Nonlinear and Adaptive Control Design (Adaptive and Learning Systems for Signal Processing, Communications and Control Series)

564 Pages·1995·15.96 MB·English

by  Miroslav Krstic

Adaptive and Learning Systems for Signal Processing,
Communications. and Control
Editor Simon Haykln
Werbos I THE ROOTS OF BACKPROPAGATION: FROM ORDERED
DERIVATIVES TO NEURAL NETWORKS AND POLITICAL FORECASTING
Krstic. Kanellakopoulos. and Kokotovlc I NONLINEAR AND
ADAPTIVE CONTROL DESIGN
Nlklas and Shoo / SIGNAL PROCESSING WITH ALPHA-STABLE
DISTRIBUTIONS AND APPLICATIONS
NONLINEAR
AND ADAPTIVE
CONTROL DESIGN
Mlroslav Krstlc
loannis Kanellakopoulos
Petar Kokotovic
A WILEV-JNTERSCIENCE PUBLICATION
JOHN WILEY & SONS, INC.
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Krstic. Miroslnv.
Nonlineor and adaptive control design / MirosJav Krstic. loannis
K;anell'lkopoulos. Petar Kokolovic.
p. cm. - (Adaptive lind learning systems for signlll
processing. communications, and control)
"A Wiley-Intcrscience publication."
Includes bibliographical references and index,
ISBN 0-471-12732-9
1. Automatic control. 2. Nonlinear conlrol tlteory. 3. Adaptive
conlrol systems. I. KancllakopouJos. Ionnnis. II. Koklltovic.
Petnr III. Title. IV. Series.
TJ213.K748 1995
629.R-dc.2U 95-10082
ell'
Printed in the Unilcd States of America
10 9 R 7 6 S 4 3 .2 I
Preface
This book opens a view to the largely une~"'Plored hmdscape of nonlinear sys
tems with un('ertainties, Its main subject is feedback design for sllch syst.pm!;.
New design t.oo]s and systcmatic desigll procedures arc developed which guar
antee t.hat. t.he designed feedback systems will possess desired properties not.
ouly locally, hut. also globally 01' in a Rpecjfjed region of the state spacc.
Backstepping. Compared with other books on nonlim-'ar control, the:' mAjor
novelty of this book is a recursive design methodology: backstepP·il1D. \Vith
t.his mcthodology the constl'uctioll of bot,h feedback ('out,rol laws aud assod
at-ed Lyapullov fUllctions is systenutti('. Strong propf'rt.ies of global 01' regional
sta.bility and tracking arc built into the llol1line~tr system in u. number of steps,
which is never higher than the system order. While fecdback linellIization
methods require precise models and onen cauccl s011le useful llonliuel:tritips.
hackstepping designs offpr a choice of design tools for accommodation of tln
('el'tain nonlinearit.ies and can a.void wa."teflll cancellations.
Classes of systelns. By the very nature of the modeling pr[)('f'SS, llolllineal'
models are morc structured dmn their linear offsprillg. IVhlllY llonlinetlrities
introduced by physical laws (e.g., centrifugal [orces amI chemical kinetics) CUll
not. be confined to a lineal' sector, because, with increasing magnitudes, their
growth is po1ynomial 01' even exponentia1. Systems with sitch nonlillenritics
are prOllP to explosive iustabilities which 111ust be Rnti['ipaterl and prevented
by fl'edbuck control.
N on1ille~u-itics usually appear multiplied with physi('al COllstlll1ts, often
poorly known or dependent ou the slmvly cbanging environlllent. This commOll
form of uneertainty is cttptured by llonlillettr models with lluknown COllsttlnt
para,nu.lters. In ~'parametric pure-feedbttck." systems studied in this book the
t
growth of llonlillearities is ullrestri('tE'd~ whilE' the lIokoO\vn pltrtUlleters tlppear
linearly. \\Then only 8U output is measllred the llonlillcaritics are assumed t.o
t
be fuud-iolls of the output. The cousidercd dass of nonliucar systems is broad
ttlld Pllcompasses liuetu' systems as a spf'cia] c~tse.
Adaptive control. The control of nonlinear s.),stP.IlU; with unknown ptU-alll
f'ters is traditionally app)'oached as an adaptive control problem. separate from
the rest of nonlinear controJ theory. Here we bridge t:his gap and let. adaptive
vi PREFACE
control be what it is-a paradigm for constructing nonlinear dynamic feed
back for both nonlinear and linear systems, We depart from the traditional
"certainty equivalence" approach and design stronger nonline81' control laws
that achieve the desired objectives either by interacting with parameter up
date laws or by a.ttenua.ting the effect of parameter estimation errors. In this
way we achieve not only stronger stability properties, but also quantifiable
improvements of transient performance.
Applications. Strollg regional or global properties achieved by the design
methods presented in this book have the potential to expand the operating
range of feedback controllers and make tllem applicable to physical plants fol'
which linear or other local controllers are inadequate. In this text, application
exaJnples of this type include automotive suspensions, jet engine stall and surge
control, biochemical processes, aircraft wing rock control. induction motors,
robotic manipulators, and magnetic levitation.
Organization of the book. Although Illost of the results in this book
are new! they are presented at a level accessible to audiences with a standard
undergraduate background in control theory and a basic knowledge of stability
concepts. Each chapter is written in a pedagogical style, with detailed proofs
and illustrative e.xRmples. !vIore inyolved extensions and generalizations are
only outlined, Basic tools for nOlladaptive backstepping design with state
feedback are given ill Chapter 2, and with output feedback in Chapter 7, while
adaptive backstepping is introduced in Chapter 3. CJlapters 1, 2, 3, and 7 are
written as a short te..xt for a part of either a course on nonlinear systems or
on adaptive control. The remaining chapters, although more advanced, are
accessible without Chapters 2, 3, and 7. The core of the controller design
is the tuning functions method in Chapters 4 and 8. The modular designs
in Chapters 5, 6, and 9 achieve the independence of the controller and the
identifier. The new adaptive nonlinear designs are applied to linear systems in
Chapter 10 and are shown to outperform the traditional linear designs. Tbis
chapter can be read independently from t,Jle rest of the book. Thus, depending
on the reader's interest, the book can be read in several ways:
Chapters
• Nonlinear stabilization without adaptation ... , ... . 2, 7
• Introduction to adaptive backstepping .. , ... , .... . 2,3,7
• Adaptive Lyapunov design with tuning fUllctions ., 2,4,8, 10
• Estimatiou-based design with modularity ... , .... . 2, 5, 6, 9, 10
• State feedback adaptive designs ...... , .... ,., .... . 2,4,5,6
• Adaptive control of linear systems ............... . 10
PREFACE vii
This book is a joint effort based on recent research of its authors. Chap
ters 4,5,6,8,9, and 10 and Appendices A-F were written by the first author,
and Chapters 2,3, and 7 and Appendix G were written by the second author.
Their labor was arduous due to frequent revisions by the third author, their
former Ph.D. advisor, who also wrote Chapter 1.
Acknowledgments. At the initial stage of this research, Riccardo Ivlarino
introduced us to nonlinear geometric methods and collaborated Wit11 us and
David Taylor on the early adaptive nonlinear schemes, contributing a key ex
tended matching idea. A separate line of research with Hector Sussmann and
Ali Saberi led us to a nonadaptive recursive design. About a year later, ill a
joint effort with Steve Morse, we developed the first adaptive recursive design
procedure, which we named adaptive backst.epping. We express our deep
est gratitude to these colleagues. We are also thankful to Randy Freeman,
Petros Ioannou, Laurent Praty, and Andy Teel for continuous exchange and
discussion of current results. Among many other colleagues who helped us
with frequent debates on broader issues of nonlinear and adaptive control are
Brian Anderson, Karl A.strom, Tamer B~ar, Gil Blankenship, Bill Boothby,
lvlohammed Dahleh, Reza Ghanadan, Graham Goodwin, Jessy Grizzle, Al
berto !sidori, Nlrdjan Jankovic, Hassan Khalil, Art Krener, P. R. Kumar, loan
Landau, David Mayne, Rick NIiddleton, Carl Nett, Shankar Sastry, Rodolphe
Sepulchre, Eduardo Sontag, !vIar], Spong, Jillg Sun, and Gang Tao. And above
all, our warmest thanks go to the most generous cOlltributOl"S t.o this project,
our wives, Angela, Georgia, and Anna.
The writing of this book was supported in part by the National Science
Foundation under Grants ECS-9203401 and RIA ECS-9309402, by the Air
Force Office of Scientific Research under Grant F-49620-92-.l-0495, by UCLA
through the SEAS Dean's Fund, and also by grants from Ford Motor Company
and Rockwell International.
lvlIROSLAV KRSTIC
IOANNIS KANELLAl(OPOULOS
PETAR I(OJ<OTOVIC
Santa. Barbara, Californ,ia
Contents
1 Introduction 1
1.1 Adaptive Linear Control 1
1.1.1 Emergence of adaptiye control 1
1.1.2 Acbievempl1ts of adapj;jve linear cont.rol 2
1.1.3 Ad~tptive control as dynamic llouJinear fepdbuck 3
1.1.-1 Lyapunov-based design 4
1.1.5 Esthmttioll-hased design 5
1.2 Ad~tpt.ive Nonlinear Cout.rol 7
1.2.1 A nonlinear clutllenge 7
1.2.2 A structural obst~tcle 9
1.2.3 Early results 10
1.3 Preview of t.he lvlaill Topics 11
1.3.1 Classes of nonlinear systems 11
1.3.2 Adaptive backsteppillg and tuning function" 12
1.3.3 rVIoclulal' designs 14
1.3.'1 Outpul.-l'eedback designs 15
1.3.5 Linear syst.ems 15
1.3.6 Transient. performance 16
1.3.7 Caveat: robustness 17
1.4 Notation 17
Not.es and References 17
I State Feedback 19
2 Design Tools for Stabilizatioll 21
2.1 Stability 22
2.1.1 rvlllin st,ability tbeorems 22
2.1.2 Control Lyapunm' fUllctions (elf) 25
2.2 Ba.ckst.opping 29
2.2.1 Integrator ba.ckstepping 29
2.2.2 Example: actiye suspension (paraUel) 37
2.2.3 Feedba.ck linea.rization ~tl1d zero dynamics 39
2.2.4 Stabilizat.ion of cascade systems 42
2.2.5 Block backsteppillg with zero dynamics ,19
2.2.6 Exatnple: active suspension (sedes) 55
2.3 Recursive Design Procedures 58
2.3.1 Strict-feedhack systems 58
2.3.2 Pure-feedback systems 61
2.3.3 Block-strict-feedbnck systems 64
2.4 Design Flexibility: .let Engine Example 66
2.4.1 .1et engine stall and surge 6i
2.4.2 A two-step design 68
2.4.3 Avoiding cancellations iO
2.5 St.abilization with Uncertainty i2
2.5.1 Nonlinear damping 73
2.5.2 Backstcpping with unccrtaint:y 80
2.5.3 Robust strict-feedback systems 84
Notes and References 86
Adaptive Backstepping Design 87
3.1 Adaptation as Dynamic Feedback 88
3.2 Ada]lUve Backstepping 92
3.2.1 Adaptive iutegrator backstepping 92
3.2.2 Adaptive block backstepping 98
3.3 Recursive Design Procedures 99
3.3.1 Parametric strict-feedhack systems 99
3.3.2 NIlllti~input systems 103
3.3.3 Parametric block-strict-feedback systcms 105
3.4 Extended :Matching Design 110
3.4.1 Reducing the overpal'ametrizat.ioll 110
3.4.2 Exa'mple: biocbmnical process 113
3.4.3 1'ransient performance improvement 115
Notes and References 121
Tuning Functions Design 123
4.1 Adaptive Contl'ol Lyapullov FunctiOlls 124
4.1.1 Departure from certainty equivalence 124
4.1.2 Certainty equivalence for a modified syst.em 128
4.1.3 Adaptive hackstepping via aclf 134
4.2 Set-Point Rcgulation 139
4.2.1 Design procedure 140
4.2.2 Stability and convergence 151
4.2.3 Passivity 154
4.3 Tracking 156
4.3.1 Design procedure 157
4.3.2 Trajectory initialization 162
CONTENTS xi
4.4 Transient Performance 165
4.4.1 £2 performance 165
4.4.2 £00 performance 166
4.5 E~"tensions HiS
"1.5.1 Unknowll virtual control coefficients 108
4.5.2 Block-strict-feedback systems 173
4.5.3 Pure-feedback systems 175
4.6 Example: Ail'craft \Ving Rock 180
Notes and References 183
5 Modular Design with Passive Identifiers 185
5.1 \Veakness of Certainty Equivalence 186
5.2 ISS-Control Lyapullov Funrtions 189
5.3 ISS-Controller Design 198
5.4 Observers for StriC/: Passivity 206
5.5 z-Passive Schmne 209
5.6 a;-Passive Scheme 212
5.7 Transient Performance 218
5.8 SG-Scheme (\Veak iVlodularit.y) 222
5.8.1 COll troller design 223
5.8.2 Scheme with strengtbened identifier 225
5.9 Unknown Virtual Control Coefficients 229
5.9.1 Controller design 229
5.9.2 Passive adaptive scheme 232
Notes and References 233
6 Modular Design with Swapping Identifiers 235
6.1 ISS-Controller 235
6') Swapping and Static Para.met.ric Ivlodels 237
6.3 z-Swapping Scheme 239
6.4 x-Swapping Scheme 248
6.5 Transient Performance 254
6.5.1 Simulation c-~amples 259
6.6 SG-Scheme 265
6.7 Schemes with Weak ISS-Controller 271
0.8 Unknown Virtual Coutrol Coefficients 277
Notes and References 282
II Output Feedback 283
7 Output-Feedback Design Tools 285
7.1 Observer Backsteppil1g 285
7.1.1 U nrneasured states 285