Optimal Control of a Double Integrator: A Primer on Maximum Principle

313 Pages·2017·3.365 MB·English

by  Arturo Locatelli

Studies in Systems, Decision and Control 68
Arturo Locatelli
Optimal Control
of a Double
Integrator
A Primer on Maximum Principle
Studies in Systems, Decision and Control
Volume 68
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: [email protected]
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Arturo Locatelli
Optimal Control of a Double
Integrator
A Primer on Maximum Principle
123
ArturoLocatelli
Dipartimento di Elettronica, Informazione e
Bioingegneria
Politecnico di Milano
Milan
Italy
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-319-42125-4 ISBN978-3-319-42126-1 (eBook)
DOI 10.1007/978-3-319-42126-1
LibraryofCongressControlNumber:2016945869
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Preface
OptimalControlTheoryhasbeenverypopularalongafairlylargenumberofyears,
startingfromthelate1950swhensomefundamentalresultshavebeenestablished.
AmongthemthereisnodoubtthattheMaximumPrinciplemustbeconsideredasa
cornerstone.However,the(possibly)excessiveenthusiasmforitsreputedcapability
ofsolvinganykindofproblem,whichhascharacterizedthebeginningofitsstory,
has been followed by an (equally) unjustified rejection by considering it as purely
abstractconceptswithnorealutility.Inrecentyearsithasbeenrecognizedthatthe
truthliessomewherebetweenthesetwoopinions,andoptimalcontrolhasfoundits
(appropriate yet limited) space within any curriculum where system and control
theory plays a significant role. Consistently, this book is intended to supply an
introductory but fairly comprehensive treatment of the founding issues of
Pontryagin’s Maximum Principle.
The book is suited for students who are already familiar with the basics of
system and control theory and possess the calculus background usually taught in
undergraduateengineeringcurricula.Furthermoreitsstructureallowsdifferentways
of reading it and teaching its contents. A first level presentation of the Maximum
Principle can be carried on by referring to Chap. 1, Sects. 2.1 and 2.2 (restraining
the attention to Sects. 2.2.1 and 2.2.2) and to Chaps. 3–5. Subsequently, the
materialofSect.2.2.3andChap.6canbeexploredandafurtherin-depthstudycan
be achieved through Sect. 2.2.4 and Chap. 7. A second level presentation which
suppliesadeeperinsightmakesreferencetothemorecomplexproblemsdealtwith
in Sect. 2.3 and Chaps. 8–10. Their treatment is almost self-consistent so that the
fruitionofthismaterialneedsSects.2.1–2.3,whereasitdoesnotnecessarilyrequire
consideringChaps.3–7.Finally,thetopicsinChaps.11and12canbeaddedboth
to the first and the second level presentations.
Milan, Italy Arturo Locatelli
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 The Maximum Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Statement of the Optimal Control Problems. . . . . . . . . . . . . . . 3
2.2 Necessary Conditions: Simple Constraints. . . . . . . . . . . . . . . . 6
2.2.1 Purely Integral Performance Index . . . . . . . . . . . . . . . 9
2.2.2 Performance Index Function of the Final Event . . . . . . 11
2.2.3 Non-standard Constraints on the Final State . . . . . . . . 13
2.2.4 Minimum Time Problems . . . . . . . . . . . . . . . . . . . . . 14
2.3 Necessary Conditions: Complex Constraints . . . . . . . . . . . . . . 20
2.3.1 Description of Complex Constraints. . . . . . . . . . . . . . 20
2.3.2 Integral Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Punctual and Isolated Constraints. . . . . . . . . . . . . . . . 22
2.3.4 Punctual and Global Constraints . . . . . . . . . . . . . . . . 23
2.4 Necessary Conditions: Singular Arcs . . . . . . . . . . . . . . . . . . . 26
2.5 The Considered Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Simple Constraints: J =∫, x(t ) = Given . . . . . . . . . . . . . . . . . . . . 31
0
3.1 ðxðt Þ; t ; t Þ¼Given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
f 0 f
3.2 ðxðt Þ; t Þ¼ Given; t ¼ Free . . . . . . . . . . . . . . . . . . . . . . . . 38
f 0 f
3.3 xðt Þ¼Not Given; ðt ;t Þ¼ Given . . . . . . . . . . . . . . . . . . . . 48
f 0 f
3.4 xðt Þ¼Not Given; t ¼ Given; t ¼Free. . . . . . . . . . . . . . . . 62
f 0 f
4 Simple Constraints: J =∫, x(t ) = Not Given. . . . . . . . . . . . . . . . . 77
0
4.1 ðxðt Þ; t ; t Þ¼ Given . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
f 0 f
4.2 xðt Þ¼Not Given; ðt ; t Þ¼ Given. . . . . . . . . . . . . . . . . . . . 84
f 0 f
4.3 xðt Þ¼Not Given; ðt ; t Þ¼Free . . . . . . . . . . . . . . . . . . . . . 92
f 0 f
5 Simple Constraints: J =∫+ m, x(t ) = Given,
0
x(tf) = Not Given . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1 (t ; t Þ¼ Given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
0 f
5.2 t ¼ Given; t ¼Free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
0 f
ix
x Contents
6 Nonstandard Constraints on the Final State . . . . . . . . . . . . . . . . . 127
7 Minimum Time Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8 Integral Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.1 Integral Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2 Integral Inequality Constraints. . . . . . . . . . . . . . . . . . . . . . . . 178
9 Punctual and Isolated Constrains. . . . . . . . . . . . . . . . . . . . . . . . . 193
10 Punctual and Global Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.1 Punctual and Global Equality Constraints . . . . . . . . . . . . . . . . 227
10.2 Punctual and Global Inequality Constraints. . . . . . . . . . . . . . . 240
11 Singular Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
12 Local Sufficient Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
12.1 xðt Þ¼PartiallyGiven; t ¼Free. . . . . . . . . . . . . . . . . . . . . . 299
f f
12.2 xðt Þ¼Given; t ¼Given. . . . . . . . . . . . . . . . . . . . . . . . . . . 303
f f
12.3 xðt Þ¼Free; t ¼Free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
f f
12.4 xðt Þ¼Free; t ¼Given. . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
f f
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Chapter 1
Introduction
Since its appearance in the early sixties, the Pontryagin’s Maximum Principle is a
fundamental part of Control Theory. Indeed it is the basic tool when we have to
cope with Optimal Control problems, in particular those stated in terms of finite-
dimensional,continuous-timedynamicalsystems.
Aswellknown,theMaximumPrincipleisrootedintheCalculusofVariationsand
manypeoplelegitimatelyseeitasanaturalextensionofthischapterofmathematics,
specifically oriented to control problems. Thus it should not be surprising that the
supplied results intrinsically have the nature and present the limits of necessary
conditions.Asamatteroffactstheseconditionsstemfromacustomaryandsimple
idea.Specifically,asolutionoftheproblemmustentailthatanychangeofitsfree
parameters does not cause an improvement of the value of the performance index
(theindextobeoptimized)ifthechangeisaccomplishedstartingfromthesituation
whichisclaimedtobeoptimal.Theconditionswhichresultbyexploitingfirstorder
variationalmethods,namelybyimposingthattheimprovementdoesnottakeplace
whenfirstordervariationsoccur,arenecessaryoptimalityconditions.Howeverthis
simpleideaisoftendevelopedinafairlycomplexmathematicalcontext,especially
whentheproblemisfeaturedwithnontrivialconstraintsonthestateand/orcontrol
variables.
Theaimofthisbookistopresentinasimpleandfriendlymannertheachievements
resultingfromfirstordervariationalmethods.Theyareillustratedbyalargenumber
of problems which, almost without exceptions, refer to a particular second-order,
linearandtime-invariantdynamicalsystem,theso-calleddoubleintegrator.There
is no doubt that it is the simplest system which supplies a fairly comprehensive
overview of the topics under consideration. Furthermore a deeper insight on the
significance of the solutions can frequently be gained by recalling that the double
integrator constitutes the mathematical model of a body with unitary mass which
isconstrainedtomovealongastraightlineundertheactionofaforce.Finally,the
required computational burden is made as small as possible by the choice of this
particularsystem,sothatthepresentedmaterialcaneasilybegrasped.Consistently,
wehavechosennottogiveaformalproofofthevariousnecessaryconditionsand
©SpringerInternationalPublishingSwitzerland2017 1
A.Locatelli,OptimalControlofaDoubleIntegrator,StudiesinSystems,
DecisionandControl68,DOI10.1007/978-3-319-42126-1_1