Stochastic Systems: The Mathematics of Filtering and Identification and Applications: Proceedings of the NATO Advanced Study Institute held at Les Arcs, Savoie, France, June 22 – July 5, 1980

654 Pages·1981·38.009 MB·English

by  Michiel Hazewinkel, Jan C. Willems (auth.), Michiel Hazewinkel, Jan C. Willems (eds.)

Stochastic Systems: The Mathematics of Filtering and Identification and Applications
NATO ADVANCED STUDY INSTITUTES SERIES
Proceedings of the Advanced Study Institute Programme, which aims
at the dissemination ofa dvanced knowledge and
the formation of contacts among scientists from different countries
The series is published by an international board of publishers in conjunction
with NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical and D. Reidel Publishing Company
Physical Sciences Dordrecht, Boston and London
D Behavioural and Sijthoff & Noordhoff International
Social Sciences Publishers
E Applied Sciences Alphen aan den Rijn and Germantown
U.S.A.
Series C - Mathematical and Physical Sciences
Volume 78 - Stochastic Systems: The Mathematics of Filtering and Identification
and Applications
Stochastic Systems:
The Mathematics of
Filtering and Identification
and Applications
Proceedings of the NATO Advanced Study Institute
held at Les Arcs, Savoie, France, June 22 -July 5,1980
edited by
MICHIEL HAZEWINKEL
Erasmus Universiteit, Rotterdam, The Netherlands
and
JAN C. WILLEMS
Rijksuniversiteit, Groningen, The Netherlands
D. Reidel Publishing Company
Dordrecht: Holland / Boston: U.S.A. / London: England
Published in cooperation with NATO Scientific Affairs Division
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute (1980: Savoie, France)
Stochastic systems: The mathematics of filtering and identification
and applications.
(NATO advanced study institutes series.
Series C, Mathematical and physical sciences; v. 78)
Includes index.
1. Stochastic systems-Congresses. 2. Filters (Mathematics)
Congresses. 3. System identification-Congresses. 1. Hazewinke1,
Michie!. II. Willems, Jan C. III. North Atlantic Treaty Organization.
Division of Scientific Affairs. IV. Title. V. Series.
QA402.N33 1980 003 81-11944
AACR2
ISBN-13: 978-94-009-8548-3 e-ISBN-13: 978-94-009-8546-9
DOl: 10.1007/978-94-009-8546-9
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Copyright © 1981 by D. Reidel Publishing Company, Dordrecht, Holland
Sotkover reprint of the hardcover 1st edition 1981
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TABLE OF CONTENTS
PREFACE ix
LEITFADEN: Interdependence Flowchart for the Tutorials
and Invited Lectures xii
I. The Main Themes of the ASI "Stochastic Systems":
The Mathematics of Filtering and Identification and
Applications
1.1. M. Hazewinkel and J. C. Willems / Introduction:
An Annotated Navigation Chart and Sketch of the
Main Themes in This Volume and their Interrelations 3
2. Introductory Tutorial Material for Identification
and Filt~ring Problems
2.1. J. C. Willems / The LQG-Problem: A Brief Tutorial
Exposition 29
2.2. R. F. Curtain / A Tutorial Article on the Ito
Integral and the Stochastic Calculus 45
2.3. M. H. A. Davis and S. I. Marcus / An Introduction
to Nonlinear Filtering 53
2.4. M. Hazewinkel / A Tutorial Introduction to
Differentiable Manifolds and Vector Fields 77
2.5. M. Hazewinkel / A Short Tutorial on Lie Algebras 95
3. Foundations of Stochastic Systems and Modeling Issues
and Applications
3.1. L. Arnold / Mathematical Models of Chemical Reactions*) III
3.2. L. Arnold / Qualitative Theory of Real Noise Excited
Systems*) 135
**)Invited main series of lectures
. *)Invited lectures
vi TABLE OF CONTENTS
3.3. R. Boel / Stochastic Models of Computer Networks*) 141
3.4. A. Lindquist and G. Picci / State Space Models for
Gaussian Stochastic Processes*) 169
3.5. V. Wihstutz / Ergodic Theory of Linaar Pararoeter-
excited systems 205
4. Identification
4.1. E. J. Hannan / System Identification*) 221
4.2. L. Ljung / Recursive Identification*) 247
4.3. M. Deistler / The Properties of the Parameterization
of ARMAX Systems in Structural Specification and
their Relevance for Estimation 283
4.4. D. Guegan / Tests of Adequacy for ARMA Models and
Tests of Separated Hypotheses 291
4.5. P. S. Krishnaprasad and S. I. Marcus / Some Nonlinear
Filtering Problems Arising in Recursive
Identification 299
5. Linear Filtering
5.1. T. Kailath / Some Topics in Linear Estimation**) 307
5.2. P. Dewilde, J. T. Fokkema and I. Widya / Inverse
Scattering and Linear Prediction, the Time
Continuous Case*) 351
5.3. M. Hazewinkel / Harmonic Analysis and the Mathematics
Behind Filtering and Prediction (d'apres P. Cartier
d'apres N. Wiener, A. N. Kolmogorov)*)
Abstract only. 383
6. Adaptive Control
6.1. Y. D. Landau / Deterministic and Stochastic Model
Reference Adaptive Control*) 387
6.2. P. E. Caines and D. Dorer / Adaptive Control of
Systems Subject to a Class of Random Parameter
Variations and Disturbances*) 421
6.3. J.-J. J. Fuchs / On Stochastic Self-Tuning Methods 433
7. Nonlinear Filtering
7.1. R. W. Brockett / Nonlinear systems and Nonlinear
Estimation Theory**) 441
7.2. S. K. Mitter / Non-linear Filtering and Stochastic
Mechanics**) 479
7.3. M. H. A. Davis / Pathwise Non-linear Filtering*) 505
TABLE OF CONTENTS vii
7.4. E. Pardoux / Non-linear Filtering, Prediction and
Smoothing*) 529
7.5. J. Baillieul / Estimation Problems with Low Dimensional
Filters 559
7.6. J. S. Baras / Group Invariance Methods in Nonlinear
Filtering of Diffusion Processes 565
7.7. B. Z. Bobrovsky and M. Zakai / Asymptotic Bounds
on the Minimal Error of Non-linear Filtering 573
7.8. G. B. Di Masi and W. J. Runggaldier / An Approximation
to Optical Nonlinear Filtering with Discontinuous
Observations 583
7.9. M. Hazewinkel and S. 1. Marcus / Some Results and
Speculations on the Role of Lie Algebras in
Filtering 591
7.10. O. Hijab / Deterministic Estimation and Asymptotic
Stochastic Estimation 605
7.11. F. Le Gland / Non-linear Filtering Equation and a
Problem of Parametric Estimation 613
7.12. D. Michel/Regularity of Conditional Laws in
Non-linear Filtering Theory and Stochastic
Calculus of Variations 621
7.13. D. Ocone / Finite Dimensional Estimation Algebras
in Nonlinear Filtering 629
7.14. H. J. Sussmann / Rigorous Results on the Cubic
Sensor Problem 637
7. 15 •. M. Zakai / A Footnote to the Papers which Prove the
Non-Existence of Finite Dimensional Filters 649
List of Other Contributed Papers 651
List of Participants 655
Index 657
PREFACE
In the last five years or so there has been an important
renaissance in the area of (mathematical) modeling, identification
and (stochastic) control. It was the purpose of the Advanced
Study Institute of which the present volume constitutes the
proceedings to review recent developments in this area with par
ticular emphasis on identification and filtering and to do so in
such a manner that the material is accessible to a wide variety
of both embryo scientists and the various breeds of established
researchers to whom identification, filtering, etc. are important
(such as control engineers, time series analysts, econometricians,
probabilists, mathematical geologists, and various kinds of pure
and applied mathematicians; all of these were represented at the
ASI).
For these proceedings we have taken particular care to see
to it that the material presented will be understandable for a
quite diverse audience. To that end we have added a fifth tutorial
section (besides the four presented at the meeting) and have also
included an extensive introduction which explains in detail the
main problem areas and themes of these proceedings and which
outlines how the various contributions fit together to form a
coherent, integrated whole. The prerequisites needed to understand
the material in this volume are modest and most graduate students
in e.g. mathematical systems theory, applied mathematics, econo
metrics or control engineering will qualify. And if one finds
that one does not have the prerequisites necessary to understand
the prerequisites (to quote Paul Halmos) there is no need to get
discouraged; much can be done (particularly in this field) by
means of osmosis or diffusion coupled with the ability to believe
(results of others).
The mathematical tools and results used in the area of
mathematical systems theory centering around filtering and iden
tification have evolved considerably in the recent past. They do
not remain restricted to the obviously relevant parts of statis
tics, probability and stochastic differential equations but also
involve techniques, concepts and results from such fields as
topology and geometry of manifolds, Lie algebras (or differential
operators), Lie groups, functional analysis (evolution semigroups),
ix
M. Hazewinkel and J. C. Willems (eds.), Stochastic Systems: The Mathematics of Filtering and Identification
and Applications, ix-xi.
Copyright © 1981 by D. Reidel Publishing Company.
PREFACE
x
functional integration (Feynman-Kac formulae, path integrals),
differential geometry (curvature), quantum field theory (Heisen
berg commutation relations, Stone-v. Neumann theorem, stochastic
quantization, stochastic mechanics) and in the not too distant
future probably such topics as (harmonic) analysis on nil- and
solv-manifolds, theta functions, ••••••••
As indicated above there are diverse groups of scientists
to whom it is of importance to become acquainted woth the various
mathematical tools available and above all to see how they work
in more or less applied situations.
These groups do not always communicate perfectly with each
other and we hope that these proceedings will serve as a forum
for a dialogue and facilitate interspecialist appreciation and
understanding. To this end we aimed to present and discuss the
new trends in filtering and identification in an integrated
manner together with the underlying mathematics and its applica
tions (both actual and potential) and in relation with the
mathematical problems thrown up or suggested by the various
engineering and time series approaches to filtering, identifica
tion, adaptive control and modeling.
We leave to the reader to judge whether these proceedings
fulfill these noble intentions.
In any case the NATO division of scientific affairs agreed
with us that it was definitely worth trying and supplied a
substantial grant from their ASI program which put the meeting
on a sound financial basis. It is a pleasure to record here our
indebtedness to Dr T. Kester of International Transfer of Science
and Technology and Dr M. diLullo of the NATO Scientific Affairs
division for their help, advice, and encouragement also in the
earlier planning stages of this ASI.
In addition we gave a rather large number of big industrial
corporations and other organizations the opportunity to contribute
s~mething financial towards the success of the conference. Most
resolutely refused to help in any way (a list is available on
request) but it is pleasure to mention the favourable exceptions:
ESSO Nederland BV which supplied support which enabled us to
assist a number of participants from Non-NATO countries; NSF which
supplied the air ticket for one junior participant fron the USA
and Shell Nederland BV which donated full expenses for three
junior participants from The Netherlands. In addition North
Holland Publ. Co., Prentice Hall and especially D. Reidel Publ. Co.
and Pitman Publ. Co. helped us with a modest book display, thus
making the ASI more attractive.
Finally it remains for us to thank the many people in our
respective institutes and the staff at Les Arcs for their manifold
efforts towards the success of the meeting.
In addition to the tutorial and invited papers (marked
with * or ** in the table of contents) these proceedings contain
a number of contributed papers. Quite apart from the intrinsic
merits of these papers their selection mainly reflects the
PREFACE
idiosyncratic tastes of the editors, though we also believe that
they fit the main themes of the conference well and together
with the invited papers give a very fair picture of the emerging
new trends in modeling, identification and adaptive control and
linear and nonlinear filtering. This last topic, practically a
new emerging field, which seems to have an unusually bright
immediate future, has especially received much attention both
because of the wealth of recent fundamental results and its
glittering prospects.