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Kalman Filtering
Kalman Filtering: Theory and Practice Using MATLAB, Second Edition,
Mohinder S. Grewal, Angus P. Andrews
Copyright # 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-39254-5 (Hardback); 0-471-26638-8 (Electronic)
Kalman Filtering:
Theory and Practice
Using MATLAB
Second Edition
MOHINDER S. GREWAL
California State University at Fullerton
ANGUS P. ANDREWS
Rockwell Science Center
A Wiley-Interscience Publication
John Wiley & Sons, Inc.
NEW YORK CHICHESTER WEINHEIM BRISBANE SINGAPORE TORONTO
Copyright # 2001 by John Wiley & Sons, Inc. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form
or by any means, electronic or mechanical, including uploading, downloading, printing, decompiling,
recording or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States
Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for
permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third
Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008,
E-Mail: PERMREQ @ WILEY.COM.
This publication is designed to provide accurate and authoritative information in regard to the subject
matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional
services. If professional advice or other expert assistance is required, the services of a competent
professional person should be sought.
ISBN 0-471-26638-8.
This title is also available in print as ISBN 0-471-39254-5.
For more information about Wiley products, visit our web site at www.Wiley.com.
Contents
PREFACE ix
ACKNOWLEDGMENTS xiii
1 General Information 1
1.1 On Kalman Filtering 1
1.2 On Estimation Methods 5
1.3 On the Notation Used in This Book 20
1.4 Summary 22
Problems 23
2 Linear Dynamic Systems 25
2.1 Chapter Focus 25
2.2 Dynamic Systems 26
2.3 Continuous Linear Systems and Their Solutions 30
2.4 Discrete Linear Systems and Their Solutions 41
2.5 Observability of Linear Dynamic System Models 42
2.6 Procedures for Computing Matrix Exponentials 48
2.7 Summary 50
Problems 53
3 Random Processes and Stochastic Systems 56
3.1 Chapter Focus 56
3.2 Probability and Random Variables 58
3.3 Statistical Properties of Random Variables 66
v
3.4 Statistical Properties of Random Processes 68
3.5 Linear System Models of Random Processes and Sequences 76
3.6 Shaping Filters and State Augmentation 84
3.7 Covariance Propagation Equations 88
3.8 Orthogonality Principle 97
3.9 Summary 102
Problems 104
4 Linear Optimal Filters and Predictors 114
4.1 Chapter Focus 114
4.2 Kalman Filter 116
4.3 Kalman±Bucy Filter 126
4.4 Optimal Linear Predictors 128
4.5 Correlated Noise Sources 129
4.6 Relationships between Kalman and Wiener Filters 130
4.7 Quadratic Loss Functions 131
4.8 Matrix Riccati Differential Equation 133
4.9 Matrix Riccati Equation in Discrete Time 148
4.10 Relationships between Continuous and Discrete Riccati Equations 153
4.11 Model Equations for Transformed State Variables 154
4.12 Application of Kalman Filters 155
4.13 Smoothers 160
4.14 Summary 164
Problems 165
5 Nonlinear Applications 169
5.1 Chapter Focus 169
5.2 Problem Statement 170
5.3 Linearization Methods 171
5.4 Linearization about a Nominal Trajectory 171
5.5 Linearization about the Estimated Trajectory 175
5.6 Discrete Linearized and Extended Filtering 176
5.7 Discrete Extended Kalman Filter 178
5.8 Continuous Linearized and Extended Filters 181
5.9 Biased Errors in Quadratic Measurements 182
5.10 Application of Nonlinear Filters 184
5.11 Summary 198
Problems 200
6 Implementation Methods 202
6.1 Chapter Focus 202
6.2 Computer Roundoff 204
6.3 Effects of Roundoff Errors on Kalman Filters 209
6.4 Factorization Methods for Kalman Filtering 216
vi CONTENTS
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