Optimal Control

211 Pages·2016·3.27 MB·English

by  Leonid T. Aschepkov

Leonid T. Aschepkov · Dmitriy V. Dolgy
Taekyun Kim · Ravi P. Agarwal
Optimal
Control
Optimal Control
Leonid T. Aschepkov Dmitriy V. Dolgy
(cid:129)
Taekyun Kim Ravi P. Agarwal
(cid:129)
Optimal Control
123
LeonidT. Aschepkov Taekyun Kim
Department ofMathematics Department ofMathematics
FarEastern FederalUniversity KwangwoonUniversity
Vladivostok Seoul
Russia Korea (Republicof)
Dmitriy V.Dolgy RaviP. Agarwal
Department ofMathematical Methods Department ofMathematics
inEconomy Texas A&MUniversity–Kingsville
FarEastern FederalUniversity Kingsville, TX
Vladivostok USA
Russia
ISBN978-3-319-49780-8 ISBN978-3-319-49781-5 (eBook)
DOI 10.1007/978-3-319-49781-5
LibraryofCongressControlNumber:2016959253
©SpringerInternationalPublishingAG2016
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This material is based on lectures from a one-year
courseattheFarEasternFederalUniversity(Vladivostok,
Russia) as well as on workshops on optimal control
offeredtostudentsatvariousmathematicaldepartmentsat
the university level. The main themes of the theory of
linear and nonlinear systems areconsidered, includingthe
basic problem of establishing the necessary and sufficient
conditions of optimal processes.
In the first part of the course, the theory of linear control
systems is constructed on the basis of the separation
theorem and the concept of a reachability set. We prove
the closure of a reachability set in the class of piecewise
continuous controls, and the problems of controllability,
observability, identification, performance and terminal
control are also considered. The second part of the course
is devoted to non-linear control systems. Using the
method of variations and the Lagrange multipliers rule of
non-linear problems, we prove the Pontryagin maximum
principle for problems with mobile ends of trajectories.
Further exercises and a large number of additional tasks
are provided for use as practical training in order for the
reader to consolidate the theoretical material.
Preface
This manual is written based on the lectures and practical exercises of the optimal
control course conducted by the authors at Irkutsk and Far Eastern Federal
Universities for students in the Faculty of Mathematics.
The course begins with optimal control theory of linear systems. First, the fun-
damentalconceptsofthemathematicalmodelsofcontrolledobjectsareintroduced,
andthencontrolandtrajectoryareusedtointroducetheconceptofareachablesetof
alinearsystem.Thepropertiesofthesystemaredetermined,includingwhetheritis
convex, has boundaries, is closed and is continuous according to time. Almost
all control problems of linear systems are formulated in terms of the reachable
set—from the problem of controllability to the problem of identification. This
concept is capacious in meaning and has a clear geometric interpretation, and it is
natural to use the separation theorem of convex sets to obtain some results. The
Cauchy formula, the concept of reachability and the separation theorem form the
basis for thewhole theory [8–10]of linear controlled systems.
Thesecondpartofthecourseisthendevotedtothetheoryofnonlinearsystems.
Thematerialispresentedfollowingariseincomplexity—fromthesimplestoptimal
control problem to the nonlinear control problem. The focus of this theory is the
Pontryagin maximum principle [13], and its justification, analysis, application and
modification depend on the type of problem of general nonlinear systems.
Itappearsfrommanyknownproofsofthemaximum principlethatthesimplest
formula that uses small increments of the trajectory is chosen. There is only one
step from here to the formula of small increments of a functional, as proposed by
Rozonoer [14], and to the maximum principle for the simplest problem of optimal
control.Theproblemsofoptimalcontrolwithconstraintsontheendsofapathare
studiedonthebasisofthewell-knownformulaofincrementsofthefunctionaland
the Lagrange multiplier rule [1,4] for nonlinear problems. The use of nonlinear
methods of optimization for optimal control is attractive not only to simplify the
technique used to obtain the proof of the maximum principle, but also to improve
the methodology. The continuity of a finite-dimensional and infinite-dimensional
optimization is established, and there is confidence in that simple and clear ideas
underpin complex constructions. This methodology has been successfully applied
vii
viii Preface
[2] to the study of control systems of differential equations with discontinuous
right-hand sides [5].
Forthesakeofcompleteness,thesufficientoptimalityconditionsofKrotov[11]
are concisely presented in the course, and the relation of these conditions is given
using the method of dynamic programming introduced by Bellman [3].
The appendix provides auxiliary information concerning multi-dimensional
geometry, convex analysis and the theory of extremal problems, all of which are
usedinthemainpartofthetext.Thecontentintheappendixallowsforthecourse
to be self-contained, without the need to handle other textbooks.
A considerable part of textbook is devoted to exercises and tasks. The purpose
of the exercises is to reinforce the assimilation of the theoretical material and to
independently apply new knowledge to solve similar or more complex theoretical
problems. The practical training material aims to develop theskillsand techniques
required to obtain analytical solutions for certain classes of optimal control prob-
lems. Most of these were used directly in practical classes and as homework.
Our teachers have significantly impacted the selection of the material and
structure of the course, including Profs. Gabasov, Kirillova and Vasiliev. The
communication with Profs. Srochko and Tyatyushkin, our colleagues from the
University of Irkutsk, was also extremely useful. We would like to take this
opportunity to express our sincere appreciation and thanks to all.
Vladivostok, Russia Leonid T. Aschepkov
Vladivostok, Russia Dmitriy V. Dolgy
Seoul, Korea (Republic of) Taekyun Kim
Kingsville, USA Ravi P. Agarwal
Contents
Part I Introduction
1 The Subject of Optimal Control.... .... .... .... .... ..... .... 3
1.1 “Mass-Spring” Example . .... .... .... .... .... ..... .... 3
1.2 Subject and Problems of Optimal Control.... .... ..... .... 5
1.3 Place of Optimal Control .... .... .... .... .... ..... .... 6
2 Mathematical Model for Controlled Object... .... .... ..... .... 7
2.1 Controlled Object . ..... .... .... .... .... .... ..... .... 7
2.2 Control and Trajectory .. .... .... .... .... .... ..... .... 7
2.3 Mathematical Model.... .... .... .... .... .... ..... .... 8
2.4 Existence and Uniqueness of a Process.. .... .... ..... .... 9
2.5 Linear Models.... ..... .... .... .... .... .... ..... .... 10
2.6 Example .... .... ..... .... .... .... .... .... ..... .... 11
Part II Control of Linear Systems
3 Reachability Set.... .... ..... .... .... .... .... .... ..... .... 17
3.1 Cauchy Formula.. ..... .... .... .... .... .... ..... .... 17
3.2 Properties of the Fundamental Matrix... .... .... ..... .... 19
3.3 Examples ... .... ..... .... .... .... .... .... ..... .... 21
3.4 Definition of a Reachability Set ... .... .... .... ..... .... 23
3.5 Limitation and Convexity.... .... .... .... .... ..... .... 25
3.6 Closure. .... .... ..... .... .... .... .... .... ..... .... 27
3.7 Continuity... .... ..... .... .... .... .... .... ..... .... 30
3.8 Extreme Principle. ..... .... .... .... .... .... ..... .... 33
3.9 Application of the Extreme Principle ... .... .... ..... .... 36
Exercise Set ... .... .... ..... .... .... .... .... .... ..... .... 38
ix
x Contents
4 Controllability of Linear Systems... .... .... .... .... ..... .... 41
4.1 Point-to-Point Controllability . .... .... .... .... ..... .... 41
4.2 Analysis of the Point-to-Point Controllability Criteria.... .... 42
4.3 Auxiliary Lemma . ..... .... .... .... .... .... ..... .... 45
4.4 Kalman Theorem . ..... .... .... .... .... .... ..... .... 47
4.5 Control with Minimal Norm.. .... .... .... .... ..... .... 48
4.6 Construction of Control with Minimum Norm .... ..... .... 49
4.7 Total Controllability of Linear System .. .... .... ..... .... 51
4.8 Synthesis of Control with a Minimal Norm .. .... ..... .... 52
4.9 Krasovskii Theorem .... .... .... .... .... .... ..... .... 54
4.10 Total Controllability of Stationary System ... .... ..... .... 55
4.11 Geometry of a Non-controllable System. .... .... ..... .... 56
4.12 Transformation of Non-controllable System .. .... ..... .... 57
4.13 Controllability of Transformed System.. .... .... ..... .... 59
Excercise Set... .... .... ..... .... .... .... .... .... ..... .... 61
5 Minimum Time Problem. ..... .... .... .... .... .... ..... .... 63
5.1 Statement of the Problem .... .... .... .... .... ..... .... 63
5.2 Existence of a Solution of the Minimum Time Problem.. .... 64
5.3 Criterion of Optimality.. .... .... .... .... .... ..... .... 65
5.4 Maximum Principle for the Minimum Time Problem.... .... 67
5.5 Stationary Minimum Time Problem .... .... .... ..... .... 69
Exercise Set ... .... .... ..... .... .... .... .... .... ..... .... 74
6 Synthesis of the Optimal System Performance .... .... ..... .... 77
6.1 General Scheme to Apply the Maximum Principle. ..... .... 77
6.2 Control of Acceleration of a Material Point .. .... ..... .... 79
6.3 Concept of Optimal Control Synthesis .. .... .... ..... .... 81
6.4 Examples of Synthesis of Optimal Systems Performance. .... 82
Exercise Set ... .... .... ..... .... .... .... .... .... ..... .... 90
7 The Observability Problem.... .... .... .... .... .... ..... .... 91
7.1 Statement of the Problem .... .... .... .... .... ..... .... 91
7.2 Criterion of Observability.... .... .... .... .... ..... .... 92
7.3 Observability in Homogeneous System.. .... .... ..... .... 93
7.4 Observability in Nonhomogeneous System... .... ..... .... 95
7.5 Observability of an Initial State ... .... .... .... ..... .... 96
7.6 Relation Between Controllability and Observability ..... .... 98
7.7 Total Observability of a Stationary System... .... ..... .... 99
Exercise Set ... .... .... ..... .... .... .... .... .... ..... .... 99
8 Identification Problem... ..... .... .... .... .... .... ..... .... 101
8.1 Statement of the Problem .... .... .... .... .... ..... .... 101
8.2 Criterion of Identifiability.... .... .... .... .... ..... .... 102
Contents xi
8.3 Restoring the Parameter Vector.... .... .... .... ..... .... 103
8.4 Total Identification of Stationary System .... .... ..... .... 104
Exercise Set ... .... .... ..... .... .... .... .... .... ..... .... 105
Part III Control of Nonlinear Systems
9 Types of Optimal Control Problems .... .... .... .... ..... .... 109
9.1 General Characteristics.. .... .... .... .... .... ..... .... 109
9.2 Objective Functionals... .... .... .... .... .... ..... .... 110
9.3 Constraints on the Ends of a Trajectory, Terminology ... .... 112
9.4 The Simplest Problem... .... .... .... .... .... ..... .... 112
9.5 Two-Point Minimum Time Problem.... .... .... ..... .... 113
9.6 General Optimal Control Problem.. .... .... .... ..... .... 113
10 Small Increments of a Trajectory... .... .... .... .... ..... .... 115
10.1 Statement of a Problem.. .... .... .... .... .... ..... .... 115
10.2 Evaluation of the Increment of Trajectory.... .... ..... .... 115
10.3 Representation of Small Increments of Trajectory.. ..... .... 120
10.4 Relation of the Ends of Trajectories.... .... .... ..... .... 122
11 The Simplest Problem of Optimal Control ... .... .... ..... .... 125
11.1 Formula of the Increment of a Functional.... .... ..... .... 126
11.2 Maximum Principle for the Simplest Problem .... ..... .... 128
11.3 Boundary Value Problem of the Maximum Principle.... .... 129
11.4 Continuity of the Hamiltonian .... .... .... .... ..... .... 129
11.5 Sufficiency of the Maximum Principle .. .... .... ..... .... 131
11.6 Applying the Maximum Principle to the Linear Problem . .... 133
11.7 Solution of the Mass-Spring Example... .... .... ..... .... 134
Exercise Set ... .... .... ..... .... .... .... .... .... ..... .... 136
12 General Optimal Control Problem.. .... .... .... .... ..... .... 139
12.1 Formula of the Increment of Functional . .... .... ..... .... 140
12.2 Variation of the Process . .... .... .... .... .... ..... .... 141
12.3 Necessary Conditions of Optimality .... .... .... ..... .... 144
12.4 Lagrange Multiplier Rule .... .... .... .... .... ..... .... 147
12.5 Universal Lagrange Multipliers.... .... .... .... ..... .... 150
12.6 Maximum Principle for the General Problem . .... ..... .... 151
12.7 Comments... .... ..... .... .... .... .... .... ..... .... 153
12.8 Sufficiency of the Maximum Principle .. .... .... ..... .... 154
12.9 Maximum Principle for Minimum Time Problem.. ..... .... 156
12.10 Maximum Principle and Euler-Lagrange Equation . ..... .... 158
12.11 Maximum Principle and Optimality of a Process .. ..... .... 161
Exercise Set ... .... .... ..... .... .... .... .... .... ..... .... 161