optimal filtering.pdf

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Cover

Title Page

Contents

Preface

1. Introduction

1.1. Filtering

1.2. History of Signal Filtering

1.3. Subject Matter of this Book

1.4. Outline of the Book

References

2. Filtering, Linear Systems, and Estimation

2.1. Systems, Noise, Filtering, Smoothing, and Prediction

2.2. The Gauss-Markov Discrete-time Model

2.3. Estimation Criteria

References

3. The Discrete-time Kalman Filter

3.1. The Kalman Filter

3.2. Best Linear Estimator Property of the Kalman Filter

3.3. Identification as a Kalman Filtering Problem

3.4. Application of Kalman Filters

References

4. Time-invariant Filters

4.1. Background to Time Invariance of the Filter

4.2. Stability Properties of Linear, Discrete-time Systems

4.3. Stationary Behavious of Linear Systems

4.4. Time Invariance and Symptotic Stability of the Filter

4.5. Frequency Domain Formulas

References

5. Kalman Filter Properties

5.1. Introduction

5.2. Minimum Variance and Linear Minimum Variance Estimation; Orthogonality and Projection

5.3. The Innovations Sequence

5.4. The Kalman Filter

5.5. True Filtered Estimates and the Signal-to-noise Ratio Improvement Property

5.6. Inverse Problems; When is a Filter Optimal?

References

6. Computational Aspects

6.1. Signal Model Errors, Filter Divergence, and Data Saturation

6.2. Exponential Data Weighting—A Filter With Prescribed Degree of Stability

6.3. The Matrix Inversion Lemma and the Information Filter

6.4. Sequential Processing

6.5. Square Root Filtering

6.6. The High Measurement Noise Case

6.7. Chandrasekhar-type, Doubling, and Nonrecursive Algorithms

References

7. Smoothing of Discrete-time Signals

7.1. Introduction to Smoothing

7.2. Fixed-point Smoothing

7.3. Fixed-lag Smoothing

7.4. Fixel-interval Smoothing

References

8. Applications in Nonlinear Filtering

8.1. Nonlinear Filtering

8.2. The Extended Kalman Filter

8.3. A Bound Optimal Filter

8.4. Gaussian Sum Estimators

References

9. Innovations Representations, Spectral Factorization, Wiener and Levinson Filtering

9.1. Introduction

9.2. Kalman Filter Design From Covariance Data

9.3. Innovations Representations With Finite Initial Time

9.4. Stationary Innovations Representations and Spectral Factorization

9.5. Wiener Filtering

9.6. Levinson Filters

References

10. Parameter Identifications and Adaptive Estimation

10.1. Adaptive Estimation Via Parallel Processing

10.2. Adaptive Estimation Via Extended Least Squares

References

11. Colored Noise and Suboptimal Reduced Order Filters

11.1. General Approaches to Dealing with Colored Noise

11.2. Filter Design with Markov Output Noise

11.3. Filter Design with Singular or Near-singular Output Noise

11.4. Suboptimal Design Given Colored Input or Measurement Noise

11.5. Suboptimal Filter Design by Model Order Reduction

References

A. Brief Review of Results of Probability Theory

A.1. Pure Probability Theory

A.2. Stochasitic Processes

A.3. Gaussian Random Variables, Vectors, and Processes

References

B. Brief Review of Some Results of Matrix Theory

References

C. Brief Review of Several Major Results of Linear System Theory

References

D. Lyapunov Stability

References

Author Index

Subject Index

BRIAN D. O. ANDERSON JOHN B. MOORE Optimal Filtering INFORMATION AND SYSTEM SCIENCES SERIES i Thomas Kailath Editor

PRENTICE-HALL AND SYSTEM SCIENCES INFORMATION SERIES Thomas Kailath, Editor ANDERSON& MOORE BERGER DI FRANCO& Rmm DOWNING Duwx FRANKS GLORIOSO GOLOMB,ETAL. KAILATH LINDSEY LINDSEY& SIMON MELSA& SAGE PATRICK RAEMER STIFFLER VANDERZEL Systems and Noise Systems Optimal Filtering Rate Distortion Theory: A Mathematical Basis for Data Compression Radar Detection Modulation The Theory of Applied Probability Signal Theory Engineering Cybernetics Digital Communications with Space Applications Linear System Theory Synchronization in Communication and Control Telecommunication An Introduction to Probability and Stochastic Processes Fundamentals Statistical C~mmunication Theory and Applications Theory of Synchronous Communications Noise: Sources, Characterization, Measurement of Pattern Recognition Systems Engineering

OPTIMAL FILW?IIVG Brian D. O. Anderson John B. Moore Professors of Electrical Engineering University of Newcastle New South Wales, Australia Englewood PRENTICE-HALL, INC. Cliffs, New Jersey 07632

Llbra?y of Con8?essCafalo8fngin Publication Data ANDERSON,BRIAND O optimalfiltering. (Informationcnd systemsciemw series) IncludesbibliographleYandindex. 1. Sial processing. 2 Electricfilters. I. Moore,JohnBmratt, date jointauthor. 33. Title. TKSI02.5.A53 ISBN (+13-638122-7 621.381S’32 78-8938 @ 1979 by Prentice-Hall, Inc., Englewood Cliffs, N.J. 07632 All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the publisher. Printed in the United States of America 10987654321 PRENTICE-HALL PRENTICE-HALL PRENTICE-HALL INTERNATIONAL, OF AUSTRALIA PTY. LIMITED, INC., London Sydney OF CANADA, LTD., Toronto OF INDIA PRIVATE LIMITED, New Delhi PRENTICE-HALL INC., Tokyo OF SOUTHEAST &31A PTE. LTD., Singapore BOOKS LIMITED, Wellington, New Zealand OF JAPAN, PRENTICE-HALL PRENTICE-HALL WHITEHALL

PREFACE 1 INTRODUCTION 1 1.1 Filtering 1.2 History of Signal Filtering 1.3 Subject Matter ofthis Book 1.4 Outline of the Book 6 2 4 References 7 2 FILTERING, AND ESTIMATION LINEAR SYSTEMS, 2.1 Systems, Noise, Filtering, 2.2 2.3 Smoothing, and Prediction The Gauss -Markov Discrete-time Model Estimation Criteria References 34 23 9 12 v

vi CONTENTS 3 THE DISCRETE-TIME KALMAN FILTER 36 3.1 3.2 3.3 3.4 36 The Kalman Filter Best Linear Estimator Property of the Kalman Filter 46 Identification as a Kalman Filtering Problem Application of Kalman Filters References 50 59 53 4 TIME-INVARIANT FILTERS 4.1 4.2 4.3 4.4 4.5 62 Background to Time Invariance of the Filter Stability Properties of Linear, Discrete-time Systems 63 Stationary Behaviour of Linear Systems Time Invariance and Asymptotic Stability of the Filter Frequency Domain Formulas References 76 88 85 5 KALMAN FILTER PROPERTIES 5.1 5.2 5.3 5.4 5.5 5.6 92 90 Introduction Minimum Variance and Linear Minimum Variance Estimation; Orthogonality and Projection The Innovations Sequence 105 The Kalman Kilter True Filtered Estimates and the Signal-to -Noise Ratio Improvement Property Inverse Problems: When is a Filter Optimal? References 100 122 127 115 68 62 90 6 COMPUTATIONAL ASPECTS 129 6.1 6,2 Signal Model Errors, Filter Divergence, 129 and Data Saturation Exponential Data Weighting-- A Filter with Prescribed Degree of Stability 135

6.3 The Matrix Inversion Lemma CONTENTS vii 6.4 6.5 6.6 6.7 138 142 747 and the Information Filter Sequential Processing Square Root Filtering The High Measurement Noise Case Chandrasekhar - Type. Doubling. and Nonrecursive Algorithms References 762 155 153 SMOOTHING OF DISCRETE-TIME SIGNALS 165 Fixed-point Smoothing 7.1 Introduction to Smoothing 7.2 7.3 Fixed-fag Smoothing 7.4 Fixed-interval Smoothing References 190 165 170 176 187 APPLICATIONS IN NONLINEAR FILTERING 193 193 Nonlinear Filtering The Extended Kalman Filter 8.1 8.2 8.3 A Bound Optimal Filter 8.4 Gaussian Sum Estimators 795 205 211 References 221 INNOVATIONS SPECTRAL WIENER REPRESENTATIONS, FACTORIZATION, AND LEVINSON FILTERING 7 8 9 9.1 Introduction 9.2 9.3 223 Kalman Filter Design from Covariance Data Innovations Representations with Finite Initial Time Stationary Innovations Representations and Spectral Factorization 230 238 9.4 227 9.5 Wiener Filtering Levinson Filters 9.6 References 264 254 258 10 PARAMETER AND ADAPTIVE IDENTIFICATION ESTIMATION Adaptive Estimation via Parallel Processing 10.1 10.2 Adaptive Estimation via Extended Least Squares 267 References 286 223 267 279

viii CONTENTS COLORED REDUCED NOISE AND SUBOPTIMAL ORDER FILTERS 11 288 11.1 General Approaches to Dealing with Colored Noise 288 11.2 Filter Design with Markov Output Noise 11.3 290 292 11.4 11.5 Filter Design with Singular or Near-singular Output Noise Suboptimal Design Given Colored Input or Measurement Noise Suboptimal Filter Design by Model Order Reduction References 301 296 304 APPENDIXES A BRIEF REVIEW OF RESULTS OF PROBABILITY THEORY A.1 Pure Probability Theory A.2 316 A,3 Gaussian Random Variables. Stochastic Processes 308 Vectors, and Processes References 323 320 B BRIEF REVIEW OF SOME RESULTS OF MATRIX THEORY References 339 c BRIEF REVIEW OF SEVERAL MAJOR OF LINEAR SYSTEM THEORY RESULTS References 346 D LYAPUNOV STABILITY References 349 AUTHOR INDEX SUBJECT INDEX 307 324 340 347 351 354

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