Identification of Linear Systems. A Practical Guideline to Accurate Modeling

339 Pages·1991·15.89 MB·English

by  J. Schoukens and R. Pintelon (Auth.)

Pergamon Titles of Related Interest
AULIN
Foundations of Mathematical System Dynamics
CALVAER
Power Systems: Modelling and Control Applications
DAVIES
Protection of Industrial Power Systems
DUMMER & WINTON
Elementary Guide to Reliability, 4th edition
FRIEDMAN
Exhuming Induction
HALL
Metasystems Methodology
ISIDORI
Nonlinear Control Systems Design
KAMPIS
Self-modifying Systems in Biology and Cognitive Sciences
ROSEN
Anticipatory Systems
Pergamon Related Journals
SIGUERDIDJANE & BERNHARD
C(froenet roslp eAcipmpelni catcioopnys ogfl adNlyo nlsineneta r oPn rogrreaqmuemsti)n g and Optimization
Annual Review in Automatic Programming
Automatica
Computers and Electrical Engineering
Computers and Operations Research
Mechanism and Machine Theory
Mechatronics
Neural Networks
Problems of Control and Information Theory
Pergamon Titles of Related Interest
AULIN
Foundations of Mathematical System Dynamics
CALVAER
Power Systems: Modelling and Control Applications
DAVIES
Protection of Industrial Power Systems
DUMMER & WINTON
Elementary Guide to Reliability, 4th edition
FRIEDMAN
Exhuming Induction
HALL
Metasystems Methodology
ISIDORI
Nonlinear Control Systems Design
KAMPIS
Self-modifying Systems in Biology and Cognitive Sciences
ROSEN
Anticipatory Systems
Pergamon Related Journals
SIGUERDIDJANE & BERNHARD
C(froenet roslp eAcipmpelni catcioopnys ogfl adNlyo nlsineneta r oPn rogrreaqmuemsti)n g and Optimization
Annual Review in Automatic Programming
Automatica
Computers and Electrical Engineering
Computers and Operations Research
Mechanism and Machine Theory
Mechatronics
Neural Networks
Problems of Control and Information Theory
Identification
of
Linear Systems
A Practical Guideline
to Accurate Modeling
J. SCHOUKENS
and
R. PINTELON
Vrije Universiteit Brussel, Brussels, Belgium
PERGAMON PRESS
Member of Maxwell Macnnillan Pergamon Publishing Corporation
OXFORD · NEW YORK · BEIJING · FRANKFURT
SAO PAULO SYDNEY · TOKYO · TORONTO
U.K. Pergamon Press pIc, Headington Hill Hall,
Oxford 0X3 OBW, England
U.S.A. Pergamon Press Inc., Maxwell House, Fairview Park,
Elmsford, New York 10523, U.S.A.
PEOPLE'S REPUBLIC Pergamon Press, Room 4037, Qianmen Hotel, Beijing,
OF CHINA People's Republic of China
Pergamon Press GmbH, Hammerweg 6,
GERMANY D-6242 Kronberg, Germany
Pergamon Editora Ltda, Rua Ega de Queiros, 346,
BRAZIL CEP 04011, Paraiso, Sao Paulo, Brazil
Pergamon Press (Australia) Pty Ltd, P.O. Box 544,
AUSTRALIA
Potts Point, NSW 2011, Australia
Pergamon Press, 5th Floor, Matsuoka Central Building,
JAPAN 1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan
Pergamon Press Canada Ltd., Suite No. 271,
CANADA
253 College Street, Toronto, Ontario, Canada M5T 1 R5
Copyright © 1991 J. Schoukens and R. Pintelon
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, electrostatic, magnetic
tape, mechanical, photocopying, recording or otherwise,
without permission in writing from the publishers.
First edition 1991
Library of Congress Cataloging in Publication Data
Schoukens, J.
Identification of linear systems.
J. Schoukens and R. Pintelon.
1st ed.
p. cm.
Includes index.
I. Control theory. 2. System analysis. I. Pintelon, R.
II. Title.
aA402.3.S345 1991 003'.74—dc20 91-2508
British Library Cataloguing in Publication Data
Schoukens, J.
Identification of linear systems.
1. Systems
I. Title II. Pintelon R.
003.74
ISBN 0-08-040734-X
Printed in Great Britain by BPCC Wheatons Ltd, Exeter
To my wifeAnnick and my children
Johan
To my wife Khadia, my children Yacine, Korka and Laura,
and my parents mams and paps
Rik
Acknowledgement
This book is the culmination of 10 years of research in the department of
Fundamental Electricity and Instrumentation (ELEC) of the Vrije Universiteit
Brussel (VUB). It is the result of a close and harmoneous cooperation between
many of the workers in the department, and we should like to thank them all for
exchanges of ideas and important discussions which have taken place over the
gestation period of this, the end result. We are gready indebted to ELEC, which is
directed by professors Jean Renneboog and Alain Barel, to the R&D department of
the VUB and to the Belgian National Fund for Scientific Research (NFWO). Without
their sustained support this work would never have seen the light of day.
A number of specialists in the field have helped us to substantially improve the
book's quality by their critical comments. Among these we should like to explicitiy
mention prof. Pieter EykhofF (Technical University of Eindhoven), prof. Keith
Godfrey (University of Warwick) and prof. Istv^n. KoUar (Technical University of
Budapest).
We should also like to thank Yves Rolain (ELEC) and Ron Cottam (IMEC) for their
enthousiastic support in improving the quality of the artwork and the language of
the book respectively.
Xlll
Preface
Identification is a powerful technique for building accurate models of complex
systems from noisy data. It consists of three basic steps which are interrelated: the
design of an experiment, the construction of a model from physical laws, and the
estimation of the model parameters from the measurements. The art of modeling
from physical laws lies in proper use of the skills and specialized knowledge of
experts in the field of study, who decide what approximations can be made,
suggest how to manipulate the system, reveal the important aspects, etc. Conse-
quentiy, modeling should preferably be executed by these experts themselves.
They naturally require relevant tools for extracting information of interest. How­
ever, most of them will not be familiar with identification theory, and will struggle
in each new situation with the same difficulties while developing their own
identification techniques, losing time over problems already solved in the literature
of identification.
This book presents a thorough description of a method of modeling a linear
dynamic time invariant system by its transfer function. The relations between the
transfer function and the physical parameters of the system are very dependent
upon the specific problem: since transfer function models are generally valid, we
have restricted the scope of the book to these alone, so as to develop a general
purpose estimation/identification technique. This should not be unnecessarily
restricting for readers who are more interested in the physical parameters of a
system: the transfer function still contains all the information which is available in
the measurements, and it can be considered to be an intermediate model between
the measurements and the physical parameters.
Transfer function models can be identified in both the time and frequency
domains, using discrete (z-domain) or continuous (Laplace domain) representa­
tions. Since the late sixties, much attention has been paid to the identification of
discrete time models in the time domain, particularly for control purposes because
they are well suited to on-line implementation. Frequency domain methods are
more suitable for building very accurate models of complex systems which permit
physical interpretation of the parameters : in this case computing time is not so
important, and on-line estimation is not usually required.
The structure of the book is as follows. The first chapter >1 General Introduction to
Parameter Estimation and the second chapter Λ Review of Some Estimation
Methods are intended for readers who are unfamiliar with identification theory.
These chapters provide sufficient background for understanding the methods
presented later.
The third, fourth, and fifth chapters form the kernel of the book. In Λ Maximum
Likelihood Estimatorfor Linear Time Invariant Systemsihe basic method we use
xv
XVi PREFACE
to estimate the parameters of the transfer function model is developed. The chapter
Design of Excitation Signals shows how it is possible to optimize the excitation
signals so as to obtain the maximum information in the minimum time. The fifth
chapter on Model Selection studies the problem of choosing between different
models.
Estimation of Linear Time Invariant Systems with Delay ma Phase Correction of
Linear Time Invariant Systems with Digital Allpass Filters extend the estimation
method proposed in chapter 3.
Application of ELiS to Measurement Problems illustrates use of the method in a
wide variety of situations, for example in vibrational analysis of the wings of an
aiφlane, modeling of a synchronous electric motor, modeling and correction of a
data acquisition system, and the design of digital filters.
The book concludes with A Guidelinefor Transfer Function Estimation, in which
the complete identification procedure is illustrated for an elaborated example, and
some rules-of-thumb are given for constructing a transfer function model.
This book is written for researchers and practicing engineers who have to deal with
the modeling of linear dynamic systems. Those who are not really interested in the
theory can skip the proofs without losing a general view of the proposed
techniques. We hope that specialists in identification theory will also find some
interesting and new ideas in the kernel chapters 3 to 5.
The following prerequisites will be sufficient for reading the book :
- basic knowledge of system theory, spectral analysis and associated mathematics
(convolution integral, complex numbers, linear algebra, Fourier transform,
Laplace transform, and z-transform)
- basic knowledge about discrete time systems and sampled signals (discrete
Fourier transform, alias and leakage effects)
- basic knowledge of probability theory, elementary statistics and stochastics
(mathematical expectation, covariance matrix, probability density function, cor­
relation, power spectral density etc. are assumed to be familiar)
A large amount of literature dealing with identification is available. A first group of
books gives a general introduction to identification theory not directed to a specific
class of problems. Among these are the books by Eykhoff (System Identification,
Parameter and State Estimation, 1974), Goodwin and Payne (Dynamic Identifica­
tion : Experiment Design and Data Analysis, 1977), Beck and Arnold (Parameter
Estimation in Engineering and Science, 1977), Sorenson (Parameter Estimation,
Principles and Problems, 1980), Eykhoff (Trends and Progress in System Identifi­
cation, 1981), Norton (An Introduction to Identification, 1986), and Godfrey and
PREFACE XVll
Jones (Signal Processing for Control, 198©. Each of these gives a good treatment
of general identification theory.
A second group of books also gives an introduction to identification theory but
concentrates mainly on discrete time models and/or recursive techniques for on­
line implementation. These includes those by Ljung (System Identification, Theory
for the User, 1987), Ljung and Soderstrom (Theory and Practice of Recursive
Identification, 1983) and Soderstrom and Stoica (System Identification, 1989).
Together with Ljung's book there is a Matlab toolbox available, so the reader can
apply the proposed methods to his own problems with minimal programming
eflfoiL
There are not many books on die accurate modeling of linear dynamic systems
from noisy measurements leading to physical interpretation. We try to fill this gap
in the literature with our publication.
A complete Frequency Domain System Identification toolbox has been developed
in MATLAB™ including the basic routines, the examples of the book, and
additional illustrations. The toolbox is matched directly to the theory and
techniques in this book, and can be used
- for the design and the execution of a complete identification procedure
- for the processing of measured data
- for a better understanding of the presented methods
- for experimentation with variants of the presented procedures
- as a classroom package when using the book as a textbook.
The routines include: excitation signal design - data preprocessing -identification
of model parameters - plots with uncertainties - model validation -model
conversions - data export/import - allpass filter design.
For further information on the Frequency Domain System Identification toolbox,
please contact the authors at the Vrije Universiteit Brussel, Department ELEC,
Pleinlaan 2, B-1050 Brussels, Belgium. Telefax 32 2 641-2850.
Brussels, December 1990
Notations
matrix and vector bold face.
Euclidian norm.
Ε mathematical expectation.
Ρ probability.
superscript * complex conjugate.
superscript t transposition of a matrix (or veaor).
superscript Η Hermittian transposition of a matrix (or vector).
suffix e exact (true) value.
suffix R denotes the real part of complex vector (or number),
suffix I denotes the imaginary part of a complex vector (or
number).
subscript m measured variable,
subscript ML estimated variable.
subscript χ with respea to the input of the device under test,
subscript y with respect to the output of the device under test,
diagixj,... xj diagonal matrix
0
plim limit in probability.
l.i.m. limit in mean - limit in mean square.
μ mean.
variance.
Ε(μ,σ2) exponential distribution with mean μ, and variance
L(μ,σ2) Laplace distribution with mean μ, and variance .
Ν(μ,σ2) normal distribution with mean μ, and variance .
R(H,a2) Rayleigh distribution with mean μ, and variance .
υ(μ,σ2) uniform distribution with mean μ, and variance .
A amplitude of the transfer function.
Φ phase of the transfer funaion.
Fi Fisher information matrix.
covariance matrix of x.
autocorrelation of x.
crosscorrelation of χ and y.
fc autopower spectrum of x.
crosspower spectrum correlation of χ and y.
xix