Nonlinear Control Under Nonconstant Delays

295 Pages·3.413 MB·English

by  Bekiaris-Liberis N., Krstic M.

NoNliNear CoNtrol
UNder
NoNCoNstaNt delays
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Advances in Design and Control
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Bekiaris-Liberis, Nikolaos and Krstic, Miroslav, Nonlinear Control Under Nonconstant Delays
Osmolovskii, Nikolai P. and Maurer, Helmut, Applications to Regular and Bang-Bang Control: Second-Order
Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control
Biegler, Lorenz T., Campbell, Stephen L., and Mehrmann, Volker, eds., Control and Optimization with
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Hovakimyan, Naira and Cao, Chengyu, L Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation
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Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory
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Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications
Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design
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Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and
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Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-
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Ioannou, Petros and Fidan, Barı¸s, Adaptive Control Tutorial
Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems
Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming
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Huang, J., Nonlinear Output Regulation: Theory and Applications
Haslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation
Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems
Gunzburger, Max D., Perspectives in Flow Control and Optimization
Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization
Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming
El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control
Helton, J. William and James, Matthew R., Extending H∞ Control to Nonlinear Systems: Control of Nonlinear
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NoNliNear CoNtrol
UNder
NoNCoNstaNt delays
Nikolaos Bekiaris-liberis
University of California at San Diego
La Jolla, California
Miroslav Krstic
University of California at San Diego
La Jolla, California
Society for Industrial and Applied Mathematics
Philadelphia
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Copyright © 2013 by the Society for Industrial and Applied Mathematics
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Library of Congress Cataloging-in-Publication Data
Bekiaris-Liberis, Nikolaos.
Nonlinear control under nonconstant delays / Nikolaos Bekiaris-Liberis, University of California at San
Diego, La Jolla, California, Miroslav Krstic, University of California at San Diego, La Jolla, California.
pages cm -- (Advances in design and control ; 25)
Includes bibliographical references and index.
ISBN 978-1-611973-17-4
1. Automatic control. I. Krstic, Miroslav. II. Title.
TJ213.B4175 2013
629.8’312--dc23
2013020352

is a registered trademark.
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Contents
Preface ix
1 Introduction 1
1.1 DelaySystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ControlviaDelayCompensation . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Varyingvs. ConstantDelays . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 FuturePaths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 OrganizationoftheBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 SpacesandSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
I ConstantDelays 9
2 LinearSystemswithInputandStateDelays 11
2.1 PredictorFeedbackDesignforSystemswithInputDelay . . . . . . . 12
2.2 Strict-FeedbackSystemswithDelayedIntegrators . . . . . . . . . . . . 19
2.3 AdaptiveControlofFeedforwardSystemswithUnknownDelays. . 35
2.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 LinearSystemswithDistributedDelays 51
3.1 SystemswithKnownPlantParameters . . . . . . . . . . . . . . . . . . . 51
3.2 AdaptiveControlforSystemswithUncertainPlantParameters . . . 70
3.3 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Application: AutomotiveCatalysts 87
4.1 ModelDevelopment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 AnalysisoftheModelunderaSquareWaveAir-to-FuelRatioInput. 92
4.3 Observer-BasedOutputFeedback . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 NonlinearSystemswithInputDelay 105
5.1 NonlinearPredictorFeedbackDesignforConstantDelay . . . . . . . 105
5.2 NonlinearInfinite-DimensionalBacksteppingTransformation . . . . 106
5.3 Lyapunov-BasedStabilityAnalysis . . . . . . . . . . . . . . . . . . . . . . 108
5.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
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vi Contents
II Time-VaryingDelays 115
6 LinearSystemswithTime-VaryingInputDelay 117
6.1 PredictorFeedbackDesignforTime-VaryingInputDelay . . . . . . . 117
6.2 StabilityAnalysisforTime-VaryingDelays . . . . . . . . . . . . . . . . . 120
6.3 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7 Robustness of Linear Constant-Delay Predictors to Time-Varying Delay
Perturbations 129
7.1 RobustnesstoTime-VaryingPerturbationforLinearSystems. . . . . 129
7.2 ATeleoperation-LikeExample . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.3 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8 NonlinearSystemswithTime-VaryingInputDelay 141
8.1 NonlinearPredictorFeedbackDesignforTime-VaryingDelay . . . . 141
8.2 StabilityAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.3 StabilizationoftheNonholonomicUnicyclewithTime-Varying
Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9 Nonlinear Systems withSimultaneous Time-Varying Delays on theInput
andtheState 155
9.1 Predictor Feedback Design for Systems with Both Input and State
Delays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.2 StabilityAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
III State-DependentDelays 171
10 PredictorFeedbackDesignWhentheDelayIsaFunctionoftheState 173
10.1 NonlinearPredictorFeedbackDesignforState-DependentDelay . . 173
10.2 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
11 StabilityAnalysisforForward-CompleteSystemswithInputDelay 179
11.1 StabilityAnalysisforForward-CompleteNonlinearSystems . . . . . 179
11.2 Example: Nonholonomic UnicycleSubject toDistance-Dependent
InputDelay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
11.3 GlobalStabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
11.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12 StabilityAnalysisforLocally Stabilizable SystemswithInputDelay 195
12.1 StabilityAnalysisforLocallyStabilizableNonlinearSystems . . . . . 195
12.2 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
13 NonlinearSystemswithStateDelay 207
13.1 ProblemFormulationandControllerDesign . . . . . . . . . . . . . . . 207
13.2 StabilityAnalysisforForward-CompleteSystems . . . . . . . . . . . . 208
13.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
13.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
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14 Robustness of Nonlinear Constant-Delay Predictors to Time- and State-
DependentDelayPerturbations 221
14.1 Robustness to Time- and State-Dependent Delay Perturbations for
NonlinearSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
14.2 Example: ControlofaDCMotoroveraNetwork . . . . . . . . . . . . 230
14.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
14.4 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
15 State-DependentDelaysThatDependonDelayedStates 243
15.1 PredictorFeedbackunderInputDelays . . . . . . . . . . . . . . . . . . . 243
15.2 StabilityAnalysisunderInputDelays . . . . . . . . . . . . . . . . . . . . 246
15.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
15.4 StabilizationunderStateDelays . . . . . . . . . . . . . . . . . . . . . . . . 253
15.5 NotesandReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
A BasicInequalities 265
B Input-to-OutputStability 267
C LyapunovStability,Forward-Completeness,andInput-to-StateStability 273
C.1 LyapunovStabilityand(cid:2) Functions. . . . . . . . . . . . . . . . . . . . 273
∞
C.2 Forward-Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
C.3 Input-to-StateStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
D ParameterProjection 283
Bibliography 287
Index 299
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Preface
Considerthetimeperiodsrequired
• foradrivertoreactinresponsetolargedisturbancesinroadtraffic;
• forasignalsentfromHoustontoreachasatelliteinorbit;
• foracoolanttobedistributedamongallair-conditionersinaresidentialbuilding;
• forajobtobeexecutedonaserver;
• foracelltoreachacertainmaturationlevel;
• forarelativisticparticletofeeltheelectromagneticforcefromanotherparticle;
• foracuttingtooltoperformtwosucceedingcuts.
Thecommonattributeofallthesetimeperiodsisthattheydonotremainconstant.
Anotherfeatureincommontothedynamicsoftrafficflow,coolingsystems,networks
andqueuingsystems,populationgrowth,andcuttingprocessesisthattheyareallnonlin-
ear. Althoughaplethoraoftechniquesexistforthecontrolofnonlinearsystemswithout
delays,controldesignfornonlinearsystemsinthepresenceoflongdelayswithlargeand
rapidvariationintheactuationorsensingpath,ordelaysaffectingtheinternalstatesofa
system,introducessignificantfeedbackdesignchallengesthathave,heretofore,remained
largelyuntackled.
Inthisbookwepresentsystematicdesigntechniquesapplicabletogeneralnonlinear
systemswithlong,nonconstantdelays. Whilethereisanearlyinexhaustiblenumberof
combinationsin whichoneormultipledelays(as wellasdiscreteordistributed delays)
can enter a dynamical system, we focus our attention on problems with input delays.
Arguably, ifthesystem hasonlyasinglediscretedelay, thecase wherethedelayaffects
theinput(ratherthansomeofthestatecomponentsthatappearonthesystemmodel’s
right-handside)isthemostchallengingcaseforcontrol,andinparticularforstabilization.
Hence,ourfocusoninputdelayproblemsiswithoutmuchlossofgenerality.
InODEsystemswithinputdelays,theoverallstateofthedynamicalsystemconsists
ofthevectorstateoftheODEandthefunctionalstateoftheinputdelay. (Ifthedelayis
nonconstant,thesupportofthefunctionalpartofthestateisnonconstantaswell.) Fora
problemwithsucha(relatively)“unusual”state,thecontroldesigncanbeapproached—in
principle—inanabstractsettingwheretheparticularstructureofthesystemandofthe
stateisdeemphasizedandthedesignisperformedinanabstractinfinite-dimensionalset-
ting.However,atpresent,methodsthatfitsuchanabstractapproachexistonlywhenthe
ODEplantislinear(andwhenthedelayisconstant),butnotwhentheplantisnonlinear.
To develop designs that are applicable to both linear and nonlinear plants, a much
morestructure-specific approach isneeded. Thisapproach exploits thestructureofthe
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