Delay Compensation for Nonlinear, Adaptive, and PDE Systems

458 Pages·2009·5.57 MB·English

by  Miroslav Krstic (auth.)

Systems & Control: Foundations & Applications

Series Editor
Tamer Başar, University of Illinois at Urbana-Champaign

Editorial Board
Karl Johan Åström, Lund University of Technology, Lund, Sweden
Han-Fu Chen, Academia Sinica, Beijing
William Helton, University of California, San Diego
Alberto Isidori, University of Rome (Italy) and
Washington University, St. Louis
Petar V. Kokotović, University of California, Santa Barbara
Alexander Kurzhanski, Russian Academy of Sciences, Moscow
and University of California, Berkeley
H. Vincent Poor, Princeton University
Mete Soner, Koç University, Istanbul
Miroslav Krstic
Delay Compensation
for Nonlinear, Adaptive,
and PDE Systems
Birkha¨user
Boston • Basel • Berlin
Miroslav Krstic
Department of Mechanical and
Aerospace Engineering
University of California, San Diego
La Jolla, CA 92093-0411
[email protected]

ISBN 978-0-8176-4876-3 e-ISBN 978-0-8176–4877-0
DOI10.1007/978-0-8176-4877-0
LibraryofCongressControlNumber:2009933091
MathematicsSubjectClassification (2000): 34H05, 34Kxx, 34K35, 35B37, 49J25, 70Q05,
93–XX, 93C10, 93C20, 93C23, 93C40, 93D05
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Preface
This year is the 50th anniversaryof Otto J. Smith’s 1959 publication of a control
design idea commonly referred as the Smith predictor for the compensation of
actuatordelays.Actuatorandsensordelaysareamongthemostcommondynamic
phenomenathatariseinengineeringpracticebutfalloutsidethescopeofthestan-
dardfinite-dimensionalsystems.
Predictor-based feedbacks and other controllers for systems with input and
outputdelayshavebeen(andcontinueto be)anactiveareaofresearchduringthe
lastfivedecades.Severalbooksexistthatfocusonthemathematicalandengineer-
ingproblemsinthisarea.Thegoalsofthisbookarenottoduplicatethematerialin
thosebooksnortopresentacomprehensiveaccountaboutcontrolofsystemswith
inputandoutputdelays.Instead,thebook’sgoalistoshedlightonnewopportuni-
tiesforpredictorfeedback,throughextensionstononlinearsystems,delay-adaptive
control,andactuatordynamicsmodeledbyPDEsmorecomplexthantransport(pure
delay)PDEs.
* * *
WhatDoestheBookCover?Thisbookisaresearchmonographthatintroducesthe
treatmentofsystemswithinputdelaysasPDE-ODEcascadesystemswithbound-
arycontrol.ThePDE-basedapproachyieldsLyapunov–Krasovskiifunctionalsthat
make the control design constructive and enables stability analysis with quantita-
tive estimates, which leads to the resolution of several long-standing problems in
predictor feedback for linear time-invariant (LTI) systems. More importantly, the
PDE-based approach enables the extension of predictor feedback design to non-
linearsystemsandtoadaptivecontrolforsystemswithunknowndelays.
However, the book’s treatment of input and output delays as transport PDEs
allowsittoaimevenfurther,inexpandingthepredictorfeedbackideastosystems
with other types of infinite-dimensional actuator dynamics and sensor dynamics.
We develop methods for compensating heat PDE and wave PDE dynamics at the
input of an arbitrary, possibly unstable, LTI-ODE plant. Similarly, we develop
observersforLTI-ODEsystemswithsimilartypesofsensordynamics.Finally,we
introduce problemsfor PDE-PDE cascades, such as, for example, the notoriously
v
vi Preface
ThestandardproblemoflinearODEplantwithinputdelay,leadingtoconventionalpredictor
feedbackcontrol.
Theproblemsconsideredinthisbook.
difficultproblemofawavePDEwithinputdelaywhere,ifthedelayisleftuncom-
pensated,anarbitrarilyshortdelaydestroystheclosed-loopstability(asshownby
Datkoin1988).
* * *
WhoIstheBookFor?Thebookshouldbeofinteresttoallresearchersworkingon
controlofdelaysystems—engineers,graduatestudents,anddelaysystemsspecial-
istsinacademia.Thelattergroupwillespeciallybenefitfromthisbook,asitopens
severalnew paradigmsfor delay research.Many opportunitiespresentthemselves
to extend the present results to systems that contain state delays (discrete and/or
distributed)inadditiontoinputdelays.
Mathematicianswithinterestinthebroadareaofcontrolofdistributedparameter
systems, and PDEs in particular, will find the book stimulating because it tackles
nonlinearODEs simultaneouslywith linearPDEs, as wellas PDEs fromdifferent
classes.Theseproblemspresentmanystimulatingchallengesforfurtherresearchon
thestabilizationofever-expandingclassesofunstableinfinite-dimensionalsystems.
Chemical engineers and process dynamics researchers, who have traditionally
been users of the Smith predictor and related approaches, should find the vari-
ous extensions of this methodology that the book presents (adaptive, nonlinear,
otherPDEs) to be usefuland exciting.Engineersfromotherareas—electricaland
computerengineering(telecommunicationsystemsandnetworks),mechanicaland
aerospace engineering (combustion systems and machining), and civil/structural
engineering—havenodoubtfacedproblemswithactuatordelaysandotherdistrib-
utedparameterinputdynamicsandwillappreciatetheadvancesintroducedbythis
book.
This book is not meant to be a standalone textbook for any individual gradu-
ate course. However, its parts can be used as supplementalmaterial in lectures or
projectsinmanygraduatecourses:
• generaldistributedparametersystems(Chapters2,3,6,14–20),
• lineardelaysystems(Chapters2,3,6,18,and19),
• partialdifferentialequations(Chapters14–20),
• nonlinearcontrol(Chapters10–13),
Preface vii
• stateestimators/observers(Chapters3and17),
• adaptivecontrol(Chapters7–9),and
• robustcontrol(Chapters4and5),
• lineartime-varying(LTV)systems(Chapter6).
The backgroundrequired to read this book includes little beyond the basics of
functionspacesandLyapunovtheoryforODEs.However,thebasicsofthePoincare´
andAgmoninequalities,Lyapunovandinput-to-statestability,parameterprojection
for adaptive control, and Bessell functions are summarized in appendices for the
reader’sconvenience.
I hope that the reader will not view the book as a collection of problems that
have been solved, but will focus on it as a collection of tools and techniquesthat
areapplicableinopenproblems,manymoreofwhichexistthanhavebeensolved
inthisbook,particularlyintheareasofinterconnectedsystemsofODEsandPDEs,
systems with simultaneous input and state delays, nonlinear delay systems, and
systemswithunknowndelays.
In no book are all chapters equal in value for the reader. My personal recom-
mendations to a reader on a time budget are Chapters 7, 10, 16, and 18 if the
reader is interested mainly in feedback design problems and tools. A reader pri-
marilyinterestedinanalysisandrobustnessproblemsfordelaysystemsmightalso
enjoyChapter5.
Acknowledgments. I would like to thank Delphine Bresch-Pietri, Andrey
Smyshlyaev, and Rafael Vazquez for their contributions in Chapters 8, 9, 11, 14,
andelsewhere.
I am also gratefulto Mrdjan Jankovic for exchangesof ideas and his guidance
throughtheareaofcontrolofdelaysystems.IfitwerenotforMrdjan’ssuperband
innovativepaperson controlof nonlineardelay systems, I would neverhave been
enticed to start to work on these problems. I am also pleased to express special
gratitudetoIassonKarafyllysforsomehelpfulandinspiringdiscussions.
ManythankstoManfredMorari,SilviuNiculescu,GalipUlsoy,andQing-Chang
ZhongfordiscussionsondelaysystemsandontheSmithpredictor.Iwouldalsolike
to thank Anu Annaswammy for getting me intrigued with her papers on adaptive
controlofdelaysystems.
Finally,PetarKokotovic’sencouragementandinterestinnewresearchresultsare
priceless—oftenakeydifferencebetweendecidingwhetherornottospendtimeon
writinganewbook.
Overthecourseofwritingthisbook,IhadthepleasuretomeetOttoJ.M.Smith
on the occasion of my visitto the Universityof California atBerkeley in October
2008. “Predictably,” I chose the results on predictor feedback as the topic of my
Nokia Distinguished Lecture. Otto Smith was a professor at Berkeley from 1947
until his retirement in 1988. I have never met a 91-year-oldperson with as sharp
a mindas Otto Smith’s. Truthbetold, I havemetfew 30-year-oldswhowouldbe
worth a comparison. Even at this age, Otto Smith was every bit the inventor and
creativeengineerashislistofpatentsindicates.Ihadthepleasureofhearingabout
hisfavoritedesigns,fromtheHPfunctiongeneratortohiscurrentinterestinsolar
viii Preface
energy turbine power plants with controlled focusing via heliostats. We never got
todiscussthe“Smithpredictor”—therewassomuchelseworthhearingaboutfrom
OttoSmith’sbankofengineeringknowledge.IamgratefultoAlexBayenandDean
ShankarSastryforarrangingforOttoSmithtocometocampusthatday.Ialsothank
MasayoshiTomizukaforsharingmanythoughtsonOttoSmithduringmySpringer
ProfessorsabbaticalstayatBerkeleyinthefallof2007.
Sadly,OttoSmithpassedawayonMay10,2009asaresultofafallathishome.
In the five decades since the publication of his influential paper on compensation
ofdeadtime,hehadseenhisideabecomeoneofthemostcommonlyusedtoolsin
controlpractice.
IamgratefultoCymer(BobAkins,DannyBrown,andotherfriends)andGeneral
Atomics(MikeReed,SamGurol,BogdanBorowy,DickThome,LindenBlue,and
otherfriends)fortheirsupportthroughtheCymerCenterforControlSystemsand
DynamicsatUCSanDiego.IalsoverymuchappreciatethesupportbyBosch(Nalin
Chaturvedi and Aleksandar Kojic) and the National Science Foundation (Kishan
BahetiandSuhadaJayasuriya).
Finally,forallthehundredsofeveningandweekendhoursthatwerespentonthis
bookandnotwithmyfamily,forallthemathematicshomeworkthatIwasexcused
fromhelpingwith,mygratitudeandlovegotoAlexandra,Victoria,andAngela.
LaJolla,California MiroslavKrstic
May2009
Contents
Preface............................................................ v
1 Introduction................................................... 1
1.1 DelaySystems............................................ 1
1.2 HowDoestheDifficultyofDelaySystemsComparewithPDEs? . 2
1.3 AShortHistoryofBackstepping............................. 3
1.4 FromPredictorFeedbacksforLTI-ODESystemstotheResults
inThisBook ............................................. 4
1.5 OrganizationoftheBook................................... 4
1.6 UseofExamples.......................................... 5
1.7 KrasovskiiTheoremorDirectStabilityEstimates?.............. 7
1.8 DDEorTransportPDERepresentationoftheActuator/Sensor
State? ................................................... 9
1.9 Notation,Spaces,Norms,andSolutions....................... 9
1.10 BeyondThisBook ........................................ 11
PartI LinearDelay-ODECascades
2 BasicPredictorFeedback ....................................... 17
2.1 BasicIdeaofPredictorFeedbackDesignforODESystemswith
ActuatorDelay ........................................... 18
2.2 BacksteppingDesignViatheTransportPDE................... 19
2.3 On the Relation Among the Backstepping Design, the
FSA/ReductionDesign,andtheOriginalSmithController ....... 22
2.4 StabilityofPredictorFeedback .............................. 23
2.5 ExamplesofPredictorFeedbackDesign ...................... 27
2.6 StabilityProofWithoutaLyapunovFunction .................. 30
2.7 BacksteppingTransformationintheStandardDelayNotation .... 36
2.8 NotesandReferences...................................... 39
ix