Control perspectives on numerical algorithms and matrix problems PDF
Preview Control perspectives on numerical algorithms and matrix problems
Control Perspectives on Numerical Algorithms and Matrix Problems Advances in Design and Control SIAM's Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines. Editor-in-Chief Ralph C. Smith, North Carolina State University Editorial Board Athanasios C. Antoulas, Rice University Siva Banda, Air Force Research Laboratory Belinda A. Batten, Oregon State University John Betts, The Boeing Company Christopher Byrnes, Washington University Stephen L. Campbell, North Carolina State University Eugene M. Cliff, Virginia Polytechnic Institute and State University Michel C. Delfour, University of Montreal Max D. Gunzburger, Florida State University J. William Helton, University of California, San Diego Mary Ann Horn, Vanderbilt University Arthur J. Krener, University of California, Davis Kirsten Morris, University of Waterloo Richard Murray, California Institute of Technology Anthony Patera, Massachusetts Institute of Technology Ekkehard Sachs, University of Trier Allen Tannenbaum, Georgia Institute of Technology Series Volumes Bhaya, Amit, and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical Systems Huang, J., Nonlinear Output Regulation: Theory and Applications Haslinger, J. and Makinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems Gunzburger, Max D., Perspectives in Flow Control and Optimization Delfour, M. C. and Zolesio, J.-R, Shapes and Geometries: Analysis, Differential Calculus, and Optimization Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming El Ghaoui, Laurent and Niculescu, Silviu-lulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J. William and James, Matthew R., Extending H°°'Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives Control Perspectives on Numerical Algorithms and Matrix Problems Amit Bhaya Eugenius Kaszkurewicz Federal University of Rio de Janeiro Rio de Janeiro, Brazil siauTL Society for Industrial and Applied Mathematics Philadelphia Copyright © 2006 by the Society for Industrial and Applied Mathematics. 10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Ghostscript is a registered trademark of Artifex Software, Inc., San Rafael, California. PostScript is a registered trademark of Adobe Systems Incorporated in the United States and/or other countries. Library of Congress Cataloging-in-Publication Data Bhaya, Amit. Control perspectives on numerical algorithms and matrix problems / Amit Bhaya, Eugenius Kaszkurewicz. p. cm. — (Advances in design and control) Includes bibliographical references and index. ISBN 0-89871-602-0 (pbk.) 1. Control theory. 2. Numerical analysis. 3. Algorithms. 4. Mathematical optimization. 5. Matrices. I. Kaszkurewicz, Eugenius. II. Title. III. Series. QA402.3.B49 2006 515'.642-dc22 2005057551 siam. is a registered trademark. Dedication The authors dedicate this book to all those who have given them the incentive to work, including teachers, students, colleagues, and family. We cannot name one without naming them all, so they shall remain unnamed but sincerely appreciated just the same. This page intentionally left blank Contents List of Figures xi List of Tables xvii Preface xix 1 Brief Review of Control and Stability Theory 1 1.1 Control Theory Basics 1 1.1.1 Feedback control terminology 1 1.1.2 PID control for discrete-time systems 6 1.2 Optimal Control Theory 8 1.3 Linear Systems, Transfer Functions, Realization Theory 10 1.4 Basics of Stability of Dynamical Systems 15 1.5 Variable Structure Control Systems 27 1.6 Gradient Dynamical Systems 31 1.6.1 Nonsmooth GDSs: Persidskii-type results 35 1.7 Notes and References 38 2 Algorithms as Dynamical Systems with Feedback 41 2.1 Continuous-Time Dynamical Systems that Find Zeros 42 2.2 Iterative Zero Finding Algorithms as Discrete-Time Dynamical Systems 56 2.3 Iterative Methods for Linear Systems as Feedback Control Systems . . 70 2.3.1 CLF/LOC derivation of minimal residual and Krylov subspace methods 75 2.3.2 The conjugate gradient method derived from a proportional-derivative controller 77 2.3.3 Continuous algorithms for finding optima and the continuous conjugate gradient algorithm 85 2.4 Notes and References 90 3 Optimal Control and Variable Structure Design of Iterative Methods 93 3.1 Optimally Controlled Zero Finding Methods 94 3.1.1 An optimal control-based Newton-type method 94 3.2 Variable Structure Zero Finding Methods 96 vii viii Contents 3.2.1 A variable structure Newton method to find zeros of a polynomial function 99 3.2.2 The spurt method 107 3.3 Optimal Control Approach to Unconstrained Optimization Problems . 109 3.4 Differential Dynamic Programming Applied to Unconstrained Minimization Problems 118 3.5 Notes and References 124 4 Neural-Gradient Dynamical Systems for Linear and Quadratic Programming Problems 127 4.1 GDSs, Neural Networks, and Iterative Methods 128 4.2 GDSs that Solve Linear Systems of Equations 140 4.3 GDSs that Solve Convex Programming Problems 146 4.3.1 Stability analysis of a class of discontinuous GDSs .... 150 4.4 GDSs that Solve Linear Programming Problems 154 4.4.1 GDSs as linear programming solvers 159 4.5 Quadratic Programming and Support Vector Machines 167 4.5.1 v-support vector classifiers for nonlinear separation via GDSs 170 4.6 Further Applications: Least Squares Support Vector Machines, K-Winners-Take-All Problem 173 4.6.1 A least squares support vector machine implemented by a CDS 173 4.6.2 A GDS that solves the k-winners-take-all problem . . .. 174 4.7 Notes and References 177 5 Control Tools in the Numerical Solution of Ordinary Differential Equations and in Matrix Problems 179 5.1 Stepsize Control for ODEs 179 5.1.1 Stepsize control as a linear feedback system 182 5.1.2 Optimal Stepsize control for ODEs 185 5.2 A Feedback Control Perspective on the Shooting Method for ODEs . . 191 5.2.1 A state space representation of the shooting method ... 192 5.2.2 Error dynamics of the iterative shooting scheme 195 5.3 A Decentralized Control Perspective on Diagonal Preconditioning . . . 199 5.3.1 Perfect diagonal preconditioning 203 5.3.2 LQ perspective on optimal diagonal preconditioners . . .208 5.4 Characterization of Matrix D-Stability Using Positive Realness of a Feedback System 210 5.4.1 A feedback control approach to the D-stability problem via strictly positive real functions 213 5.4.2 D-stability conditions for matrices of orders 2 and 3 . . .216 5.5 Finding Zeros of Two Polynomial Equations in Two Variables via Controllability and Observability 218 5.6 Notes and References 224 Contents ix 6 Epilogue 227 Bibliography 233 Index 255
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