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Algebraic Geometry Applied Mathematical Sciences Volume 169 Editors S.S. Antman J.E. Marsden L. Sirovich Advisors J. Hale P. Holmes J. Keener J. Keller R. Laubenbacher B.J. Matkowsky A. Mielke C.S. Peskin K.R. Sreenivasan A. Stevens For further volumes: http://www.springer.com/series/34 This page intentionally left blank David L. Elliott Bilinear Control Systems Matrices in Action 123 Prof. David Elliott University of Maryland Inst. Systems Research College Park MD 20742 USA Editors: S.S. Antman J.E. Marsden Department of Mathematics Control and Dynamical and Systems, 107-81 Institute for Physical California Institute of Science and Technology Technology University of Maryland Pasadena, CA 91125 College Park, MD 20742-4015 USA USA [email protected] [email protected] L. Sirovich Laboratory of Applied Mathematics Department of Biomathematical Sciences Mount Sinai School of Medicine New York, NY 10029-6574 [email protected] ISSN 0066-5452 ISBN 978-1-4020-9612-9 e-ISBN 978-1-4020-9613-6 DOI 10.1007/978-1-4020-9613-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009920095 Mathematics Subject Classification (2000): 93B05, 1502, 57R27, 22E99, 37C10 c Springer Science+Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The mathematical theory of control became a field of study half a century ago in attempts to clarify and organize some challenging practical problems and the methods used to solve them. It is known for the breadth of the mathematics it uses and its cross-disciplinary vigor. Its literature, which can be found in Section 93 of Mathematical Reviews, was at one time dominated by the theory of linear control systems, which mathematically are described by linear differential equations forced by additive control inputs. That theory led to well-regarded numerical and symbolic computational packages for control analysis and design. Nonlinear control problems are also important; in these either the un- derlying dynamical system is nonlinear or the controls are applied in a non- additive way. The last four decades have seen the development of theoretical work on nonlinear control problems based on differential manifold theory, nonlinear analysis, and several other mathematical disciplines. Many of the problems that had been solved in linear control theory, plus others that are new and distinctly nonlinear, have been addressed; some resulting general definitions and theorems are adapted in this book to the bilinear case. A nonlinear control system is called bilinear if it is described by linear dif- ferential equations in which the control inputs appear as coefficients. Such multiplicative controls (valves, interest rates, switches, catalysts, etc.) are common in engineering design and also are used as models of natural phe- nomena with variable growth rates. Their study began in the late 1960s and has continued from its need in applications, as a source of understandable examples in nonlinear control, and for its mathematical beauty. Recent work connects bilinear systems to such diverse disciplines as switching systems, spin control in quantum physics, and Lie semigroup theory; the field needs an expository introduction and a guide, even if incomplete, to its literature. The control of continuous-time bilinear systems is based on properties of matrix Lie groups and Lie semigroups. For that reason, much of the first half of the book is based on matrix analysis including the Campbell–Baker– Hausdorff Theorem. (The usual approach would be to specialize geometric v vi Preface methods of nonlinear control based on Frobenius’s Theorem in manifold the- ory.) Other topics such as discrete-time systems, observability and realiza- tion, applications, linearization of nonlinear systems with finite-dimensional Lie algebras, and input–output analysis have chapters of their own. The intended readers for this book includes graduate mathematicians and engineers preparing for research or application work. If short and helpful, proofs are given; otherwise they are sketched or cited from the mathematical literature. The discussions are amplified by examples, exercises, and also a few Mathematica scripts to show how easily software can be written for bilinear control problems. Before you begin to read, please turn to the very end of the book to see that the Index lists in bold type the pages where symbols and concepts are first defined. Then please glance at the Appendices A–D; they are referred to often for standard facts about matrices, Lie algebras, and Lie groups. Throughout the book, sections whose titles have an asterisk (*) invoke less familiar mathematics — come back to them later. The mark indicates the end of a proof, while indicates the end of a definition, remark, exercise, or example. △ My thanks go to A. V. Balakrishnan, my thesis director, who introduced me to bilinear systems; my Washington University collaborators in bilinear control theory — William M. Boothby, James G-S. Cheng, Tsuyoshi Goka, Ellen S. Livingston, Jackson L. Sedwick, Tzyh-Jong Tarn, Edward N. Wilson, and the late John Zaborsky; the National Science Foundation, for support of our research; Michiel Hazewinkel, who asked for this book; Linus Kramer; Ron Mohler; Luiz A. B. San Martin; Michael Margaliot, for many helpful comments; and to the Institute for Systems Research of the University of Maryland for its hospitality since 1992. This book could not have existed without the help of Pauline W. Tang, my wife. College Park, Maryland, David L. Elliott Contents 1 Introduction .................................................. 1 1.1 MatricesinAction........................................ 2 1.2 Stability: Linear Dynamics . ............................... 7 1.3 LinearControlSystems.................................... 8 1.4 WhatIsaBilinearControlSystem?......................... 10 1.5 TransitionMatrices ....................................... 13 1.6 Controllability . .......................................... 20 1.7 Stability: Nonlinear Dynamics . ........................... 24 1.8 From Continuous to Discrete . ........................... 28 1.9 Exercises................................................. 30 2 Symmetric Systems: Lie Theory ................................ 33 2.1 Introduction.............................................. 33 2.2 LieAlgebras.............................................. 34 2.3 LieGroups............................................... 44 2.4 Orbits,Transitivity,andLieRank........................... 54 2.5 AlgebraicGeometryComputations......................... 60 2.6 Low-DimensionalExamples............................... 68 2.7 GroupsandCosetSpaces.................................. 70 2.8 CanonicalCoordinates.................................... 72 2.9 ConstructingTransitionMatrices........................... 74 2.10ComplexBilinearSystems................................. 77 2.11 Generic Generation . ...................................... 79 2.12Exercises................................................. 81 3 Systems with Drift ............................................ 83 3.1 Introduction.............................................. 83 3.2 Stabilization with Constant Control . .................... 85 3.3 Controllability . .......................................... 89 3.4 Accessibility . ..........................................100 3.5 SmallControls............................................104 vii viii Contents 3.6 Stabilization by State-Dependent Inputs ....................107 3.7 LieSemigroups...........................................116 3.8 BiaffineSystems..........................................119 3.9 Exercises.................................................124 4 Discrete-Time Bilinear Systems ................................127 4.1 DynamicalSystems:Discrete-Time.........................128 4.2 Discrete-TimeControl.....................................129 4.3 Stabilization by Constant Inputs ...........................131 4.4 Controllability . ..........................................132 4.5 ACautionaryTale ........................................141 5 Systems with Outputs .........................................143 5.1 CompositionsofSystems..................................144 5.2 Observability . ..........................................146 5.3 StateObservers...........................................151 5.4 IdentificationbyParameterEstimation.....................153 5.5 Realization...............................................154 5.6 Volterra Series . ..........................................161 5.7 Approximation with Bilinear Systems . ....................163 6 Examples .....................................................165 6.1 PositiveBilinearSystems..................................165 6.2 CompartmentalModels...................................170 6.3 Switching................................................172 6.4 PathConstructionandOptimization.......................179 6.5 QuantumSystems........................................184 7 Linearization .................................................187 7.1 EquivalentDynamicalSystems ............................188
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