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APPLIED OPTIMAL ESTIMATION

witten by

Technical Staff THE ANALYTIC SCIENCES CORPORATION Arthur Gelb

editedby

principal authors

Arthur Gelb Joseph F. Kasper, Jr. Raymond A. Nash, Jr. Charles F. Price Arthur A. Sutherland, Jr.

THE M.I.T. PRESS

Massachusetts Institute of Technology Carnbrid~, Massachusetts,and London, England

FOREWORD

Sixteenthprinting, ZOO1

CowriLt 1974 by The ArUlyticSzk-rC0~pontim

AU rights reserved. No put of this bmk may be reproduced in my form or by any means, clef+xonic or mechanical, including photocopying, recording, or by any information stowe and retrieval rystan, without pamission in witin% from The Analytic Seienecs Corporation. This back was printed and bmnd in the United States of America.

Lib,@ of C o n p s r Comlq Cord Number: 74-1604 ISBN 0-262-20027-9 (hard) 0-262-57048-3 (paper)

This book rm d.ri9n.d by TASC.

Estimtion is the process of extracting information from data - data which can be used to infer the desired information and niay contain errors. Modern estimation methods use known relationships to compute the deslred information from the measurements. taking account of measurement errors, the effects of disturbances and control actions on the system, and prior knowledge of the information. Diverse measurements car! be blended to form "best" estimates, and information which is unavailable for measurement can be approximated in an optimal fashion. The intent of this book is to enable readers to achieve a level of competence that will permit their participation in the des~gnand evaluation of practical estimators. Therefore, the text is oriented to the applied rather than theoretical aspects of optimal estimation. It is our intent throughout to provide a simple and interesting picture of the central issues underlying modern estimation theory and practice. Heuristic, rather than theoretically elegant, arguments are used extensively, with emphasis on physical insights and key questlons of practical importance. The text is organized into three principal parts. Part I introduces the subject matter and provides brief treatments of the underlying mathematics. Chapter 1 presents a brief overview, including a historical perspectwe; Chapters 2 and 3 treat the mathematics underlying random process theory and state-space characterization of linear dynam~c systems, both of which are essential prerequisites to understanding optimal estimation theory. Part 11 provides derivations, interpretations and examples pertinent to the theory of optimal estimation. Thus. Chapters 4 and 5 address optimal linear filtering and

smoothing, respectively, while Chapter 6 addresses the subject of nonlinear fdtering and smoothing. Part 111 treats those practical issues which often mean the difference between success or failure of the implemented optimal estimator. The practical and often pivotal issues of suboptimal filtering, sensitivity analysis and implementation considerations are discussed at some length in Chapters 7 and 8. Additional topics of practical value are presented in Chapter 9; these include refinements and other viewpoints of estimation theory, and the close connection of the mathematics which underly both optimal linear estimation theory and optimal linear control theory. Many illustrative examples have been interspersed throughout the text to assist in effective presentation of the theoretical material. Additionally, problems with "built-in" answers have been included at the end of each chapter, t o further enable self-study of the subject matter. This hook is the outgrowth of a course taught by The Analytic Sciences Corporation (TASC) at a number of US. Government facilities. The course notes were, in turn, based on the considerable practical experience of TASC in applying modern estimation theory t o large-scale systems of diverse nature. Thus, virtually all Members of the Technical Staff of TASC have, at one time or another, contributed t o the material contained herein. It is a pleasure to specifically acknowledge those current members of TASC who, in addition to the principal authors, have directly contributed t o the writing of this hook. Bard S. Crawford provided a complete section; Julian L. Center, Jr., Joseph A. D'Appolito, and Ronald S. Warren contributed through providing text additions, technical comments and insights of a very diverse nature. Other individuals whose contributions are acknowledged are Robert G. Bellaire, Norman H. Josephy, William F. O'Halloran, Jr. and Bahar J. Uttam. William R. Sullivan and Vicky M. Koaerga created all the artwork. Renwick E. Curry, John J. Deyst, Jr. and Professor Wallace E. Vander Velde of M.I.T. contributed through technical discussions and by providing some problems for inclusion at the end of several chapters. Professor Charles E. Hutchinson, University of Massachusetts. contributed through his participation in early TASC work which set the stage for this book. Special acknowledgement is due Harry B. Silverman of TASC for his encouragement of the project from its inception. Finally, this most recent printing benefits considerably from the careful reading given earlier printings by those individuals who have provided us with comments and corrections.

THE ANALYTIC SCIENCES CORPORATION Reading, Mauachusenr

Arthur Gelb 16 July 1979

CONTENTS

Foreword v Chapter 1 Introduction 1 Chapter 2 Review of Underlying Mathematical Techniques 10

2,l 2.2

Vectors, Matrices, and Least Squares 10 Probability and Random Processes 24

Chapter 3 Linear Dynamic Systems 51

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Stateapace Notation 51 Transition Matrix 57 Matrix Superposition Integral 63 Discrete Formulation 66 System Obsemabidity and Controllability 67 Covariance Matrix 72 Propagation of Errors 75 Modeling and State Vector Augmentation 78 Empirical Model Identification 84

Chapter 4 Optimal Limar Filtering 102

4.1 4.2

Recursive Filters 105 Discrete Kalman Filter 107

4.3 4.4 4.5 4.6 4.7

Continuous Kalman Filter 119 Intuitive Concepts 127 Correlated Measurement Ernrs 133 Solution of the Riccati Equation 136 Statistical Steady State - The Wiener ~ i l 142 ~ t ~

APPLIED OPTIMAL ESTIMATION

Chapter5 Optimal Limar Smoothing 156 5.1 5.2 5.3 5.4

Form of the Optimal Smoother 157 Optimal Fixed-Interval Smoother 160 Optimal Fixed-Point Smoother 170 Optimal Fixed-Lag Smoother 173

Chapter 6 Nonlinesr Estimation 180 6.1 Nonlinear Minimum Variance Estimation 182 6.2 Nonlinear Estimation by Statistical Linearization 203 6.3 Nonlinear Least-SquaresEstimation 214 6.4 Direct Statistical Analysis of Nonlinear Systems - CADET'^ 216 Chapter 7 Suboptimal Finer Design and Sensitivity Analysis 229 7.1 Suboptimal Fdter Design 230 7.2 Sensitivity Analysis: Kalman Filte~ 246 7.3 Sensitivity Analysis Examples 255 7.4 Developingan Error Budget 260 7.5 Sensitivity Analysis: Optimal Smoother 266 7.6 Organization of a Computer Rogram for Covariance Analysis 268 Chapter 8 Implementation Considerations 277 8.1 Modeling Problems 278 8.2 Constraints Imposed by the Computer 288 8.3 The Inherently Finite Nature of the Computer 292 8.4 Algorithms and Computer Loading Analysis 303 Chapter 9 Additional Topics 316 9.1 Adaptive Kalman Filtering 31 7 9.2 Observers 320 9.3 Stochastic Approximation 335 9.4 Real-TimeParameter Identification 348 9.5 Optimal Control of Linear Systems 356

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