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S Y S T E M I D E N T I F I C A T I O N :Theory for the User

Lennart LjungUniversity of LinkpingSweden

Prentice-Hall, Inc., Engiewood Cliffs, New Jersey 07632

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C O N T E N T SPREFACE x iA C K N O W L E D G M E N T S x vO P E R A T O R S A N D N O T A T I O N A L C O N V E N T I O N S XV

1 . INTRODUCTION 11.1 Dyn am ical Systems 11.2 M ode ls 31.3 Th e System Identif icat ion Proc edu re 71.4 Or gan izatio n of the Bo ok 81.5 BibH ogra phy 10

part i: Systems and m odeis2 . TIME-INVARIANT LINEAR SYSTE MS 1 3

2.1 Impulse Respon ses , Dis turbances and 13Transfer Functions

2 .2 Frequency-dom ain Express ions 22

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2.32.42.52.62.72.8

Signal SpectraSingle Realizat ion Behavior andE rgodici tyResults *)Multivariable Systems *)SummaryBibliographyProblemsAppendix 2 .A: ProofofT h e o r e m2.2Appendix 2 .B: Proof ofTheorem2.3Appendix 2 .C: Covar iance Formulas

SIMULATION, PREDICTION, AND CONTROL3.13.23.33.43.53.63.7

Simulat ionPredict ionObserversCon t ro l *)SummaryBibliographyProblems

2634

35363738434549

5151525962656566

MODELSOFLINEAR TIME-INVARIANT SYSTEM S 694.1 Line ar Mo delsandSetsof Linear Models 694.2 A FamilyofTransfer-function M odels 714.3 State-space M odels 814.4 Dis t r ibuted -Param eter Mod els *) 904.5 Mo del Sets, M odel Structure s,and 93Identif iabi l i ty: Some Formal Aspects *)4.6 Identifiability of Some Model St ructures 1014.7 Summary 1064.8 Bibl iography 164.9 Problems 108Append ix 4 .A: Identifiability ofBlack-box 115Multivariable Model Structures

MODELS FOR TIME-VARYING 127AND NONLINEAR SYSTEMS5.1 Line ar Time-varying Mo dels 1275.2 Non l inear Mode lsasLine ar R egressions 1305.3 No nline ar State-space M odels 1325.4 Form al Cha racteriz at ion of Models *) 1345.5 Summary 1375.6 Bibl iography 1385.7 Problems 138

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pa r i i i : m ethod s6. NONPARAMETRIC TIME- 1 41AND FREQUENCY-DOMAIN METHODS

6.1 Tran sient Re spon se Analysis and Co rrelat ion 141Analysis

6.2 Freq uen cy-res pon se Analysis 1436.3 Fo urie r An alysis 1466.4 Spe ctrat An alysis 1516.5 Est im ating the Distu rban ce Spec trum *) 1606.6 Sum ma ry 1626.7 Bib liogra phy 1626.8 Prob lem s 163

Ap pen dix 6.A: De rivat ion of the Asy mp totic 167Propert ies of the SpectralAnalysis Est imate

7. PARAMETER ESTIMATION METHODS 1 697.1 Guiding Principles beh ind Pa ram eter Est im ation 169Methods

7.2 Minimizing Predict ion Er ror s 1717.3 Line ar Regre ssions and the Le ast-squ ares 176

M e t h o d7.4 A Stat ist ical Fram ew ork for Pa ram eter 181

Est imat ion and the Maximum Likel ihoodM e t h o d7.5 Co rrelat ing Predic t ion Er ror s with Past D ata 1907.6 Instrum ental-va riable M etho ds 1927.7 Sum ma ry 1957.8 Biblio grap hy 1967.9 Prob lem s 197

Ap pend ix 7 . A : Proof of the Cr am er- Ra o 206Inequali ty

8. CONVER GENCEAN DCON SISTENCY 2088.1 Introd uction 2088.2 Con dit ions on the D ata Set 2108.3 Predict ion-error Ap proac h 2148.4 Co nsisten cy an d Identifiability 2188.5 Line ar Time -invariant M odels: A Frequ ency- 224domain Descript ion of the Limit Model8 .6 The Correlat ion Ap proac h 2298.7 Sum m ary 2338.8 Bibl iography 2338.9 Prob lem s 234

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9. ASYM PTOTIC DISTRIBUTIONOF PARAMETER ESTIMATES9.1 Introdu ction 2399.2 The Predict ion-error App roach: Basic The orem 2409.3 Expressions for the Asy mp totic Variance 2429.4 Frequ ency-do ma in Expression s for the 248Asym ptot ic Var iance9.5 The Correlat ion Ap proa ch 2549.6 Use and Relevance of Asy mp totic Variance 258Express ions9.7 Sum ma ry 2629.8 Bibliogra phy 2639.9 Proble ms 264Ap pen dix 9. A : Proof of Th eo rem 9.1 266App endix 9 .B: The Asym ptot ic Param eter 270

Variance

1 0. COMPU TING THE ESTIMATE 2 7410.1 Linea r Regressions and Least Squ ares 27410.2 Nu m erica l Solution by Iterativ e Search 282

Methods10.3 Com puting Gra dien ts 28510.4 Two-stage and Mu lt istage M ethod s 28810.5 Local Solution s and Initial Values 29210.6 Sum mary 29410.7 Bibliogra phy 29410.8 Pro blem s 296

1 1 . RECURSIVE ESTIMATION METHODS 30 311.1 Intro duc tion 30311.2 The Recursive Lea st-squares Algori thm 30511.3 The Recursive IV M cthod 31111.4 Recursive Pred ict ion-E rror M ethod s 31111.5 Recursive Pseu dolinear Regressions 31611.6 Th e Cho ice of U pda ting Step 31811.7 Im plem enta t ion 32211.8 Sum ma ry 32611.9 Bibliog raphy 32711.10 Pro blem s 328App endix IL A : Techniques for Asym ptot ic 329Analysis of RecursiveAlgor i thms

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part i i i : user s choices1 2 . OPTIONS AND OBJECTIVES 339

12.1 Options 33912.2 Object ives 34112.3 BiasandVariance 34512.4 Summary 34712.5 Bibography 34712.6 Problems 347

1 3 . AFFECTING THE BIAS DISTRIBUTION 349OFTRANSFER FUNCTION ESTIMATES

13.1 Som e Basic Expres sions 34913.2 Heu rist ic DiscussionofTransfer-function Fit 350

in Open- loop Operat ion13.3 Som e Solut ionstoFormal D esign Problems 35413.4 Summary 35613.5 Bibography 35613.6 Problems 357

1 4 . EXPERIMENT DESIGN 35814.1 Some Ge neral Considerat ions 35914.2 Informat ive Exp er imen ts 36114.3 Optim al Inpu t Design *) 36914.4 Opt imal Exp er imen t Design for High-order 375

Black-box Models *)14.5 Choice of Sampling Interval and Presampl ing 378Filters14.6 Pret reatmentofD a t a 38614.7 Summary 38914.8 Bibography 39014.9 Problems 391

1 5 . CHOICE OF IDENTIFICATION CRITERION 39415.1 Ge neral Aspects 394L5.2 ChoiceofNorm: Robus tness 39615.3 Var iance: Opt imal Ins t rumen ts 40215.4 Summary 40515.5 Bibography 40615.6 Problems 406

Contents X

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1 6 . MODEL STRUCTURE SELECTIONAND MODEL VALIDATION16.1 Ge nera l Aspe cts of the Choice of Mo del 408

Structure16.2 A Priori Co nsid era tions 41116.3 Mo del Structure Select ion Base d 413

on Preminary Data Analysis16.4 Com paring Mo del Structures 4161.6.5 M od el Va lidation 42416.6 Sum ma ry 43016.7 Bibliogra phy 43116.8 Pro blem s 431

1 7 . SYSTEM IDENTIFICATION IN PRACTICE 43 417.1 Th e Tool: Intera ctive Software 43417.2 A Lab oratory-sc aie Ap plicat ion 44017.3 Identification of Ship-stee ring Dy nam ics 44917.4 W hat D oe s System Identification Ha ve 454

to Offer?17.5 Biblio grap hy 456A PP EN D IX I: Some Concep t s from Probab i ty Theory 4 5 7A PP EN D IX II: Some Stati s tical Techniques for Linear 4 6 1RegressionsREFERENCES 482AUTHOR INDEX 5 5SUBJECT INDEX 511

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