Control Theory: Multivariable and Nonlinear Methods

468 Pages·2000·11.193 MB·English

by  Torkel Glad, Lennart Ljung

Control Theory
Multivariable and Nonlinear Methods
Torkel Glad and Lennart Ljung
London and New York
First published 2000 by Taylor & Francis
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada
by Taylor & Francis
29 West 35th Street, New York, NY 10001
Taylor & Francis is an imprillt of the Taylor & Francis Group
© 2000 Torkel Glad and Lennart Ljung
Translated by the authors from the Swedish Reglerteod: flellJariabla Dell
olinjiira metoder by Torkel Glad and Lennart Ljung. Published by
Studenlitteratur 1997.
Printed and bound in Great Britain by MPG Books Ltd, Bodmin
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publisher nor the authors can accept any legal responsibility or liability for
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Publisher's Note
This book has been prepared from camera-ready copy provided by the
authors
British Library Cataloguing in Publication Data
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ISBN 0-7484-0878-9 (pbk)
ISBN 0-7484-0877-0 (hbk)
Contents
Preface xi
Notation xiii
1 Introd uction 1
1.1 Iv! ultival'iable Systems . . . . . . . . . . . . . . . . . .. 2
1.2 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . 5
1.3 Discrete and Continuous Time Models and Controllers 6
1.4 Some Basic Concepts in Control ... 8
1.5 Gain and Signal Size . . . . . . . . . . 15
1.6 Stability and the Sn1an Gain Theorem 18
1.7 Comments................ 22
PART I LINEAR SYSTEMS 25
2 Representation of Linear Systems 27
2.1 Impulse Response and Weighting Function .. 27
2.2 Transfer Function Matrices 28
2.3 Transfer Operator .... 30
2.4 Input-Output Equations. 30
2.5 State Space Fornl .... 31
2.6 Discrete Time Systems. 37
2.7 Comments ....... . 41
3 Properties of Linear Systems 43
3.1 Solving the Systelll Equations . 43
3.2 Controllability and Observability 44
3.3 Poles and Zeros . . . . . . . . . . 48
304 Stability ............. . 53
3.5 Frequency Response and Frequency Functions. 58
vi Contents
3.6 Model Reduction .... 63
3.7 Discrete Time Systems .. 68
3.8 Comments... 77
3A Proofs ........ . 79
4 Sampled Data Systems 83
4.1 Approximating Continuous Time Systems 83
4.2 System Sampling . . . . . . . . . . 85
4.3 Poles and Zeros . . . . . . . . . . 90
4.4 Controllability and Observability 93
4.5 Frequency Functions 96
4.6 Comments ............. . 99
5 Disturbance Models 101
5.1 Disturbances ............ . . 101
5.2 Signal Size and Scaling. . . . . . . . 103
5.3 Spectral Description of Disturbances 106
5.4 Description of Disturbances in the Time Domain 112
5.5 Estimation of One Signal from Another 114
5.6 Measurement and System Disturbances 116
5.7 Observers and Kalman Filters ..... . 124
5.8 Discrete Time Systems ......... . 131
5.9 Practical Aspects of Signal Sampling .. 139
5.10 Comments. 140
5A Proofs .................. . 141
PART II LINEAR CONTROL THEORY 145
6 The Closed Loop System 147
6.1 The Transfer Functions of the Closed Loop System . 147
6.2 Stability of the Closed System. . . . . . . 150
6.3 Sensitivity and Robustness .... 152
6.4 Specifications . . . . . . . . . . . . 155
6.5 Specifications in the Time Domain 156
6.6 Specifications in the Frequency Domain 159
6.7 Sampled Data Controllers 164
6.8 Comments .............. . 169
7 Basic Limitations in Control Design 171
7.1 Scaling of Variables. . . . . . . . . . . 171
7.2 Intuitive Analysis ......... ....... 172
.,
7.3 Loop Gain Limitations . . . . . . 176
7.4 Sand T: Individual Limitations 179
7.5 Consequences for the System Performance . 186
7.6 Effects of Control Signal Bounds 192
7.7 The M ultivariable Case 195
« .......
7.8 Some Examples .. 201
7.9 Comments .. 208
.. " .... .,
7A Proofs ........... ...... 209
.. .... « • "
B Controller Structures and Control Design 215
8.1 Main Ideas 216
............ " ............. 41;
8.2 Configuration of Mu ltivariable Controllers . 219
8.3 Internal Model Control. . . . . . . . . 228
804 Feedback from Reconstructed States 233
8.5 Comments ............... .............. 237
9 Minimization of Quadratic Criteria: LQG 239
9.1 The Criterion and the Main Ideas ... 239
9.2 The Optimal Controller: Main Results 242
9.3 Some Practical Aspects ..... 247
9.4 Robustness for LQG Controllers 257
9.5 Discrete Time Systems .. 264
9.6 Comments. 271
9A Proofs .......... . 273
10 Loop Shaping 277
10.1 Direct Methods . 277
10.2 Formalization of the Requirements 278
10.3 Optimal1-l2 Control . 284
IDA Optimal 1-loo Control ... 285
10.5 Robust Loop Shaping .. 293
10.6 Discrete Time Systems .. 300
10.7 Comments . . . . . . . . 301
IDA To Decrease the Spread of the Singular Values 303
PART III NONLINEAR CONTROL THEORY 305
11 Describing Nonlinear Systems 307
11.1 Linear Versus Nonlinear . . . . 308
viii Contents
11.2 Examples of Nonlinear Systems . 312
11.3 Mathematical Description . . . 314
11.4 Equilibria and Linearization . . 317
11.5 Comments. . . . . . « . . . . . 319
12 Stability of Nonlinear Systems 321
12.1 Stability of Equilibria .. . .. 321
12.2 Stability and Lyapunov Functions 323
12.3 The Circle Criterion 329
12.4 Comments . . 335
12A Proofs ....... . 337
13 Phase Plane Analysis 339
13.1 Phase Planes for Linear Systems 339
13.2 Phase Planes for Nonlinear Systems 346
13.3 Comments .............. . 352
14 Oscillations and Describing Functions 355
14.1 The Describing Function Method ...... « .. « . 356
14.2 Computing Amplitude and Frequency of Oscillations 360
14.3 Comments ...... « . « . 364
14A Some Describing Functions ......... . 366
15 Controller Synthesis for Nonlinear Systems 371
15.1 Linear Design and Nonlinear Verification. 371
15.2 Nonlinear Internal Model Control. 372
15.3 Parametric Optimization ... . 374
15.4 Other Approaches ...... . 377
15.5 State Feedback and Observer 378
15.6 Comments« ...... . 382
16 Model Predictive Control 383
16.1 Basic Idea: Predict the Output 383
16.2 k-step Prediction for Linear Systems 384
16.3 The Criterion and the Controller 385
16.4 Comments. . . . . . . . . . . . . . . 391
17 To Compensate Exactly for Nonlinearities 393
17.1 Examples of Exact Linearization 393
17.2 Relative Degree ...... . 395
17.3 Input-Output Linearization .... 397
Contents IX
17.4 Exact State Linearization 401
17.5 Comments. . . . . . . . . 405
18 Optimal Control 407
18.1 The Goddard Rocket Problem. 407
18.2 The Maximum Principle . . . . . . . . .. 410
18.3 Solution of the Goddard Rocket Problem 424
18.4 Minimum Time Problems 428
18.5 Optimal Feedback .. . 436
18.6 Numerical Methods ... . 442
18.7 Comments . . . . . . . . . 444
18A Proof of the Maximum Principle 445
19 Some Final Words 451
Preface
This book is intended for someone with basic knowledge of control
theory who wants only one additional book on control in the bookshelf.
Control theory is subject to rapid development and many new
methods have been introduced in recent years. There are now also
many new control design packages that implement advanced control
synthesis techniques. One objective with this book is to provide
sufficient background and understanding so that such packages call
be used in an insightful manner. Another objective is to describe
and discuss basic aspects of control design, the approaches, the key
quantities to consider and the fundamental limitations inherent in the
designs.
The book can be used for self study and as a conlplement to program
manuals, but is primarily intended as a text book for one or several
undergraduate or graduate control courses. It talms off where a typical
undergraduate control course ends. This means that we assulne the
reader to be familiar with continuous time, single-input-single-output
systems, with frequency functions and elementary state space theory.
This corresponds to the material in e.g.,
Franklin, G. F., Powell, J. D. & Amami-Naeini, A. (1994), Feedback
Control of Dynamic Syst.ems, 3rd edn, Addison-Wesley, Reading,
MA.
or any equivalent text book.
The material can be studied in several different ways. It is, for
exalnple, possible to leave out the parallel treatment of discrete tinle
systenls, since the basic properties are described in detail for continuous
time systems. Part III, on non-linear systems can be used separately,
etc. Please visit the home page of the book
http://www.control.isy.liu.se/controlbook
xii
for ideas on course curricula, and for exercises and software.
Many people have helped us in our work. Professor Dan Rivera
taught us IMC and MPC and also helped us with the examples on
the distillation column. Professor Anders Helmersson taught us Hoo
methods and helped us with Example 10.5. Dr. Lars Rundqwist, SAAB
AB .. supplied us with the Gripen aircraft dynamics and "approved"
our design. Professor Per-Olof Gutman shared his knowledge of QFT.
Professor Keith Glover has given valuable comments on Chapter 10.
Many students in Linkoping have supplied important comments on
the material during its development. The book has also been used
at the Royal Institute of Technology in Stockholm and Professor Alf
Isaksson and Dr. Anders Hansson have shared their experiences in a
very useful way.
A number of colleagues have commented the manuscript. We
want to mention especially Dr. Bo Bernhardsson, Dr. Claes Bretiholz,
Professor Bo Egardt, Dr. Svante Gunnarsson, Professor Per Hagander,
Dr. Kjell Nordstrom, M.Sc. Per Spangeus and Professor Torsten
Soderstroill.
Mrs. Ulla Salaneck has helped us with the translation of the
book and given us invaluable support with many other things in the
preparation of the material.
We thank all these people for their help.