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IET conTrol EngInEErIng sErIEs 65
Series Editors: Professor D.P. Atherton
Professor G.W. Irwin
Professor S. Spurgeon
Model ing and
Parameter Estimation
of Dynamic Systems

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Other volumes in this series:
Volume 2
Elevator traffic analysis, design and control, 2nd edition G.C. Barney and
S.M. dos Santos
Volume 8
A history of control engineering, 1800–1930 S. Bennett
Volume 14 Optimal relay and saturating control system synthesis E.P. Ryan
Volume 18 Applied control theory, 2nd edition J.R. Leigh
Volume 20 Design of modern control systems D.J. Bell, P.A. Cook and N. Munro (Editors)
Volume 28 Robots and automated manufacture J. Billingsley (Editor)
Volume 30 Electromagnetic suspension: dynamics and control P.K. Sinha
Volume 32 Multivariable control for industrial applications J. O’Reilly (Editor)
Volume 33 Temperature measurement and control J.R. Leigh
Volume 34 Singular perturbation methodology in control systems D.S. Naidu
Volume 35 Implementation of self-tuning controllers K. Warwick (Editor)
Volume 37 Industrial digital control systems, 2nd edition K. Warwick and D. Rees (Editors)
Volume 38 Parallel processing in control P.J. Fleming (Editor)
Volume 39 Continuous time controller design R. Balasubramanian
Volume 40 Deterministic control of uncertain systems A.S.I. Zinober (Editor)
Volume 41 Computer control of real-time processes S. Bennett and G.S. Virk (Editors)
Volume 42 Digital signal processing: principles, devices and applications N.B. Jones
and J.D.McK. Watson (Editors)
Volume 43 Trends in information technology D.A. Linkens and R.I. Nicolson (Editors)
Volume 44 Knowledge-based systems for industrial control J. McGhee, M.J. Grimble and
A. Mowforth (Editors)
Volume 47 A history of control engineering, 1930–1956 S. Bennett
Volume 49 Polynomial methods in optimal control and filtering K.J. Hunt (Editor)
Volume 50 Programming industrial control systems using IEC 1131-3 R.W. Lewis
Volume 51 Advanced robotics and intelligent machines J.O. Gray and D.G. Caldwell
(Editors)
Volume 52 Adaptive prediction and predictive control P.P. Kanjilal
Volume 53 Neural network applications in control G.W. Irwin, K. Warwick and K.J. Hunt
(Editors)
Volume 54 Control engineering solutions: a practical approach P. Albertos, R. Strietzel
and N. Mort (Editors)
Volume 55 Genetic algorithms in engineering systems A.M.S. Zalzala and P.J. Fleming
(Editors)
Volume 56 Symbolic methods in control system analysis and design N. Munro (Editor)
Volume 57 Flight control systems R.W. Pratt (Editor)
Volume 58 Power-plant control and instrumentation D. Lindsley
Volume 59 Modelling control systems using IEC 61499 R. Lewis
Volume 60 People in control: human factors in control room design J. Noyes and
M. Bransby (Editors)
Volume 61 Nonlinear predictive control: theory and practice B. Kouvaritakis and
M. Cannon (Editors)
Volume 62 Active sound and vibration control M.O. Tokhi and S.M. Veres
Volume 63 Stepping motors: a guide to theory and practice, 4th edition P.P. Acarnley
Volume 64 Control theory, 2nd edition J. R. Leigh
Volume 65 Modelling and parameter estimation of dynamic systems J.R. Raol, G. Girija
and J. Singh
Volume 66 Variable structure systems: from principles to implementation
A. Sabanovic, L. Fridman and S. Spurgeon (Editors)
Volume 67 Motion vision: design of compact motion sensing solution for autonomous
systems J. Kolodko and L. Vlacic
Volume 69 Unmanned marine vehicles G. Roberts and R. Sutton (Editors)
Volume 70 Intelligent control systems using computational intelligence techniques
A. Ruano (Editor)

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Model ing and
Parameter Estimation
of Dynamic Systems
J.R. Raol, G. Girija and J. Singh
The Institution of Engineering and Technology

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Published by The Institution of Engineering and Technology, London, United Kingdom
First edition © 2004 The Institution of Electrical Engineers
First published 2004
This publication is copyright under the Berne Convention and the Universal Copyright
Convention. All rights reserved. Apart from any fair dealing for the purposes of research
or private study, or criticism or review, as permitted under the Copyright, Designs and
Patents Act, 1988, this publication may be reproduced, stored or transmitted, in any
form or by any means, only with the prior permission in writing of the publishers, or in
the case of reprographic reproduction in accordance with the terms of licences issued
by the Copyright Licensing Agency. Inquiries concerning reproduction outside those
terms should be sent to the publishers at the undermentioned address:
The Institution of Engineering and Technology
Michael Faraday House
Six Hills Way, Stevenage
Herts, SG1 2AY, United Kingdom
www.theiet.org
While the author and the publishers believe that the information and guidance given
in this work are correct, all parties must rely upon their own skill and judgement when
making use of them. Neither the author nor the publishers assume any liability to
anyone for any loss or damage caused by any error or omission in the work, whether
such error or omission is the result of negligence or any other cause. Any and all such
liability is disclaimed.
The moral rights of the author to be identified as author of this work have been
asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
British Library Cataloguing in Publication Data
Raol, J.R.
Modelling and parameter estimation of dynamic systems
(Control engineering series no. 65)
1. Parameter estimation 2. Mathematical models
I. Title II. Girija, G. III. Singh, J. IV. Institution of Electrical Engineers
519.5
ISBN (10 digit) 0 86341 363 3
ISBN (13 digit) 978-0-86341-363-6
Typeset in India by Newgen Imaging Systems (P) Ltd, Chennai
Printed in the UK by MPG Books Ltd, Bodmin, Cornwall
Reprinted in the UK by Lightning Source UK Ltd, Milton Keynes

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The book is dedicated, in loving memory, to:
Rinky – (Jatinder Singh)
Shree M. G. Narayanaswamy – (G. Girija)
Shree Ratansinh Motisinh Raol – (J. R. Raol)

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Contents
Preface
xiii
Acknowledgements
xv
1
Introduction
1
1.1
A brief summary
7
1.2
References
10
2
Least squares methods
13
2.1
Introduction
13
2.2
Principle of least squares
14
2.2.1
Properties of the least squares estimates
15
2.3
Generalised least squares
19
2.3.1 A probabilistic version of the LS
19
2.4
Nonlinear least squares
20
2.5
Equation error method
23
2.6
Gaussian least squares differential correction method
27
2.7
Epilogue
33
2.8
References
35
2.9
Exercises
35
3
Output error method
37
3.1
Introduction
37
3.2
Principle of maximum likelihood
38
3.3
Cramer-Rao lower bound
39
3.3.1
The maximum likelihood estimate is efficient
42
3.4
Maximum likelihood estimation for dynamic system
42
3.4.1
Derivation of the likelihood function
43
3.5
Accuracy aspects
45
3.6
Output error method
47

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viii
Contents
3.7
Features and numerical aspects
49
3.8
Epilogue
62
3.9
References
62
3.10
Exercises
63
4
Filtering methods
65
4.1
Introduction
65
4.2
Kalman filtering
66
4.2.1
Covariance matrix
67
4.2.2
Discrete-time filtering algorithm
68
4.2.3
Continuous-time Kalman filter
71
4.2.4
Interpretation and features of the Kalman filter
71
4.3
Kalman UD factorisation filtering algorithm
73
4.4
Extended Kalman filtering
77
4.5
Adaptive methods for process noise
84
4.5.1
Heuristic method
86
4.5.2
Optimal state estimate based method
87
4.5.3
Fuzzy logic based method
88
4.6
Sensor data fusion based on filtering algorithms
92
4.6.1
Kalman filter based fusion algorithm
93
4.6.2
Data sharing fusion algorithm
94
4.6.3
Square-root information sensor fusion
95
4.7
Epilogue
98
4.8
References
100
4.9
Exercises
102
5
Filter error method
105
5.1
Introduction
105
5.2
Process noise algorithms for linear systems
106
5.3
Process noise algorithms for nonlinear systems
111
5.3.1
Steady state filter
112
5.3.2
Time varying filter
114
5.4
Epilogue
121
5.5
References
121
5.6
Exercises
122
6
Determination of model order and structure
123
6.1
Introduction
123
6.2
Time-series models
123
6.2.1
Time-series model identification
127
6.2.2
Human-operator modelling
128
6.3
Model (order) selection criteria
130
6.3.1
Fit error criteria (FEC)
130

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Contents
ix
6.3.2
Criteria based on fit error and number of model
parameters
132
6.3.3
Tests based on whiteness of residuals
134
6.3.4
F-ratio statistics
134
6.3.5
Tests based on process/parameter information
135
6.3.6
Bayesian approach
136
6.3.7
Complexity (COMP)
136
6.3.8
Pole-zero cancellation
137
6.4
Model selection procedures
137
6.5
Epilogue
144
6.6
References
145
6.7
Exercises
146
7
Estimation before modelling approach
149
7.1
Introduction
149
7.2
Two-step procedure
149
7.2.1
Extended Kalman filter/fixed interval smoother
150
7.2.2
Regression for parameter estimation
153
7.2.3
Model parameter selection procedure
153
7.3
Computation of dimensional force and moment using the
Gauss-Markov process
161
7.4
Epilogue
163
7.5
References
163
7.6
Exercises
164
8
Approach based on the concept of model error
165
8.1
Introduction
165
8.2
Model error philosophy
166
8.2.1
Pontryagin’s conditions
167
8.3
Invariant embedding
169
8.4
Continuous-time algorithm
171
8.5
Discrete-time algorithm
173
8.6
Model fitting to the discrepancy or model error
175
8.7
Features of the model error algorithms
181
8.8
Epilogue
182
8.9
References
182
8.10
Exercises
183
9
Parameter estimation approaches for unstable/augmented
systems
185
9.1
Introduction
185
9.2
Problems of unstable/closed loop identification
187
9.3
Extended UD factorisation based Kalman filter for unstable
systems
189

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Document Outline

• Contents
• Preface
• Acknowledgements
• 1 Introduction
  • 1.1 A brief summary
  • 1.2 References
• 2 Least squares methods
  • 2.1 Introduction
  • 2.2 Principle of least squares
    • 2.2.1 Properties of the least squares estimates
  • 2.3 Generalised least squares
    • 2.3.1 A probabilistic version of the LS
  • 2.4 Nonlinear least squares
  • 2.5 Equation error method
  • 2.6 Gaussian least squares differential correction method
  • 2.7 Epilogue
  • 2.8 References
  • 2.9 Exercises
• 3 Output error method
  • 3.1 Introduction
  • 3.2 Principle of maximum likelihood
  • 3.3 Cramer-Rao lower bound
    • 3.3.1 The maximum likelihood estimate is efficient
  • 3.4 Maximum likelihood estimation for dynamic system
    • 3.4.1 Derivation of the likelihood function
  • 3.5 Accuracy aspects
  • 3.6 Output error method
  • 3.7 Features and numerical aspects
  • 3.8 Epilogue
  • 3.9 References
  • 3.10 Exercises
• 4 Filtering methods
  • 4.1 Introduction
  • 4.2 Kalman filtering
    • 4.2.1 Covariance matrix
    • 4.2.2 Discrete-time filtering algorithm
    • 4.2.3 Continuous-time Kalman filter
    • 4.2.4 Interpretation and features of the Kalman filter
  • 4.3 Kalman UD factorisation filtering algorithm
  • 4.4 Extended Kalman filtering
  • 4.5 Adaptive methods for process noise
    • 4.5.1 Heuristic method
    • 4.5.2 Optimal state estimate based method
    • 4.5.3 Fuzzy logic based method
  • 4.6 Sensor data fusion based on filtering algorithms
    • 4.6.1 Kalman filter based fusion algorithm
    • 4.6.2 Data sharing fusion algorithm
    • 4.6.3 Square-root information sensor fusion
  • 4.7 Epilogue
  • 4.8 References
  • 4.9 Exercise
• 5 Filter error method
  • 5.1 Introduction
  • 5.2 Process noise algorithms for linear systems
  • 5.3 Process noise algorithms for nonlinear systems
    • 5.3.1 Steady state filter
    • 5.3.2 Time varying filter
  • 5.4 Epilogue
  • 5.5 References
  • 5.6 Exercises
• 6 Determination of model order and structure
  • 6.1 Introduction
  • 6.2 Time-series models
    • 6.2.1 Time-series model identification
    • 6.2.2 Human-operator modelling
  • 6.3 Model (order) selection criteria
    • 6.3.1 Fit error criteria (FEC)
    • 6.3.2 Criteria based on fit error and number of model parameters
    • 6.3.3 Tests based on whiteness of residuals
    • 6.3.4 F-ratio statistics
    • 6.3.5 Tests based on process/parameter information
    • 6.3.6 Bayesian approach
    • 6.3.7 Complexity (COMP)
    • 6.3.8 Pole-zero cancellation
  • 6.4 Model selection procedures
  • 6.5 Epilogue
  • 6.6 References
  • 6.7 Exercises
• 7 Estimation before modelling approach
  • 7.1 Introduction
  • 7.2 Two-step procedure
    • 7.2.1 Extended Kalman filter/fixed interval smoother
    • 7.2.2 Regression for parameter estimation
    • 7.2.3 Model parameter selection procedure
  • 7.3 Computation of dimensional force and moment using the Gauss-Markov process
  • 7.4 Epilogue
  • 7.5 References
  • 7.6 Exercises
• 8 Approach based on the concept of model error
  • 8.1 Introduction
  • 8.2 Model error philosophy
    • 8.2.1 Pontryagins conditions
  • 8.3 Invariant embedding
  • 8.4 Continuous-time algorithm
  • 8.5 Discrete-time algorithm
  • 8.6 Model fitting to the discrepancy or model error
  • 8.7 Features of the model error algorithms
  • 8.8 Epilogue
  • 8.9 References
  • 8.10 Exercises
• 9 Parameter estimation approaches for unstable/augmented systems
  • 9.1 Introduction
  • 9.2 Problems of unstable/closed loop identification
  • 9.3 Extended UD factorisation based Kalman filter for unstable systems
  • 9.4 Eigenvalue transformation method for unstable systems
  • 9.5 Methods for detection of data collinearity
  • 9.6 Methods for parameter estimation of unstable/augmented systems
    • 9.6.1 Feedback-in-model method
    • 9.6.2 Mixed estimation method
    • 9.6.3 Recursive mixed estimation method
  • 9.7 Stabilised output error methods (SOEMs)
    • 9.7.1 Asymptotic theory of SOEM
  • 9.8 Total least squares method and its generalisation
  • 9.9 Controller information based methods
    • 9.9.1 Equivalent parameter estimation/retrieval approach
    • 9.9.2 Controller augmented modelling approach
    • 9.9.3 Covariance analysis of system operating under feedback
    • 9.9.4 Two-step bootstrap method
  • 9.10 Filter error method for unstable/augmented aircraft
  • 9.11 Parameter estimation methods for determining drag polars of an unstable/augmented aircraft
    • 9.11.1 Model based approach for determination of drag polar
    • 9.11.2 Non-model based approach for drag polar determination
    • 9.11.3 Extended forgetting factor recursive least squares method
  • 9.12 Epilogue
  • 9.13 References
  • 9.14 Exercises
• 10 Parameter estimation using artificial neural networks and genetic algorithms
  • 10.1 Introduction
  • 10.2 Feed forward neural networks
    • 10.2.1 Back propagation algorithm for training
    • 10.2.2 Back propagation recursive least squares filtering algorithms
  • 10.3 Parameter estimation using feed forward neural network
  • 10.4 Recurrent neural networks
    • 10.4.1 Variants of recurrent neural networks
    • 10.4.2 Parameter estimation with Hopfield neural networks
    • 10.4.3 Relationship between various parameter estimation schemes
  • 10.5 Genetic algorithms
    • 10.5.1 Operations in a typical genetic algorithm
    • 10.5.2 Simple genetic algorithm illustration
    • 10.5.3 Parameter estimation using genetic algorithms
  • 10.6 Epilogue
  • 10.7 References
  • 10.8 Exercises
• 11 Real-time parameter estimation
  • 11.1 Introduction
  • 11.2 UD filter
  • 11.3 Recursive information processing scheme
  • 11.4 Frequency domain technique
    • 11.4.1 Technique based on the Fourier transform
    • 11.4.2 Recursive Fourier transform
  • 11.5 Implementation aspects of real-time estimation algorithms
  • 11.6 Need for real-time parameter estimation for atmospheric vehicles
  • 11.7 Epilogue
  • 11.8 References
  • 11.9 Exercises
• Bibliography
• Appendix A: Properties of signals, matrices, estimators and estimates
• Appendix B: Aircraft models for parameter estimation
• Appendix C: Solutions to exercises
• Index

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