Control, Identification, and Input Optimization

428 Pages·1982·7.991 MB·English

by  Robert Kalaba, Karl Spingarn (auth.)

Control, Identification,
and Input Optimization
MATHEMATICAL CONCEPTS AND METHODS
IN SCIENCE AND ENGINEERING
Series EdItor: Angelo Miele
Mechanical Engineering and Mathematical Sciences
Rice University
Recent volumes in the series:
11 INTEGRAL TRANSFORMS IN SCIENCE AND ENGINEERING
• Kurt Bernado Wolf
12 APPLIED MATHEMATICS: An Intellectual Orientation. Francis J. Murray
14 PRINCIPLES AND PROCEDURES OF NUMERICAL ANALYSIS
• Ferenc Szidarovszky and Sidney Yakowitz
16 MATHEMATICAL PRINCIPLES IN MECHANICS AND ELECI'ROMAGNETISM,
Part A: Analytical and Continuum Mechanics. C. C. Wang
17 MATHEMATICAL PRINCIPLES IN MECHANICS AND ELECI'ROMAGNETISM,
Part B: Electrom. ...e tlsm and Gnvltation • C. C. Wang
18 SOLUTION METHODS FOR INTEGRAL EQUATIONS: Theory and Applications
• Edited by Michael A. Golberg
19 DYNAMIC OPTIMIZATION AND MATHEMATICAL ECONOMICS
• Edited by Pan-Tai Liu
20 DYNAMICAL SYSTEMS AND EVOLUTION EQUATIONS: Theory and Applications
• J. A. Walker
11 ADVANCES IN GEOMETRIC PROGRAMMING. Edited by Mordecai A vriel
11 APPLICATIONS OF FUNCTIONAL ANALYSIS IN ENGINEERING. J. L. Nowinski
13 APPLIED PROBABILITY. Frank A. Haight
14 THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL: An Introduction
• George L~Umann
15 CONTROL, IDENTIFICATION, AND INPUT OPTIMIZATION
• Robert Kalaba and Karl Spingarn
A Continuation Order Plan is available for this series. A continuation order will bring
delivery of each new volume immediately upon publication. Volumes are billed only upon
actual shipment. For further information please contact the publisher.
Control, Identification,
and Input Optimization
Robert Kalaba
University of Southern California
Los Angeles, California
and
Karl Spingarn
Hughes Aircraft Company
Los Angeles, California
PLENUM PRESS • NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
Kalaba, Robert E.
Control, identification, and input optimization.
(Mathematical concepts and methods in science and engineering; 2S)
Includes bibliographical references and index.
1. Control theory. 2. System identification. I. Spingarn, Karl. II. Title. III. Series.
QA402.3.K27 629.8'312 81-23404
AACR2
ISBN 978-1-4684-7664-4 ISBN 978-1-4684-7662-0 (ebook)
DOl 10.1007/978-1-4684-7662-0
© 1982 Plenum Press, New York
Softcover reprint of the hardcover 15t edition 1982
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photOC()pying, microfilming,
recording, or otherwise, without written permission from the Publisher
Preface
This book is a self-contained text devoted to the numerical determination
of optimal inputs for system identification. It presents the current state
of optimal inputs with extensive background material on optimization
and system identification. The field of optimal inputs has been an area of
considerable research recently with important advances by R. Mehra, G.
c. Goodwin, M. Aoki, and N. E. Nahi, to name just a few eminent in
vestigators. The authors' interest in optimal inputs first developed when
F. E. Yates, an eminent physiologist, expressed the need for optimal or
preferred inputs to estimate physiological parameters.
The text assumes no previous knowledge of optimal control theory,
numerical methods for solving two-point boundary-value problems, or
system identification. As such it should be of interest to students as well
as researchers in control engineering, computer science, biomedical en
gineering, operations research, and economics. In addition the sections
on beam theory should be of special interest to mechanical and civil en
gineers and the sections on eigenvalues should be of interest to numerical
analysts. The authors have tried to present a balanced viewpoint; however,
primary emphasis is on those methods in which they have had first-hand
experience. Their work has been influenced by many authors. Special
acknowledgment should go to those listed above as well as R. Bellman,
A. Miele, G. A. Bekey, and A. P. Sage.
The book can be used for a two-semester course in control theory,
system identification, and optimal inputs. The first semester would cover
optimal control theory and system identification, and the second semester
would cover optimal inputs and applications. Alternatively, for those stu
dents who have already been introduced to optimal control theory, the
v
vi Preface
book can be used for a one-semester course in system identification and
optimal inputs. The text can also be used as an introduction to control
theory or to system identification.
The desired purpose of the text is to provide upper-division under
graduate and graduate students, as well as engineers and scientists in in
dustry, with the analytical and computational tools required to compute
optimal inputs for system identification. System identification is concerned
with the estimation of the parameters of dynamic system models. The ac
curacy of the parameter estimates is enhanced by the use of optimal inputs
to increase the sensitivity of the observations to the parameters being es
timated.
The determination of optimal inputs for system identification requires
a knowledge of dynamic system optimization techniques, numerical methods
of solution, and methods of system identification. All of these topics are
covered in the text.
Part I of the text is an introduction to the subject. Part II is Optimal
Control and Methods of Numerical Solutions. The chapters on optimal
control include the Euler-Lagrange equations, dynamic programming,
Pontryagin's maximum principle, and the Hamilton-Jacobi equations. The
numerical methods for linear and nonlinear two-point boundary-value
problems include the matrix Riccati equation, method of complementary
functions, invariant imbedding, quasilinearization, and the Newton-Raph
son methods.
Part III is System Identification. The chapters in this part include the
Gauss-Newton and quasilinearization methods for system identification.
Applications of these methods are presented. Part IV is Optimal Inputs
for System Identification. The equations for the determination of optimal
inputs are derived and applications are given. Part V lists computer pro
grams.
Optimal inputs have applications in such diverse fields as biomedical
modeling and aircraft stability and control parameter estimation. The text
is intended to provide a means of computing optimal inputs by providing
the analytical and computational building blocks for this purpose, for
example, control theory, computational methods, system identification,
and optimal inputs.
The authors wish to thank the National Science Foundation, the
National Institutes of Health, and the Air Force Office of Scientific Research
for continuing support of our research efforts.
Robert Kalaba
Karl Spingarn
Contents
PART I. INTRODUCTION
1. Introduction . . . 3
1.1. Optimal Control. . . 3
1.2. System Identification . 4
1.3. Optimal Inputs . . . 5
1.4. Computational Preliminaries 7
Exercises ........ . 9
PART II. OPTIMAL CONTROL AND METHODS FOR
NUMERICAL SOLUTIONS
2. Optimal Control . 13
2.1. Simplest Problem in the Calculus of Variations 13
2.1.1. Euler-Lagrange Equations 13
2.1.2. Dynamic Programming . . 17
2.1.3. Hamilton-Jacobi Equations 19
2.2. Several Unknown Functions .. 23
2.3. Isoperimetric Problems . . . . . 23
2.4. Differential Equation Auxiliary Conditions 27
2.5. Pontryagin's Maximum Principle . . . . 31
2.6. Equilibrium of a Perfectly Flexible Inhomogeneous Suspended
Cable ..................... . 34
2.7. New Approaches to Optimal Control and Filtering. 36
2.8. Summary of Commonly Used Equations . 43
Exercises ....... . 43
vii
viii Contents
3. Numerical Solutions for Linear Two-Point Boundary-Value Problems. 45
3.1. Numerical Solution Methods . . . . . . . 45
3.1.1. Matrix Riccati Equation . . . . . . 47
3.1.2. Method of Complementary Functions 48
3.1.3. Invariant Imbedding . . . . . . . . 51
3.1.4. Analytical Solution. . . . . . . . . 53
3.2. An Optimal Control Problem for a First-Order System 54
3.2.1. The Euler-Lagrange Equations . 55
3.2.2. Pontryagin's Maximum Principle 58
3.2.3. Dynamic Programming. . . . 59
3.2.4. Kalaba's Initial-Value Method 61
3.2.5. Analytical Solution. . . . . . 65
3.2.6. Numerical Results . . . . . . 68
3.3. An Optimal Control Problem for a Second-Order System 68
3.3.1. Numerical Methods . . . . . . . 69
3.3.2. Analytical Solution. . . . . . . . 70
3.3.3. Numerical Results and Discussion . 72
Exercises . . . . . . . . . . . . . . . 74
4. Numerical Solutions for Nonlinear Two-Point Boundary-Value Problems 77
4.1. Numerical Solution Methods . . 77
4.1.1. Quasilinearization 79
4.1.2. Newton-Raphson Method 80
4.2. Examples of Problems Yielding Nonlinear Two-Point Boundary-
Value Problems . . . . . . . . . . . . . . . . . . . . .. 81
4.2.1. A First-Order Nonlinear Optimal Control Problem . " 81
4.2.2. Optimization of Functionals Subject to Integral Constraints 86
4.2.3. Design of Linear Regulators with Energy Constraints 96
4.3. Examples Using Integral Equation and Imbedding Methods 105
4.3.1. Integral Equation Method for Buckling Loads . . 105
4.3.2. An Imbedding Method for Buckling Loads 113
4.3.3. An Imbedding Method for a Nonlinear Two-Point
Boundary-Value Problem . . . . . . . . . . . . 123
4.3.4. Post-Buckling Beam Configurations via an Imbedding
Method. . . . . . . . . . . . . . . . . . 129
4.3.5. A Sequential Method for Nonlinear Filtering. 142
Exercises . . . . . . . . . . . . . . . . . . . . 160
PART III. SYSTEM IDENTIFICATION
5. Gauss-Newton Method for System Identification . 165
5.1. Least-Squares Estimation. . . . . . . 165
5.1.1. Scalar Least-Squares Estimation. 166
5.1.2. Linear Least-Squares Estimation. 167
Contents ix
5.2. Maximum Likelihood Estimation 167
5.3. Cramer-Rao Lower Bound. . . 169
5.4. Gauss-Newton Method 170
5.5. Examples of the Gauss-Newton Method.. 172
5.5.1. First-Order System with Single Unknown Parameter 172
5.5.2. First-Order System with Unknown Initial Condition and
Single Unknown Parameter . . . . . . . . . . . .. 173
5.5.3. Second-Order System with Two Unknown Parameters
and Vector Measurement . . . . . . . . . . . . .. 174
5.5.4. Second-Order System with Two Unknown Parameters
and Scalar Measurement 175
Exercises . . . . . . . . . . . . . . . . . . . . . . .. 175
6. Quasilinearization Method for System Identification 179
6.1. System Identification via Quasilinearization. . 179
6.2. Examples of the Quasilinearization Method . 182
6.2.1. First-Order System with Single Unknown Parameter 182
6.2.2. First-Order System with Unknown Initial Condition and
Single Unknown Parameter . . . . . . . . . . . .. 184
6.2.3. Second-Order System with Two Unknown Parameters
and Vector Measurement . . . . . . . . . . . . .. 187
6.2.4. Second-Order System with Two Unknown Parameters
and Scalar Measurement 191
Exercises ......... . 192
7. Applications of System Identification . . . . . . . . 195
7.1. Blood Glucose Regulation Parameter Estimation 195
7.1.1. Introduction. . . . . . . 195
7.1.2. Physiological Experiments. 198
7.1.3. Computational Methods . 200
7.1.4. Numerical Results .... 207
7.1.5. Discussion and Conclusions. 210
7.2. Fitting of Nonlinear Models of Drug Metabolism to
Experimental Data. . . . . . . . . . . . . . . . 211
7.2.1. Introduction. . . . . . . . . . . . . . . . 211
7.2.2. A Model Employing Michaelis and Menten Kinetics for
Metabolism . . . . . . 211
7.2.3. An Estimation Problem 212
7.2.4. Quasilinearization 213
7.2.5. Numerical Results 217
7.2.6. Discussion. 221
Exercises . . . . . . . 221
x Contents
PART IV. OPTIMAL INPUTS FOR SYSTEM IDENTIFICATION
8. Optimal Inputs 225
8.1. Historical Background 225
8.2. Linear Optimal Inputs 227
8.2.1. Optimal Inputs and Sensitivities for Parameter Estimation 228
8.2.2. Sensitivity of Parameter Estimates to Observations 240
8.2.3. Optimal Inputs for a Second-Order Linear System .. 245
8.2.4. Optimal Inputs Using Mehra's Method . . . . . .. 253
8.2.5. Comparison of Optimal Inputs for Homogeneous and
Nonhomogeneous Boundary Conditions . . . . . 258
8.3. Nonlinear Optimal Inputs . . . . . . . . . . . . . . 262
8.3.1. Optimal Input System Identification for Nonlinear
Dynamic Systems . . . . . . . . . . . . . . . 263
8.3.2. General Equations for Optimal Inputs for Nonlinear
Process Parameter Estimation 274
Exercises .......... . 278
9. Additional Topics for Optimal Inputs . 281
9.1. An Improved Method for the Numerical Determination of
Optimal Inputs . . . . . . 281
9.1.1. Introduction . . . . . . . . . . . . . 282
9.1.2. A Nonlinear Example . . . . . . . . 284
9.1.3. Solution via Newton-Raphson Method 285
9.1.4. Numerical Results and Discussion. . . 287
9.2. Multiparameter Optimal Inputs . . . . . . . 290
9.2.1. Optimal Inputs for Vector Parameter Estimation 290
9.2.2. Example of Optimal Inputs for Two-Parameter Estimation 293
9.2.3. Example of Optimal Inputs for a Single-Input, Two-
Output System. . . . . . . . . . . . . 296
9.2.4. Example of Weighted Optimal Inputs . . 298
9.3. Observability, Controllability, and Identifiability 299
9.4. Optimal Inputs for Systems with Process Noise 301
9.5. Eigenvalue Problems. . . . . . . . . . . . . 302
9.5.1. Convergence of the Gauss-Seidel Method 303
9.5.2. Determining the Eigenvalues of Saaty's Matrices for
Fuzzy Sets .. .. .. .. .. .. .. .. .. . 309
9.5.3. Comparison of Methods for Determining the Weights of
Belonging to Fuzzy Sets . . . . . . . . . . 318
9.5.4. Variational Equations for the Eigenvalues and
Eigenvectors of Nonsymmetric Matrices . . . 323
9.5.5. Individual Tracking of an Eigenvalue and Eigenvector of
a Parametrized Matrix . . . . . . . . . . . . . .. 331
9.5.6. A New Differential Equation Method for Finding the
Perron Root of a Positive Matrix 335
Exercises . . . . . . . . . . . . . . . . . . . . . . .. 340