Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control

389 Pages·2012·3.065 MB·English

by  Nikolai P. Osmolovskii, Helmut Maurer

Applications to Regular
and Bang-Bang Control
DC24_Osmolovskii-MaurerFM.indd 1 10/8/2012 9:29:39 AM
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Osmolovskii, Nikolai P. and Maurer, Helmut, Applications to Regular and Bang-Bang Control: Second-Order
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Applications to Regular
and Bang-Bang Control
Second-Order Necessary and Sufficient
Optimality Conditions in Calculus
of Variations and Optimal Control
Nikolai P. Osmolovskii
Systems Research Institute
Warszawa, Poland
University of Technology and Humanities in Radom
Radom, Poland
University of Natural Sciences and Humanities in Siedlce
Siedlce, Poland
Moscow State University
Moscow, Russia
Helmut Maurer
Institute of Computational and Applied Mathematics
Westfälische Wilhelms-Universität Münster
Münster, Germany
Society for Industrial and Applied Mathematics
Philadelphia
DC24_Osmolovskii-MaurerFM.indd 3 10/8/2012 9:29:39 AM
Copyright © 2012 by the Society for Industrial and Applied Mathematics
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Library of Congress Cataloging-in-Publication Data
Osmolovskii, N. P. (Nikolai Pavlovich), 1948-
Applications to regular and bang-bang control : second-order necessary and sufficient optimality condi-
tions in calculus of variations and optimal control / Nikolai P. Osmolovskii, Helmut Maurer.
p. cm. -- (Advances in design and control ; 24)
Includes bibliographical references and index.
ISBN 978-1-611972-35-1
1. Calculus of variations. 2. Control theory. 3. Mathematical optimization. 4. Switching theory. I.
Maurer, Helmut. II. Title.
QA315.O86 2012
515’.64--dc23 2012025629
is a registered trademark.
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For our wives, Alla and Gisela
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Contents
ListofFigures xi
Notation xiii
Preface xvii
Introduction 1
I Second-Order Optimality Conditions for Broken Extremals in the Calculus
ofVariations 7
1 AbstractSchemeforObtainingHigher-OrderConditionsinSmooth
ExtremalProblemswithConstraints 9
1.1 MainConceptsandMainTheorem . . . . . . . . . . . . . . . . . . . 9
1.2 ProofoftheMainTheorem . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 SimpleApplicationsoftheAbstractScheme . . . . . . . . . . . . . . 21
2 QuadraticConditionsintheGeneralProblemoftheCalculus
ofVariations 27
2.1 StatementsofQuadraticConditionsforaPontryaginMinimum . . . . 27
2.2 BasicConstantandtheProblemofItsDecoding . . . . . . . . . . . . 34
2.3 Local Sequences, Higher Order γ, Representation of the Lagrange
FunctiononLocalSequenceswithAccuracyuptoo(γ) . . . . . . . . 39
2.4 EstimationoftheBasicConstantfromAbove . . . . . . . . . . . . . 54
2.5 EstimationoftheBasicConstantfromBelow . . . . . . . . . . . . . 75
2.6 CompletingtheProofofTheorem2.4 . . . . . . . . . . . . . . . . . . 102
2.7 Sufficient Conditions for Bounded Strong and Strong Minima in the
ProblemonaFixedTimeInterval . . . . . . . . . . . . . . . . . . . . 115
3 QuadraticConditionsforOptimalControlProblemswithMixed
Control-StateConstraints 127
3.1 QuadraticNecessaryConditionsintheProblemwithMixedControl-
StateEqualityConstraintsonaFixedTimeInterval . . . . . . . . . . 127
3.2 Quadratic Sufficient Conditions in the Problem with Mixed Control-
StateEqualityConstraintsonaFixedTimeInterval . . . . . . . . . . 138
vii
viii Contents
3.3 QuadraticConditionsintheProblemwithMixedControl-StateEqual-
ityConstraintsonaVariableTimeInterval . . . . . . . . . . . . . . . 150
3.4 QuadraticConditionsforOptimalControlProblemswithMixed
Control-StateEqualityandInequalityConstraints . . . . . . . . . . . 164
4 Jacobi-TypeConditionsandRiccatiEquationforBrokenExtremals 183
4.1 Jacobi-TypeConditionsandRiccatiEquationforBrokenExtremalsin
theSimplestProblemoftheCalculusofVariations . . . . . . . . . . . 183
4.2 RiccatiEquationforBrokenExtremalintheGeneralProblemofthe
CalculusofVariations . . . . . . . . . . . . . . . . . . . . . . . . . . 214
II Second-OrderOptimalityConditionsinOptimalBang-BangControl
Problems 221
5 Second-OrderOptimalityConditionsinOptimalControlProblems
LinearinaPartofControls 223
5.1 QuadraticOptimalityConditionsintheProblemonaFixedTime
Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.2 Quadratic Optimality Conditions in the Problem on a Variable Time
Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.3 RiccatiApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.4 NumericalExample: OptimalControlofProductionand
Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
6 Second-OrderOptimalityConditionsforBang-BangControl 255
6.1 Bang-BangControlProblemsonNonfixedTimeIntervals . . . . . . . 255
6.2 QuadraticNecessaryandSufficientOptimalityConditions . . . . . . . 259
6.3 SufficientConditionsforPositiveDefinitenessoftheQuadraticForm
(cid:3)ontheCriticalConeK . . . . . . . . . . . . . . . . . . . . . . . . 266
6.4 Example: MinimalFuelConsumptionofaCar . . . . . . . . . . . . . 272
6.5 QuadraticOptimalityConditionsinTime-OptimalBang-BangControl
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
6.6 SufficientConditionsforPositiveDefinitenessoftheQuadraticForm
(cid:3)ontheCriticalSubspaceK forTime-OptimalControlProblems . . 281
6.7 NumericalExamplesofTime-OptimalControlProblems . . . . . . . 286
6.8 Time-Optimal Control Problems for Linear Systems with Constant
Entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
7 Bang-BangControlProblemandItsInducedOptimizationProblem 299
7.1 MainResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
7.2 First-Order Derivatives of x(t ;t ,x ,θ) with Respect to t , t , x ,
f 0 0 0 f 0
andθ. LagrangeMultipliersandCriticalCones . . . . . . . . . . . . 305
7.3 Second-OrderDerivativesof x(t ;t ,x ,θ)withRespecttot ,t ,x ,
f 0 0 0 f 0
andθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
7.4 Explicit Representation of the Quadratic Form for the Induced Opti-
mizationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Contents ix
7.5 Equivalence of the Quadratic Forms in the Basic and Induced Opti-
mizationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
8 NumericalMethodsforSolvingtheInducedOptimizationProblemand
Applications 339
8.1 TheArc-ParametrizationMethod . . . . . . . . . . . . . . . . . . . . 339
8.2 Time-OptimalControloftheRayleighEquationRevisited . . . . . . . 344
8.3 Time-OptimalControlofaTwo-LinkRobot . . . . . . . . . . . . . . 346
8.4 Time-OptimalControlofaSingleModeSemiconductorLaser. . . . . 353
8.5 OptimalControlofaBatch-Reactor . . . . . . . . . . . . . . . . . . . 357
8.6 OptimalProductionandMaintenancewithL1-Functional . . . . . . . 361
8.7 VanderPolOscillatorwithBang-SingularControl . . . . . . . . . . . 365
Bibliography 367
Index 377
List of Figures
2.1 Neighborhoodsofthecontrolatapointt ofdiscontinuity. . . . . . . . . 40
1
2.2 Definitionoffunctions(cid:5)(t,v)onneighborhoodsofdiscontinuitypoints.. 51
4.1 Tunnel-diodeoscillator. I denotesinductivity,C capacity,Rresistance,
I electriccurrent,andDdiode. . . . . . . . . . . . . . . . . . . . . . . 203
4.2 Rayleighproblemwithregularcontrol. (a)Statevariables. (b)Control.
(c)Adjointvariables. (d)SolutionsoftheRiccatiequation(4.140). . . . 204
4.3 Top left: Extremals x(1),x(2) (lower graph). Top right: Variational so-
lutions y(1) and y(2) (lower graph) to (4.145). Bottom: Envelope of
neighboringextremalsillustratingtheconjugatepointt =0.674437. . . 206
c
5.1 Optimalproductionandmaintenance,finaltimet =0.9. (a)Statevari-
f
ablesx ,x . (b)Regularproductioncontrolvandbang-bangmaintenance
1 2
controlm. (c)Adjointvariablesψ ,ψ . (d)Maintenancecontrolmwith
1 2
switchingfunctionφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
m
5.2 Optimalproductionandmaintenance,finaltimet =1.1. (a)Statevari-
f
ables x ,x . (b) Regular production control v and bang-singular-bang
1 2
maintenance control m. (c)Adjoint variables ψ ,ψ . (d) Maintenance
1 2
controlmwithswitchingfunctionφ . . . . . . . . . . . . . . . . . . . . 252
m
6.1 Minimalfuelconsumptionofacar. (a)Statevariablesx ,x . (b)Bang-
1 2
bangcontrolu. (c)Adjointvariableψ . (d)Switchingfunctionφ. . . . . 274
2
6.2 Time-optimalsolutionofthevanderPoloscillator,fixedterminalstate
(6.120). (a) State variables x and x (dashed line). (b) Control u and
1 2
switchingfunctionψ (dashedline). (c)Phaseportrait(x ,x ). (d)Ad-
2 1 2
jointvariablesψ andψ (dashedline). . . . . . . . . . . . . . . . . . . 287
1 2
6.3 Time-optimalsolutionofthevanderPoloscillator,nonlinearboundary
condition(6.129). (a)Statevariablesx andx (dashedline). (b)Control
1 2
u and switching function ψ (dashed line). (c) Phase portrait (x ,x ).
2 1 2
(d)Adjointvariablesψ andψ (dashedline). . . . . . . . . . . . . . . 289
1 2
6.4 Time-optimal control of the Rayleigh equation. (a) State variables x
1
and x (dashed line). (b) Control u and switching function φ (dashed
2
line). (c)Phaseportrait(x ,x ). (d)Adjointvariablesψ andψ (dashed
1 2 1 2
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
xi
xii ListofFigures
8.1 Time-optimalcontroloftheRayleighequationwithboundaryconditions
(8.31). (a) Bang-bang control and scaled switching function (cid:8)(×4),
(b)Statevariablesx ,andx . . . . . . . . . . . . . . . . . . . . . . . . 345
1 2
8.2 Time-optimalcontroloftheRayleighequationwithboundarycondition
(8.40). (a)Bang-bangcontroluandscaledswitchingfunctionφ(dashed
line). (b)Statevariablesx ,x .. . . . . . . . . . . . . . . . . . . . . . . 346
1 2
(cid:2) (cid:2)
8.3 Two-link robot [67]: upper arm OQ, lower arm OP, and angles q
1
andq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
2
8.4 Control of the two-link robot (8.44)–(8.47). (a) Control u and scaled
1
switchingfunctionφ (dashedline). (b)Controlu andscaledswitching
1 2
function φ (dashed line). (c)Angle q and velocity ω . (d)Angle q
2 1 1 2
andvelocityω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
2
8.5 Control of the two-link robot (8.53)–(8.57). (a) Control u .
1
(b)Controlu . (c)Angleq andvelocityω . (d)Angleq andvelocity
2 1 1 2
ω [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
2
8.6 Controlofthetwo-linkrobot(8.53)–(8.57): Secondsolution. (a)Control
u . (b)Controlu . (c)Angleq andvelocityω . (d)Angleq andvelocity
1 2 1 1 2
ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
2
8.7 Time-optimal control of a semiconductor laser. (a) Normalized pho-
ton density S(t)×10−5. (b) Normalized photon density N(t)×10−8.
(c) Electric current (control) I(t) with I(t)=I =20.5 for t <0 and
0
I(t)=I∞=42.5fort >tf. (d)AdjointvariablesψS(t),ψN(t). . . . . . 356
8.8 Normalized photon number S(t) for I(t) ≡ 42.5 mA and optimal
I(t)[46]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
8.9 Schematicsofabatch-reactionwithtwocontrolvariables. . . . . . . . . 357
8.10 Control of a batch reactor with functional (8.79). Top row: Control
u=(F ,Q)andscaledswitchingfunctions. Middlerow: Molarconcen-
B
trationsM andM . Bottomrow: Molarconcentrations(M ,M )and
A B C D
energyholdupH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
8.11 Control of a batch reactor with functional (8.89). Top row: Control
u=(F ,Q) and scaled switching functions. Middle row: Molar con-
B
centrationsM ,M . Bottomrow: Molarconcentrations(M ,M )and
A B C D
energyholdupH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
8.12 OptimalproductionandmaintenancewithL1-functional(8.95). (a)State
variablesxandy. (b)Controlvariablesvandm. (c),(d)Controlvariables
andswitchingfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . 364
8.13 ControlofthevanderPoloscillatorwithregulatorfunctional. (a)Bang-
singularcontrolu. (b)Statevariablesx andx . . . . . . . . . . . . . . 366
1 2