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E-raamat: Optimal Control

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Jan-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319497815
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Optimal ControlSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Jan-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319497815
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This book is based on lectures from a one-year course at the Far Eastern Federal University (Vladivostok, Russia) as well as on workshops on optimal control offered to students at various mathematical departments at the university level. The main themes of the theory of linear and nonlinear systems are considered, including the 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. The authors 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 nonlinear control systems. Using the method of variations and the Lagrange multipliers rule of nonlinear problems, the authors prove the Pontry

agin 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. 

NOTATIONS.- PREFACE.- INTRODUCTION.- 1. Subject of optimal control.- 2. Mathematical model of controlled object.- 3. Reachability set.- 4. Controllability of linear systems.- 5. Minimum time problem.- 6. Synthesis of optimal system performance.- 7. The observability problem.- 8. Identification problem.- 9. Types of optimal control problems.- 10. Small increments of a trajectory.- 11. The simplest problem of optimal control.- 12. General optimal control problem.- 13. Sufficient optimality conditions.- CONCLUSION.- APPENDIX.- EXAMPLES OF TASKS AND SOLUTIONS.- LITERATURE.
Part I Introduction
1 The Subject of Optimal Control
3(4)
1.1 "Mass-Spring" Example
3(2)
1.2 Subject and Problems of Optimal Control
5(1)
1.3 Place of Optimal Control
6(1)
2 Mathematical Model for Controlled Object
7(10)
2.1 Controlled Object
7(1)
2.2 Control and Trajectory
7(1)
2.3 Mathematical Model
8(1)
2.4 Existence and Uniqueness of a Process
9(1)
2.5 Linear Models
10
2.6 Example
11(6)
Part II Control of Linear Systems
3 Reachability Set
17(24)
3.1 Cauchy Formula
17(2)
3.2 Properties of the Fundamental Matrix
19(2)
3.3 Examples
21(2)
3.4 Definition of a Reachability Set
23(2)
3.5 Limitation and Convexity
25(2)
3.6 Closure
27(3)
3.7 Continuity
30(3)
3.8 Extreme Principle
33(3)
3.9 Application of the Extreme Principle
36(5)
Exercise Set
38(3)
4 Controllability of Linear Systems
41(22)
4.1 Point-to-Point Controllability
41(1)
4.2 Analysis of the Point-to-Point Controllability Criteria
42(3)
4.3 Auxiliary Lemma
45(2)
4.4 Kalman Theorem
47(1)
4.5 Control with Minimal Norm
48(1)
4.6 Construction of Control with Minimum Norm
49(2)
4.7 Total Controllability of Linear System
51(1)
4.8 Synthesis of Control with a Minimal Norm
52(2)
4.9 Krasovskii Theorem
54(1)
4.10 Total Controllability of Stationary System
55(1)
4.11 Geometry of a Non-controllable System
56(1)
4.12 Transformation of Non-controllable System
57(2)
4.13 Controllability of Transformed System
59(4)
Excercise Set
61(2)
5 Minimum Time Problem
63(14)
5.1 Statement of the Problem
63(1)
5.2 Existence of a Solution of the Minimum Time Problem
64(1)
5.3 Criterion of Optimality
65(2)
5.4 Maximum Principle for the Minimum Time Problem
67(2)
5.5 Stationary Minimum Time Problem
69(8)
Exercise Set
74(3)
6 Synthesis of the Optimal System Performance
77(14)
6.1 General Scheme to Apply the Maximum Principle
77(2)
6.2 Control of Acceleration of a Material Point
79(2)
6.3 Concept of Optimal Control Synthesis
81(1)
6.4 Examples of Synthesis of Optimal Systems Performance
82(9)
Exercise Set
90(1)
7 The Observability Problem
91(10)
7.1 Statement of the Problem
91(1)
7.2 Criterion of Observability
92(1)
7.3 Observability in Homogeneous System
93(2)
7.4 Observability in Nonhomogeneous System
95(1)
7.5 Observability of an Initial State
96(2)
7.6 Relation Between Controllability and Observability
98(1)
7.7 Total Observability of a Stationary System
99(2)
Exercise Set
99(2)
8 Identification Problem
101(8)
8.1 Statement of the Problem
101(1)
8.2 Criterion of Identifiability
102(1)
8.3 Restoring the Parameter Vector
103(1)
8.4 Total Identification of Stationary System
104(5)
Exercise Set
105(4)
Part III Control of Nonlinear Systems
9 Types of Optimal Control Problems
109(6)
9.1 General Characteristics
109(1)
9.2 Objective Functionals
110(2)
9.3 Constraints on the Ends of a Trajectory, Terminology
112(1)
9.4 The Simplest Problem
112(1)
9.5 Two-Point Minimum Time Problem
113(1)
9.6 General Optimal Control Problem
113(2)
10 Small Increments of a Trajectory
115(10)
10.1 Statement of a Problem
115(1)
10.2 Evaluation of the Increment of Trajectory
115(5)
10.3 Representation of Small Increments of Trajectory
120(2)
10.4 Relation of the Ends of Trajectories
122(3)
11 The Simplest Problem of Optimal Control
125(14)
11.1 Formula of the Increment of a Functional
126(2)
11.2 Maximum Principle for the Simplest Problem
128(1)
11.3 Boundary Value Problem of the Maximum Principle
129(1)
11.4 Continuity of the Hamiltonian
129(2)
11.5 Sufficiency of the Maximum Principle
131(2)
11.6 Applying the Maximum Principle to the Linear Problem
133(1)
11.7 Solution of the Mass-Spring Example
134(5)
Exercise Set
136(3)
12 General Optimal Control Problem
139(24)
12.1 Formula of the Increment of Functional
140(1)
12.2 Variation of the Process
141(3)
12.3 Necessary Conditions of Optimality
144(3)
12.4 Lagrange Multiplier Rule
147(3)
12.5 Universal Lagrange Multipliers
150(1)
12.6 Maximum Principle for the General Problem
151(2)
12.7 Comments
153(1)
12.8 Sufficiency of the Maximum Principle
154(2)
12.9 Maximum Principle for Minimum Time Problem
156(2)
12.10 Maximum Principle and Euler-Lagrange Equation
158(3)
12.11 Maximum Principle and Optimality of a Process
161(2)
Exercise Set
161(2)
13 Sufficient Optimality Conditions
163(10)
13.1 Common Problem of Optimal Control
163(1)
13.2 Basic Theorems
164(3)
13.3 Analytical Construction of the Controller
167(3)
13.4 Relation with Dynamic Programming
170(3)
Exercise Set
172(1)
Conclusion 173(2)
Appendix 175(14)
Examples of Tasks and Solution 189(20)
Literature 209
Leonid Aschepkov is a professor in the Department of Mathematical Methods of Economy at Far Eastern Federal University. Dmitriy V. Dolgy is a professor at the Institute of Natural Sciences at Far Eastern Federal University in Vladivolstok, Russia and at Hanrimwon, Kwangwoon University in Seoul, Republic of Korea. Taekyun Kim is a professor in the Department of Mathematics at the College of Natural Science at Kwangwoon University. 

Ravi P. Agarwal is a professor and the chair of the Department of Mathematics at Texas A&M University. 
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