions ........ . 325
14 Linear Systems with Non-Actuator Saturation 327
14.1 Introduction ....... . ......... . .. . 327
14.2 Planar Linear Systems under State Saturation-
Continuous-Time Systems . . . . . . . . . . . . . 328
14.2.1 System Description and Problem Statement. 328
14.2.2 Main Results on Global Asymptotic Stability 328
14.2.3 Outline of the Proof . . . . . . . . . . . . 330
14.3 Planar Linear Systems under State Saturation-
Discrete-Time Systems . . . . . . . . . . . . . . . 344
14.3.1 System Description and Problem Statement. 344
14.3.2 Main Results on Global Asymptotic Stability 344
14.3.3 Outline of the Proof . . . . . . . . . 347
14.4 Semi-Global Stabilization of Linear Systems
Subject to Sensor Saturation 362
14.4.1 Introduction 362
14.4.2 Main Results 363
14.4.3 An Example 370
14.5 Conclusions 371
Bibliography 375
Index 387