Predictor Feedback for Delay Systems: Implementations and Approximations

309 Pages·2017·4.64 MB·English

by  Iasson Karafyllis

Systems & Control: Foundations & Applications
Iasson Karafyllis
Miroslav Krstic
Predictor Feedback
for Delay Systems:
Implementations
and Approximations

Systems & Control: Foundations & Applications
SeriesEditor
TamerBas¸ar,UniversityofIllinoisatUrbana-Champaign,Urbana,IL,USA
EditorialBoard
KarlJohanÅstr€om,LundUniversityofTechnology,Lund,Sweden
Han-FuChen,AcademiaSinica,Beijing,China
BillHelton,UniversityofCalifornia,SanDiego,CA,USA
AlbertoIsidori,SapienzaUniversityofRome,Rome,Italy
MiroslavKrstic,UniversityofCalifornia,SanDiego,CA,USA
H.VincentPoor,PrincetonUniversity,Princeton,NJ,USA
MeteSoner,ETHZürich,Zürich,Switzerland;
SwissFinanceInstitute,Zürich,Switzerland
RobertoTempo,CNR-IEIIT,PolitecnicodiTorino,Italy
Moreinformationaboutthisseriesathttp://www.springer.com/series/4895
Iasson Karafyllis (cid:129) Miroslav Krstic
Predictor Feedback for Delay
Systems: Implementations
and Approximations
IassonKarafyllis MiroslavKrstic
DepartmentofMathematics Mechanical&AerospaceEngineering
NationalTechnicalUniversityofAthens UniversityofCalifornia,SanDiego
Athens,Greece LaJolla,CA,USA
ISSN2324-9749 ISSN2324-9757 (electronic)
Systems&Control:Foundations&Applications
ISBN978-3-319-42377-7 ISBN978-3-319-42378-4 (eBook)
DOI10.1007/978-3-319-42378-4
LibraryofCongressControlNumber:2016949408
MathematicsSubjectClassification(2010):93B52,93C23,93D15,93C10
©SpringerInternationalPublishingAG,CH2017
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Preface
Time delays abound in dynamical systems. And long values of such delays often
induce instability. The idea of using “predictors” of the future state or output,
initiatedwiththe“Smithpredictor”inthelate1950sforcompensatinglongdelays
inactuation,sensing,computation,orcommunication,preventssuchinstabilities—
forlinearsystems.NexttothePIDcontrol,theSmithpredictorisarguablythemost
frequentlyusedfeedbackstrategy.
AbouthalfacenturyaftertheSmithpredictor—inthelate2000s—thepredictor
idea was extended to nonlinear systems. In principle, any stabilizable nonlinear
systemcanbestabilizedinthepresenceofanarbitrarilylonginputdelay.
The reason why it took half a century to make the advance from linear to
nonlinear systems has largely to do with the distinctions between the solvability
of linear and nonlinear systems. Predictors employ model-based solutions of
differential equations to produce a value of the future state, which the feedback
law employs to compensate the delay. Model-based and explicit solutions are
available for all linear systems. Nonlinear systems, in turn, are generically
unsolvableinclosedform.
Thenonlinearpredictorresultsofthelate2000s,usheredinbythesecondauthor
of this book, do not tackle the question of solvability of nonlinear systems. They
only employ the properties of such solutions, without presenting ways in which
thosesolutionswouldbegeneratedinrealtime,toprovestabilization.
This book bridges the chasm between the nonlinear predictor as a concept and
thenonlinearpredictorasapracticaltool.
Since nonlinear predictors are not solutions of nonlinear differential equations
thathavetobeperformedjustonce—off-line—buthavetobeperformedcontinu-
ously, in real time, developing methods to generate nonlinear predictors is not
merelyaboutsolvingdifferentialequationsbutaboutensuringstabilitybyapplying
feedbackgeneratedbyimperfectsolutionstodifferentialequationsinrealtime.
This book supplies several methods for generating such solutions. We refer to
them as approximate predictors. The solution methods we offer are diverse and
v
vi Preface
include both numerical methods and methods that employ iterative analytical
integrations.
Sincethisbookdealswithbringingthenonlinearpredictormethodologycloser
to reality, we address, in addition to delays on the input and output, a number of
other“reality-induced”inconveniences:
(cid:129) partialstatemeasurement,
(cid:129) sampledmeasurement,
(cid:129) uncertaintyinthesamplingschedule,
(cid:129) inputapplicationviaZeroOrderHold,
(cid:129) thepresenceofmeasurementnoiseandmodelingerrors.
What Does the Book Cover?
Thebookcoversallaspectsofthesolutionofrobust,globalstabilizationproblems
bymeansofpredictorfeedbackforsystemswithconstantdelays.
Thebookisdividedintothreeparts.PartIofthebookisdevotedtoLinearTime-
Invariant (LTI) systems. The reason for the study of the application of predictor
feedbacktoLTIsystemsinabookthatismostly“nonlinear”ismainlypedagogical.
Allconceptsandnoveltiesintroducedinthebookareclearlyshowninthecontext
ofLTIsystems,andthereaderistaughtthefollowingnotionswithoutthetechnical
difficultiesthatarepresentfornonlinearsystems:
(cid:129) theconceptofanapproximatepredictor
(cid:129) closed-looprobustnesstomeasurementnoiseandmodelingerrorsunderpredic-
torfeedback.
Chapter 2 deals with predictor feedback for LTI systems with state measure-
ment. The reader is introduced to three ways of implementing the predictor
feedback:
1) thedirectimplementation,
2) thedynamicimplementation,and
3) thehybridimplementation.
Alladvantagesanddisadvantagesofeachwayofimplementationarediscussed,
anditisemphasizedthateach wayofimplementingpredictorfeedback leads toa
differentkindofclosed-loopsystem:
a) the direct implementation leads to a system described by Integral Delay Equa-
tions(IDEs),
b) the dynamic implementation leads to a standard time-delayed system with
distributedstatedelays,and
c) thehybridimplementationleadstoacomplicatedhybridsystemwithdelays.
Preface vii
Chapter 2 shows the robustness properties of each implementation but also
emphasizes the fact that there are severe disturbance attenuation limitations due
todelays.Morespecifically,novelresultsareprovidedforthespecificationofthe
disturbanceattenuationlimitationforeverylinearandnonlinearsystemunderany
kind of controller. Chapter 2 discusses the robustness of predictor feedback with
respect todelayperturbations,andsimple(butconservative) explicitformulas are
provided. Finally, Chapter 2 discusses the concept of the approximate predictor
(i.e.,amappingthatapproximatesthefuturevalueofthestatevector)anditsusein
predictorfeedback.
Chapter 3 is dedicated to the predictor feedback for LTI systems with partial
state measurement. The reader is introduced to the Inter-Sample Predictor-
Observer-Predictor-Delay Free Controller (ISP-O-P-DFC) control scheme in the
contextofLTIsystems.Indeed,whendelayedoutputmeasurementisavailable,we
can no longer use the (approximate) predictor mapping directly. First, we have to
use an observer (and an inter-sample predictor if the measurement is sampled) in
order to obtain a continuous estimation of the past value of the state vector. Only
then we are in a position to use the (approximate) predictor mapping, which can
provideusanestimationofthefuturevalueofthestate.
Part II of the book is devoted to nonlinear time-invariant systems. All notions
introducedinapedagogicalwayforLTIsystemsarenowusedinfullextent.The
use of approximate predictors is almost always necessary because the predictor
mapping is available explicitly only for limited classes of nonlinear systems.
Chapters 4 and 5 show that predictor feedback can be applied successfully with
approximatepredictorstowideclassesofnonlinearsystems.
Part III of the book is devoted to extensions of predictor feedback either in a
conceptual level or to different classes of systems. The conceptual extension of
predictorfeedbackleadsustothenotionofasystemdescribedbyIDEs,whichare
analyzedindetailinChapter7.Theextensionofpredictorfeedbacktootherclasses
of systems leads us to the application of predictor feedback methodologies to
discrete-timesystems:thisisthetopicofChapter8.
Who Is the Book for?
This book should be of interest to researchers working on control of time-delay
systems.Manyengineers,mathematicians,andstudentsareworkingonimportant
control-theoretic aspects of time-delay systems, and a significant number of them
havebecomeusersofpredictorfeedbackmethodologies.APh.D.studentcanfind
thecurrentstateoftheartinpredictorfeedbackdesigninthisbook.
Time-Delaysystemsareabundantinmanysciences:physics,biology,engineer-
ing and economics. Stabilization problems arise naturally in many cases, and the
implementation of globally stabilizing predictor feedback laws can solve such
problemselegantlyandrobustly.
viii Preface
MathematiciansmayalsobeinterestedinthematerialofChapter7onIDEs.It
has been shown that IDEs are closely related to systems described by first-order
hyperbolic partial differential equations arising in mathematical physics, mathe-
maticalbiology,andengineering.
The book starts from the already fairly advanced state of the art in stability
theory and feedback design for nonlinear ordinary differential equations. We
assume that the reader is familiar with stability and feedback stabilization theory
foruncertainnonlinearfinite-dimensionalsystemsatleastatamoderatelyadvanced
level.Themathematicalbackgroundneededforthecompletecomprehensionofthe
resultsinthepresentbookisprovidedinChapter1,wheremanyusefulnotionsin
nonlinear mathematical control theory are reviewed for the reader’s convenience.
The reader who is not familiar with these notions can return to Chapter 1 when
needed.
Athens,Greece IassonKarafyllis
LaJolla,CA,USA MiroslavKrstic
September14,2015
Acknowledgments
MiroslavKrsticthanksNikosBekiaris-Liberis,DelphineBresch-Pietri,andMrdjan
Jankovicfortheircollaborationandinspirationintheareaofdelaysystems,aswell
as the following collaborators on related topics in control of hyperbolic PDEs:
RafaelVazquez,Jean-MichelCoron,AndreySmyshlyaev,FlorentDiMeglio,and
Federico Bribiesca Argomedo. Miroslav also warmly thanks his daughters
Alexandra and Victoria and his wife Angela for the joint daily experiences in
delaycompensationandpunctuality.
IassonKarafylliswouldliketothankallthepeoplewhounderstoodwhathehas
beenthroughthelastfiveyearsofhislifeanddedicatesthisbooktothreewomen
who have tortured himina very sweet and lovingway:Xριστίνα, Kατερίνα, and
Oλυμπία.
ix