Table Of ContentLecture Notes ni
Control and
nottamrofnI Sciences
Edited by .V.A and Balakrishnan .M Thoma
57
IIIIIIII IHIIIIIIII IIIIIIIIIIIIIII IHIIIIIIIII
.T SSderstrSm
.G.P Stoica
Instrumental Variable Methods
for System Identification
II IIIIIIIIIII II I
galreV-regnirpS
Berlin Heidelberg New York oykoT 1983
Series Editors
A. V, Balakrishnan • M. Thoma
Advisory Board
L D. Davisson • ,~ G..1. MacFarlane. H. Kwakernaak
.J L Massey Ya. • Z. Tsypkin • A. .J Viterbi
Authors
Torsten SSderstr~m
Dept. of Automatic Control and Systems Analysis
Inst. of Technology
Uppsala University
Uppsala, Sweden
Petre Gheorghe Stoica
Dept. of Automatic Control
Polytechnical Inst. of Bucharest
Bucharest, Romania
AMS Subject snoitacifissalC :)0891( 93 E12, 93 B 30, 90A16, 90A ,91 62 M 01
ISBN 3-540-12814-X galreV-regnirpS New Heidelberg Berlin kroY oykoT
ISBN 0-387-12814-X galreV-regnirpS weN Berlin Heidelberg York oykoT
Library of Congress Cataloging in Publication Data
S6derstr6m, Torsten.
Instrumental variable methods for system identification.
(Lecture notesi n control and information sciences ; 57)
Bibliography: p.
Includes index.
.1 System identification. 2. Parameter estimation.
.I Stoica, .P G. (Petre Gheorghe)
.1I Title. .1II Series.
CIA402.$693 1983 003 83-13525
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
206113020-543210
DROWEROF DNA STNEMEGDELWONKCA
This book is about system identification dna parameter estimation using the
instrumental variable (IV) approach. It sah grown out of our joint research in
this field during the last five years. ruO attempt sah been not only to ezirammus
our findings in a unified form dna extend them to emos degree but also to include
practical hints for the users. eW have nesohc a mathematical level of the text
which ew hope will appeal both to those interested in the technical details dna
those merely interested in woh to apply VI sdohtem in practice.
oT understand the book the reader is demussa to have emos basic egdelwonk of
stochastic systems or time series analysis dna also to wonk a little about parameter
estimation in models. dynamic In places ew emussa the reader to eb familiar with
structures of multivariable systems. eW have included various other dnuorgkcab
material in the appendices to ekam the book reasonably self-contained. roF readers
ohw are only practically interested ew dnemmocer a first reading of Part I dna a
careful reading of Part III, thus skipping the erem theoretical Part II where the
analysis is developed.
eW have deliberately used rather wef references in the text. Instead ew evah
included emos bibliographical notes in the dne of the chapters. Since instrumental
variable sdohtem have u~ed been for quite a long time dna in diverse areas it
should eb needless to yas that yna attempt to give a full dna perfect (historical)
list of references is demood to fail. eW hope that ew have dedeeccus in giving
the yek references dna emos more. Moreover, our purpose sah been to present a
unified analysis rather than tracing the exact historical background of every
single variant of the instrumental variable approach.
eW wish to express our sincere thanks to the ynam persons ohw helped ekam this
koob possible.
Several colleagues detnemmoc no the text. eW would like to mention Professor
Lennart Ljung in particular ohw edam valuable suggestions based no a preliminary
version of the manuscript. eW also thank Professor Peter gnuoY for his helpful
stnemmoc no our work dna for providing several references. Professor namroN hguoG
sah suggested ynam stnemevorpmi to the English. ehT responsibility for the re-
maining errors, English or technical, remains of course ours. eW also thank
VI
Professor Manfred ,amohT the editor of this series, ohw after preliminary discussions
sah degaruocne su to write the .koob
Most of our joint publications over the past years have been expertly typed by
Miss Ingrid RingArd. eW are very grateful to her dna srM Lis Timner ohw jointly
have typed the text os nicely with great care dna patience. srM Timner sah also
skilfully prepared the figures.
eW thank rD reP Erik n~doM ohw provided su with the drum drier data which ew evah
desu in Section 9.2.
ehT first author saw given a scholarship yb the Swedish Institute within na
official exchange margorp between nedewS dna .ainamoR This grant edam possible a
very fruitful stay in Bucharest. There ew worked efficiently together no the spot
thus avoiding the more emosrebmuc communication yb mail.
Last, but not least ew thank our families for their moral support during our work
no this book.
ELBAT FO STNETNOC
TRAP I: SEIRANIMILERP DNA SNOITINIFED
retpahC i .N OITCUDORTNI
retpahC 2. SEIRANIMILERP
.2 I Mod~ er~t~u~t~ 6
2.2 Sgst~ 21
3.2 latnemirepxE condit~n 81
2.4 emoS hOt--hal snoitnevnoc 02
2.5 Remarks and 51JoZiog~p~ not~ 12
retpahC 3. CISAB DNA DEDNETXE S VIMHTIROGLA
1.3 Basic VI varian~ 32
3.2 dedne~xE VI u~ia~t~ 13
3.3 Boo~t~p IV ua~ 73
3.4 R~k~ dna 6ibliographical not~ 83
PART : I I ANALYSIS
Chapter 4. YCNETSISNOC
.4 I Gcn~al co~ideratio~ 74
4.2 Analys~ of VI varian~ for ~alac~ ~yst~ 55
4.3 Anat4~/~ of IV v ~ for m~v~able ~yste~ 06
4.4 Converg~ce ~aZysi~ of bootstrap IV variant~ 76
4.5 skran1PR dna bib~.ioga~pl~te(zt not~ 47
Chapter 5. YCARUCCA
1.5 A~pt~t~ dist~ution of VI ~ t ~ 57
5.2 A~ympt~t~ d~tri6ut~n of bootstrap VI e~t~r~ 58
5.3 R~ar~ dna 6i61iographical not~ 88
retpahC 6. LAMITPO VISROTAMITSE
1.6 tncmet~t& fo opt/~y proble~ 98
6.2 emoS opti~iz~n exampl~ 39
6.3 Optimal IV ~t~o~ and app~m~tz implem~ntatio~ 69
lV
6.4 Analys~ of the approxi~e noitu~t~emelpm~ of o~cimal
IV ~t~r6 601
6.5 Re~r~ dna bibliographical note6 t19
retpahC .7 LAMITPO TUPNI NGISED
I 7. St~tcmc~t of ~¢ melborp 021
2.7 Inpu~ noitmzLcct~m~uJep 221
3.7 ¢cn~J~euoC rccUzmtion 521
4.7 cinu/tJJ~og~JA ~tJeep~ 231
5.7 R ~ dna bib~Log~pkiccZ seton 141
TRAP III: LACITCARP STCEPSA DNA ESAC SEIDUTS
retpahC .8 YRAMMUS FO EHT SISYLANA DNA STNIH ROF EHT RESU
1.8 Basic fact~ no IV ~t~m~tLon 541
8.2 Hint~ for choosing the IV variant 051
8.3 Hints for input d~ign 651
4.8 Hin~ for ledom structure selection 751
retpahC .9 ESAC SEIDUTS
.9 I Z~Je~G ~Ioce~ 561'
2.9 A murd ~L~Jd 861
3.9 nA cimonoc¢ p~oc~s 271
4.9 k ~m9 f~e 571
5.9 k tu~bo-~nctor 181
SECIDNEPPA
xidneppA I. YLTNETSISREP GNITICXE SLANGIS 981
xidneppA .2 EMOS STLUSER S LNAOIMONYLOP DNA CITYLANA SNOITCNUF 591
xidneppA .3 EMOS XIRTAM STLUSER 002
xidneppA .4 EMOS YTILIBABORP STLUSER 312
xidneppA .5 NEIHBTRUD-NOSNIVEL MHTIROGLA 612
xidneppA .6 EMO SYGOLONIMRET DETAICOSSA HTIW LANOITAR SECIRTAM 225
iiV
REFERENCES 722
TCEJBUS XEDNI 932
iI o
I
m l B m~ z rrl cs~ IT imw4 z tim4 o z
retpahC I
I NTRODUCT I NO
ehT area of "system identification", in particular "parameter estimation in
dynamical models", is na important eno in ynam fields, such sa control engineering,
econometrics dna signal processing. There is certainly na overwhelming literature
in this area. roF na introduction to dna a general survey of the area metsys
identification ew refer the reader to Astr~m dna Eykhoff (1971), xoB dna Jenkins
(1976), Eykhoff (1974), niwdooG dna enyaP (1977), arheM dna Lainiotis (1976).
ehT present state of the art nac eb studied for example in Eykhoff (1981) dna
recent proceedings of the CAFI aisopmyS no Identification dna metsyS retemaraP
Estimation.
ehT area of identification is, sa already indicated, related to other areas such
sa time series analysis, statistics dna econometrics. A lot of various identification
sdohtem have proposed been dna are in current use. A systematic yaw of nosirapmoc
neewteb various sdohtem sah been given to emos degree but it is far from complete.
There smees to eb na increasing interest in parametric methods, i.e. identification
sdohtem which sa a crucial part include parameter estimation from experimental data.
owT well-known parametric sdohtem are the least-squares dohtem )MSL( dna the
prediction error dohtem (PEM). ehT MSL nac eb traced back to ssuaG (1809). It is
desab no a simple algorithm. In fact, na analytic formo f the estimate nac eb
given once the data are .nwonk A disadvantage of the MSL is that na asymptotic bias
often occurs. This snaem that the estimated parameters often contain systematic
errors that od not vanish even if the data series semoceb infinitly long.
oT avoid the asymptotic bias eno possibility is to esu a MEP in a model structure
that also describes the properties of the noise. This leads to a more xelpmoc
dohtem dna requires a numerical optimization of a nonlinear function that sdneped
no the recorded data. ehT advantage of using such a dohtem is that the estimated
ledom often will give quite doog a description of the data. In particular the bias
obstacle is on longer a problem sa for the simple .MSL
oS far the discussion sah indicated that the user must face a trade-off neewteb
algorithm complexity dna the properties of the estimated parameters. It would of
course eb suoegatnavda to have estimation sdohtem that enibmoc the small algorithmic
