Table Of ContentP1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
This page intentionally left blank
P1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
NonparametricSystemIdentification
Presenting a thorough overview of the theoretical foundations of nonparametric sys-
tems identification for nonlinear block-oriented systems, Włodzimierz Greblicki and
Mirosław Pawlak show that nonparametric regression can be successfully applied to
systemidentification,andtheyhighlightwhatyoucanachieveindoingso.
Startingwiththebasicideasbehind nonparametric methods, variousalgorithmsfor
nonlinearblock-orientedsystemsofcascadeandparallelformsarediscussedindetail.
Emphasisisplacedonthemostpopularsystems,HammersteinandWiener,whichhave
applicationsinengineering,biology,andfinancialmodeling.
Algorithmsusingtrigonometric,Legendre,Laguerre,andHermiteseriesareinvesti-
gated,andthekernelalgorithm,itssemirecursiveversions,andfullyrecursivemodifica-
tionsarecovered.Thetheoriesofmodernnonparametricregression,approximation,and
orthogonalexpansionsarealsoprovided,asarenewapproachestosystemidentification.
Theauthorsshowhowtoidentifynonlinearsubsystemssothattheircharacteristicscan
beobtained even whenlittleinformationexists,whichisofparticular significance for
practical application. Detailed information about all the tools used is provided in the
appendices.
Thisbookisaimedatresearchersandpractitionersinsystemstheory,signalprocess-
ing,andcommunications.Itwillalsoappealtoresearchersinfieldssuchasmechanics,
economics,andbiology,whereexperimentaldataareusedtoobtainmodelsofsystems.
WłodzimierzGreblickiisaprofessorattheInstituteofComputerEngineering,Control,
andRoboticsattheWrocławUniversityofTechnology,Poland.
MirosławPawlakisaprofessorintheDepartmentofElectricalandComputerEngineer-
ingattheUniversityofManitoba,Canada.HewasawardedhisPh.D.fromtheWrocław
UniversityofTechnology,Poland.
Both authors have published extensively over the years in the area of nonparametric
theoryandapplications.
P1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
P1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
Nonparametric System
Identification
WŁODZIMIERZ GREBLICKI
WrocławUniversityofTechnology
MIROSŁAW PAWLAK
UniversityofManitoba,Canada
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521868044
© Cambridge University Press 2008
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2008
ISBN-13 978-0-511-40982-0 eBook (NetLibrary)
ISBN-13 978-0-521-86804-4 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
P1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
Contents
Preface pageix
1 Introduction 1
2 Discrete-timeHammersteinsystems 3
2.1 Thesystem 3
2.2 Nonlinearsubsystem 4
2.3 Dynamicsubsystemidentification 8
2.4 Bibliographicnotes 9
3 Kernelalgorithms 11
3.1 Motivation 11
3.2 Consistency 13
3.3 Applicablekernels 14
3.4 Convergencerate 16
3.5 Themean-squarederror 21
3.6 Simulationexample 21
3.7 Lemmasandproofs 24
3.8 Bibliographicnotes 29
4 Semirecursivekernelalgorithms 30
4.1 Introduction 30
4.2 Consistencyandconvergencerate 31
4.3 Simulationexample 34
4.4 Proofsandlemmas 35
4.5 Bibliographicnotes 43
5 Recursivekernelalgorithms 44
5.1 Introduction 44
5.2 Relationtostochasticapproximation 44
5.3 Consistencyandconvergencerate 46
5.4 Simulationexample 49
5.5 Auxiliaryresults,lemmas,andproofs 51
5.6 Bibliographicnotes 58
P1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
vi Contents
6 Orthogonalseriesalgorithms 59
6.1 Introduction 59
6.2 Fourierseriesestimate 61
6.3 Legendreseriesestimate 64
6.4 Laguerreseriesestimate 66
6.5 Hermiteseriesestimate 68
6.6 Waveletestimate 69
6.7 Localandglobalerrors 70
6.8 Simulationexample 71
6.9 Lemmasandproofs 72
6.10 Bibliographicnotes 78
7 Algorithmswithorderedobservations 80
7.1 Introduction 80
7.2 Kernelestimates 81
7.3 Orthogonalseriesestimates 85
7.4 Lemmasandproofs 89
7.5 Bibliographicnotes 99
8 Continuous-timeHammersteinsystems 101
8.1 Identificationproblem 101
8.2 Kernelalgorithm 103
8.3 Orthogonalseriesalgorithms 106
8.4 Lemmasandproofs 108
8.5 Bibliographicnotes 112
9 Discrete-timeWienersystems 113
9.1 Thesystem 113
9.2 Nonlinearsubsystem 114
9.3 Dynamicsubsystemidentification 119
9.4 Lemmas 121
9.5 Bibliographicnotes 122
10 Kernelandorthogonalseriesalgorithms 123
10.1 Kernelalgorithms 123
10.2 Orthogonalseriesalgorithms 126
10.3 Simulationexample 129
10.4 Lemmasandproofs 130
10.5 Bibliographicnotes 142
11 Continuous-timeWienersystem 143
11.1 Identificationproblem 143
11.2 Nonlinearsubsystem 144
11.3 Dynamicsubsystem 146
11.4 Lemmas 146
11.5 Bibliographicnotes 148
P1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
Contents vii
12 Otherblock-orientednonlinearsystems 149
12.1 Series-parallel,block-orientedsystems 149
12.2 Block-orientedsystemswithnonlineardynamics 173
12.3 Concludingremarks 218
12.4 Bibliographicalnotes 220
13 Multivariatenonlinearblock-orientedsystems 222
13.1 Multivariatenonparametricregression 222
13.2 Additivemodelingandregressionanalysis 228
13.3 Multivariatesystems 242
13.4 Concludingremarks 248
13.5 Bibliographicnotes 248
14 Semiparametricidentification 250
14.1 Introduction 250
14.2 Semiparametricmodels 252
14.3 Statisticalinferenceforsemiparametricmodels 255
14.4 StatisticalinferenceforsemiparametricWienermodels 264
14.5 StatisticalinferenceforsemiparametricHammersteinmodels 286
14.6 Statisticalinferenceforsemiparametricparallelmodels 287
14.7 Directestimatorsforsemiparametricsystems 290
14.8 Concludingremarks 309
14.9 Auxiliaryresults,lemmas,andproofs 310
14.10 Bibliographicalnotes 316
A Convolutionandkernelfunctions 319
A.1 Introduction 319
A.2 Convergence 320
A.3 Applicationstoprobability 328
A.4 Lemmas 329
B Orthogonalfunctions 331
B.1 Introduction 331
B.2 Fourierseries 333
B.3 Legendreseries 340
B.4 Laguerreseries 345
B.5 Hermiteseries 351
B.6 Wavelets 355
C Probabilityandstatistics 359
C.1 Whitenoise 359
C.2 Convergenceofrandomvariables 361
C.3 Stochasticapproximation 364
C.4 Orderstatistics 365
References 371
Index 387
P1:IBE
Main CUFX280D/Greblicki 9780521868044 April28,2008 8:11
Tomywife,Helena,andmychildren,Jerzy,Maria,andMagdalena–WG
TomyparentsandfamilyandthosewhomIlove–MP
