Optimal Control with Engineering Applications.pdf-资料库

jiangdmdr-5425231-4744302543002438196.pdf-第1页.png

第1页 / 共141页

jiangdmdr-5425231-4744302543002438196.pdf-第2页.png

第2页 / 共141页

jiangdmdr-5425231-4744302543002438196.pdf-第3页.png

第3页 / 共141页

jiangdmdr-5425231-4744302543002438196.pdf-第4页.png

第4页 / 共141页

jiangdmdr-5425231-4744302543002438196.pdf-第5页.png

第5页 / 共141页

jiangdmdr-5425231-4744302543002438196.pdf-第6页.png

第6页 / 共141页

jiangdmdr-5425231-4744302543002438196.pdf-第7页.png

第7页 / 共141页

jiangdmdr-5425231-4744302543002438196.pdf-第8页.png

第8页 / 共141页

Optimal Control with Engineering Applications Hans P. Geering

Hans P. Geering Optimal Control with Engineering Applications With 12 Figures 123

Hans P. Geering, Ph.D. Professor of Automatic Control and Mechatronics Measurement and Control Laboratory Department of Mechanical and Process Engineering ETH Zurich Sonneggstrasse 3 CH-8092 Zurich, Switzerland Library of Congress Control Number: 2007920933 ISBN 978-3-540-69437-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by author Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: eStudi Calamar Steinen-Bro o S.L. F. , o , Girona, Spain SPIN 11880127 7/3100/YL - 5 4 3 2 1 0 Printed on acid-free paper

Foreword This book is based on the lecture material for a one-semester senior-year undergraduate or ﬁrst-year graduate course in optimal control which I have taught at the Swiss Federal Institute of Technology (ETH Zurich) for more than twenty years. The students taking this course are mostly students in mechanical engineering and electrical engineering taking a major in control. But there also are students in computer science and mathematics taking this course for credit. The only prerequisites for this book are: The reader should be familiar with dynamics in general and with the state space description of dynamic systems in particular. Furthermore, the reader should have a fairly sound understand- ing of diﬀerential calculus. The text mainly covers the design of open-loop optimal controls with the help of Pontryagin’s Minimum Principle, the conversion of optimal open-loop to optimal closed-loop controls, and the direct design of optimal closed-loop optimal controls using the Hamilton-Jacobi-Bellman theory. In theses areas, the text also covers two special topics which are not usually found in textbooks: the extension of optimal control theory to matrix-valued performance criteria and Lukes’ method for the iterative design of approxi- matively optimal controllers. Furthermore, an introduction to the phantastic, but incredibly intricate ﬁeld of diﬀerential games is given. The only reason for doing this lies in the fact that the diﬀerential games theory has (exactly) one simple application, namely the LQ diﬀerential game. It can be solved completely and it has a very attractive connection to the H∞ method for the design of robust linear time-invariant controllers for linear time-invariant plants. — This route is the easiest entry into H∞ theory. And I believe that every student majoring in control should become an expert in H∞ control design, too. The book contains a rather large variety of optimal control problems. Many of these problems are solved completely and in detail in the body of the text. Additional problems are given as exercises at the end of the chapters. The solutions to all of these exercises are sketched in the Solution section at the end of the book.

vi Foreword Acknowledgements First, my thanks go to Michael Athans for elucidating me on the background of optimal control in the ﬁrst semester of my graduate studies at M.I.T. and for allowing me to teach his course in my third year while he was on sabbatical leave. I am very grateful that Stephan A. R. Hepner pushed me from teaching the geometric version of Pontryagin’s Minimum Principle along the lines of [2], [20], and [14] (which almost no student understood because it is so easy, but requires 3D vision) to teaching the variational approach as presented in this text (which almost every student understands because it is so easy and does not require any 3D vision). I am indebted to Lorenz M. Schumann for his contributions to the material on the Hamilton-Jacobi-Bellman theory and to Roberto Cirillo for explaining Lukes’ method to me. Furthermore, a large number of persons have supported me over the years. I cannot mention all of them here. But certainly, I appreciate the continuous support by Gabriel A. Dondi, Florian Herzog, Simon T. Keel, Christoph M. Sch¨ar, Esfandiar Shafai, and Oliver Tanner over many years in all aspects of my course on optimal control. — Last but not least, I like to mention my secretary Brigitte Rohrbach who has always eagle-eyed my texts for errors and silly faults. Finally, I thank my wife Rosmarie for not killing me or doing any other harm to me during the very intensive phase of turning this manuscript into a printable form. Hans P. Geering Fall 2006

Contents List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . 1.1 Problem Statements . . . . . . 1.1.1 The Optimal Control Problem . 1.1.2 The Diﬀerential Game Problem . . . . . . . . . 1.3.1 Unconstrained Static Optimization . 1.3.2 Static Optimization under Constraints . 1.2 Examples . 1.3 Static Optimization . 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Optimal Control . . . . . . . . . . . . . . . . . . 2.1 Optimal Control Problems with a Fixed Final State . . . . 2.1.1 The Optimal Control Problem of Type A . . 2.1.2 Pontryagin’s Minimum Principle . . 2.1.3 Proof . . . 2.1.4 Time-Optimal, Frictionless, . . . . . . . . . . . . . . . . . . . Horizontal Motion of a Mass Point . . . 2.1.5 Fuel-Optimal, Frictionless, 2.2 Some Fine Points Horizontal Motion of a Mass Point . . 2.2.1 Strong Control Variation and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . global Minimization of the Hamiltonian . . 2.2.2 Evolution of the Hamiltonian . . 2.2.3 Special Case: Cost Functional J(u) = ±xi(tb) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 . 3 . 3 . 4 . . 5 . 18 . 18 . 19 . 22 . 23 . 24 . 24 . 25 . 25 28 32 . 35 35 . 36 . 36

viii Contents 2.3 Optimal Control Problems with a Free Final State 2.3.1 The Optimal Control Problem of Type C . . 2.3.2 Pontryagin’s Minimum Principle . 2.3.3 Proof . . . . . . 2.3.4 The LQ Regulator Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optimal Control Problems with a . . . . . . . . . . . . Partially Constrained Final State . 2.4.1 The Optimal Control Problem of Type B . . 2.4.2 Pontryagin’s Minimum Principle . . 2.4.3 Proof . . . . . 2.4.4 Energy-Optimal Control . . . . . 2.5 Optimal Control Problems with State Constraints . . . . . . . . . . . 2.5.1 The Optimal Control Problem of Type D . . . 2.5.2 Pontryagin’s Minimum Principle . 2.5.3 Proof . . . . 2.5.4 Time-Optimal, Frictionless, Horizontal Motion of a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Singular Optimal Control Mass Point with a Velocity Constraint . . . . 2.6.1 Problem Solving Technique 2.6.2 Goh’s Fishing Problem . 2.6.3 Fuel-Optimal Atmospheric Flight of a Rocket . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Existence Theorems . . 2.8 Optimal Control Problems . . . . . . . . . . with a Non-Scalar-Valued Cost Functional . . 2.8.1 Introduction . . . 2.8.2 Problem Statement . . . 2.8.3 Geering’s Inﬁmum Principle . . . 2.8.4 The Kalman-Bucy Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Optimal State Feedback Control . . . 3.1 The Principle of Optimality . 3.2 Hamilton-Jacobi-Bellman Theory . . . . 3.2.1 Suﬃcient Conditions for the Optimality of a Solution . 3.2.2 Plausibility Arguments about the HJB Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 . 38 . 38 . 39 . 41 . 43 . 43 . 43 . 44 . 46 . 48 . 48 . 49 . 51 54 . 59 . 59 . 60 . 62 . 65 . 67 . 67 . 68 . 68 . 69 . 72 . 75 . 75 . 78 . 78 . 80

资料库

Optimal Control with Engineering Applications.pdf

相关推荐

课程资源

热门标签

最新资料