frequency-domain system identification for modal analysis

SREVINU

ITEIT

EJIR

V

BRUSSE

L

ECNIVRE TENE

BR

AS

AI

TN

EIC

S

VRIJE UNIVERSITEIT BRUSSEL

FACULTEIT TOEGEPASTE WETENSCHAPPEN

VAKGROEP WERKTUIGKUNDE

Pleinlaan 2, B-1050 Brussels, Belgium

FREQUENCY-DOMAIN SYSTEM IDENTIFICATION

FOR MODAL ANALYSIS

Peter VERBOVEN

Promotor:

Prof. dr. ir. P. Guillaume

Copromotor:

Prof. dr. ir. M. Van Overmeire

Proefschrift ingediend tot

het behalen van de academische

graad van doctor in de

toegepaste wetenschappen

May 2002

SREVINU

ITEIT

EJIR

V

BRUSSE

L

ECNIVRE TENE

BR

AS

AI

TN

EIC

S

VRIJE UNIVERSITEIT BRUSSEL

FACULTEIT TOEGEPASTE WETENSCHAPPEN

VAKGROEP WERKTUIGKUNDE

Pleinlaan 2, B-1050 Brussels, Belgium

FREQUENCY-DOMAIN SYSTEM IDENTIFICATION

FOR MODAL ANALYSIS

Peter VERBOVEN

Jury:

Prof. dr. ir. D. Lefeber, voorzitter

Prof. dr. ir. J. Vereecken, vice-voorzitter

Prof. dr. ir. P. Guillaume, promotor

Prof. dr. ir. M. Van Overmeire, copromotor

Prof. dr. ir. B. De Moor (KULeuven)

Prof. dr. ir. J. Schoukens

Prof. dr. ir. J. Swevers (KULeuven)

Dr. ir. H. Van der Auweraer (LMS International)

Dr. lic. S. Vanlanduit

Proefschrift ingediend tot

het behalen van de academische

graad van doctor in de

toegepaste wetenschappen

May 2002

c©Vrije Universiteit Brussel – Faculteit Toegepaste WetenschappenPleinlaan 2, B-1050 Brussel (Belgium)

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Voorwoord

Tijdens mijn ruim vier jaren van onderzoek aan de vakgroep Werktuigkunde hebik de gelegenheid gehad met vele mensen samen te werken. Graag zou ik dan ookiedereen willen bedanken die rechtstreeks of onrechtstreeks heeft bijgedragen totde realisatie van dit doctoraatswerk.

In de eerste plaats dank ik mijn promotor, prof. dr. ir. Patrick Guillaume encopromotor prof. dr. ir. Marc Van Overmeire. Alleerst omdat beiden me motiveer-den om na mijn ingenieursstudies het doctoraat te beginnen en me het nodigevertrouwen en steun hebben gegeven om in een vrije en inspirerende omgevingaan onderzoek te doen. De nauwe samenwerking met prof. dr. ir. Patrick Guil-laume was een uitermate leerrijke en stimulerende ervaring en heeft me vele nieuweinzichten en methodes voor onderzoek bijgebracht. Niet alleen was hij een aange-name bureaugenoot, maar bracht hij me ook de basis van multivariabele systeemi-dentificatie en modale analyse bij. De vele technische discussies en zijn assistentiebij de experimenten hebben mede geleid tot het verhogen van het wetenschap-pelijk niveau van het geleverde werk. Tevens was het een voorrecht om zijn eerstedoctoraatsstudent te zijn.

Daarnaast dank ik de leden van de jury voor hun enthousiasme en interesse inmijn onderzoek en hun kostbare tijd die zij besteedden aan het nalezen van ditproefschrift. Ik hoop dat, door de realisatie van dit doctoraat, de reeds bestaandecontacten die ik met elk van hun heb kunnen leiden tot de verdere ontwikkelingvan nieuwe onderzoeksideeen en mogelijke samenwerking.

Mede doordat het onderwerp van dit onderzoek direct gerelateerd was met eenaantal onderzoeksprojecten binnen de onderzoeksgroep, heb ik de kans gehad ommet verschillende bedrijven in contact te komen. Dit liet toe van de toepasbaarheidvan de onderzoeksresultaten te toetsen voor vele practische probleemstellingen.Daarom dank ik meer specifiek LMS International en ASCO aero industries voorde nauwe en constructieve samenwerking en het verschaffen van de vele datasetsen teststructuren.

Prof. dr. ir. J. Schoukens en prof. dr. ir. Rik Pintelon van de vakgroep Al-

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gemene Elektriciteit en Instrumentatie ben ik erkentelijk voor het delen van hunenorme kennis op het gebied van frequentiedomein systeemidentificatie en hun lei-dinggevende rol voor het initieren van nieuwe ideeen welke ik kon toepassen binnenhet domein van de modale analyse.

Verder wil ik Prof. dr. ir. D. Lefeber, vakgroepsvoorzitter en alle medewerkersvan de vakgroep Werktuigkunde bedanken voor de toffe en stimulerende werks-feer. Hierbij denk ik dan vooral aan de directe collega’s van de onderzoeksgroepEli Parloo, Bart Cauberghe, Steve Vanlanduit en Gert De Sitter voor hun enormemedewerking en het samen publiceren van onderzoekswerk alsmede de vele in-spirerende discussies over de mogelijke toepassing van mijn resultaten binnen hunonderzoek. Ook een speciaal woord van dank voor de secretaresse Carine Vaere-mans voor de vele administratieve taken gedurende afgelopen jaren, Frank Daerdenvoor het oplossen van vele van mijn LaTeX vragen, Andre Plasschaert en Jean-Paul Schepens voor de uitmuntende technische bijstand in het labo en ThierryLenoir voor de steeds ongewenste informatica problemen. Vele collega’s wareneveneens vrienden waarmee ik ook naast het werk prettige tijden heb beleefd.

Tevens ben ik het Instituut voor de aanmoediging van Innovatie door Weten-schap en Technologie in Vlaanderen (IWT-Vlaanderen) en VUB Onderzoeksraaderkentlijk voor de financiele steun.

Tenslotte wil ik benadrukken dat de realisatie van dit proefschrift slechts mo-gelijk was dankzij de onvoorwaardelijke steun, begrip en raad van mijn oudersop elke mogelijke wijze. Helaas dat mijn vader niet meer kan genieten van ditmoment, waarop ik dit werk succesvol heb kunnen beeindigen. Aan hem en mijnmoeder draag ik dit werk op. Daarnaast wens ik ook mijn vriendin Els, hoewelze de laatste maanden niet veel aandacht van mij kreeg, te bedanken voor haarvoortreffelijke steun en begrip.

Peter Verboven

Brussel, Mei 2002

Abstract

Experimental modal analysis has become a commonly-used technique for studyingthe dynamical behaviour of mechanical and civil structures, such as for examplecars, aircrafts, bridges, offshore platforms and industrial machinery. During amodal test, both the applied forces and vibration responses of the structure aremeasured when excited in one or more locations. Based on this data, a modalmodel of the structure, that essentially contains the same information as the orig-inal vibration data, is derived by means of system identification. The appropri-ateness of the modal model for physical interpretation makes that scientists andengineers most often prefer this type of model for their structural dynamics re-search. Recent advances in the field of frequency-domain system identification arethe development of the so-called stochastic Total Least Squares and MaximumLikelihood methods. Since the noise on the measured data is taken into accountduring the estimation process, these methods yield very accurate parameter esti-mates with confidence bounds, even in the case of a poor quality of the data.

Recently, the need for improved system identification methods within the do-main of modal analysis results from the broadening field of applications, e.g. dam-age detection and vibro-acoustics, and the increased complexity of today’s testedstructures. The aim of this thesis was to use the stochastic methods as a startingpoint to develop more advanced modal parameter identification techniques. Thishowever required an extensive revision of both the initial formulation of thesemethods with respect to computational efficiency and the modal analysis processwith respect to the integration of the noise on the measured data.

First of all, an improved nonparametric processing has been proposed in orderto obtain accurate frequency response function (FRF) estimates as well as theuncertainty on this data, required by the stochastic methods. Based on an errors-in-variables framework a methodology was derived that yields FRF estimates withmaximum likelihood properties and at the same time the noise covariance matrix.Comparing this so-called instrumental variables approach with the current state-of-the art, pinpoints the important benefits of this method, combining accuracyand computational efficiency during the nonparametric data processing. At thesame time, by using signal windowing techniques, this approach has been optimized

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to reduce the effects of leakage resulting from the use of random noise excitationsignals.

The adaptation of the stochastic Total Least Squares and Maximum Likeli-hood methods for modal analysis starts from the numerical optimization of thelinear Least Squares formulation. Different types of parameterization were as-sessed to obtained an improved numerical conditioning for a high modal densityas well as computational efficiency for large data sets. A normal equations formu-lation using a discrete-time common-denominator model with complex coefficientsdemonstrates an important robustness for high model orders up to say 100 modesand more, while different representations for a stabilization diagram are possible.Based on the fast formulation and the knowledge of the noise on the FRF data,the extension of the optimized Least Squares formulation to the stochastic TotalLeast Squares and Maximum Likelihood methods is possible. In the case thatonly very limited data sequences are available, it is preferable to start from theInput/Output Fourier data for reasons related to both leakage and frequency reso-lution. It has been demonstrated that in this case, the efficient formulation of theLeast Squares problem is still possible, although the generalization to the stochas-tic Total Least Squares methods is not straightforward anymore. Moreover, theInput/Output methods allow the parametric compensation for effects of leakage,possibly present in the short data sequences.

Based on the improved accuracy of the parameter estimates and the availabilityof confidence bounds, stochastic model validation criteria and the automationof the modal parameter extraction are developed. This results in an importantreduction of the user-time as well as the possibility for non-expert users to applymodal analysis techniques. At the same time, a method for tracking the modalparameters over subsequent monitoring instances is proposed, providing a basictool for structural health monitoring applications.

The application of the proposed system identification methods to many real-life and benchmark case studies clearly illustrates that these techniques performvery well and indicates their great potential within the domain of modal analysis.

Korte Inhoud

Experimentele modale analyse is geevolueerd tot een algemeen gebruikte techniekvoor het bestuderen van het dynamisch gedrag van mechanische en bouwkundigestructuren, zoals bijvoorbeeld auto’s, vliegtuigen, bruggen, boorplatformen en in-dustriele machines. Tijdens een modale test worden beide de krachten, aangelegdin een of meerdere lokaties, en de trillingsresponses opgemeten. Op basis van dezedata wordt vervolgens een “modal model” van de structuur, dat in essentie dezelfdeinformatie bevat als de oorspronkelijke data, bekomen aan de hand van systeemi-dentificatie. De uitermate geschiktheid van dit model voor fysiche interpretatiemaakt dat wetenschappers en ingenieurs dit model meestal verkiezen voor structu-urdynamica onderzoek. Recente vooruitgang binnen het domein van de frequentie-domein systeemidentificatie zijn de ontwikkeling van de zogenaamde stochastische“Total Least Squares” en “Maximum Likelihood” methodes. Aangezien de onze-kerheid op de meetdata in beschouwing wordt genomen tijdens het schattingspro-ces, resulteren deze methodes in een zeer nauwkeurige schatting van de parameterssamen met onzekerheidsintervallen, zelfs in het geval dat de meetdata van slechtekwaliteit is.

Gedurende de laatse jaren is er een toenemende behoefte aan verbeterde sys-teemidentificatie methods binnen het domein van modale analyse als gevolg van hetverruimde toepassingsgebied, zoals bijvoorbeeld schadedetectie en vibro-akoestiek,en de toenemende complexiteit van de huidig bestudeerde structuren. Het doel vandeze thesis was, vertrekkende van de stochatische methodes, geavanceerde modaleparameter identificatietechnieken te ontwikellen. Dit vereiste echter een uitge-breide doorlichting van zowel de oorspronkelijke formulering van deze methodesmet nadruk op de numerieke aspecten als het modal analyse proces met betrekkingtot het integreren van de meetruis.

Alleerst is een verbeterde niet-parametrische verwerking voorgesteld voor denauwkeurige schatting van frequenctie respons functies (FRF) alsmede de onze-kerheid op deze data, vereist voor de stochastische methodes. In het “errors-in-variables” kader is het mogelijk een methodologie af te leiden voor het bepalen vanFRF schattingen met “maximum likelihood” eigenschappen en terzelfdertijd eenruisanalyse voor de meetdata. Vergelijking van deze zogenaamde “instrumentele

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variabelen” aanpak met de huidige “state-of-the-art” benadrukt de belangrijkevoordelen van deze methode welke nauwkeurigheid en numerieke efficientie combi-neert gedurende de niet-parametrische verwerking. Verder is deze aanpak, door detoepassing van signaal vensters, geoptimaliseert voor het reduceren van “leakage”effecten als gevolg van het gebruik van ruis excitatiesignalen.

De aanpassing van de stochastische “Total Least Squares” en “Maximum Like-lihood” methodes voor modale analyse start van de numerieke optimizatie van delineaire “Least Squares” formulatie op basis van FRF data. Verscheidene parame-terizaties zijn geevalueerd voor enerzijds een verbeterde numerieke conditioneringvoor een hoge modale densiteit en anderzijds een gereduceerde rekentijd voor uit-gebreide datasets. Een formulering vertrekkende van de normaal vergelijkingengebruikmakende van een discreet-tijdsdomein “common denominator” model metcomplexe coefficienten blijkt zeer robuust voor een hoge model orde van 100 modesen meer. Op basis van deze formulatie en de kennis van onzekerheid op de data,volgt eveneens de numerieke optimizatie van de stochastische methodes. In gevaldat slechts zeer korte datasequenties zijn opgemeten, is het echter verkiesbaarde modale parameters te schatten rechstreeks vanaf de ingang/uitgang Fouriersequenties. Hoewel een efficiente formulatie van het “Least Squares” probleemnog steeds mogelijk is, is de veralgemening naar de stochastische “Total LeastSquares” niet meer vanzelfsprekend. Verder, maken de ingang/uitgang methodeseen parametrische compensatie voor “leakage” effecten, vaak aanwezig in de kortedatasequenties, mogelijk.

Op basis van de verbeterde nauwkeurigheid van de geschatte parameters ende bijhorende onzekerheidsintervallen, zijn stochastische mode validatie criteriaand een geautomatiseerde parameter schatting ontwikkeld. Dit resulteert in eenbelangrijke reductie van de gebruikertijd alsmede de mogelijkheid voor niet-expertgebruikers modale analyse toe te passen. Terzelfdertijd, is een methode voorgesteldvoor de opvolging van de modale parameters als functie van de tijd.

In deze thesis worden het nut en potentieel van de voorgestelde systeemiden-tificatie methodes voor modale analyse duidelijk geıllustreerd door de toepassingvan deze algoritmen op talrijke simulatie en praktische datasets.

Nomenclature

List of Operators

i i2 = −1∗ convolution operatorY outline uppercase font denotes a set, e.g R, C

are, respectively, real and complex numbers(·)T matrix transpose(·)−1 matrix inverse(·)∗ complex conjugate(·)H Hermitian transpose

complex conjugate transpose of matrix(·)† Moore-Penrose pseudo-inverse(·)−T transpose of the inverse matrix(·)−H Hermitian transpose of the inverse matrix(·)⊥ orthogonal complement of a subspace or matrixIm (m×m) identity matrix⊗ Kronecker matrix productδij Kronecker deltaB[i] ith entry of vector BA[i,j] i, jth entry of matrix AA[:,j] jth column of matrix AA[i,:] ith row of matrix AX [m](k) mth realization of a stochastic process X(k)vec(A) a column vector formed by stacking the columns

of the matrix A on top of each othertrace(A) =

∑ni=1A[i,i] trace of an (n× n) matrix A

det(A) determinant of matrix Adiag(A1, A2, . . . , AL) block diagonal matrix with blocks Al, l = 1, . . . , Lrank(A) rank of (n×m) matrix A, maximum number of

linear independent rows (columns) of A

‖A‖F =√

trace(AHA) Frobenius norm of an (n×m) matrix A

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λ(A) eigenvalue of a square matrix Aσ(A) singular value of an (n×m) matrix Aκ(A) condition number of an (n×m) matrix A

κ(A) = (maxiσi(A)/miniσi(A))Re(·) real part ofIm(·) imaginary part ofherm(A) = (A+AH)/2 Hermitian symmetric part of an (n×m) matrix A

ARE ARE =

[

Re(A)Im(A)

]

|x| =√

(Re(x))2 + Im(x))2 absolute value of a complex number xθ0 true value of θ

θ estimated value of θEX mathematical expectation of stochastic variable XbX = X − EX bias of the estimate XCov(X,Y ) cross-covariance matrix of X and Y

Cov(X,Y ) = E(X − EX)(Y − EY )Hcovar(x, y) covariance of scalars x and y

covar(x, y) = E(x− Ex)(y − Ey)∗var(x) = E|x− Ex|2 variance of a scalar xCX = Cov(X) = Cov(X,X)CXY = Cov(X,Y )σ2x = var(x)µx = Ex mean value of a scalar x

MSE(θ) mean square error of the estimate X

MSE(θ) = E(θ − θ0)(θ − θ0)HRxx(τ) = Ex(t)xH(t− τ) auto-correlation function of x(t)Rxy(τ) = Ex(t)yH(t− τ) cross-correlation function of x(t) and y(t)GXX(iω) = GX(iω) Fourier transform of Rxx(τ)

(auto-power spectrum of x(t))GXY (iω) Fourier transform of Rxy(τ)

(cross-power spectrum of x(t) and y(t))a.s.lim almost sure limit, limit with probability onep.lim limit in probabilityN(µ, σ) Gaussian distribution with

mean µ and standard deviation σ

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List of Symbols

Ni number of inputsNo number of outputsnθ number of parametersNm number of modest continuous-time variableTs sampling periodtn discrete time variable tn = nTsa[n] nth sample of the discrete-time signal a(tn)Ntot total number of time samplesNs number of time samples within one recordM number of experimentsfs sampling frequencyNf number of DFT frequenciesf frequencyω = 2πf angular frequencyωf angular frequency at frequency fU(eiωTs) Discrete Fourier transform of u(fTs)s = iω Laplace transform variablesf Laplace transform variable at DFT frequency fz = esTs z-domain transform variablezf z-domain transform variable at DFT frequency f

zf = eiωfTs

h(t) impulse response functionδ(t) Dirac deltaH(iω) frequency response functionH(s) transfer function in the Laplace domainH1 noise-on-output estimatorH2 noise-on-input estimatorHv Total Least Squares estimatorHiv Instrumental Variables estimatorHev Errors-in-Variables estimator

or Hiv using periodic excitationHgtls Generalized Total Least Squares estimatorr(t), s(t), f(t), x(t) generator, shaker, input, output time signalsns(t), nf (t), nx(t) disturbing time-domain noise on

the shaker, input and output time signalsR(iω), S(iω), F (iω), X(iω) Fourier transform of r(t), s(t), f(t) and x(t)NS(iω), NF (iω), NX(iω) Fourier transform of ns(t), nf (t), nx(t)Z(ωf ) = [F (ωf )

HX(ωf )H ]H data vector containing the measured input

and output DFT spectra at frequency fM ,C,K mass, damping and stiffness matricesθ column vector of model parameters

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Ω generalized transform variableLaplace domain Ω = s, Z-domain Ω = z−1

Ωf Ω evaluated at DFT frequency fΩf = iωf (Laplace), Ωf = e−iωfTs (Z)

A(Ω, θ) =∑na

j=1 ajΩj denominator polynomial expanded in basis Ω

B(Ω, θ) =∑nb

j=1 bjΩj numerator polynomial expanded in basis Ω

n order of the systemp system polezoi transfer function zero

fd damped natural frequency fd = Im(p)2π

ζ damping ratio ζ = Re(p)|p|

R residue matrix R = ΨLT

Ψ mode shape vectorL participation factor vectorUR Upper Residual matrixLR Lower Residual matrix

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List of Abbreviations

ARMA AutoRegressive Moving AverageARMAX AutoRegressive Moving Average with eXternal inputARX AutoRegressive with eXternal inputBL Band LimitedBTLS Bootstrapped Total Least SquaresCMIF Complex Mode Indicator FunctionCRB Cramer Rao lower BoundDFT Discrete Fourier TransformDOF Degree Of FreedomERA Eigensystem Realization AlgorithmEV Errors-in-VariablesEVD Eigen Value DecompositionFEM Finite Element MethodFFT Fast Fourier TransformFRF Frequency Response FunctionFIR Finite Impulse ResponseFDPI Frequency-domain Direct Parameter IdentificationGSVD Generalized Singular Value DecompositionGTLS Generalized Total Least SquaresIFFT Inverse Fast Fourier TransformI/O Input/OutputIRF Impulse Response FunctionITD Ibrahim Time-DomainIV Instrumental VariablesIWLS Iterative Weighted Least SquaresIQML Iterative Quadratic Maximum LikelihoodLOG LOGarithmicLS Least SquaresLSCE Least Squares Complex ExponentialLSFD Least Squares Frequency-DomainLTI Linear Time InvariantMAC Modal Assurance CriteriumMDOF Multiple Degree of FreedomMIMO Multiple Input Multiple OutputMISO Multiple Input Single OutputML Maximum LikelihoodMOV Mode OVercomplexityMPC Modal Phase CollinearityMPD Mean Phase DeviationMPE Modal Parameter EstimationMQI Mode Quality indexNLS Nonlinear Least SquaresPDF Probability Density Function

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PEM Prediction Error MethodPLSCE Polyreference Least Squares Complex ExponentialPSD Power Spectral DensityPS Power Spectrumrms Root Mean SquareRLDS Related Linear Dynamic SystemSDOF Single Degree of FreedomSISO Single Input Single OutputSLDV Scanning Laser Doppler VibrometerSNR Signal-to-Noise RatioSNS Stochastic Noise SourceSVD Singular Value DecompositionTDPI Time-domain Direct Parameter AlgorithmTLS Total Least SquaresWGTLS Weighted Generalized Total Least SquaresWLS Weighted Least SquaresZOH Zero Order Hold

Contents

Voorwoord i

Abstract iii

Korte Inhoud v

Nomenclature vii

Contents xiii

1 Introduction 1

1.1 Research Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Focus and Outline of the Thesis . . . . . . . . . . . . . . . . . . . . 5

2 FRF Identification in the Presence of Measurement Noise 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Errors-in-Variables Stochastic Framework . . . . . . . . . . . . . . 13

2.3 Hgtls – a Maximum Likelihood Approach . . . . . . . . . . . . . . . 16

2.4 Practical Issues of EMA . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1 Modal Testing . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.2 Effects of Environmental Noise . . . . . . . . . . . . . . . . 20

2.5 FRF identification for EMA . . . . . . . . . . . . . . . . . . . . . . 22

2.5.1 Hv – TLS estimator . . . . . . . . . . . . . . . . . . . . . . 22

2.5.2 H1 and H2 – LS estimators . . . . . . . . . . . . . . . . . . 23

2.5.3 Hiv estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.4 Hev estimator – Hiv Using Periodic Excitation . . . . . . . 28

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2.6 Effects of Non-linear Distortions on FRF Identification . . . . . . . 31

2.7 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.9 Appendix A – The GTLS Estimator per Output . . . . . . . . . . 51

2.10 Appendix B – Unbiased Estimate of GEX . . . . . . . . . . . . . . 51

3 FRF Identification in the Presence of Leakage 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Simulated Vibration Experiment . . . . . . . . . . . . . . . . . . . 55

3.3 Power Spectra Estimators . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.1 Periodogram PS Estimator . . . . . . . . . . . . . . . . . . 57

3.3.2 Correlogram PS Estimator . . . . . . . . . . . . . . . . . . 59

3.4 Exponential Windowing of FRF Estimates . . . . . . . . . . . . . . 64

3.4.1 Hexp FRF Estimator . . . . . . . . . . . . . . . . . . . . . . 64

3.4.2 Nonparametric Least Squares IRF Estimator . . . . . . . . 67

3.5 Iterative Nonparametric FRF Estimator . . . . . . . . . . . . . . . 69

3.6 Cyclic Averaging Approach . . . . . . . . . . . . . . . . . . . . . . 70

3.7 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . . . 72

3.8 Experimental Case Study . . . . . . . . . . . . . . . . . . . . . . . 73

3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Introduction to Modal Parameter Estimation 85

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3 Overview of MPE Algorithms . . . . . . . . . . . . . . . . . . . . . 90

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 Frequency-domain MPE from FRF Data 99

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Parametric model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3 (Weighted) Linear Least Squares . . . . . . . . . . . . . . . . . . . 102

5.3.1 LS Formulation based on Jacobian Matrix . . . . . . . . . . 102

Contents xv

5.3.2 LS Formulation based on Normal Matrix . . . . . . . . . . 107

5.3.3 Optimal LS Implementation for MPE . . . . . . . . . . . . 109

5.3.4 Transformation to Modal Model . . . . . . . . . . . . . . . 124

5.3.5 Constructing Stabilization Charts . . . . . . . . . . . . . . 125

5.3.6 LSCF Estimator . . . . . . . . . . . . . . . . . . . . . . . . 129

5.3.7 LS Residue Estimator . . . . . . . . . . . . . . . . . . . . . 130

5.3.8 Comparison of LSCF with LSCE . . . . . . . . . . . . . . . 131

5.4 Weighted Generalized Total Least Squares . . . . . . . . . . . . . . 133

5.4.1 Cost Function Formulation . . . . . . . . . . . . . . . . . . 133

5.4.2 Generalized Eigenvalue Problem Formulation . . . . . . . . 135

5.5 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.5.1 ML Equations . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.5.2 Parameter Uncertainty Bounds . . . . . . . . . . . . . . . . 140

5.5.3 ML Stabilization Chart . . . . . . . . . . . . . . . . . . . . 141

5.5.4 Logarithmic ML . . . . . . . . . . . . . . . . . . . . . . . . 141

5.6 Choice of Frequency Weighting . . . . . . . . . . . . . . . . . . . . 142

5.7 Validation of Stochastic Estimators for Experimental Data . . . . . 148

5.8 Asymptotical Properties of Stochastic Estimators . . . . . . . . . . 148

5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.10 Appendix A – Jacobian and Normal Matrix based LS Algorithms . 152

5.11 Appendix B – Fast Calculation of Normal Matrix Entries . . . . . 153

5.12 Appendix C – Equivalency of WGTLS Solutions . . . . . . . . . . 154

6 Frequency-domain MPE from Input/Ouput Data 155

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2 Parametric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.3 Errors-in-Variables Noise Model . . . . . . . . . . . . . . . . . . . . 158

6.4 (Weighted) Linear Least Squares . . . . . . . . . . . . . . . . . . . 158

6.5 Weighted Generalized Total Least Squares . . . . . . . . . . . . . . 161

6.5.1 Cost Function Formulation . . . . . . . . . . . . . . . . . . 161

6.5.2 Generalized Eigenvalue Problem Formulation . . . . . . . . 161

xvi Contents

6.5.3 Validation of Linear Approximation . . . . . . . . . . . . . 164

6.6 Validation of (WG)TLS for Experimental Data . . . . . . . . . . . 167

6.7 Maximum Likelihood Estimator . . . . . . . . . . . . . . . . . . . . 171

6.8 MPE in the Presence of Leakage Phenomena . . . . . . . . . . . . 173

6.9 MPE from Auto and Cross Power Spectra . . . . . . . . . . . . . . 175

6.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7 Automated Modal Parameter Estimation and Tracking 181

7.1 Overview of “State-of-the-Art” . . . . . . . . . . . . . . . . . . . . 182

7.2 Autonomous Modal Parameter Identification . . . . . . . . . . . . 183

7.2.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.2.2 Mode Validation Criteria . . . . . . . . . . . . . . . . . . . 184

7.2.3 Physical Mode Selection . . . . . . . . . . . . . . . . . . . . 187

7.3 Case Study I: Airbus A320 Slat Track . . . . . . . . . . . . . . . . 189

7.3.1 Test Structure and Experiments . . . . . . . . . . . . . . . 189

7.3.2 Autonomous Estimation Results for Slat Track . . . . . . . 192


7.4 Case Study II: I40 Bridge . . . . . . . . . . . . . . . . . . . . . . . 199

7.4.1 Test Structure and Experiments . . . . . . . . . . . . . . . 199

7.4.2 Model Order Reduction . . . . . . . . . . . . . . . . . . . . 200

7.4.3 Autonomous Estimation Results for I40 Bridge . . . . . . . 203


7.5 Mode Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.5.1 Tracking Approach . . . . . . . . . . . . . . . . . . . . . . . 206

7.5.2 Tracking Algorithm – Problem of Missing Modes . . . . . . 208

7.5.3 Tracking Results for Slat Track . . . . . . . . . . . . . . . . 209

7.5.4 Damage Assessment during Tracking . . . . . . . . . . . . . 212

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8 Conclusions 217

8.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Contents xvii

8.2 Ideas for Further Research . . . . . . . . . . . . . . . . . . . . . . . 219

Bibliography 221

Chapter 1

Introduction

The first chapter contains the general introduction of and motivation for this thesis.First the research context – experimental modal analysis – and its applications arediscussed in Section 1.1. The focus and main contributions as well as the outlineof the thesis are presented in Section 1.2.

1

2 Chapter 1. Introduction

1.1 Research Context

During the last two decades, there has been a growing interest for the domainof modal analysis. Evolved from a simple technique for trouble shooting, modalanalysis has become an established technique to analyze the dynamical behaviourof complex mechanical structures. Important examples are found in the automo-tive (cars, trucks, motorcycles), railway, maritime, aerospace (aircrafts, satellites,space shuttle), civil (bridges, buildings, offshore platforms) and heavy equipmentindustry.

The vibration of a structure originates from its resonance modes that are in-herent properties of the structure. Small forces exciting one or more of thesemodes can result in an important deformation and possibly damage in the struc-ture. These modes are determined by the material properties (mass, dampingand stiffness) and the boundary conditions of the structure. If either the materialproperties or boundary conditions change, the structural modes change.

The modal model expresses the behaviour of a linear time-invariant mechanicalsystem as a linear combination of these different resonance modes. Each mode isdefined by a damped resonance frequency, modal damping and mode shape, i.e.the so-called modal parameters. Given the physical interpretation of the modes,engineers often prefer this mathematical model.

When the material properties and boundary conditions of a structure are fairlyknown, the finite element method (FEM) can be used to compute the modalparameters. The structure is discretized into a finite number of geometric elementsthat are defined by the known material properties and boundary conditions. Basedon this information, the mass and stiffness matrices can be calculated [93, 27, 152],from which the modal parameters are derived. During the last decade, finiteelement analysis has been integrated in the design process since it can significantlyreduce the number of prototypes and expensive experiments and at the same timereduce the time-to-market. This was mainly possible thanks to the availabilityof extensive computation power. However, for complex structures the materialproperties and boundary conditions are often not well-known, especially in thecase multiple materials with anisotropic properties such as composites. In addition,including general damping in the finite element analysis is still a main topic forresearch.

Another approach to obtain the modal model is the so-called ExperimentalModal Analysis (EMA), which is based on the measured forces and vibrationresponses of the structure when excited in one or more locations. Since this gen-erally results in large amounts of data, there is the need to compress the amountof data by estimating an experimental parametric model of the studied structurethat essentially contains the same information as the original vibration data. Moregeneral, the process of finding a model from the data is called system identification,

1.1. Research Context 3

originating from the domain of electrical engineering, where [85] is a prominentreference. Recent advances in the field are the development of subspace methods[141] and frequency-domainmaximum likelihood methods [109]. The application ofsystem identification to identify the experimental modal model of vibrating struc-tures, has been important research domain in mechanical engineering, especiallyduring the last two decades. Moreover, as a direct result of the emerging informa-tion technology significant advances have been available to both modal test andanalysis equipment, explaining the important gain of interest for EMA. A num-ber of textbooks gives a good overview of the theory and current practice in thedomain of experimental modal analysis [62, 36, 88].

Consider a modal experiment, where the structure is excited at Ni locationswith a predefined signal while measuring the excitation forces f(t) at the inputlocations and the structural responses (displacements, velocities, accelerations)x(t) at No output locations. For a linear time-invariant system the input-outputrelation is then given by

x(t) = h(t) ∗ f(t) (1.1)

In the frequency-domain the convolution is replaced by a matrix multiplication

X(ω) = H(ω)F (ω) (1.2)

where h(t) and H(ω) are respectively the No×Ni impulse response and frequencyresponse matrices, which can be decomposed as [62]

h(t) = ΨeΛtL (1.3)

H(ω) = Ψ(jωI − Λ)−1L (1.4)

The 2Nm columns (Nm number of modes in the model) of the modal matrix Ψcontain the mode shapes and the complex conjugates of the mode shapes, whilethe modal participation matrix L is a measure for the contribution of the differentinput forces to the total vibration. The diagonal matrix Λ contains the complexsystem poles from which damped resonance frequencies fn and damping ratios ζncan be computed using (n = 1, . . . , Nm)

fn =1

2πIm(Λ(n, n)) (1.5)

ζn =−Re(Λ(n, n))| Λ(n, n) | (1.6)

In practice, an experimental modal analysis typically consists of three phases:

1. Performing the experiments: concerns the experimental setup (i.e. actuatorsand sensors, boundary conditions,...) and the acquisition parameters, (i.e.instrumentation, excitation signal, frequency band and resolution, numberof records,...).


2. Processing of the measured data: the measured time signals are first con-verted into Impulse Response (IRF) or Frequency Response Functions (FRF)using signal processing and nonparametric identification techniques. Next,based on this IRF or FRF data, the modal parameters Ψ,Λ and L can beextracted in the time-domain (cf. Eq. 1.3) or the frequency-domain (cf.Eq. 1.4) using a parametric identification approach.

3. Validating the model : the extracted modal model must be assessed for itsphysical representation of the dynamical behaviour of the structure in thestudied frequency band.

For engineers, the modal modal is very often not the final goal, but only ameans to get condensed experimental data used for a wide range of applicationsbased on a modal model:

• Model Updating : since the reliability of initial finite element models is oftennot guaranteed, modal updating (correlation, correction and verification) bymeans of experimental data is necessary. This results in a finite elementmodel that is more reliable for predicting the dynamical behavior of thestructure under various loading and boundary conditions.

• Damage Detection and Structural Health Monitoring : since the dynamicalbehaviour of structures is influenced by damage, one is able to detect the oc-currence and evolution of damage by monitoring the modal parameters [37].Besides the detection of structural damage, modal analysis also enables thelocalization and quantification of the damage since also spatial informationis present in the model (mode shapes). Compared to nondestructive testing(ultrasound techniques, analysis of magnetic fields, radiology, thermal meth-ods) vibration-based analysis allows a more global analysis of the structurewith the potential of being applied in situ. However, varying environmentaland operational conditions affect the modal parameters as well which canseriously complicate in situ structural health monitoring.

• Vibro-Acoustics: the existence of a vibro-acoustical coupling between struc-tural vibrations and radiated noise is an important aspect for the designprocess in terms of comfort. The noise radiated in an aircraft’s fuselagedue to the vibrations induced by the engines is one of the many examples.Based on the modal parameters it is possible to calculate the sound intensityradiated from the vibrating structure [120] without the need for expensiveacoustical holography experiments.

• Forced Response Analysis: is mainly applied to study the behavior of thestructure under operating conditions such as for example, a bridge withroad traffic and wind load, a car on the road or an aircraft during flight.This way designers can check the system under normal and extreme loadingconditions and detect structural weaknesses subject to design modification.

1.2. Focus and Outline of the Thesis 5

• Sensitivity Analysis and Structural Modification Prediction: computing thesensitivity of the modal parameters for a certain change of mass, stiffness ordamping at each response location assists the design engineer by indicatingthe type and location for structural changes that mostly affects the modethat is subject for modification [142]. The actual quantitative change of themodal model induced by the most promising modifications is next estimatedduring the modification prediction.

• Substructuring : given the modal model of different components of a struc-ture, the dynamical behaviour of the complete structure can be computedusing substructuring techniques. This application is quickly gaining greatinterest for studying large complex structures such as cars, aircrafts andsatellites.

Obviously, a successful applicability of the above discussed engineering methodsgreatly depends on the quality of the modal model as well as on computationalefficiency of the identification algorithms used.

1.2 Focus and Outline of the Thesis

Due to the advances in measurement technology and new EMA applications, ex-tensive data sets, obtained by multiple input excitation and measuring at a largenumber of response locations, are common practice today. Given the increasingcomplexity of the structures under test, this data is also often characterized by ahigh frequency resolution, high dynamical range and close-coupled modes. More-over, the measured signals f(t) and x(t) (in Eq. 1.1) are disturbed by errors thatinterfere with the physical content of the system in the signals. These errors origi-nate from different sources in the modal experiment: mechanical setup errors (massloading, interfering boundary conditions, structure-shaker interaction,...), excita-tion and acquisition errors (excitation in node, spatial aliasing, calibration sensors,...), signal processing errors (leakage, temporal aliasing, imperfect compensationfor time windows), non-linear distortions and measurement noise (environmen-tal noise, instrumentation noise, digitalization noise). These errors, can seriouslyhamper the extraction of the modal parameters and introduce both systematic andstochastic errors (i.e. a certain degree of uncertainty) in the final model. Fromthis it must be clear that the estimation of the modal model is not an obviousproblem at all.

As a result, the need for improved system identification methods within thedomain of modal analysis increases under impulse of the broadening field of appli-cations, e.g. damage detection and vibro-acoustics, and the increased complexityof today’s tested structures. Recent advances in the field of frequency-domainsystem identification are the development of the so-called stochastic Total Least


Squares and Maximum Likelihood methods [109, 41]. Since the noise on the mea-sured data is taken into account during the estimation process, these methodsyield very accurate parameter estimates with confidence bounds, even in the caseof a poor quality of the data.

The aim of this thesis is to use the stochastic methods as a starting point todevelop more advanced modal parameter identification techniques. This howeverrequires an extensive revision of both the initial formulation of these methods withrespect to computational efficiency and the modal analysis process with respectto the integration of the noise on the measured data. So, this thesis focusses onthe current shortcomings related to the application of these stochastic methods formodal parameter identification. Three main parts are considered and discussedin more detail in the next three sections, while a chapter-by-chapter outline ispresented in Figure 1.1.

Nonparametric Identification of FRF Data

The first part of this thesis deals with the nonparametric processing of modal data.Most often frequency response functions (FRFs) are estimated by averaging themeasurements using a nonparametric FRF estimator. The accuracy (in terms ofsystematic and stochastic errors) of these algorithms greatly depends the measure-ment noise as well as the excitation signal used (random or burst random noise,multisine, periodic random,...). In the domain of experimental modal analysis, it iscommon practice to consider only the errors on the measured response signals x(t),although this assumption is often not satisfied, resulting in systematic errors onthe FRF estimates. However, the application of the accurate stochastic methodsfor modal parameter estimation is only meaningful when starting from accurateFRF data. Therefore, an approach for the identification of the FRF matrix basedon an errors-in-variables (EV) framework will be presented in Chapter 2. AnEV model considers the measurement noise on both the input and output signals,yielding a better noise model for a practical modal test setup compared to theclassical noise assumptions. Besides obtaining unbiased FRF estimates, it is justas important to know the uncertainty (noise) on the measured data, required bythe stochastic methods. A methodology for deriving the noise covariance matrix ofthe FRF estimates without the need for additional measurements will be presentedas well. This uncertainty information can be used to check the quality of the dataand will be taken into account during the modal parameter estimation process.On the other hand, leakage effects can as well result in important systematic er-rors on the FRF estimates. When applying arbitrary excitation signals, such asrandom noise, effects of leakage occur as a result of the finite measurement time.Although periodic signals can simply eliminate leakage errors, arbitrary signals arestill frequently used for modal testing purposes. Through the use of appropriatesignal windowing techniques, the effects of leakage on the FRF estimation can bereduced as will be discussed in Chapter 3.


Parametric identification of Modal Parameters

Although significant research efforts during the last two decades have resulted inan extensive number of parametric identification algorithms, most of them arecertainly not directly applicable for modal parameter extraction. Chapter 4 in-troduces the parametric identification part of this thesis by giving an overview ofthe most important parametric identification methods with specific attention fortheir potential adequacy for modal parameter estimation. Although estimatorssuch as the LSCE, LSFD, FDPI, ERA, and ITD were specifically developed forEMA [62, 36, 88], these estimators tend to fail in cases where the modal data ischaracterized by low signal-to-noise ratios and acquired from systems with highdynamical range and modal density. In this thesis, the focus will be on the classof stochastic frequency-domain methods that estimate the parameters of rationalfraction transfer function models using a (Total) Least Squares [108, 109] or Max-imum Likelihood approach [41, 109]. These methods were initially developed in aerrors-in-variables framework, under the assumption that the measured data arethe input and output Fourier sequences acquired using periodic excitation. Theresults presented in Chapter 2, already facilitate FRF and noise identification inthe same EV framework. Nevertheless, the initial formulation of these stochasticmethods requires a thoroughly revision with respect to the numerical performancefor analyzing typical modal data sets.

Hence, adapting these algorithms for modal parameter estimation representsthe main subject of the second part of this thesis. Typical for modal analysis is thelarge number of output measurements and a high modal density. As will be shownin Chapter 5, the extensive amount of data requires optimized algorithms thatbalance between accuracy and memory/computation efficiency. A multivariableimplementation for frequency-domain estimators, based on a common denominatortransfer function model, will be given. First, a numerical improved implementationof the frequency-domain linear Least Squares (LS) approach will be presented forextracting the modal parameters from FRF data. Aspects, such as robustnessfor high modal density and numerical performance (computation time, memoryusage) are studied for different possible parameterizations. The final result isa fast LS solver, i.e. the so-called Least Squares Complex Frequency (LSCF)method. This estimator also offers the user, in analogy with the well-known LSCEmethod, a stabilization chart. Furthermore, it is shown in this chapter how thefast implementation can also be extended to the class of Total Least Squares (TLS)estimators and the Maximum Likelihood (ML) method.

For some applications, such as flight flutter testing for instance, only shortdata sequences are measured. In that case, nonparametric FRF identificationbased on averaging techniques becomes less applicable. Sufficient time samplesmust be available to have an acceptable frequency resolution to detect closelyspaced modes. Therefore an alternative approach consists of estimating the pa-rameters based on the input and output (I/O) Fourier sequences for which a fast


implementation of the frequency-domain TLS and ML methods will be discussedin Chapter 6. Moreover, by generalizing the results presented in [111] for modalparameter estimation, the effects of leakage (or transients in general) can be takinginto account by estimating the system and transient parameters simultaneously.This approach can also be applied to estimate the parameters from auto and crosspower spectra, which can be preferable when sufficient data is available and aparametric compensation for possible transient effects is desired.

Automation of Modal Parameter Estimation

Traditional experimental modal identification can still be user-time intensive andas a result costly. Most important reasons for this are related to the constraints inanalyzing large, complex structures, which are discussed in [4, 136]. More specificfor the modal identification step, noisy data, a high modal density and the lackof error bounds on the estimated parameters, can result in a very difficult modelorder and mode selection process. Based on the availability of the advanced sys-tem identification techniques, the third and last part of this thesis, presented inChapter 7, focuses on the development of improved model validation criteria andthe automation of the modal parameter extraction. An automated approach willcertainly result in an important reduction of the user-time as well as the possibil-ity for non-expert users to apply modal analysis techniques. At the same time, itis a basic tool for structural health monitoring applications, which however, alsorequire a method for tracking the modal parameters over subsequent monitoringinstances. Hence, a method for an automated estimation and tracking of the modalparameters was developed as well.

A summary and the main conclusions as well as indications for further researchare given in Chapter 8.


Modal Testing

Modal Applications

FRF & Noise FRF & Leakage

Chapter 2 Chapter 3

I/O-based MPE

Chapter 6

FRF-based MPE

Chapter 5

Overview MPE

Chapter 4

Autonomous MPE

Chapter 7

Figure 1.1: Organization of the text. Three main parts can be distinguished: the non-parametric identification of FRF data, the parametric identification of modal parametersand the automation of parameter estimation process.

Chapter 2

FRF Identification in thePresence of MeasurementNoise

Estimating frequency response functions from the measured data is most oftenthe first step of experimental modal analysis. In this chapter, nonparametric FRFidentification has been studied for a maximum likelihood approach based the errors-in-variables framework, introduced in § 2.2 and § 2.3. Based on this result andconsidering the practical issues related to experimental modal analysis discussed inSections 2.4, the approximations and possible errors made by the FRF estimatorsused in practice are studied in Section 2.5. Next, a methodology for the accurateestimation of the FRF and corresponding noise covariance matrix from the sameset of measurements is derived for the class of arbitrary excitation signals. Itis shown that this so-called “Instrumental Variables” nonparametric estimator hasmaximum likelihood properties, while remaining consistent in the case of nonlineardistortions originating from the excitation devices (cf. § 2.6). In order to illustratethe results of this chapter, the simulation of modal test setup is introduced.

11

12 Chapter 2. FRF Identification in the Presence of Measurement Noise

2.1 Introduction

As for many scientific investigations, traditional modal analysis typically startsfrom frequency response functions (or impulse response functions). For this reason,much attention has been paid to nonparametric estimators in the literature [7, 85,14, 21, 80]. Although most of these estimators were originally developed for single-input, single-output (SISO) systems, some of them where later generalized for themulti-input, multi-output (MIMO) case.

As was already explained in Chapter 1, the nonparametric FRF identificationplays a crucial role in the overall quality of an experimental modal analysis (EMA).Two important sources of errors, encountered during the nonparametric identifi-cation process, are noise on the measured data and spectral leakage due to a finitemeasurement time in the case of random excitation. This chapter concentrateson the effects of measurement noise and possible nonlinear distortions due to theexcitation devices on the estimated frequency response functions. Although, untiltoday, the H1 FRF estimator is still most often used, it is well-known that this esti-mator generally yields erroneous results due to much simplified noise assumptions,i.e. only the output errors are considered.

For this purpose, first the stochastic errors-in-variables (EV) framework is in-troduced for a multivariable (MIMO) system. Based on this it is possible to studyand compare nonparametric FRF estimators as a function of the assumptionsregarding the measurement noise and the type of excitation signal (random orperiodic).

Given the MIMO character of EMA, multivariable expressions are derived forthe FRF matrix estimate and its corresponding covariance matrix. The uncer-tainty on the measured data is an important element for the evaluation of the testsetup and the measured data (i.e check of SNR, sensor working, ...) prior to theparametric estimation step. Moreover, knowledge about the measurement noisewill be taken into account when using the parametric frequency-domain estima-tors developed in a stochastic framework [109]. This can significantly improve thequality of the parameter estimates compared to e.g. a classical Least Squares ap-proach, while confidence intervals for parameter estimates can be computed basedon the knowledge of the noise covariance matrix of the FRF estimates.

Based on the multivariable expressions, the asymptotic (statistical) propertiesof the different FRF estimators are studied. It is shown under which assumptionsfor the noise model and excitation conditions it is possible to derive consistentFRF estimates in a statistically efficient manner (i.e. with maximum likelihoodproperties).

Considering the excitation-side of traditional modal testing, a final part of thischapter is related to the influence of the exciter setup (i.e. the power amplifierand electro-dynamic shaker) on the estimated FRF and covariance matrix.

2.2. Errors-in-Variables Stochastic Framework 13

s0[1](t)

s0[2](t)

s0[Ni](t)

[h0(t)]No×Ni

.

.

....

r(t)[m]

[Ni]-

r(t)[m]

[2]-

r(t)[m]

[1]-

f[m]

0[Ni](t)

f[m]

[Ni](t)

n[m]

f[Ni](t)

n[m]

s[Ni](t)

l+- -?

¾ l+ ¾?

f[m]

0[2](t)

f[m]

[2](t)

n[m]

f[2](t)

n[m]

s[2](t)

l+- -?

¾ l+ ¾?

f[m]

0[1](t)

f[m]

[1](t)

n[m]

f[1](t)

n[m]

s[1](t)

l+- -?

¾ l+ ¾?

x[m]

0[No](t)

x[m]

[No](t)

n[m]

x[No](t)

l+- -?

x[m]

0[2](t)

x[m]

[2](t)

n[m]

x[2](t)

l+- -?

x[m]

0[1](t)

x[m]

[1](t)

n[m]

x[1](t)

l+- -?

Figure 2.1: Time-domain errors-in-variables noise model.

It should be noticed however that for some specific applications, it can bepreferable to estimate the modal parameters directly from the Input/Output Fourierdata. This can be the case when only short data sequences are available and asufficient frequency resolution is required in order to have sufficient nonparametricinformation, especially for low damped modes (cf. Chapter 6). Examples are flightflutter testing and condition monitoring of a rocket during the launch.

2.2 Errors-in-Variables Stochastic Framework

Consider a linear time-invariant multivariable system with Ni inputs and No out-puts. In order to account for the most important sources of errors encountered dur-ing the non-parametric identification process, the modal test setup is schematicallyrepresented by the time-domain errors-in-variables noise model shown in Figure2.1. Following time-domain signals are observed during M measured records


f [m](tn) = f[m]0 (tn) + e

[m]f (tn)

x[m](tn) = x[m]0 (tn) + e[m]

x (tn) (2.1)

with tn = nTs , n = 0, 1, . . . , Ns − 1 and Ts the sampling period. The vectorsf [m](tn) ∈ RNi×1 and x[m](tn) ∈ RNo×1 respectively stand for the measuredforces (inputs) and accelerations, velocities or displacements (outputs) signals ofthe multivariable system h0(t) observed at time t = tn during the mth measured

record. f[m]0 (tn) and x

[m]0 (tn) are the corresponding true but unknown values.

Using the notations as defined in the Nomenclature, f[m][i] (tn) denotes the sample

n measured at input location i during record m. The noise sources e[m]f (tn) and

e[m]x (tn) on the measurement signals comprise of the disturbing time-domain noise

sources n[m]f (tn), n

[m]x (tn), n

[m]s (tn).

Using the Discrete-time Fourier transform (DFT), the time-domain vector se-quences f [m](tn) and x[m](tn) , n = 0, 1, . . . , Ns − 1 are transformed into thefrequency-domain

F [m](ωf ) =

Ns−1∑

n=0

f [m](tn)e(−i2πnf/Ns)

X [m](ωf ) =

Ns−1∑

n=0

x[m](tn)e(−i2πnf/Ns) (2.2)

with angular frequency ωf = 2πf/NsTs and f = 0, 1, . . . , Ns/2− 1.

The relation in the frequency-domain between the noisy input vector F [m](ωf ) ∈CNi×1 and noisy output vector X(ωf ) ∈ CNo×1, and the true ones is given by thefollowing frequency-domain errors-in-variables stochastic model (Figure 2.2)

F [m](ωf ) = F[m]0 (ωf ) + E

[m]F (ωf )

X [m](ωf ) = X[m]0 (ωf ) + E

[m]X (ωf ) (2.3)

The relation between the ”true”input and output Fourier vectors, F[m]0 (ωf ) ∈

CNi×1 and X[m]0 (ωf ) ∈ CNo×1, respectively, is given by

X[m]0 (ωf ) = H0(ωf )F

[m]0 (ωf ) (2.4)

where the complex valued matrix H0(ωf ) ∈ CNo×Ni stands for the FrequencyResponse Function Matrix of the system at spectral line f .

The vectors E[m]F (ωf ) ∈ CNi×1 and E

[m]X (ωf ) ∈ CNo×1 occurring in the

frequency-domain errors-in-variables (EV) model (2.3) are a general represen-tation of all possible disturbing spectral noise sources. Referring to Figure 2.2,

2.2. Errors-in-Variables Stochastic Framework 15

S0[1](ω)

S0[2](ω)

S0[Ni](ω)

[H0(ω)]No×Ni

.

.

....

R[m]

[Ni](ω) -

R[m]

[2](ω) -

R[m]

[1](ω) -

F[m]

0[Ni](ω)

F[m]

[Ni](ω)

N[m]

F [Ni](ω)

N[m]

S[Ni](ω)

l+- -?

¾ l+ ¾?

F[m]

0[2](ω)

F[m]

[2](ω)

N[m]

F [2](ω)

N[m]

S[2](ω)

l+- -?

¾ l+ ¾?

F[m]

0[1](ω)

F[m]

[1](ω)

N[m]

F [1](ω)

N[m]

S[1](ω)

l+- -?

¾ l+ ¾?

X[m]

0[No](ω)

X[m]

[No](ω)

N[m]

X[No](ω)

l+- -?

X[m]

0[2](ω)

X[m]

[2](ω)

N[m]

X[2](ω)

l+- -?

X[m]

0[1](ω)

X[m]

[1](ω)

N[m]

X[1](ω)

l+- -?

Figure 2.2: Frequency-domain errors-in-variables noise model.

N[m]F (ωf ) and N

[m]X (ωf ) represent the measurement channel noise mainly due to

instrumentation and digitizer noise. In the case of modal testing, the structure isoften excited by a shaker. The excitation units (i.e. power amplifier, shaker) arecharacterized by the transfer functions S0[1](ω), . . . , S0[Ni](ω) denoted by the diag-

onal matrix S0(ω) ∈ CNi×Ni . Instrumentation noise due to these excitation units,

adds random errors, N[m]S (ωf ) ∈ CNi×1, resulting in the ”true”Fourier vectors

F[m]0 (ωf ) ∈ CNi×1 with disturbing shaker noise N

[m]S (ωf ).

Hence, the frequency-domain EV stochastic model can also be written underthe form

F [m](ωf ) = S0(ωf )R[m](ωf ) + E

[m]F (ωf )

X [m](ωf ) = H0(ωf )S0(ωf )R[m](ωf ) + E

[m]X (ωf ) (2.5)

with

E[m]F (ωf ) = N

[m]S (ωf ) +N

[m]F (ωf )

E[m]X (ωf ) = H0(ωf )N

[m]S (ωf ) +N

[m]X (ωf )


Assuming that the random errors E[m]Z (ωf ) = [E

[m]F (ωf )

H , E[m]X (ωf )

H ]H arecomplex normally distributed, with

Cov(

E[m]Z (ωf ), E

[n]Z (ωf )

)

= CEZ (ωf )δmn , m,n = 1, . . . ,M (2.6)

where δmn the Kronecker delta and CEZ ∈ C (Ni+No)×(Ni+No) an a priori knownHermitian symmetric noise covariance matrix. Hence, the disturbing noise is as-sumed to be (second-order) stationary and uncorrelated for different measuredrecords (m 6= n). Also notice that CEZ accounts for possible correlations among

the noise sources E[m]F (ωf ) and E

[m]X (ωf ) on the input/output Fourier coefficients,

while EE[m]F (ωf ) = 0 and EE[m]

X (ωf ) = 0.

Furthermore, it is proven by Brillinger [21] that for a wide class of probability

density functions of the time-domain noise sequences e[m]f (nTs) and e

[m]x (nTs),

n = 0, . . . , Ns − 1, the Fourier coefficients of the noise E[m]F (ωf ) and E

[m]X (ωf ),

f = 1, . . . , Nf , are asymptotically complex multivariate normally distributed andindependent, as the number of time samples Ns used to calculate the DiscreteFourier Transform goes to infinity for a fixed number of DFT frequencies. In [124]it is demonstrated that these properties are reached for practical values of Ns (e.g.,Ns = 512). Since the measurements for different spectral lines are asymptoticallyuncorrelated, (ωf ) is omitted in all the following equations of this chapter.

2.3 Hgtls – a Maximum Likelihood Approach

Given the errors-in-variables framework, defined in Section 2.2, accurate nonpara-metric FRF identification can be derived starting from a maximum likelihoodapproach. The assumptions concerning the measurement noise in the EV model,motivate the use of complex Gaussian probability density function to constructa maximum likelihood FRF estimator (MLE). The (asymptotical) independencyover the frequencies allows to write the ML equations for a single frequency with-out the loss of (statistical) efficiency. This results in the (negative) log-likelihoodfunction

`(H, F ) = trace

C−1EZ

[

Z −

INi

H

F

][

Z −

INi

H

F

]H

(2.7)

The matrices H and F = [F [1], . . . , F [M ]] are independent variables, while Z =[Z [1], . . . , Z [M ]] represent the measurements, where Z [m] = [F [m]H , X [m]H ]H . In[50] it has been shown that F can be eliminated from (2.7) using the method ofthe Lagrangian multipliers, resulting in the following cost function

`(H) = trace(

[

BCEZ BH]−1[

BGZBH]

)

(2.8)

2.3. Hgtls – a Maximum Likelihood Approach 17

with GZ = 1MZZH = 1

M

∑Mm=1 Z

[m]Z [m]H and B = [H,−INo ]. The matrix GZ

contains the cross and auto power spectra between the measured input (force) andoutput (response) signals.

The minimization of (2.8) with respect to B is equivalent to the generalizedtotal least squares (GTLS) approach [138] (p.347), which boils down to solving ageneralized eigenvalue problem

GZV = CEZV Λ (2.9)

where Λ represents a diagonal matrix containing the eigenvalues and V the matrixof corresponding eigenvectors. This can be written more explicitly under theformulation

[

GF GFX

GXF GX

]

[

HH

−INo

]

T =

[

CEF CEFEXCEXEF CEX

]

[

HH

−INo

]

TΛ (2.10)

since V = BHT where T represents a transformation matrix with dimensions(No × No). Notice that Eqs. (2.8) and (2.9) are scale invariant. As a result it issufficient to know the covariance matrix CEZ up to a certain scaling factor.

The GTLS solution Bgtls is obtained by choosing the columns of BH to be

linear combinations of the No eigenvectors (contained by V ) of GZ in the metricCEZ that are associated with the No smallest eigenvalues of Λ. Finally, the FRFmatrix estimate Hgtls is given by

Hgtls = −Bgtls(:, Ni + 1 : Ni +No)−1Bgtls(:, 1 : Ni) (2.11)

while `(Hgtls) =∑No

o=1 λNi+o. The generalized singular value decomposition pro-vides a numerical stable way for computing the Hgtls [40]. In the modal analysissociety this estimator is better known as the scaled total least squares estimator[82].

It has been proven that this estimator, which belongs to the class of the max-imum likelihood estimators, is consistent and asymptotically efficient with theCramer-Rao lower bound given by

CCR( ˆvec(H)) =1

MG−TF0⊗ Cε (2.12)

with Cε = BCEZBH (with B = [H0,−INo ]) and vec and ⊗ the Vector and Kro-

necker product operators.

In Appendix A it is proven that under the assumption that either BCEZ BH or

BGZBH in Eq. (2.8) are diagonal matrices, the FRF matrix obtained by solving

a GTLS problems for each output separately is the same as the one derived fromEq. (2.11).


Notice however, that CEZ (i.e. the noise covariance matrix of the input andoutput errors) should be a priori known to get consistent FRF estimates. As formany real-life applications this represents one of the practical constraints for theapplication of the Hgtls for modal analysis.

2.4 Practical Issues of EMA

2.4.1 Modal Testing

Nowadays, both the requirements of new EMA applications (cf. § 1.1) and theavailability of multi-channel data acquisition units (typically 48 up to 128 measure-ment channels) often result in extensive amounts of measurement data from whichthe modal parameters have to be derived. Structures being excited by 4 shak-ers and measured in 500 response locations by means of a multi-patch (tri-axial)accelerometer (MPA) setup are not uncommon today. Also novel optical mea-surement techniques, such as Scanning Laser Doppler Vibrometer (SLDV) andElectronic Speckle Pattern Interferometry (ESPI) can be applied for plate-likestructures.

However, the large number of output measurements introduces two sorts ofconstraints on the nonparametric identification process:

• although information technology has entered a new era, computation timeand memory requirements often remain important issues in practice since of-ten only a limited time is available for processing and analyzing an increasingamount of test data.

• a consistent (i.e. full rank) estimate of the complete (No×No) output covari-ance matrix requires at least No response measurements at each output lo-cation. Obviously, this would drastically increase the time needed for modaltesting, which is often not feasibly in practice. Depending on the type ofexcitation signal and the signal-to-noise ratios (SNR) the number of recordsM measured at each location does not often exceeds 10 – 25 records.

Considering the configuration of the modal test setup, it is possible to takea priori knowledge about the noise model into account, as discussed now for twooften used test setups

• Multi-patch accelerometer (MPA) setup: When the number of avail-able accelerometers or acquisition channels is smaller than the number ofresponse locations, the structure is measured in P different patches, whereeach patch consists of one configuration of the accelerometers. In this case

2.4. Practical Issues of EMA 19

( )i oN N P+

oN~

oN~

iN

iN

oN~

oN~

iN

iN( )i oN N P+

Figure 2.3: Noise covariance matrix for a multi-patch accelerometer setup.

it is clear that only the noise on the No responses belonging to a same patchcan be correlated to a certain extend as shown in Figure 2.3. At the sametime this also implies that a different set of input measurements correspondsto each patch of responses. As a result, in the nonparametric case, it is pos-sible to process the measured data for each patch separately, which resultsin significant smaller matrix dimensions (No is typically 10 – 50). However,even for this reduced problem, a full rank (No × No) output covariance ma-trix still requires at least No measurements at each response location. Inpractice, the time available for testing is often too limited to measure a largenumber of records M and as a result only the output variances (diagonalelements) are considered. This leads to a further reduction of the computa-tional load since then each output can be processed separately. On the otherhand, since the number of input measurements is small, typically not morethan 5, the full input covariance matrix can be determined. For the FRFmatrix, this boils down to estimating the FRFs between the inputs and eachoutput, which corresponds to each row of this matrix. The noise covariancematrix CEZ is then constructed by estimating the input covariance matrixfor each patch, and for each output its variance as well as the covariancesbetween the particular output and the corresponding set of inputs. This


results in a much faster approach than estimating the full covariance matrix.In general, the price paid by omitting possible correlations between the out-puts most often does not countervail against the gain in computation time,especially when the number of measurements is large. A possible drawbackof a multi-patch setup relies in the mass-loading influence of the sensors onthe structure. Because the accelerometers are placed in different locationsfor each patch the dynamic characteristics of the measured structure are alsoslightly differently from patch to patch. The data of the different patches aretherefore inconsistent to each other [136, 24]. An optimal sensor positioning(i.e. in practice usually a random spread) can significantly minimize the in-fluence of mass-loading on the structure, which, in general, is also beneficialfor minimizing the level of correlation between the output measurements.This avoids that a complete patch of sensors would be located near DOFs ofmaximum amplitude for some modes resulting in important correlations anddata inconsistencies. Mass-loading inconsistencies can be avoided by usingfor example a laser measurement setup. However, these systems are expen-sive and more difficult to use for measuring three-dimensional structures.

• Scanning laser doppler vibrometer (SLDV) setup: can be consideredas a special case of the MPA setup where every patch consists of 1 responsesensor. Each response DOF and its corresponding set of input measurementsis measured separately, which is equivalent to measuring No times a MISOsystem. Acquiring each output separately, implies that no correlation existsbetween any of the measured responses (No = 1) and consequently the EVnoise model can be completely satisfied by this type of modal test setup.In [143] adapted system identification techniques were proposed to processSLDV or ESPI data sets, requiring data reduction prior to the parametricestimation of the modal parameters.

2.4.2 Effects of Environmental Noise

Depending on the application and test environment, another source for stochasticerrors on the measured data originates from a stochastic and unmeasured exci-tation of the system by the environment, called process or environmental noise.Important examples in the case of modal testing are the wind excitation on a bridgeor the influence of the road traffic nearby a lab. Another example is found in flightflutter testing, where an unmeasured part of the total excitation of the wings ofan airplane stems from the wind and turbulence effects. Since this excitation isnot measured it is also observed as stochastic disturbing noise.

Although not explicitly shown in Figure 2.2, the general formulation of theerrors-in-variables model Eq. (2.3) also includes process noise when present in themeasurement setup. The general EV setup in the case of process noise is presented

2.4. Practical Issues of EMA 21

S0(ω) H0(ω)

G0(ω)

R[m](ω)

-

F[m]0 (ω)

F [m](ω)

N[m]F (ω)

N[m]S (ω)

-l+- - l+ -?

6

®©

l+-

X[m]0 (ω)

X[m](ω)

N[m]X (ω)

N[m]P (ω)

l+-

¾

l+-

?

Figure 2.4: Frequency-domain errors-in-variables noise model for a system operating ina closed loop.

in Figure 2.4. Depending on the type of excitation device, the effect of this processnoise differs:

• When an impact hammer is used, there is no fixed interaction between thestructure and the excitation device. In this case, the effect of process noisewill only result in some mutually correlated noise on the output measure-ments since there is no feedback to the input side (open loop case).

• In the case of shaker excitation, the EV setup operates in a closed loop. Thestructure-shaker interactions will result in a feedback of the process noiseto the input side through the system G0 characterized by the dynamicalstiffness of the excitation devices. As a result, mutually correlated noiseis, in addition to the output measurements, also introduced on each of themeasured input signals.

For the case that process noise is present in a shaker setup, the multivari-able frequency-domain EV stochastic model (2.5) is then given, for the angularfrequency ωf , as

F [m] = [INi −G0H0]−1S0R

[m] + E[m]F

X [m] = H0[INi −G0H0]−1S0R

[m] + E[m]X (2.13)


with

E[m]F = N

[m]S +N

[m]F − [INi −G0H0]

−1G0N[m]P

E[m]X = H0N

[m]S +N

[m]X −H0[INi −G0H0]

−1G0N[m]P

where N[m]P (ωf ) ∈ CNo×1 represents the process noise vector and G0 ∈ CNi×No

the dynamical stiffness matrix of the excitation devices.

In § 2.7, the effects of process noise on the correlations between the FRF esti-mates, will be illustrated by means of a simulated multivariable modal experiment.In practice, both input channel noise and process noise are most often not con-sidered, which however can result in important errors depending on the type ofapplication.

2.5 FRF identification for EMA

In this section, different nonparametric FRF estimators will be discussed for themultivariable case and expressions for the noise covariance information are derived.In addition, the estimators are also assessed for the specific case of modal analysiswhich often requires a MISO approach, for which the effect on the asymptoticalproperties is studied as well.

2.5.1 Hv – TLS estimator

Assuming the input and output noise sources to be uncorrelated and of equalamplitude results in the so-called Hv FRF estimator [80]. This estimator does notrequire prior knowledge of the noise covariance matrix CEZ and yields the totalleast squares (TLS) estimate for the FRF matrix by solving following eigenvalueproblem (cf. Eq. 2.10)

[

GF GFX

GXF GX

]

[

HH

−INo

]

T =

[

INi 00 INo

]

[

HH

−INo

]

TΛ (2.14)

In general however, this estimator is not consistent since the specific noise assump-tions are not satisfied. Indeed, the noise covariance matrix can only be scaled to aunity matrix, when all the noise sources are uncorrelated and of equal amplitude.In practice, however, the signal-to-noise ratios for the different signals generallydiffer depending on the input and response locations, while the actual noise sourcesthemselves can be of equal amplitude. As a result, this would require the properscaling of each signal in order to obtain a unity noise covariance matrix, whichobviously is not feasible in practice since the scaling factors are not a priori known.

2.5. FRF identification for EMA 23

Referring to Appendix A, it follows that the FRF matrix yielded by processingeach output separately is equal to the one obtained from solving Eq. (2.14) underthe assumption that BCEZ B

H = HHH + INo is diagonal, which is satisfied inthe case that HHH is diagonal. However, as there is no reason for HHH to bediagonal, the MIMO and MISO Hv estimator generally differ.

2.5.2 H1 and H2 – LS estimators

H1 estimator

In the domain of modal analysis, the most commonly-used FRF estimator is theso-called H1 estimator [62, 88], which considers only the errors EX on the outputmeasurements. The inputs are assumed to be error-free. The TLS problem (2.14)now reduces to a classical least squares (LS) problem

[

GF GFX

GXF GX

]

[

HH

−INo

]

T =

[

0 00 INo

]

[

HH

−INo

]

TΛ (2.15)

for which the estimate of the FRF matrix is derived from the first Ni rows givenby

H1 = GXF G−1F (2.16)

Under the specific noise assumptions, the H1 estimator is consistent without theneed for the a priori knowledge of the noise covariance matrix CEZ . Since inthis case the matrix BCEZ B

H = INo is diagonal by construction, the MIMO LSproblem (2.15) and solving the MISO problem for each output separately alwaysyield the same FRF matrix.

Covariance matrix of H1 estimates

An estimate of the output noise covariance matrix CEX is derived from the lastNo rows of (2.15)

TΛT−1 = GEX = GX − GXF G−1F GFX (2.17)

An expression for the covariance matrix of the H1 FRF estimates can be obtainedby means of a sensitivity analysis (with δFm = 0 since no input noise)

δH1 =

(

1

M

M∑

m=1

δX [m]F [m]H

)

G−1F (2.18)

Using the Vector and the Kronecker operator [20], equation (2.18) can be rewrittenas

vec(δH1) =1

M

M∑

m=1

Q[m]δX [m] (2.19)


withQ[m] =

(

G−TF F [m]∗ ⊗ INo

)

(2.20)

Assuming stationarity and no correlation over the different measurements, thecovariance matrix becomes

CH1= Cov(vecδH1) =

1

M2

M∑

m=1

Q[m]E

δX [m]δX [m]H

Q[m]H (2.21)

and since for a Hermitian matrix the Transpose and Complex Conjugate operatorsare equivalent, this can be rewritten as

CH1=

1

M2

M∑

m=1

(

G−TF F [m]∗F [m]T G−T

F

)

⊗ CX

=1

MG−TF ⊗ CX (2.22)

with an estimate of CX given by expression (2.17), i.e

GEX = GX − GXF G−1F GFX (2.23)

ForM →∞ Eq. (2.22) becomes the Cramer Rao lower bound (2.12), which provesthat the H1 estimator is asymptotically efficient when there is no input noise.Consequently, this estimator, under this specific noise assumption, also belongs tothe class of maximum likelihood estimators.

Remark : As proven in appendix B, an unbiased estimate of GunbiasedEX

for a finitenumber of experiments M is given by

GunbiasedEX =

M

(M −Ni)GEX (2.24)

In practice however, the input measurements are never free of noise and to copewith these errors, the EV model presented in Section 2.2 has to be considered. Themodel equations are then given by Eq. (2.5) and the probability limit (plim) forthe number of measurements going tot infinity (M → ∞) results in biased H1

estimates

plimM→∞

H1 = H0

[

1 +GNF (S0GRSH0 +GNS )

−1]−1

= H0

[

1 +GNFG−1F0

]−1

(2.25)where the bias term is related to the signal-to-noise ratios of the input signals.

H2 estimator

As for the H1 estimator, similar equations can be derived for the H2 estimator,which assumes that errors are only present on the input measurements. The


outputs are now assumed to be error-free. The TLS problem (2.14) again reducesto a classical least squares (LS) problem

[

GF GFX

GXF GX

]

[

HH

−INo

]

T =

[

INi 00 0

]

[

HH

−INo

]

TΛ (2.26)

for which the H2 estimate of the FRF matrix is derived from the last No rowsgiven by

H2 = GXG−1FX (2.27)

Again in practice, however, this generally results in biased FRF estimates

plimM→∞

H2 = H0

[

1 +GNXG−1X0

]

(2.28)

where the bias term is now related to the signal-to-noise ratios of the outputsignals. Furthermore, the H2 technique is constrained in practice to the casewhere No = Ni in order to compute G−1

FX . When analyzing the FRF data peroutput, the H2 can only be applied in the case of a single excitation source.

Since, in many practical cases, especially when accuracy is important, the noisesources on the input and output signals can not be neglected, both the H1 and H2

techniques generally yield biased FRF estimates, which will also be illustrated in§ 2.7.

2.5.3 Hiv estimator

Unlike the H1 or Hv estimators, a nonparametric approach based on the instrumen-tal variables (IV) principle has been proven to be unbiased for SISO systems andis referred to in the literature as Hs or Hc or 3-channel FRF estimator [95, 26]. Ingeneral, the instruments can be constructed from the measured force signal basedupon the technique of delayed observations. In practice, however, it is preferableto use the exactly known generator (electrical) signals that are sent through thepower amplifier-shaker units. In that case, the assumption that the noise on thereference signals has to be mutually uncorrelated with the noise on input and out-put signals is obviously satisfied. Considering this approach in the framework ofthe MIMO errors-in-variables model (cf. § 2.2) this estimator is defined as

HXFiv = GXRG

−1FR = GXRG

−1R GRG

−1FR = HXR

1 HFR−1

1 (2.29)

with GXR ∈ CNo×Ni , GFR ∈ CNi×Ni and GRR ∈ CNi×Ni .

Notice that in practice, the HXR1 matrix estimate can be computed for each

output or a patch of outputs separately (cf. § 2.5.2), while the HFR1 matrix estimate

is computed for all corresponding inputs.


Consistency of Hiv estimates

It can be proven under mild conditions that this Hiv FRF estimator yields consis-tent FRF estimates without the need for an a priori known noise covariance matrixCEZ . To prove asymptotic consistency, consider the almost sure limit (a.s.lim) ofEq (2.29)

a.s.limM→∞

HXFiv = a.s.lim

M→∞

(

GXRG−1FR

)

(2.30)

with HXFiv (M) the Hiv FRF estimate for M measured records (the number of

averages). Since a continuous function and a.s.lim may be interchanged this canbe rewritten as

a.s.limM→∞

HXFiv (M) (2.31)

=

a.s.limM→∞

GXR

a.s.limM→∞

GFR

−1

=

a.s.limM→∞

1

M

M∑

m=1

X [m]R[m]H

a.s.limM→∞

1

M

M∑

m=1

F [m]R[m]H

−1

Under the Assumptions that the random variables X [m] , F [m] independent andidentically distributed [87] and uniformly bounded [16], the a.s.lim and the ex-pected value operator E may be interchanged. Applying the expected value op-erator to equation (2.31) after substituting (2.5), assuming the noise sources NF ,NX and NS to be uncorrelated with the reference signal R up to the second ordermoments with these moments being bounded, yields

limM→∞

EHXFiv (2.32)

= H0S0E

R[m]R[m]H

(

S0E

R[m]R[m]H

)−1

For the number of averages going to infinity this results in

limM→∞

EHXFiv = H0S0GRG

−1R S−1

0 = H0 (2.33)

which proves that the Hiv estimator is consistent.


Covariance matrix of Hiv estimates

An expression for the covariance matrix C(HXFiv ) of this consistent HXF

iv estimatecan be found again by starting from a sensitivity analysis

δ(HXFiv ) = δ(HXR

1 )HFR−1

1 + HXR1 δ(HFR−1

1 ) (2.34)

with δ(HFR−1

1 ) = −HFR−1

1 δ(HFR1 )HFR−1

1 . Using the Vector and Kronecker operatorthis is written as

vec(δHXFiv ) =

(

HFR−T

1 ⊗ INo)

vec(δHXR1 )−

(

HFR−T

1 ⊗ HXR1 HFR−1

1

)

vec(δHFR1 )

(2.35)or

vec(δHXFiv ) =

[

M K]

[

vec(δHFR1 )

vec(δHXR1 )

]

(2.36)

withM = −

(

HFR−T

1 ⊗ HXR1 HFR−1

1

)

, K =(

HFR−T

1 ⊗ INo)

(2.37)

Assuming all signals to be stationary and the noise to be uncorrelated for differentmeasurements, the covariance matrix is given by

Cov(

HXFiv

)

= E

vec(δHXFiv )vec(δHXF

iv )H

(2.38)

= MCov(HFR1 )MH +KCov(HXR

1 )KH

−2herm(

MCov(HFR1 , HXR

1 )KH)

Given that

HZR1 =

[

HFR1

HXR1

]

and Z =

F

X

the covariance matrix of vec(δHFR1 ) and vec(δHXR

1 ) can be determined from

Cov(

HZR1

)

=

[

Cov(HFR1 ) Cov(HFR

1 , HXR1 )

Cov(HXR1 , HFR

1 ) Cov(HXR1 )

]

=1

MG−TR ⊗ CEZ (2.39)

where an estimate of the covariance matrix CEZ of the signals X and F is easilydetermined using again Eq. (2.17)

GEZ = GZ − GZRG−1R GRZ (2.40)

This equation illustrates that in practice the data covariance matrix is determinedby calculating the Auto and Cross Power Spectra of the input and output signalsobtained from multiple measurements under arbitrary excitation.


Remark : In modal analysis, the so-called coherence function is better known thanthe variance as a measure for the quality of the FRF data, although both functionsare related as will now be shown. Reformulating Eq. (2.40) as

GEZ = GZ

[

INi+No − G−1Z GZRG

−1R GRZ

]

(2.41)

the diagonal elements of the matrix G−1Z GZRG

−1R GRZ are the multiple coherence

functions for respectively the coherence between each force signal and all the gen-erator signals

mγ2i =1

GFi

GFR[i,:]G−1R GRF[:,i]

(2.42)

and the coherence between each response signal and all the generator signals

mγ2o =1

GXo

GXR[o,:]G−1R GRX[:,o]

(2.43)

The multiple coherence function is a correlation coefficient bounded between 0 and1. A value of unity indicates that the output signal is perfectly predicted by allthe reference signals. A coherence less than one can be due to one or more of thefollowing reasons:

• uncorrelated noise on the measured signals, e.g. channel noise

• non-linear distortions introduced by the system under test

• leakage errors in the signal processing (cf. Ch. 3)

2.5.4 Hev estimator – Hiv Using Periodic Excitation

The use of random excitation signals is still most common in the domain of exper-imental modal analysis since it has the advantage of averaging possible non-linearbehavior of the structure. Although this is also an advantage of periodic randomsignals, such as for example a random multisine, these signals are not generallyavailable in the signal generator of commercial measurement equipment. However,random excitation signals also have some drawbacks, such as spectral leakage er-rors and lower signal-to-noise ratios compared to periodic excitation signals.

For the case of deterministic excitation, the Instrumental Variables approachboils down to a special case, i.e. the Hev estimator. Under the condition thatrepeated observations of the same deterministic excitation signals are available,it is possible to obtain consistent estimates of the FRFs without requiring any apriori noise information as well as the need for additional instrumental variables.


Equation (2.29) gives the expression for the Hiv estimator, which can also bewritten as

HXFiv =

(

1

M

M∑

m=1

X [m]R[m]

)(

1

M

M∑

m=1

F [m]R[m]H

)−1

(2.44)

For the case that a synchronized multi-excitation measurement setup is used withNi linear independent stimuli, R0 = [R0(1), . . . , R0(Ni)] ∈ CNi×Ni , the result-ing input F ∈ CNi×Ni and output X ∈ CNo×Ni Fourier data are measuredM = M

Nitimes using the same excitation sequence R0(i). In practice, these signal

assumptions can easily be realized by applying Ni periodic broadband excitationsequences, e.g. a multisine, by measuring M periods of the signals.

Consequently, Eq. (2.44) changes to

HXFiv =

1

M

M∑

m=1

X [m]

RH0 RH−1

0

1

M

M∑

m=1

F [m]

−1

(2.45)

resulting in the so-called nonparametric errors-in-variables estimator [42]

Hev =

1

M

M∑

m=1

X [m]

1

M

M∑

m=1

F [m]

−1

(2.46)

It is clear from this equation that this estimator can also be processed for differentoutput patches or each output separately.

Like the Hgtls, this estimator belongs to the class of the maximum likelihoodestimators [75] (i.e. consistent and asymptotically efficient). Notice, however,that the covariance matrix of the errors on the measured signals is not a priorirequired to obtain consistent FRF estimates (cf. Eq. 2.46). Even in the case thatthe errors on the measured signals are not complex normally distributed, the Hev

estimator is still proven to be consistent (but not efficient anymore) [42]. The useof this estimator avoids the need for the reference signals R[m] (i.e. instrumentalvariables) but requires the use of periodic excitation signals.

Defining X = 1M

∑Mk=1 X [k] and F = 1

M

∑Ml=1 F [l], equation (2.46) can also be

written asHev = X F−1 (2.47)

Again, starting from a sensitivity analysis, an expression for the covariance matrixof the Hev FRF estimates can be found

δ(Hev) = δ(X )F−1 + X δ(F−1) (2.48)

with δ(F−1) = F−1δ(F)F−1. Using the Vector and Kronecker operators, this canalso be written as

vec(δHev) =(

F−T ⊗ INo)

vec(δX )−(

F−T ⊗ X F−1)

vec(δF) (2.49)


the covariance matrix of the Hev FRF matrix is then given by

Cov(Hev) = E

vec(δHev)vec(δHev)H

(2.50)

=(

F−T ⊗ X F−1)

E

vec(δF)vec(δF)H(

F−∗ ⊗ F−HXH)

+(

F−T ⊗ INo)

E

vec(δX )vec(δX )H(

F−∗ ⊗ INo)

−2herm(

F−T ⊗ X F−1)

E

vec(δF)vec(δX )H(

F−∗ ⊗ INo)

where for example

E

vec(δX )vec(δX )H

= E

1

M

M∑

k=1

vec(δX [k])1

M

M∑

l=1

vec(δX [l])H

(2.51)

under the assumptions that over the different measurements all signals and noiseare stationary and the noise is not correlated, reduces to

E

vec(δX )vec(δX )H

=1

M(INi ⊗ CX ) (2.52)

Consequently, the expression for the covariance matrix becomes

Cov(Hev) =1

M

(

F−T F−∗)

⊗ CX

+1

M

(

F−T F−∗)

⊗(

X F−1CF F−HXH)

− 1

M

(

F−T F−∗)

⊗(

X F−1Cov(F ,X ))

− 1

M

(

F−T F−∗)

⊗(

Cov(X ,F)F−HXH)

which, given equation (2.47), finally results in

Cov(Hev) =1

M

1

M

M∑

k=1

F [k]

∗

1

M

M∑

l=1

F [l]

T

−1

⊗ Cε (2.53)

with

Cε = BCEZ BH , B =

[

Hev,−INo]

The probability limit (p.lim) for M →∞ results in the Cramer Rao lower bound(cf. Eq. 2.12), which proves that the Hev is asymptotically efficient. In practice,the data covariance matrix CEZ is now determined from repeated measurementsunder synchronous periodic excitation. Referring to Eq. (2.40), the calculation of

2.6. Effects of Non-linear Distortions on FRF Identification 31

the Power Spectra now reduces to the calculation of the sample covariance matrixof the synchronized deterministic measurements.

In the case of asynchronous periodic measurements, it is possible to use, be-sides the Hiv estimator, FRF estimators based on non-linear averaging techniques.FRF estimators such as the Hari, Hhar and the Hlog are also able to derive the FRFmatrix with its covariance matrix under the EV noise model assumptions [48, 43].However, nowadays, modal testing equipment has built-in generators facilitatingthe synchronizing of the measurements automatically. Only in the case that anexternal generator, which cannot be triggered with the data-acquisition systemis used, the input and output signals can not be synchronized. Recently, an au-tomated spectral analysis of periodic signals was proposed in [127], for which nosynchronization between the generator and the data acquisition is needed. Thisapproach only requires that more than 2 periods of the periodic signal are presentin the acquired data .

2.6 Effects of Non-linear Distortions on FRF Iden-tification

In order to examine the effects of non-linear distortions, introduced by the exci-tation devices (i.e. power-amplifiers and shakers), these non-linear contributionshave to be included in the errors-in-variables model. Although different modelassumptions are made depending on the type of excitation signal, the derivedconclusions are valid for the following types of excitation signals

• Class of stochastic signals: i.e. random and burst random

• Class of deterministic and periodic signals: i.e. sine, stepped sine, periodicchirp and deterministic multisine, periodic random and random multisine.

Model for Nonlinear Distortions

In [13] it shown that, for the class of stochastic signals, the effects of non-lineardistortions on the FRF identification can be described by a so-called Optimumlinear system (OLS) and Revised nonlinear system (RNS). A similar approachwas presented in [121] for the case of a pseudo random excitation (i.e. multisineswith deterministic amplitudes and random phases, also called random multisine),where the model is called the RLDS (Related Linear Dynamic System) and SNS(Stochastic Noise Source) model. The latter terminology will now be used for theremaining part of this section.

Using the same approach as Bendat, the non-linear distortions, introduced by


the excitation devices, can be included in the EV model (2.5) (cf. Figure 2.2). Thetransfer function matrix SRLDS

0 can then be considered as the RLDS containingthe linear shaker transfer function S0 and the transfer function of the coherent(linear) part of the nonlinear distortions. The noise sources NS[i] , i = 1, . . . , Ni

are considered as the SNS containing the non-coherent (and thus stochastic) partof the nonlinear distortions introduced by the shaker-amplifier system.

Hence the following relation holds

F[m]0 = R[m] +N

[m]SNS (2.54)

where the related linear output R[m] = SRLDS0 R[m] and stochastic noise source

N[m]SNS are uncorrelated, i.e. E

R[m]N[m]HSNS

= 0 as well E

R[m]N[m]HSNS

= 0. Basedon this model, the effect of nonlinear distortions, introduced by the excitationdevices, on the asymptotical properties of the Hiv estimator can now be studied.

Asymptotic Properties

Referring to § 2.5.3, it is proven that the consistency is not influenced by the trans-fer function matrix S0 and thus non-linear distortions do not affect the consistencyof the Hiv estimator.

Considering the EV model (2.5), assuming that the noise sources NF , NX areuncorrelated with NS , the data covariance matrix is also given by

CEZ = E

EZEHZ

=

[

GNS + GNF GNSHH0 + GNFNX

H0GNS + GNXNF H0GNSHH0 + GNX

]

(2.55)

Substitution of (2.55) in

Cov(HXFiv ) =

1

MHFR−T

1 G−TR HFR−∗

1 ⊗(

CX + HXR1 HFR−1

1 CF HFR−H

1 HXRH

1

+HXR1 HFR−1

1 Cov(F,X) + Cov(X,F )HFR−H

1 HXRH

1

)

(2.56)

yields

Cov(HXFiv ) (2.57)

=

(

1

MHFR−T

1 G−TR HFR−∗

1

)

⊗(

(

HXR1 HFR−1

1 (GNS + GNF )HFR−H

1 HXRH

1

)

(

H0GNSHH0 + GNX

)

− 2herm(

HXR1 HFR−1

1 (GNSHH0 + GNFNX )

)

)

2.7. Validation 33

Given that GNS is an Hermitian matrix and considering the consistency propertyfinally results in

Cov(HXFiv ) (2.58)

=1

M

(

HFRT

1 GTRH

FR∗

1

)−1

⊗(

GNX +H0GNFHH0 − 2herm(H0GNFNX )

)

However, when nonlinear distortions are introduced by the excitation devicesand arbitrary excitation signals are used, the covariance matrix (2.58) will not beequal to the Cramer Rao lower bound, meaning that some efficiency will be lost.This can be explained by considering the model proposed by Bendat and the EVmodel (2.5) from which it follows that

F [m] = F[m]0 +N

[m]F = R[m] +N

[m]S +N

[m]F (2.59)

Consequently, assuming the related linear output signal R[m] and the stochasticnoise source signal NSNS to be complex normally distributed (which follows fromtheorem 4.4.1 in [21]). Then, based on Eq. (2.59), the Cramer Rao lower bound(2.12) is given by

CCR =1

M

(

GR + GNSNS

)−1

⊗ Cε (2.60)

Hence,

GR + GNSNS≥ HFRT

1 GTRH

FR∗

1 = GR (2.61)

the Cramer Rao lower bound will be lower than the covariance matrix (2.58)and some efficiency will be lost when using the IV approach in the presence ofnonlinear distortions introduced by the excitation devices. This is explained by

the fact that the stochastic noise source N[m]SNS is not coherent with the R and

thus, when applying the IV estimator, the contributions of the SNS are going tozero for the number of measurements going to infinity (i.e. are averaged out whencomputing HFR

1 ). It should be noticed that, in the case of periodic excitation,the Hiv is equivalent to the Hev estimator, and no loss of asymptotical efficiencyappears.

2.7 Validation

In order to validate the results of this chapters the 7 DOF system, shown in Fig-ure 2.5, was used in order to simulate a modal testing experiment. The simulateddata is used to compare the different nonparametric FRF estimators. Monte carlosimulations are used to study the validity of the expressions for the FRF covari-ance matrices. The influence of different possible noise sources on this matrix isstudied as well.


exciter systemme = 1.5kg ms = 10kg

ce = 75.62Ns/m cs = 6Ns/mke = 3000N/m ks = 150000N/m

Table 2.1: Mass, damping and stiffness characteristics of 7 DOF system and exciter.

The system is excited by 2 electrodynamic shakers with their body grounded.The exciter coil has mass me and is supported on a flexure of stiffness ke anddamping ce. The forces acting on the exciter coil, due to the electromagneticeffect are fe1 and fe7, while the forces actually applied to the system, i.e f1 or f7,are measured on the mass fixed to the coil (rigid stinger). The structure consists of7 subsystems with an equal effective mass ms, damping cs and stiffness ks exceptfor the fourth (middle) subsystem, that has an effective mass that equals 3ms.The characteristics of this system and the electrodynamic shakers are given inTable 2.1.

The electrodynamic shakers are considered in order to have a good approx-imation of a typical experimental modal test setup. After all, the excitors willaffect the dynamical behaviour of the structure in just the same way as a force orresponse transducer. Obviously the larger the excitation device, the greater theinfluence on the system under test. In practice, by mounting the force transduceras close as possible to the system the primary effects of the shakers can be avoidedin the measured forces. However, responses of the shaker on its suspension willstill be observed in the Auto Power Spectra of the force measurements as so-calledforce drop-outs, which can be reduced by the use of a constant current amplifierensuring the application of a constant force.

Given the effective mass, damping and stiffness matrices of the system, theexact frequency response functions can be computed using Newton’s equation ofmotion:

Mx(t) + Cx(t) +Kx(t) = f(t) (2.62)

me ms

ks

cs

ms . . . ms me

ke

ce

ks

cs

ks

cs

ke

ce

fe1

f1

x1 x7

f7fe7

fp

Figure 2.5: Simulated 7 DOF system excited by 2 electrodynamic shakers.

2.7. Validation 35

case σF σX σPa 1E-4 (∼40dB SNR) 1E-12 (∼40dB SNR) 1E-2b 1E-2 (∼20dB SNR) 1E-12 (∼40dB SNR) 1E-2c 1E-4 (∼40dB SNR) 1E-12 (∼40dB SNR) 1E-1d 1E-2 (∼20dB SNR) 1E-10 (∼20dB SNR) 1E-1

Table 2.2: Different noise cases considered for comparison of nonparametric FRF esti-mators.

with M , C and K respectively the mass, damping and stiffness matrices, and f(t)and x(t) the applied force and structural response vectors. Using the Laplacetransform, the frequency-domain equivalent is given as

G(s)X(s) = F (s) (2.63)

with G(s) = Ms2 + Cs + K the dynamic stiffness, where the transfer functionmatrix H(s), is found as

H0(s) =(

G(s))−1

(2.64)

The applied forces fe are assumed noise-free and equal to the electrical signals(sent to the power amplifier) up to a certain force/current factor. Two forces f1 andf7 and seven responses (displacements) x1, . . . , x7 were considered. Both channeland process noise are are taken into account during the simulations. The shakernoise sources NS(ωf ) (cf. Eq. 2.5) are not considered during the simulations.Based on the exactly known FRFs, all signals could be simulated directly in thefrequency-domain, with the advantage that no leakage phenomena are presentin the data and as a result, the application of time-windows, such as Hanning,was not needed. Throughout this chapter it is implicitly assumed that leakage isnegligible, while Chapter 3 will address the problem of leakage.

Example 1: A first simulation experiment was done to study the performance offollowing FRF estimators: H1, Hv and Hiv using a random noise excitation and Hev

using a multisine excitation. For a random excitation (with normal distributionN(0, 1)) and no measurement noise added on the data, Figure 2.6 shows the AutoPower Spectrum of the force f1 and response x1 obtained during 50 simulatedexperiments (M = 50) as well as the exact FRF between these DOFs with 6modes in the frequency band 5–40Hz.

The effect of measurement noise on the nonparametric FRF estimators is stud-ied by considering both channel and process noise. Gaussian distributed randomnoise is added on the force (N(0, σF )) and response (N(0, σX)) Fourier data tosimulate the effect of channel noise. The process noise is simulated by apply-ing an additional but unmeasured random noise (N(0, σP )) excitation on mass2. Table 2.2 summarizes the different noise cases used in this example, with anindication of the overall SNR on the force and response signals.


5 10 15 20 25 30 35 40−20

−10

0

10

20

Freq (Hz)

|GF| (

N2 )

5 10 15 20 25 30 35 40−260

−240

−220

−200

−180

−160

−140

Freq (Hz)

|GX| (

m2 )

5 10 15 20 25 30 35 40−150

−100

−50

Freq (Hz)

|H| (

m/N

)

Figure 2.6: Example of the Auto Power Spectra of force f1 (top) and response x1 (mid-dle) measured during 50 experiments and the exact FRF between these DOFs (bottom).The force drop-outs are present due to the effect of the excitators.

For these 4 noise cases, the FRFs estimated using the H1, Hv, Hiv and Hev,are compared with the true FRF and respectively shown in Figures 2.7, 2.8, 2.9and 2.10 for the FRF(1,1), i.e. an impedance measurement at the mass 1 during50 experiments. Since for modal analysis, one is mainly interested in the modes,a zoom around the resonance peaks is shown.

The H1 yields biased estimates for all 4 cases. The systematic errors are largestin the resonances and the bias is related to the signal-to-noise ratios of the inputsignals (cf. Eq. 2.25). Comparing cases a and b, denoted as (a À b), and (a À d)learns that the bias effect is mainly due to the input channel noise, which is nottaken into account by the H1 noise model. The comparison (aÀ c) illustrates theeffect of the process noise, since due to the feedback through the shaker-systeminteraction, a part of this noise is found in the input measurements and is nottaken into account. Notice however that, given the smaller bias error, this noisecontribution is smaller compared to the increase in channel noise (a À b). On

2.7. Validation 37

the other hand, the part of the process noise (present on the outputs) as wellas the output channel noise are included in the noise model and hence have nocontribution to the bias of the H1 estimates.

The results obtained by means of the Hv approach, shown in Figure 2.8, arevery similar to those of the H1 estimator. This can be explained by the im-portant difference in signal amplitudes between the forces and responses (i.e.rms(F )rms(X) + 10E4). This causes the Hv estimator, which is bounded by the H1

and H2 estimator, to over-emphasize the input signals, and so the input-noise,leading to a result close to the H1 estimates. By scaling the amplitude of thesignals with for instance their rms value prior to the Hv calculations, the Hv tendsin general to be more consistent as illustrated for noise case d in Figure 2.11.

As shown in Figure 2.9, the Hiv estimator yields fairly accurate FRF estimatesfor all the considered noise cases. This shows that the use of an errors-in-variablesnoise model will improve the FRF measurements by taking also the noise on theinput measurements into account. Since the process noise, present on the forceand response measurements, is uncorrelated with the (noise-free) source signalsfe, the effect of this noise is eliminated by averaging over a sufficient numberof experiments. This illustrates that the Hiv estimator, is certainly preferred tothe H1, especially in the case of low channel SNRs and a significant presence of(unmeasured) environmental noise. An important example are forced vibrationtests on a bridge where the wind load can introduce significant process noise.In practice, the Hiv estimator only requires the additional storage of the source(electrical) signals that are send to the shaker amplifiers. Since the number ofinputs is typically not more than 5, this amount of data is still small compared tothe number of response DOFs.

Comparison of the Hiv and Hev learns that both estimators perform well. Nev-ertheless, the use of a multisine excitation still has the advantage of better SNRs,since a constant amplitude is applied at each spectral line of the observed fre-quency band (in the case that no interaction exists with shaker). As a result, forthe same number of experiments, the uncertainty on the measurements will belower in the case of the multisine. Although the SNR for random noise signalsincreases rapidly for a small number of experiments, because dips in the averagedinput power spectrum (due to the random behaviour) disappear, it is shown in[109] that four experiments are needed to guarantee that 95% of the measure-ment points have a SNR corresponding to that of a well-designed, deterministicexcitation after one period. A rule of thumb for practical applications is that 5experiments with random noise are needed to have a comparable SNR to 1 ex-periment with a deterministic excitation (such as a multisine). This is certainlyan important factor with respect to the overall testing time, although multipleexperiments are required by the averaging process. On the other hand, randomnoise excitation is still often used in modal analysis since possible nonlinear dis-tortions are reduced by averaging a sufficient number of records, say 10 or more.


8.4 8.6 8.8 9−80

−75

−70

−65

−60

−55

−50

Freq (Hz)

|H| (

m/N

)

13.4 13.6 13.8 14−85

−80

−75

−70

−65

−60

Freq (Hz)

|H| (

m/N

)

24 25 26 27−130

−120

−110

−100

−90

−80

Freq (Hz)

|H| (

m/N

)

34.5 35 35.5 36 36.5−115

−110

−105

−100

−95

Freq (Hz)

|H| (

m/N

)

Figure 2.7: Effects of channel and process noise on the H1 FRF estimator. Comparisonof estimated FRF(1,1) with true FRF(1,1) (• • •) for noise cases a (—), b (· · · ), c (– ·)and d (– –). Both the process and channel noise on the input result in an importantsystematic error.

The same averaging effect is obtained by means of random multisine excitation,which, however, is not commonly available in modal testing equipment.

Using Eq. (2.22), the covariance matrix can be computed for the H1 estimateof the FRF matrix, from which the uncertainty bounds on the FRF estimates arefound. In a similar way the covariance matrix is computed for the Hiv and the Hev

(Hiv using a periodic excitation) based on Eq. (2.39). Figure 2.12 shows, for noise

2.7. Validation 39

8.4 8.6 8.8 9−80

−75

−70

−65

−60

−55

−50

Freq (Hz)

|H| (

m/N

)

13.4 13.6 13.8 14−85

−80

−75

−70

−65

−60

Freq (Hz)

|H| (

m/N

)

24 25 26 27−130

−120

−110

−100

−90

−80

−70

Freq (Hz)

|H| (

m/N

)

34.5 35 35.5 36 36.5−115

−110

−105

−100

−95

Freq (Hz)

|H| (

m/N

)

Figure 2.8: Effects of channel and process noise on the Hv FRF estimator. Comparisonof estimated FRF(1,1) with true FRF(1,1) (• • •) for noise cases a (—), b (· · · ), c (– ·)and d (– –). Both the process and channel noise on the input result in an importantsystematic error due to an important difference in signal amplitude.

case d, the amplitude difference between the true FRF(1,1) and respectively theH1 (top), Hiv (middle) and Hev (bottom) estimate as well as the corresponding99% confidence interval. This result illustrates the systematic errors in the H1

estimate around the resonance frequencies since the amplitude difference is clearlylarger than the 99% uncertainty bound. On the other hand, both the Hiv andHev estimates fall within the 99% uncertainty bounds, and hence the amplitude


8.4 8.6 8.8 9−80

−75

−70

−65

−60

−55

−50

Freq (Hz)

|H| (

m/N

)

13.4 13.6 13.8 14−85

−80

−75

−70

−65

−60

Freq (Hz)

|H| (

m/N

)

24 25 26 27−130

−120

−110

−100

−90

−80

Freq (Hz)

|H| (

m/N

)

34.5 35 35.5 36 36.5−115

−110

−105

−100

−95

Freq (Hz)

|H| (

m/N

)

Figure 2.9: Effects of channel and process noise on the Hiv FRF estimator. Comparisonof estimated FRF(1,1) with true FRF(1,1) (• • •) for noise cases a (—), b (· · · ), c (– ·)and d (– –).

difference are solely the result of the uncertainty on the measurement data. Thisconfirms the results presented in Figures 2.7, 2.9 and 2.10 and confirms that theH1 is a biased estimator, while the Hiv and Hev are unbiased. Furthermore, Figure2.12 shows a lower uncertainty on the FRF estimates for the Hev, which is explainedby the better SNRs during the multisine measurements.

After all, the bias effect of the H1 estimator will result in errors on the modalparameters, and especially the damping and mode shapes. This is illustrated in

2.7. Validation 41

8.4 8.6 8.8 9−80

−75

−70

−65

−60

−55

−50

Freq (Hz)

|H| (

m/N

)

13.4 13.6 13.8 14−85

−80

−75

−70

−65

−60

Freq (Hz)

|H| (

m/N

)

24 25 26 27−130

−120

−110

−100

−90

−80

Freq (Hz)

|H| (

m/N

)

34.5 35 35.5 36 36.5−115

−110

−105

−100

−95

Freq (Hz)

|H| (

m/N

)

Figure 2.10: Effects of channel and process noise on the Hev FRF estimator. Compar-ison of estimated FRF(1,1) with true FRF(1,1) (• • •) for noise cases a (—), b (· · · ), c (–·) and d (– –).

Figure 2.13 by comparing the operational deflection shapes (i.e. amplitude of theFRF at each response location at a resonance frequency) for the different estima-tors (noise case d). The difference between the true mode shape and the differentestimates is clearly larger for the H1 due to the bias in the FRF estimates, whilethe Hiv and Hev are in good agreement. Again, the benefit of using a multisine isobserved since the Hev has a smaller difference compared to the Hiv.


23.5 24 24.5 25 25.5 26 26.5 27 27.5−120

−115

−110

−105

−100

−95

−90

−85

−80

−75

Freq (Hz)

|H| (

m/N

)

23.5 24 24.5 25 25.5 26 26.5 27 27.5−120

−115

−110

−105

−100

−95

−90

−85

−80

−75

Freq (Hz)

|H| (

m/N

)

Figure 2.11: Scaled Hv (—) (top) and unscaled Hv (—) (bottom) compared to H1 (– ·)and Hiv (· · · ) FRF(1,1) estimate.

Example 2: In order to assess the validity of the methodology for obtaining thecovariance matrix of the Hiv FRF estimates based on Eq. (2.39), Monte carlosimulations were performed and compared to the true covariance matrix. Usingthe expressions (2.29) and (2.39), 100 experiments were simulated in the frequencyband of 6–28Hz using 20 averages (M = 20) and the noise settings of case c (cf.Table 2.2), from which a Monte Carlo estimate of the FRF covariance matrix wasobtained. The true covariance matrix can be computed based on the true FRFmatrix H0 and shaker dynamical stiffness matrix G0 as well as the knowledge ofthe added noise and the expressions for C(Hiv).

Figure 2.14 illustrates the validity of these expressions, given the good agree-ment between the true and the Monte Carlo results. The differences, for some ofthe covariances are only due to the number of 20 averages used to compute theFRF matrix estinates in each Monte Carlo run. By increasing M the uncertainty

2.7. Validation 43

8 10 12 14

−20

−10

0

10

20

Freq (Hz)

|H| (

m/N

)

24 25 26 27−50

0

50

Freq (Hz)

|H| (

m/N

)

8 10 12 14

−20

−10

0

10

20

Freq (Hz)

|H| (

m/N

)

24 25 26 27−50

0

50

Freq (Hz)

|H| (

m/N

)

8 10 12 14

−20

−10

0

10

20

Freq (Hz)

|H| (

m/N

)

24 25 26 27−50

0

50

Freq (Hz)

|H| (

m/N

)

Figure 2.12: Amplitude difference (—) between the true FRF(1,1) and respectively theH1 (top), Hiv (middle) and Hev (bottom) estimate and the 99% confidence interval (· · · ).

on the estimates will further decrease to the true values. It can be seen that theeffects of the process and input noise become important in the resonance peaks re-sulting in the higher uncertainties. This can be explained by the shaker-structureinteraction, which introduces mutually correlated noise on all the signals. Thefeedback to the input-side is largest in the resonances, since then this interactionbecomes most important, which was also indicated by the important force-dropsin Figure 2.6, also resulting in lower SNRs on the inputs.

Example 3: As discussed in the previous example, the true covariance matrixcan be computed based on the true FRF matrix H0 and shaker dynamical stiffness


1 2 3 4 5 6 7−3

−2

−1

0

1

2

3x 10−3

DOF

|H| (m

/N)

1 2 3 4 5 6 7−1

−0.5

0

0.5

1x 10−3

DOF

|H| (m

/N)

Figure 2.13: Comparison of the operational deflection shapes of the first (top) andsecond mode (bottom) obtained from the true (+ · · · ) FRFs and H1 (× · · · ), Hiv ( · · · )and Hev (¤ · · · ) FRF estimates (noise case d).

matrix G0 as well as the knowledge of the added noise. By means of this, the effectsof the different possible noise sources on the correlation between the FRFs can bestudied.

Using again the 7 DOF system (cf. Figure 2.5), the correlation matrix ofC(Hiv) is computed for three cases by considering respectively only output, inputand process noise. Figure 2.16, shows the results at a resonance frequency ofthe mode at 8.74Hz (left) and a near frequency at 9.23Hz (right) which is stillnear this mode as shown in Figure 2.6. The effect of output channel noise onthe FRF correlations can be seen in the top figures. Although this channel noiseis mutually uncorrelated, an important interaction between the shakers and thestructure, also results in a correlation between the FRFs measured between thetwo input and same output locations. Referring to Eqs. (2.37), (2.39) and (2.39),learns that, in the case that no input or process noise exists and CX and GR

2.7. Validation 45

10 15 20 25−140

−120

−100

−80

−60Var(H

11) [(m/N)2]

10 15 20 25−140

−120

−100

−80

−60Var(H

12) [(m/N)2]

10 15 20 25−160

−140

−120

−100

−80

−60Cov(H

11,H

12) [(m/N)2]

10 15 20 25

−160

−140

−120

−100

−80

−60Cov(H

11,H

21) [(m/N)2]

10 15 20 25

−160

−140

−120

−100

−80

−60Cov(H

11,H

31) [(m/N)2]

10 15 20 25

−160

−140

−120

−100

−80

−60Cov(H

11,H

41) [(m/N)2]

10 15 20 25−180

−160

−140

−120

−100

−80

−60Cov(H

72,H

62) [(m/N)2]

Freq (Hz)10 15 20 25

−180

−160

−140

−120

−100

−80

−60Cov(H

72,H

52) [(m/N)2]

Freq (Hz)10 15 20 25

−180

−160

−140

−120

−100

−80

−60Cov(H

72,H

42) [(m/N)2]

Freq (Hz)

Figure 2.14: Comparison between the true (—) and estimated (Monte Carlo) (· · · ) FRFcovariance matrix CHiv .

are diagonal, correlations can still be introduced by the non-diagonal elementsof the FRF matrix HFR

1 , which can become large due to an interaction betweenthe exciters and the structure (cf. Figure 2.15). This figure gives again the non-diagonal elements of the same FRF matrix when using smaller exciters (i.e. ke, ceand me 10 times smaller than values in Table 2.1). Since the input locations arenot a node for the considered mode, this interaction is large near the resonancefrequency resulting in the additional correlations as shown in Figure 2.16. In orderto illustrate the effects of this interaction, Figure 2.17, shows similar results forthe smaller exciters, where no correlation is introduced.


6 8 10 12 14 16 18 20 22 24 26 28

−60

−50

−40

−30

−20

−10

0

Freq (Hz)

|H| (

m/N

)

Figure 2.15: The non-diagonal elements of the FRF matrix HFR1 for the “output noise

only” case when using large (—) and small (– ·) exciters.

For the case that only input channel noise is considered, the results are shownin Figures 2.16 (middle) and 2.17 (middle). Although CF and GR are diagonal,correlations are again introduced between the FRFs corresponding to the multipleinput (and same output) locations by the shaker-structure interaction (i.e. HFR

1

non-diagonal). The correlation between the FRFs corresponding to the same inputlocation results from the fact that near a resonance the matrix HXR

1 in the matrixproductMCov(HFR

1 )MH has a rank 1. Consequently the FRFs corresponding tothe same input location will be totally correlated. In the case of the first bendingthe mode, DOF 4 is a node (cf. Figure 2.13) and hence the elements for that DOFin the transfer function matrix M will be zero. This difference of correlation for2 shaker setups can be noticed by comparing Figures 2.16 and 2.17.

Based on the EV relationship, process or environment noise introduces mutu-ally correlated noise on all the outputs. At the same time, the shaker-structureinteraction introduces a feedback to the inputs as well. Hence, since all elementsof CEZ generally differ from zero, and since only one source of process noise isconsidered in the simulations, the total correlation of all the FRFs correspondingto the same input location and partial correlation between FRFs with differentinput is explained again by Eq. (2.39).

As can be expected from the results shown Figure 2.14, the measurement noisecan certainly introduce important correlations between the different FRFs. Itis however important to remember that a high correlation between the FRFs,does not necessarily has an important influence in terms of the actual FRF noisecovariances depending on the process and measurement noise as well as on the

2.7. Validation 47

0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14 0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14 0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14

Figure 2.16: Effects of the measurement noise on the FRF correlations at the frequencycorresponding to the first bending mode at 8.74Hz (left) and at a near frequency at 9.23Hz(right). The correlation matrix shows the effect of respectively only output noise (top),input noise (middle) and process noise (bottom).


0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14 0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14 0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14 0

0.2

0.4

0.6

0.8

1

FRF

FRF

2 4 6 8 10 12 14

2

4

6

8

10

12

14

Figure 2.17: Similar results as in Figure 2.16, but using smaller excitors, reducing theshaker-structure interaction significantly.

2.8. Conclusions 49

dimensioning of the excitors. Moreover, the output correlations can be eliminatedby using a Scanning Laser Doppler Vibrometer setup (cf. § 2.4).

2.8 Conclusions

Classical modal analysis starts from the frequency response functions that areestimated using FRF estimators. The data is typically acquired using a randomexcitation. However, in general, too simplified noise models assumed by theseestimators, result in biased FRF estimates. Furthermore, the uncertainty on theFRF estimates is yet never considered in modal analysis during the parametricidentification process.

The application of parametric stochastic estimators, such as for instance thefrequency-domain ML estimator, in the domain of modal analysis requires that thenoise information for the FRF data is known. Moreover, since the accuracy of theparametric results is related to the quality of the measured data, the maximumlikelihood approach is only meaningful in the case that consistent FRF estimatescan be obtained.

This chapter has revised the nonparametric FRF identification in a maximumlikelihood framework based on an errors-in-variables stochastic noise model. Thismodel provides an accurate representation of the measurement setup taking theeffects of channel, shaker and process noise on all the inputs and outputs intoaccount. Table 2.3 summarizes the results of the comparison for the differentFRF estimators. From this, it can be concluded that the estimators currentlyused in practice, i.e. H1, H2 and Hv are generally not suited for accurate FRFidentification since no general errors-in-variables noise model is considered. If thenoise covariance matrix would be a priori known, the Hv can be generalized tothe Hgtls, which belongs to the class of maximum likelihood estimators. However,since this noise covariance matrix is never a priori known in practice, this Hgtls

estimator is not applicable.

H1 H2 Hv Hgtls Hiv Hev

EV noise model – – – – – + + + + + +ML properties – – – – – + + + + +

No a priori noise info + + + + + + – – + + + +MISO + + – – – – + + + +

General applicability + + + + + + – – + –

Table 2.3: Comparison of different nonparametric FRF estimators.

For this reason, the so-called nonparametric instrumental variables or Hiv es-timator was studied for FRF identification for arbitrary excitation signals. It has


been shown in this chapter that this estimator has maximum likelihood properties,without requiring any a priori information about the measurement noise. Never-theless, this approach uses additional information under the form of the generatorsignals, which in practice only requires the additional acquisition of a very lim-ited number of signals. Multivariable expressions were derived for both the FRFand noise covariance matrix from the same measurement data. When nonlineardistortions are introduced by the excitation devices, the consistency property re-mains valid, only resulting in some loss of the asymptotical efficiency. Moreover,the Hiv remains also consistent when a MISO approach is used to process theextensive amount of data, which is an important property for the application ofmodal analysis. The methodology is validated and compared with the classicalnonparametric approach for a simulated vibration experiment that considers theinteraction between the shaker and the structure. It is shown that both the noiseon the measurements and these interactions can result in important correlationsbetween the FRFs.

For the case of periodic signals, the errors-in-variables FRF estimator Hev isa special case of this instrumental variables Hiv estimator. The Hev remains itsML properties when nonlinear distortions are introduced by the excitation devices.Although the Hev does not require the generator signals, its practical applicabilityis constraint by the class of excitation signals, since periodic signals are still notoften used for modal analysis practices.

2.9. Appendix A – The GTLS Estimator per Output 51

2.9 Appendix A – The GTLS Estimator per Out-put

Assuming the (No × No) matrix BCEZ BH diagonal, the cost function (2.8) can

also be written as

`(H) = Mtrace(

[

diag(BCEZ BH)]−1[

BGZBH]

)

= Mtrace(

[

diag[C]]−1

[G])

(2.65)

with

[

diag[C]]−1

[G] =

c11 0 . . . 00 c22 . . . 0...

.... . .

...0 0 . . . cNoNo

m11 m12 . . . m1No

m21 m22 . . . m2No

......

. . ....

mNo1 mNo2 . . . mNoNo

=

c11m11 c11m12 . . . c11m1No

c22m21 c22m22 . . . c22m2No

......

. . ....

cNoNomNo1 cNoNomNo2 . . . cNoNomNoNo

proving that the cost function boils down to the sum of cost functions obtainedby estimating each row of B = [H,−INo ] ∈ CNo×Ni+No separately

`(H) =M(c11m11 + c22m22 + . . .+ cNoNomNoNo) (2.66)

The same result is found for the case that the (No×No) matrix BGZBH is di-

agonal. Since GZ = GZ0+CEZ , with GZ0

the true data, BGZBH is asymptotically

(M →∞) a diagonal matrix if BCEZ BH is diagonal.

From this it is also trivial that Equation (2.65) is equivalent with

`(H) =Mtrace(

[

diag(BCEZ BH)]−1[

diag(BGZBH)]

)

(2.67)

2.10 Appendix B – Unbiased Estimate of GEX

For simplicity, the scaling factor occurring in Eq. (2.24) will be proven assumingthere is only one output. The relation between the exact (noise-free) input Fouriervectors and the output Fourier coefficient at spectral line f can be written as

X [m] = HF [m] + E[m]X (2.68)


Consider the following normalized cost function

` =

M∑

m=1

∣

∣

∣E[m]X

∣

∣

∣

2

GEX0

(2.69)

with GEX0the true variance of the output noise. Assuming

E[m]Xo√GEX0

distributed

as N(0, 1) then the cost function ` is χ2(0, 1) (Chi-quadratic) distributed and itsexpected value equals

E` =M −Ni (2.70)

whereM−Ni corresponds to the number of degrees of freedom of the Chi-quadraticdistribution. M is the number of equations and Ni number of constraints whenminimizing the cost function (2.69), i.e.

∂`

∂H1= 0 . . .

∂`

∂HNi

= 0 (2.71)

An estimate of the output noise variance is given by (2.23) as

GEX =1

M

M∑

m=1

∣

∣E[m]X

∣

∣

2(2.72)

which expected value also equals, given the above derivation

E

GEX

=GEX0

ME` = (M −Ni)

MGEX0

(2.73)

Chapter 3

FRF Identification in thePresence of Leakage

In practice, the estimation of frequency response functions (FRFs) is often compli-cated by the influence of noise on the measured data as well as by spectral leakagein the case of random excitation. Chapter 2 discussed accurate FRF identificationin the presence of measurement noise by considering FRF estimators developed inan errors-in-variables framework. This chapter deals with the problems related toleakage, due to the finite measurement time, when arbitrary excitation signals areused. In modal analysis, an exponential time-window is often applied, for reducingboth the effects of leakage and measurement noise, in the case of a time-limitedexcitation such as for pulse or burst random signals. The possibility of applying anexponential time-window in combination with random noise excitation is studiedby revising the estimation of Auto and Cross Power Spectra in Section 3.3, whilethe same is done starting directly from FRFs in Section 3.4. A comparison ofthe different approaches in Section 3.7, including other methods suggested in theliterature, learns more about their effectiveness to cope with leakage phenomena.Finally, an experimental case study for a subframe of a car studies the applicabilityin practice of the most important methods derived in both chapters 2 and 3.

53

54 Chapter 3. FRF Identification in the Presence of Leakage

3.1 Introduction

Chapter 2 discussed accurate FRF identification in the presence of measurementnoise by considering FRF estimators with ML properties that were developed inan errors-in-variables framework. However, another important source of errors formodal analysis applications is spectral leakage due to a finite measurement timeof non-periodic signals. Errors-in-variables FRF identification in the presence ofleakage can still be seriously hampered since the damping of modes is usuallyoverestimated corresponding to a bias (systematic) error.

Periodic (broadband) excitation [39], certainly avoids leakage problems andcan improve the signal-to-noise ratio compared to arbitrary signals (cf. § 2.7).Arbitrary signals, such as random noise, have the advantage of reducing possi-ble nonlinear distortions by averaging over a sufficient number of measurements.As a compromise between random and deterministic excitation, periodic randomsignals can be used, however with the important drawback of an increased mea-surement time. Each new random sequence should be applied during a certaintime before starting the actual measurement in order to avoid the transient effects(such as leakage) appearing each time a new sequence is applied. This transientbehavior, however, depends on the damping of the system as well as on the desiredfrequency resolution. For low damped systems, the length of each period shouldbe sufficiently long in order to have enough information (i.e. frequency resolution)in the small resonance peaks. In that case, the additional waiting time generallyis not more than one period (i.e. typically 20 – 80% of the measurement period).The same is true for pseudo random (i.e random multisine signals), however withthe advantage that the amplitude spectrum is constant (while the phase is stillrandom), resulting in better SNRs compared to periodic random.

However, arbitrary signals, such as random noise, are still most often usedfor modal testing because of their general availability as well as the reduction ofnon-linear distortions. Furthermore, in general, arbitrary signals are a better ap-proximation for the operational conditions of a wide class of mechanical structurescompared to deterministic signals. A common approach to reduce the leakage ef-fects is the application of a time-window (e.g. a Hanning window) that certainlyhelps to reduce the effects of leakage. However the problem will not be completelyeliminated, especially for the case of lightly damped structures.

For that reason, the use of a time-limited burst random excitation (combinedwith the implicit rectangular finite time-window) has gained popularity in themodal analysis community since it allows the signals to decay to zero before theend of the time window. Analogously to impact testing, an exponential windowcan also be applied in order to amplify this decay by artificially increasing dampingby means of an exponential window. The use of an exponential window has theimportant advantage that the effect of this window can be totally compensated foron the modal parameters. However, disadvantages of burst random, and transient-

3.2. Simulated Vibration Experiment 55

5 10 15 20 25 30 35 40−50

−40

−30

−20

−10

0

10

20

30

Freq. (Hz)

|m/s

2 /N| (

dB)

Figure 3.1: The exact FRF(1,1) between the acceleration and force at DOF 1.

like excitation in general, are a worse signal-to-noise ratio and a higher crest factorcompared to random noise excitation.

Therefore, this chapter studies the possibility for reducing both the effects ofleakage and measurement noise by combining the application of a random noiseexcitation and an exponential time-window. A comparison of several approaches ismade with existing techniques available in literature. The different approaches willbe studied by means of the simulated vibration experiment that is first discussedin the next section. At the end of this chapter, an experimental case study for asubframe of a car illustrates the applicability in practice of the most importantmethods derived in both chapters 2 and 3.

3.2 Simulated Vibration Experiment

The same 7 DOF system as presented in § 2.7 is used again in order to simulate amodal testing experiment. For simplicity reasons, only a single excitor is consid-ered at DOF 1, without any loss of generality of the presented results. In orderthe study the problem of leakage, time-domain signals were now generated usinga discrete-time state space formulation based on the shaker and system charac-teristics given in Table 2.1. Using this model, the exact FRFs (cf. Figure 3.1)and modal parameters for the 6 modes in the band of 5–40Hz (cf. Table 3.1) werecomputed.


mode fd (Hz) ζ (%)1 8.6711 0.09392 13.6263 0.13003 24.2989 0.22704 26.4022 0.26455 35.1201 0.40476 35.6307 0.4185

Table 3.1: Damped natural frequency and damping ratio for the 7 DOF system.

Normal distributed noise was used to excite the system in a band of 0–50Hz.The force fe, applied to the shaker coil, is assumed noise-free and equal to theelectrical signal (sent to the power amplifier) up to a certain force/current factor,while the actual force acting on the system and the 7 measured responses arerespectively denoted as f1 and x1, . . . , x7. Given the low damping of the struc-ture, time histories with a total number of 128K (1K = 1024) time samples weremeasured (corresponding to a total measurement time of about 1300s) given thesample rate of Ts = 0.01s.

Based on these time histories, the frequency response function (FRF) data isderived using averaging-based nonparametric processing techniques. As discussedin chapter 2, when using random excitation, this FRF identification, is classicallydone using the noise-on-output H1 estimator (for angular frequency ωf )

HXF1 = GXF G

−1F (3.1)

or in an EV framework by means of the Hiv estimator

HXFiv = GXRG

−1FR = GXRG

−1R GRG

−1FR = HXR

1 HFR−1

1 (3.2)

However, given the finite total amount of data samples acquired, the useris always faced with a tradeoff in an effort to maximize the number of spectralaverages (i.e. minimizing the uncertainty) while keeping a sufficient frequencyresolution (i.e. maximizing the number of samples within the record). Whenthe number of data samples within one record becomes small, leakage errors willcertainly be introduced in the FRF estimates. Since this chapter focusses on theeffects of leakage on the nonparametric processing, measurement noise was nownot considered in the simulation experiment.

3.3 Power Spectra Estimators

Basically, two classical approaches exist to estimate the Auto and Cross PowerSpectra in Eqs. (3.2) and (3.1) [89]. The Periodogram estimator operates directly

3.3. Power Spectra Estimators 57

on the Fourier data to yield a Power Spectra (PS) estimate. On the other hand,the Correlogram estimator first estimates correlation functions, and then obtainsthe PS estimate by means of a DFT. These approaches are now revised for thepossibility of applying exponential windowing techniques, in order to reduce theeffects of leakage when using random noise excitation.

3.3.1 Periodogram PS Estimator

In practice, Auto and Cross Power Spectra are generally estimated using theWelchPeriodogram estimator [149, 89] where this approach was also implicitly used in§ 2.7, for the computation of the nonparametric FRF estimates. This PS estimatorintroduces time-windows to reduce the bias error caused by leakage and optionallysegment overlapping to increase the number of records for the (asynchronous)averaging process.

Given the time histories of the reference (i.e. force on shaker coil), force andresponse signals with a total number of Ntot samples measured at a samplingrate Ts. Dividing these histories in M adjacent records results in the time recordsr[m](tn), force f

[m](tn) and response x[m](tn) as defined by the EV model Eq. (2.1),where m = 1, . . . ,M and tn = nTs , n = 0, 1, . . . , Ns − 1 with Ns the number ofsamples within each record. In general, given the finite total number of samplesacquired Ntot = MNs, there is a trade-off between a high spectral resolution(Ns as large as possible) and minimizing the estimation variance (M as largeas possible) since the variance decreases as O(M−1) while the bias error due toleakage increases for decreasing Ns.

In order to increase the number of segments for averaging, and therefore todecrease the variance of the PSD estimate, segment overlapping can be introducedby shifting over S (S ≤ M) samples between adjacent records. With a totalnumber of time samples Ntot =MNs the maximum number of segments availablefor averaging is then given by the integer part of (Ntot − Ns)/S + 1. However,caution is needed since overlap also introduces correlations between the adjacentrecords [106].

Compared to the rectangular time-window (corresponding to the finite time ofeach record), applying for example a Hanning or Hamming window to the recordsprior to the DFT transform, reduces the effect of the sidelobes and so the leakageerror at the price of only a slight decrease in resolution. Such a time-windowforces the random signals to be periodic by reducing the discontinuities at theboundaries of the finite record. However, this also reduces the total energy in themeasured signal resulting in an attenuation of the amplitude, which is typicallya bias error. Although, the amount of attenuation solely depends on the windowshape and so this error can be compensated. However, since the estimate of theFRF matrix (3.1) or (3.2) is found as the ratio of Power Spectra, a compensation


for the attenuation of the time-window is cancelled out and hence is not consideredhere in the derivation of the expressions for the Power Spectra.

Applying a time-window to the records z[m](tn) ∈ RNi+No×1 and r[m](tn) ∈RNi×1, where z[m](tn) =

[

f [m]T (tn), x[m]T (tn)

]Tresults in the following samples

z[m]w [n] = w[n]z[m][n]

r[m]w [n] = w[n]r[m][n] (3.3)

for 0 ≤ n ≤ Ns − 1 and 1 ≤ m ≤ M , where the windowed segment Periodogrammatrix is computed

G[m]ZR(ωf ) = Z [m](ωf )R

[m](ωf )H (3.4)

over the angular frequency range ωf = 2πf/NsTs , f = 0, 1, . . . , Ns/2 − 1, whereZ [m](ωf ) and R

[m](ωf ) are the DFT (cf. Eq. 2.2) of the records (3.3). The WelchCross Power Spectra estimator is then given by the linear averaging of the segmentPeriodograms

GPZR(ωf ) =

1

M

M∑

m=1

G[m]ZR(ωf ) (3.5)

where a similar expression is found for the Auto PS GPR(ωf ). Although a wide

range of possible time-windows w(tn) can be considered for this approach (Han-ning, Hamming, Nuttall, Gaussian, Chebyshev, ...), an exponential window is notapplicable for this approach. However, since the modal parameters (i.e. damping)can be compensated for an exponential window, a possible alternative approachfor the calculation of the PS estimates is studied in the next section.

As discussed in Chapter 2, the FRF covariance matrix is found using the ex-pressions (2.22) and (2.39) respectively for the case of the H1 and Hiv estimator.

Remark 1 : Alternatively, Power Spectral Density functions, which are scaled tothe unity frequency resolution and compensated for the energy attenuation due tothe time-window, can be computed as

S[m]ZR(ωf ) =

1

UNsTsZ [m](ωf )R

[m](ωf )H (3.6)

over the angular frequency range ωf = 2πf/NsTs , f = 0, 1, . . . , Ns/2 − 1, whereU is the discrete-time window energy

U = Ts

Ns−1∑

n=0

w2(tn) (3.7)

This factor U removes the effect of the window energy bias.

Remark 2 : Depending on the considered FRF estimator, H1 or Hiv, the nota-tions used for the signals differ. For the case of the H1 estimator (3.1), the force


8.64 8.66 8.68 8.7

10

15

20

25

Freq. (Hz)

|m/s

2 N| (

dB)

M = 1,2,4,8,16

13.6 13.65 13.7

6

8

10

12

14

16

18

20

Freq. (Hz)

|m/s

2 N| (

dB)

M = 1,2,4,8,16

Figure 3.2: The H1 estimator using Periodogram approach for increasing number ofaverages M = 1, 2, 4, 8, 16 and decreasing blocksize Ns = 128, 64, 32, 16, 8K (Ntot = 128Ksamples) together with the true FRF (bold line).

signals f [m](tn) are used instead of the generator signal r[m](tn) and hence theCross GXF (ωf ) and Auto PS GF (ωf ) are only computed. This convention is usedthroughout this chapter.

Example: The effect of leakage is now illustrated for the H1 estimator usingthe Periodogram estimator to compute the FRF estimates for the 7DOF system,introduced in § 3.2. Figure 3.2 shows one of the estimated FRFs, obtained bydecreasing the blocksize Ns and increasing number of averages M , while the totalnumber of time samples was fixed to Ntot = 128K (1K=1024). A Hanning windowwas applied to reduce leakage. Zooming in on the first 2 modes clearly illustratesthe effects of leakage that are present in the data, resulting in an important biaserror. This error can be reduced by using only a small number of averages, howeverresulting in noise FRF estimates. The low damping of the considered system,explains why the leakage errors become important even in the case of the largeblocksizes.

3.3.2 Correlogram PS Estimator

Given againM adjacent records of Ns of the generator r[m](tn), force and response

z(tn) obtained by the division of the time histories. The (Ni +No)×Ni unbiasedcross correlation matrix estimate between the two signal vectors z(tn) and r(tn) ,


is computed as

gzr[j] =

1Ns−j

∑Ns−j−1n=0 z[n+ j]rT [n] for 0 ≤ j ≤ Ns − 1

1Ns−|j|

∑Ns−|j|−1n=0 z[n]rT [n+ |j|] for −(Ns − 1) ≤ j < 0

(3.8)

The biased correlation estimate uses 1/Ns rather than 1/(Ns − j).

Next, an estimate of the windowed segment Correlogram matrix is obtainedby taking the DFT of the finite sequence of the cross correlation estimate (3.8)

GCZR(ωf ) =

L∑

l=−L

w[l]gzr[l]e(−iωf lTs) (3.9)

The maximum lag index L is typically less than the number of data samples inorder to avoid the greater statistical variance associated with the higher lags ofthe correlation estimates. Although a maximum of L ≈ Ns/10 was suggestedby Blackman and Tukey [18], for mechanical systems this cut-off length stronglydepends on the damping of the considered system. Furthermore, the correlogramPS estimator is asymptotically unbiased as the number of lags increases [18]. TheFFT algorithm can be used to perform a fast computation of the correlogramestimate by the convolution method [35].

An odd-length (2L+1)-samples lag time window w[l] (e.g. Hanning, Hamming,Chebyshev, ...) over the interval −L ≤ l ≤ L and symmetric about the origin canbe applied in (3.9). This window reduces the effect of leakage due to the largesidelobes of the implicit rectangular window and therefore the bias error in theestimate. This is illustrated, for the considered 7 DOF system, in Figure 3.3, wherethe cross-correlation function gx1f (3.8) with Ns=8192 samples is shown. At thehigher lags, the greater statistical variance can be noticed. Applying for examplea Hanning window of length (2L+ 1)-samples with L = Ns results in Figure 3.3,forcing the correlation function to zero at the higher lags to reduce leakage effects.

However, when applying a time-window, the poles (especially the damping)will be affected. In the case that an exponential window is used, the poles can beexactly compensated for the added damping, which is not the case for other timewindows such as e.g. Hanning and Hamming. Hence, for the purpose of modalparameter estimation (i.e. the poles and residues), the correlogram estimatoris best combined with an exponential window. The exponential window addsartificial damping shifting the stable poles more to the left an the unstable polesmore to the right in the Nyquist plane.

Given the form of the correlogram estimate GCZR(ωf ), a double-sided exponen-

tial time-window w[l]

w[l] = e−β|l|Ts with − L ≤ l ≤ L (3.10)


−5000 0 5000−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

lag number

gxf

[l]

−5000 0 5000−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

lag number

w[l]gxf

[l] Hanning

Figure 3.3: The cross-correlation function gx1f with Ns=8192 (left) and the effect fromapplying a Hanning window (right).

−5000 0 5000−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

lag number

w[l]gxf

[l] Expon − 0.1%

−5000 0 5000−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

lag number

w[l]gxf

[l] Expon − 1%

Figure 3.4: The effect of applying an exponential window to the cross-correlation func-tion gx1f with different β factor (left: 1% at end of window) and (right: 0.1% at end ofwindow).

can be applied in order to reduce the effect of leakage (due to the implicit rectan-gular window) and noise (greater statistical variance) associated with the higherlags of the correlation estimates. Typically, for a time-limited signal, such as apulse or burst random excitation, the factor β is chosen such that amplitude atthe last time lag of the exponential window is 1% of its initial amplitude [62]. Fig-


ure 3.4 shows, for the cross-correlation function gx1f , the effect of an exponentialwindow with different values for β. As can be seen in this case, 0.1% at the end ofthe exponential window seems a more appropriate choice since the higher dampingonly reduces the greater variance at the higher lags, while the system dynamics(around the central lag) are not much affected.

Moreover, an exponential time-window offers the advantage that the extractedpoles of the modal model can be corrected for the artificial damping introducedby the window that is a priori known by the choice of the β-factor

pcorrectedr = pestimatedr + β (3.11)

Remark : In the case that the instrumental variables approach is used in com-bination with a random noise excitation, the generator signals generally are whitenoise sequences, in which case this approach can be further simplified. Given therelation in the continuous-time domain between the measured time signals z(t)

and the generator signals r(t), with h(t) ∈ R (No+Ni)×Ni the impulse responsefunction matrix

z(t) = hzr(t) ∗ r(t) (3.12)

Convolution with r(t) yields

gzr(t) = hzr(t) ∗ grr(t) (3.13)

Given that the generator signals r(t) are normalized white noise signals, the auto-correlation function matrix is then given by grr(t) = INiδ(t), with δ(t) a Diracpulse function. Hence, Eq. (3.13) indicates that the correlation functions gzr(t)are impulse response functions since for causal systems the negative lags are zero.This implies that it is sufficient to consider only the positive lags of the correlogramestimate (3.9) since they contain all the information needed to identify the modalparameters

GCZR(ωf ) = Ts

L∑

l=0

w[l]gzr[l]e(−iωf lTs) (3.14)

Figure 3.5 shows the both cross-correlations gfr and gx1r. As can be seen from thisfigure, these correlation functions have the form of an impulse response functionsand hence only the positive lags contain the dynamics of the system.

Given the generator r(tn), force f(tn) and response x(tn) signals for the M

subsequent records of Ns samples, a correlogram PS estimate GC[m]

ZR (ωf ) (3.9) canbe computed for each record (m = 1, . . . ,M). Assuming the segment correlogramsstatistically independent, then the elements of the matrix

GCZR(ωf ) =

1

M

M∑

m=1

GC[m]

ZR (ωf ) (3.15)

represent the sample mean of a set of M independent variables.


−5000 0 5000−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

lag number

w[l]gfr[l] Expon − 0.1%

−5000 0 5000−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

lag number

w[l]gxr

[l] Expon − 0.1%

Figure 3.5: The cross-correlations gfr (left) and gx1r (right) after applying an exponen-tial window (0.1% at end of window).

An estimate of the sample variance of the correlogram estimates is given as

var(

vec(GCZR(ωf ))

)

=1

M

(

1

M − 1

M∑

m=1

∣

∣

∣vec(GC[m]

ZR (ωf ))− vec(GCZR(ωf ))

∣

∣

∣

2)

(3.16)

As for the Periodogram estimator there is a trade-off between a high spectralresolution (Ns as large as possible) and minimizing the estimation variance (M aslarge as possible). However, since in this case an exponential window can be usedto reduce the effects of leakage, the user could opt for more records of a smallerblocksize.

Example: The H1 FRF estimates using the Correlogram approach estimatedfor an increasing number of averages M = 2, 4, 8 and decreasing blocksize Ns =64, 32, 16K (Ntot = 128K samples) are shown in Figure 3.6. An exponential win-dow was applied such that the window was 0.1% at the end of the correlationfunction (i.e. at lags Ns − 1 and −Ns + 1). Similar results are found for othervalues of β. Although an exponential window is applied to the correlation func-tion in order to reduce leakage, the FRF estimates found by using the correlogramapproach clearly still suffer from leakage, at least in this case of the considered lowdamped system. Moreover, taking the ratio of 2 Correlogram PS estimates thatalready include the effects of an exponential window seems less robust than theclassical H1 using the ratio of 2 Periodogram estimates.


8.5 8.6 8.7 8.8 8.9

−5

0

5

10

15

20

25

Freq. (Hz)

|m/s

2 N| (

dB)

35 35.2 35.4 35.6 35.8

−9

−8

−7

−6

−5

−4

−3

Freq. (Hz)

|m/s

2 N| (

dB)

Figure 3.6: The H1 FRF using the Correlogram PS estimates for an increasing numberof averages M = 2, 4, 8 and decreasing blocksize Ns = 64, 32, 16K (Ntot = 128K samples)and true FRF. The exponential window was 0.1% at the end of the correlation function.

3.4 Exponential Windowing of FRF Estimates

Instead of reducing the effects of leakage by acting on the calculation of the PowerSpectra, an exponential window can also be applied to the inverse DFT of theFRF estimate given by (3.1) or (3.2). In this section it is shown how exponentialwindowing can be used in combination with these FRF estimates in order to copewith the trade-off between minimal variance and bias for the final FRF estimates.

3.4.1 Hexp FRF Estimator

An obvious way to minimize the effects of leakage is to measure long enough bychoosing a sufficient number of samples Ns within a record. Hence, the basicidea of this approach consists of computing the FRF matrix estimate in Eq. (3.1)or (3.2) by means of the Periodogram PS estimator (3.5), where Ns is sufficientlylarge to ensure that both the effects of leakage and the time-window (e.g. Hanning)are sufficiently small.

The damping behavior of the system determines the significant length of thecorresponding impulse response function (IRF). An idea of the optimal value forNs can be found for instance from the inverse Fourier transform of the EmpiricalTransfer Function Estimate. This ETFE is computed as the ratio of the DFT ofthe full-length random input and output signals, which is basically an H1 estimatorusing the Periodogram with no averages (M = 1). However, this often results in

3.4. Exponential Windowing of FRF Estimates 65

very noisy FRF estimates as can be seen in Figure 3.2.

A schematic overview of this approach is shown in Figure 3.7. In practice, giventhe fixed total number of time samples Ntot, only a small number of averages, saytypicallyM= 1–4 can be used to obtain the FRF estimate in order to avoid effectsof leakage and the Hanning window. In this case, however, taking the inverse FFTof the FRF estimate H(ω), results in an impulse response function h(t). In order toimprove the quality of the IRF, by reducing the noise for the higher time samples,an exponential window can then be applied resulting hexp(t) and subsequentlyHexp(ω) by computing again the FFT.

This is equivalent to a convolution with a low pass filter LP(ω) in the frequency-domain

HXFexp

(ωf ) = HXF(ωf ) ∗ LP(ωf ) (3.17)

where LP(ωf ) is the DFT of an exponential window in the time-domain

LP(ωf ) =

Ns−1∑

n=0

w[n]e(−iωfnTs) with w[n] = e−β|n|Ts (3.18)

The FRF matrix estimate HXF is computed using the Periodogram Power Spectraestimator according the H1 (cf. Eq. 3.1) or Hiv (cf. Eq. 3.2) approach.

Since, the effect of the exponential window can be compensated for usingEq. (3.11), this can be done without introducing a bias error on the final parameterestimates. In the case of the classical approach, the Hanning window, applied tosmaller blocks in order to have a sufficient number of averages, introduces a biaserror that cannot be compensated and increases for a decreasing blocksize. Alsonotice that data compression of the initial FRF data can be obtained by cuttingthe IRF at a lower time sample (L ≤ Ns) and next applying an exponential lagwindow w[n] (0 ≤ n ≤ L) on the shorter IRF. Another possible approach for datacompression can be obtained by first applying an exponential on the full IRF andnext using cyclic averaging (cf. § 3.6) to reduce the frequency resolution.

Remark : As for any of the nonparametric methods considered in this chapter,this method requires a sufficient total number of time samples such that no leakageis present in the empirical transfer function estimate.

The covariance matrix of HXFexp

(ωf ) is found as

Cov(

HXFexp

(ωf ))

= Cov(

HXF(ωf ))

∗ |LP(ωf )|2 (3.19)

where Cov(

HXFexp

(ωf ))

is given by Eqs. (2.22) and (2.39) for respectively the caseof the H1 and Hiv FRF estimators. In [145], the validity of this expression hasbeen demonstrated by means of Monte Carlo simulations.


IFFT

0 500 1000 1500 2000 2500-10

0

10

20

30

40

50

60

H( )ω

0 500 1000 1500 2000 2500-10

0

10

20

30

40

50

60

expH ( )ω

FFT

0 1000 2000 3000 4000 5000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

exph (t)h(t)

0 1000 2000 3000 4000 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

tw(t) e β−=

=.

Figure 3.7: Schematic overview of Hexp estimator.

Example: Based on the ETFE of the FRF (i.e H1 with M = 1), the effect ofusing an exponential window was studied for an increasing value of β in Eq. (3.18).Choosing β = β1, 2β1, 4β1, 8β1, with β1 = 3.5135E-3, resulted in an exponentialwindow with a value of 1% at respectively Ns, Ns/2, Ns/4, Ns/8 and Ns/16,thus increasingly cutting off. The length of the IRF (and thus the frequencyresolution) was kept the same (Ns = 128K) in all cases. In the frequency-domain,this results in an increased smoothing effect of the low pass filter, which can beseen in Figure 3.8 by zooming on the first mode and the modes 5 and 6 thathave a higher damping. It can be seen that the noise on the ETFE is smoothed,while damping is artificially added, however, with an a priori known factor β.For the case that the exponential window is defined by increasing this β factor 16times, the added damping is approximately equal to the real part of the pole ofthe first mode, meaning that the pole is located twice as far from the imaginaryaxis of the Nyquist plane. In § 3.7, the parametric results will be compared aftercompensating the real part of the pole for the added damping.

3.4. Exponential Windowing of FRF Estimates 67

8.63 8.64 8.65 8.66 8.67 8.68 8.69 8.7 8.71

10

15

20

25

Freq. (Hz)

|m/s

2 N| (

dB)

1 β, 2β, 4β, 8β, 16β

35 35.2 35.4 35.6 35.8

−9

−8

−7

−6

−5

−4

−3

Freq. (Hz)

|m/s

2 N| (

dB)

Figure 3.8: The true FRF (bold line), ETFE (dashed line) and the Hexp1 estimator for

increasing value of β, while length of the IRF (and hence the frequency resolution) waskept the same.

3.4.2 Nonparametric Least Squares IRF Estimator

Another way for obtaining the IRF from the data is found by considering (forsimplicity) the (SISO) relation in the discrete-time domain between a measuredinput and output signal

x[m](tn) = hxf (tn) ∗ f [m](tn) (3.20)

where the input f [m](tn) and input x[m](tn) were divided overM adjacent records.The IRF hxf (tn) = hxf (nTs) for n = 0, . . . , Ns − 1 can be computed directly inthe time-domain by solving the following over-determined set of equations as aclassical least squares problem [f ]hxf = x or explicitly given by Eq. (3.21).The submatrices of [f ] are non-symmetric (Ns × Ns) with a Toeplitz structure.Solving this LS problem (3.21) yields an estimate of hxf (tn). Notice however,that the first submatrix of [f ] contains the samples of the generator signal at thenegative time instants, which in practice correspond to the initial conditions andare generally not known.

Remark : The generalization for an errors-in-variables noise model using an instru-mental variables measurement setup is straightforward, by considering the relation(3.20) between the generator and respectively the force and response signals andsubstituting the final FRF estimates in Eq. (3.2).


f0 f−1 f−2 · · · f−(Ns−1)

f1 f0 f1 · · · f−(Ns−2)

......

.... . .

...fNs−1 fNs−2 fNs−3 · · · f0

−−−− −−−− −−−− −− −−−−fNs fNs−1 fNs−2 · · · f1

fNs+1 fNs+1 fNs · · · f2

......

.... . .

...f2Ns−1 f2Ns−2 f2Ns−3 · · · fNs

−−−− −−−− −−−− −− −−−−f2Ns f2Ns−1 f2Ns−2 · · · fNs+1

......

.... . .

...fMNs−1 fMNs−2 fMNs−3 · · · f(M−1)Ns

hxf0

hxf2

hxf3

...

hxfNs−1

=

x0

x1

x2

...xNs−1

−−xNs

xNs+1

...x2Ns−1

−−x2Ns

...xMNs−1

(3.21)

In order to reduce the effect of leakage as well as noise, a one-sided exponentialwindow can be applied again to this IRF

hxfLSexp

(tn) = hxfLS

(tn).e−β|n|Ts (3.22)

with 0 ≤ n ≤ Ns − 1 and finally HLSexp(ω) is given by the FFT of (3.22).

An advantage of this approach, compared to the approach in § 3.4.1, is thatthe system IRF can be computed without having to transform to the frequency-domain. However, in order to solve the LS problem in a time-efficient manner,“Fast Toeplitz solvers” need to be used to reduce the computation time whenNs is typically 2048 or higher. This can be considered as a disadvantage of thisapproach since these solvers can be numerically bad conditioned and assume that[f ] is symmetric and positive definite. For Ns smaller than 2048 a simple matrixinversion of [f ] can be used.

Instead of using all data samples to formulate the large LS problem (3.21),M − 1 compact LS problems can be solved from the M − 1 last blocks of Ns

equations. This yields M − 1 different estimates of the IRF hLSexp[l](l=2,. . . ,M)from which a sample mean and sample variance can be computed. Assuming theM − 1 IRFs statistically independent observations, the sample mean is given by

HLSexp

(ωf ) =1

M − 1

M∑

l=2

HLSexp[l](ωf ) (3.23)

while an estimate of the sample variance is computed as

varHLSexp(ω) = 1

M − 1

(

1

M − 2

M∑

l=2

∣

∣

∣HLSexp[l](ωf )−H

LSexp(ωf )

∣

∣

∣

2)

(3.24)

3.5. Iterative Nonparametric FRF Estimator 69

3.5 Iterative Nonparametric FRF Estimator

A different approach for leakage reduction was proposed in [126]. The basic ideais to remove the effect of leakage on the FRF by iterative improvement. Althoughthe approach was first presented for the H1 estimator (for SISO systems), it isstraightforward to generalize the results to the multivariable Hiv estimator. Theproposed algorithm comprises of the following steps:

1. Calculate HZR1 (ωf ) , f = 0, 1, . . . , Ns/2 − 1 using Eq. (3.1) and the peri-

odogram PSD estimator (cf. § 3.3.1). A Hanning window is used to reducethe leakage error.

2. Calculate the inverse FFT of HZR1 : h[n] = IFFT

(

HZR1 (ωf )

)

.

3. Eliminate the offset errors in h[n]

h[n] = h[n]− δ with δ =1

Ns − n

Ns−1∑

n=N2

h[n] (3.25)

The reason for this is that the offset error, which is a systematic error, isO(√Ns) if the input signal contains no DC component, else the error becomes

O(Ns) [126]. A typical choice is N2 = 0.9Ns, since the algorithm requiresthat h[n] can be set to zero for n > Ns.

4. Determine the optimal length N1 of the IRF, by looking for the minimumof the approximation error s(N1)

s(N1) =

N3∑

n=N1+1

e2N1[n] (3.26)

with eN1[n] = z[n]− zN1

[n], zN1[n] =

∑N1

l=0 h[l]r[n− l] and N3 a user-selectednumber. Since the optimum is typically rather flat, it is sufficient to checkonly a small number of different filter lengths (e.g. Ns/4, Ns/2, 3Ns/4, Ns.If the optimal length is Ns, then it is strongly advised to increase the lengthof the subrecords Ns.

5. Calculate the new (updated) FRF estimate

HZR1 (ωf ) = HN1

+ GPeN1

R(ωf )GP−1

RR (ωf ) (3.27)

with HN1the transfer function of the finite impulse response (FIR) filter

with length N1. The error terms GPeN1

R(ωf )GP−1

RR (ωf ) can be interpreted as

a transient error as shown in [111] starting from the observation that whatis called a leakage error is essentially a transient problem.


8.63 8.64 8.65 8.66 8.67 8.68 8.69 8.7 8.71

10

15

20

25

M = 2, 4, 8, 16

13.56 13.58 13.6 13.62 13.64 13.66 13.68 13.7

6

8

10

12

14

16

18

20

Freq. (Hz)

|m/s

2 N| (

dB)

M = 2, 4, 8, 16

Figure 3.9: The iterative (H1) FRF estimator for increasing number of averages M =2, 4, 8, 16 and decreasing blocksize Ns = 64, 32, 16, 8K (Ntot = 128K samples).

6. Repeat the procedure from step 3 with

h[n] = IFFT(

HZR1 (ωf )

)

(3.28)

The iteration process is controlled by monitoring the minimum of s(N1).Once this error starts to increase, the iteration process is stopped.

Example: Using this iterative approach, the FRFs of the 7 DOF system wereestimated subsequently using M = 2, 4, 8, 16. The results, given in Figure 3.9,show that this approach improves the FRF estimates compared to the classicalH1. As cited in [126], this approach requires a large number of data samples in thecase of low damped systems as is typical for mechanical structures. Reducing Ns

in order to reduce the variance, results in a bias error, which is however smallerthan for the H1 for comparable blocksizes (cf. Figure 3.2).

3.6 Cyclic Averaging Approach

Recently, the use of cyclic averaging was proposed as a method for reducing theleakage error in the estimation process of frequency response functions for modalanalysis applications [6, 103, 104]. Cycling averaging, historically, has been usedin rotating machinery data analysis to enhance characteristics in the data thatare functions of the rotating speed. The same averaging technique has been usedfor estimating FRFs to enhance the frequency characteristics in the data thatare integer functions of the observation window while minimizing the nonperiodic

3.6. Cyclic Averaging Approach 71

characteristics and so reducing the leakage. The main idea is to use the techniqueof cyclic averaging prior to the FRF computation (e.g. H1, Hiv) by means of thePeriodogram estimator (cf. § 3.3.1).

Consider againM adjacent records of the generator r[m](tn), force f[m](tn) and

response x[m](tn) signals with Ns samples in each a record and the total number oftime samples Ntot = MNs. To illustrate the cyclic averaging technique, considerfor example a record of one of the measured response signals denoted as xo(tn).The Discrete Fourier Transform (DFT) of this record (cf. Eq. (2.2)) can also bewritten as

Xo(ωfP ) =

Ns−1∑

n=0

x(tn)e(−i2πnfP

Ns) (3.29)

with ωfP = 2πfP/NsTs and P an integer number such that fP = 0, P, . . . ≤Ns − 1. Choosing for example P = 2, Eq. (3.29) for f = 1 becomes

Xo(ω2) =

Ns−1∑

n=0

x(tn)e(−i4πn

Ns) (3.30)

=

Ns/2−1∑

n=0

x(tn)e(−i2πnNs/2

) +

Ns−1∑

n=Ns/2

x(tn+Ns/2)e(−i2πnNs/2

)

where the Fourier transform is computed by describing the unity circle twice forn = 0, 1, . . . , Ns. Using the time shift theorem of the Fourier transform this canalso be written

Xo(ω2) =

Ns/2−1∑

n=0

[

x(tn) + x(tn+Ns/2)]

e(−i2πnNs/2

) (3.31)

and in general Eq. (3.29) becomes

Xo(ωfP ) =

Ns/P−1∑

n=0

[

P−1∑

l=0

x(tn+lNs/P )

]

e(−i2πnfNs/P

) (3.32)

From this it follows that the cyclic averaging process is tuned by choosing P sincethis divides a record of Ns samples into P blocks that are averaged by describingthe unity circle P times. This results in a Fourier spectrum that has Ns/P spectrallines and thus with a frequency resolution that is P times smaller. Before cyclicaveraging, a Hanning window of length Ns is first applied to the records r[m](tn),f [m](tn) and x

[m](tn). The FRF computation is done by means of the Periodogramapproach (cf. §3.3.1) starting from the M cyclic averaged Fourier spectra.

Figure 3.10, shows the results for the classical Periodogram H1 estimator(with M = 4) combined with cyclic averaging. The data was first cyclic aver-aged using P = 1, 2, 4, 8 blocks with a corresponding decrease of the blocksize


8.63 8.64 8.65 8.66 8.67 8.68 8.69 8.7 8.71

10

15

20

25

Freq. (Hz)

|m/s

2 N| (

dB)

exactM=4,P=1M=4,P=2M=4,P=4M=4,P=8

13.56 13.58 13.6 13.62 13.64 13.66 13.68 13.7

6

8

10

12

14

16

18

20

Freq. (Hz)

|m/s

2 N| (

dB)

exactM=4,P=1M=4,P=2M=4,P=4M=4,P=8

Figure 3.10: The Periodogram H1 estimator combined with cyclic averaging withM = 4and P = 1, 2, 4, 8 resulting in a decreasing blocksize Ns = 32, 16, 8, 4K samples (Ntot =128K samples).

(Ns = 32, 16, 8, 4K) for a total of 128K samples. A Hanning window was appliedto reduce leakage.

Although cited in literature [6, 104] as an approach for the reduction of leakage,it follows from these results that cyclic averaging is merely a signal processingtechnique to reduce the computation time for deriving the FRFs. The amountof memory required to store this data is also reduced since it contains a factor Pless frequency lines compared to the large blocksize case (i.e. Ns/2). However,no gain in terms of SNR can be obtained while the measurement (testing) timeitself cannot be reduced. Moreover, the lower frequency resolution can also resultin a less accurate reading of resonance frequencies as can be seen in Figure 3.10and closely spaced modes may be less accurate estimated during the parametricidentification. Furthermore, in this case of a low damped system, this approachsuffers from leakage in the same way as the classical H1 FRF estimators (for asame number of averages M).

3.7 Comparison of Methods

Since the decrease in frequency resolution and the additional damping (bias) in-troduced by the exponential window make a nonparametric comparison a difficulttask, the comparison is based on the modal parameters. These were estimated us-ing a frequency-domain maximum likelihood estimator (cf. Chapter 5). Since thebias error mainly affects the damping of the modes, the parametric results were

3.8. Experimental Case Study 73

compared by means of the relative bias error for the damping defined as Re(pr−p0)Re(p0)

.

The results for the classical H1, Hexp1 and the iterative Hiter

1 approach are givenin Table 3.2 for the damping ratio of first mode. The Correlogram approach wasnot considered, given the poor nonparametric results.

Ns (K) H1 Hexp1 Hexp

1 Hiter1

(M = 1) (M = 2)128 -0.0068 0.0006 / /64 -0.0170 0.0008 -0.0168 -0.017232 0.0683 0.0015 -0.0167 0.021716 0.0886 0.0018 -0.0168 0.05528 0.2369 0.0020 -0.0207 0.0964

Table 3.2: Relative bias error for the damping ratio ζ (%) of the first mode for differentapproaches.

When decreasing the blocksize Ns, the H1 estimator (cf. § 3.3.1) clearly suffersfrom leakage, resulting in a systematic error up to 24% for the case of Ns = 8K,while the newly proposed FRF estimator Hexp

1 (cf. § 3.4) using the ETFE hasonly a very small bias error of 0.2%. Notice, that even in the case that the addeddamping is as large as the damping of the mode (i.e. Ns = 8K where β = 16β1),a proper compensation for the exponential window (cf. Eq. 3.18) can still be doneon the parametric results. When applying the Hexp

1 estimator with M = 2, itcan already be seen that a small systematic error exists on the initial H1 estimate,before applying the exponential window (3.17), since the relative bias error remainssimilar for an increasing Ns. The iterative approach Hiter

1 clearly performs betterthat the classical H1, although this method also introduces an important bias erroronce the blocksize becomes too small as was also explained in § 3.5.

3.8 Experimental Case Study

This section will discuss the application of the most important methods derivedin Chapters 2 and 3 for the nonparametric processing of the data obtained froma subframe of a car. Figure 3.11 shows the multi-input modal test setup usingtwo electrodynamic shakers. The subframe is supported by elastic chords in orderto approximated the free-free conditions. Time histories of 256K samples aremeasured for the two applied forces and 23 responses (accelerations in vertical or+Z direction) using a random noise excitation in a frequency band 0–1024Hz witha sampling rate of 2048Hz. Since this subframe structure is characterized by a lowdamping (ζ ∼ 0.5− 1.5%), caution is needed with respect to leakage.

The Hiv estimator was used to estimate the FRF (2.29) and corresponding


noise covariance matrix (2.39). Hereto, the time histories were divided into 32blocks (M = 32) of 8K time samples (Ns = 8K) on which a Hanning windowwas applied to reduce possible effects of leakage. Figure 3.12 shows the FRF(1,1)as well as its variance obtained by the Hiv estimator, from which it can be seenthat the uncertainty on the data is highest at the resonance frequencies. Thisis explained by considering the signal-to-noise (SNR) ratios for the correspondingforce, due to force-drops in the input at resonance frequencies. The response signalhas lower SNRs at the anti-resonance frequencies. Furthermore, the harmonics of50Hz can be noticed in both the force and response. These are introduced by thepower amplifiers, and so can be considered as a part of the excitation explainingwhy these harmonics are not present in the FRF estimates.

Next, different FRF estimators are compared for this practical case in Fig-ure 3.13 (top) by zooming in on the first mode at 55.5Hz. The signal processingparameters were the same as for the Hiv, i.e. M = 32, Ns = 8K and a Hanningwindow. As can be seen, the Hiv is situated between the H1 and H2 estimate, whilethe Hv almost coincides with the H2. Computing the bias for each frequency us-ing Eq. (2.25) and Eq. (2.28) for respectively the H1 and H2 estimator, results inFigure 3.13 (bottom), indicating that both estimators yield a biased result. Com-paring the difference between the H1 and Hiv and the same for the H2 with thetheoretically computed bias, clearly indicates that Hiv is a consistent FRF esti-mate since an errors-in-variables noise model is considered. On the contrary, thebias error is largest for the H2 explained by the poor SNR of the response signalin the lower frequency range.

Based on the knowledge of the full noise covariance matrix, it was also possibleto compute the degree of correlation between each of the Hiv FRF estimates,as shown for 3 close frequencies around the first damped resonance frequency at55.5Hz in Figure 3.14 (a). As can be seen at the resonance frequency of 55.5Hz,the FRFs corresponding to the same input have a high correlation, which can beexplained by the fact that near a resonance the matrix HXR

1 in Eq. (2.39) has arank 1. As a result, in practice, the FRFs corresponding to the same input havea high correlation, except for the DOFs corresponding to nodal points of thatmode. Furthermore, some correlation is introduced by the interaction between theshakers and the structure, explaining the medium correlation between FRFs forthe same response DOF and the two input locations. From this experimental casestudy, it also follows that higher correlations do appear at resonance frequencies,while being small even only a few DFT lines away from the resonance. Moreover,in this case the shaker-structure interaction is small, avoiding high correlationsbetween FRFs corresponding to different input and same response location.

When increasing the number of blocks M to 64 and 128 and hence decreasingthe blocksize Ns to 4K and 2K, leakage certainly poses a problem given the lowdamping of the subframe. Although the use of a Hanning window reduces theleakage effects resulting in Figure 3.15 (top), an attenuation (bias error) is present.The effect of leakage is clear from Figure 3.15 (bottom) where a rectangular window

3.9. Conclusions 75

is used resulting in important errors in the H1 FRF estimate as could be expected.On the contrary, the results obtained by the Hiv are still acceptable even for theshort blocks of 2K samples (M = 128). This indicates that the Hiv is clearlymore robust for leakage, compared to the H1, which can be explained by the factthat leakage mainly translates into stochastic errors that are non-coherent with thegenerator signals (or instruments) and hence are averaged out by the Hiv approach.

Finally, the subframe data was processed with the classical H1, Hiter1 and Hexp

1

estimators (cf. Chapter 3), for which the damping ratios estimated by the para-metric ML estimator (cf. Chapter 5) are summarized in Figure 3.16. For the H1

and Hiter1 , the number of blocks M was increased as M = 2k for k = 1, 2, . . . , 7.

From the parametric results given in Figure 3.16 based on the H1 (top) and Hiter1

(middle) it clearly follows that an overall improvement is obtain by means of thenonparametric iterative estimator, especially for the case that important leakageeffects are present in the data (i.e. M > 5). However, applying the Hexp

1 esti-mator, for an increase of the β-factor (and thus increasing cut-off) of the appliedexponential window as β = kβ1 for k = 1, 2, . . . , 7, while Ns is fixed to 128K sam-ples, results in Figure 3.16 (bottom). This clearly illustrates the robustness forleakage of this last estimator since the initial FRF estimate (cf. Figure 3.7) is com-puted based on a small number of large (and leakage free) blocks. Also notice thatthe additional damping introduced by the exponential window can be correctlycompensated on the modal damping even for the higher cut-off (β) values.

3.9 Conclusions

The problem of leakage, encountered when using arbitrary excitation for modaltesting, has been studied in this chapter. Leakage effects result from using ar-bitrary signals, such as random noise, within a finite measurement time. It isshown that, in the case of the classical approach for FRF estimation, using a Peri-odogram estimator, the severity of these errors depends on the trade-off betweenmaximizing the number of samples within a time record (and hence the frequencyresolution) to reduce the effects of leakage, and maximizing the number of recordsavailable for averaging to reduce the statistical variations on the FRF estimates.

In modal analysis, the use of an exponential window is often applied to reduceleakage in the case of a time-limited excitation such as for an impact or burstrandom excitation. In this chapter the use of an exponential window has beengeneralized for random noise excitation. Several possible approaches for leakagereduction on the estimated FRFs have been studied and compared with techniquesproposed in the literature. The most important characteristics of the differentmethods are summarized in Table 3.3.

The robustness for both leakage and measurement noise is poor for the classicalFRF estimators Hper and Hcycl given the trade-off conflict. As a result, the number


Hper Hcycl Hcorr Hexp HLS Hiter

Robustness leakage / noise – – – – – + + + + +Window Compensation – – – – + + + + + +Computation efficiency + + + + + + + – – +

Table 3.3: Comparison of different estimators for nonparametric FRF identification inthe presence of leakage.

of records M needed to have an acceptable variance will introduce problems ofleakage, especially in the case of low damped mechanical systems. For the samereason, the effect of a Hanning window, which cannot be completely compensatedfor, becomes more important for a decreasing number of samples within a record.On the other hand, these approaches are very fast, which is a requirement forprocessing the large data sets in the case of modal analysis applications.

Although, the use of the Correlogram approach for FRF estimation, makes theapplication an exponential window in combination with random noise excitationpossible, the ratio of the correlogram Power Spectra that already contain the effectsof the window, appears not robust in practice, again resulting in important errors.

On the contrary, the Hexp FRF estimator, based on the use of an exponentialwindow in combination with the classical FRF estimators (such as e.g. H1 andHiv), turns out to be a robust method with respect to both leakage and measure-ment noise. By using large blocksizes, the effects leakage on the FRF estimatesare minimized, while the exponential window is actually used to reduce the statis-tical variance on the FRF estimate. Although this results in biased nonparametricresults, the additional damping is known and its effect on the modal parameterscan be compensated, which is not the case for e.g. a Hanning window. Further-more, the noise covariances for the windowed FRF estimates Hexp can still bedetermined. The robustness for leakage of this method was clearly demonstratedcompared to the classical Hper and the iterative Hiter FRF estimators. Further-more, this method is simple to apply in practice, since commercial measurementequipment for modal analysis often only stores the measured FRF data. Thismethod requires a sufficient total number of time samples such that no leakageis present in the empirical transfer function estimate, which however is a generalrequirement for each of the discussed nonparametric FRF estimators.

An alternative method, based on the same idea, starts from the impulse re-sponse functions that are found by solving a nonparametric Least Squares problemdirectly in the time-domain. This HLS FRF estimator has similar performancewith respect to accuracy for leakage and measurement noise as the Hexp, but re-quires too much computation time when used in practice for typical modal analysisproblems.

Finally, the nonparametric iterative FRF estimator Hiter, that reduces some

3.9. Conclusions 77

of the leakage effects by iterative improvement, clearly performs better than theclassical approach. Nevertheless, this method is also confronted with the trade-offproblem, requiring sufficient records in order to yield an acceptable stochastic erroron the FRF estimates. As a result, the effects of leakage and the Hanning windowbecome important in the case that the blocksize becomes too small. Nevertheless,further optimization of this method by means of exponential windowing, can alsoresult in a robust method for FRF identification in the presence of leakage, besidesthe Hexp approach.


Figure 3.11: Multi-input modal test setup using two electrodynamic shakers. The re-sponses are measured in 23 DOFs using accelerometers.

3.9. Conclusions 79

0 100 200 300 400 500 600 700−80

−60

−40

−20

0

20

40

60

Freq (Hz)

Am

pl. (

dB)

Hiv

var(Hiv

)

0 100 200 300 400 500 600 7000

10

20

30

40

50

Freq (Hz)

Am

pl. (

dB)

SNR F1

0 100 200 300 400 500 600 7000

10

20

30

40

50

Freq (Hz)

Am

pl. (

dB)

SNR X1

Figure 3.12: FRF(1,1) and variance obtained by the Hiv estimator (top) and SNRs forcorresponding force (middle) and response signal (bottom).


54.5 55 55.5 56 56.5−10

−5

0

5

10

15

20

Freq (Hz)

Am

pl. (

dB)

H1

H2

Hv

Hiv

54.5 55 55.5 56 56.5−5

−4

−3

−2

−1

0

1

2

3

4

5

Freq (Hz)

Bia

s E

rror

(dB

)

H2

H1

Figure 3.13: Comparison of different FRF estimators for subframe data (top) and the-oretical bias for H1 and H2 result (bottom).

3.9. Conclusions 81

54 56 58 60 62

−20

−15

−10

−5

0

5

Freq (Hz)

Ampl

. (dB

)

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FRF

FRF

10 20 30 40

5

10

15

20

25

30

35

40

45

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FRF

FRF

10 20 30 40

5

10

15

20

25

30

35

40

45

(c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FRF

FRF

10 20 30 40

5

10

15

20

25

30

35

40

45

(d)

Figure 3.14: Degree of correlation between each of the Hiv FRF estimates for 3 closefrequencies around the first damped resonance frequency at 55.5Hz shown in (a), i.e. at53.5Hz (b), 55.5Hz (c) and 57.5Hz (d).


540 550 560 570 580 590 600 610 620 630 64010

15

20

25

30

35

40

45

50

55

60

Freq (Hz)

Am

pl. (

dB)

H1

Hiv

540 550 560 570 580 590 600 610 620 630 64010

15

20

25

30

35

40

45

50

55

60

Freq (Hz)

Am

pl. (

dB)

H1

Hiv

Figure 3.15: Comparison of the H1 and Hiv for increasing M to 64 (solid and dashedline) and 128 (• and ×) when using a Hanning (top) and rectangular (bottom) window.

3.9. Conclusions 83

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

k (M = 2k)

Dam

ping

ratio

(%)

mode 1mode 2mode 3

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

k (M = 2k)

Dam

ping

ratio

(%)

mode 1mode 2mode 3

1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Dam

ping

ratio

(%)

beta = k.beta1

mode 1mode 2mode 3

Figure 3.16: Comparison of the damping ratios ζ (%) estimated by the parametric MLbased on the H1, H

iter1 FRFs for a decreasing blocksize (M = 2k for k = 1, 2, . . . , 7) and

the Hexp1 for an increasing β-factor (β = kβ1 for k = 1, 2, . . . , 7).

Chapter 4

Introduction to ModalParameter Estimation

This chapter introduces the concepts of parametric system identification for modalparameter estimation. First, parametric models that can be used to describe thevibration behaviour of mechanical systems are discussed in Section 4.2. The differ-ent models are interconnected by means of a matrix polynomial description. NextSection 4.3 gives an extensive overview of parametric identification methods orig-inating from control theory and different engineering areas, with their potentialsand drawbacks from the point of view for MPE.

85

86 Chapter 4. Introduction to Modal Parameter Estimation

4.1 Introduction

The next section discusses several models that can be used to represent the in-formation about the vibration behaviour of a structure, present in the measure-ments, with a limited number of parameters. Although the modal model is mostappropriate for a physical understanding of this vibration behaviour, most iden-tification techniques do not directly identify this model since it is nonlinear-in-the-parameters. Instead, models such as ARMAVX and State Space and MatrixFraction Descriptions originating from electrical engineering and control theoryare often used. The modal parameters can be derived once a model is identifiedfrom the data, given the linear relation between the different models.

An overview of identification methods that can be used to extract the parame-ters of these models is given in Section 4.3. Based on the properties of the identifiedmodel and the algorithm implementation, a better insight in the potential for ex-tracting the model parameters from large modal data sets is obtained. A majorgrouping is usually done based on the domain in which the data is treated numeri-cally resulting in time-domain and frequency-domain methods. This overview alsogives a good introduction and motivation for the parametric approach that will beused in Chapters 5 and 6.

4.2 Parametric Models

Matrix Fraction Descriptions

A first model to describe the vibration behaviour of a mechanical structure withNm degrees of freedom is given by Newton’s equation of motion:

Mx(t) + Cx(t) +Kx(t) = f(t) (4.1)

with M , C and K respectively the mass, damping and stiffness matrices, and f(t)and x(t) the applied force and structural response vectors. Using the Laplacetransform, the frequency-domain equivalent is given as

G(s)X(s) = F (s) (4.2)

with G(s) =Ms2 + Cs+K the dynamic stiffness. A matrix inversion yields

F (s)H(s) = X(s) (4.3)

with H(s) the transfer function matrix, which can be expressed as

H(s) =(

G(s))−1

=Gadj(s)

|G(s)| (4.4)

4.2. Parametric Models 87

While the numerator Gadj(s) (the adjoint matrix) is an (Nm × Nm) matrix con-taining polynomials in s of order 2(Nm − 1), the denominator is a polynomial ins of order 2Nm. Therefore, Eq. (4.4) can also be written as

H(s) =B(s)

A(s)=

B1,1(s) . . . B1,Nm(s)...

. . ....

BNm,1(s) . . . BNm,Nm(s)

A(s)(4.5)

which is called a common denominator model or also scalar matrix fraction de-scription.

In fact, this expression can also be considered as a special case of multivariabletransfer function models described using a Matrix Fraction Description (MFD),i.e. the ratio of two matrix polynomials [74, 51]. Two kinds of MFDs are possible,i.e. a left MFD (LMFD)

H(s) =(

AL(s))−1

BL(s) (4.6)

where the matrix coefficients ALr are (No ×No) matrices and BLr are (No ×Ni)matrices or a right MFD (RMFD)

H(s) = BR(s)(

AR(s))−1

(4.7)

where the matrix coefficients ALr are (Ni ×Ni) matrices and BLr are (No ×Ni)matrices. When the frequency response function matrix H(ωf ) is used as primarydata, both descriptions can be linearized in the matrix coefficients (parameters),while for input/output Fourier data only the LMFD results in a linear relationship.

From the LMFD, Eq. (4.4) is found for the specific case that the number of out-put observationsNo equals the number of degrees of freedomN since AL(s) = G(s)and BL(s) a constant (No ×Ni) matrix. Notice that AL(s) now is a second (low)order matrix polynomial since the matrix coefficients are (Nm × Nm) matrices.Since the dynamic stiffness G(s) is a matrix polynomial of second order with thecoefficients (Nm × Nm) matrices, the characteristic equation, given by the de-terminant of G(s), is of order 2Nm. However, in the case that the number ofresponse locations No is smaller than the number of degrees of freedom (modes)Nm, this model has to be expanded to a higher order (2n) matrix polynomial withsmaller matrix coefficients in order to identify Nm modes (roots, poles or modalfrequencies)

Γ2ns2n + Γ2n−1s

2n−1 + Γ2n−2s2n−2 + . . .+ Γ0 = 0 (4.8)

of which the characteristic equation and hence the roots are the same (up to ascale factor) as for the second order matrix polynomial equation (4.2). The limitof this expansion process results in a polynomial of high order 2Nm with scalarcoefficients

α2Nms2Nm + α2Nm−1s

2Nm−1 + α2Nm−2s2Nm−2 + . . .+ α0 = 0 (4.9)


and likewise, the roots of this scalar characteristic equation are the same as forthe second order matrix polynomial equation.

Therefore, the number of characteristic values that can be determined dependsupon both the size of the (Nc×Nc) matrix coefficients and the order of the matrixpolynomial (nj) given by the following relation Nc.nj = 2Nm.

For the case of modal analysis, the number of responses No is typically muchlarger than the number of applied forces Ni. Hence, for a LMFD (4.6), AL(s)is a low order matrix polynomial with large matrix coefficients (No × No), whilefor the RMFD (4.7), AR(s) is a high order matrix polynomial with small matrixcoefficients (Ni×Ni). Based on this relation, a so-called Unified Matrix PolynomialApproach (UMPA) was proposed in [5], for the comparison of different estimationalgorithms using a common mathematical structure.

Another class of models often used for time-domain system identification thatalso uses matrix fraction descriptions, are the so-called ARMAVX models [85, 131].A commonly-used formulation for the input-output relationship is described as

x(t) +A1x(t− 1) + . . .+Anax(t− na)= B1f(t− 1) + . . .+Bnbf(t− nb) + e(t) (4.10)

where the Ar are (No×No) matrices and the Br are (No×Ni) matrices. Definingthe backward shift operator as q−1f(t) = f(t−1), the following matrix polynomialsin q−1 can be introduced

A(q) = 1 +A1q−1 + . . .+Anaq

−na

B(q) = B1q−1 + . . .+Bnbq

−nb (4.11)

and equation (4.10) becomes

x(t) = H(q)f(t) +G(q)e(t) (4.12)

withH(q) = A−1(q)B(q) , G(q) = A−1(q) (4.13)

where the factorization of H(q) in terms of two matrix polynomials is also called aLMFD. This model is called the ARX model, where AR refers to the autoregressivepart A(q)x(t) and X to the input or exogeneous part B(q)f(t). In the special casewhere na = 0, x(t) is modelled as a finite impulse response (FIR). By introducingmore matrix polynomials in the equation error model structure other variants arederived, such as ARMA, ARMAX,... [85].

Pole-Residue Model

Often used in modal analysis is the Partial-Fraction Description (PFD), also calledthe pole-residue parameterization, which is found through a partial fraction ex-

4.2. Parametric Models 89

pansion of Eq. (4.4)

H(s) =

Nm∑

r=1

Rr

s− pr+

Nm∑

r=1

R∗r

s− p∗r(4.14)

with pr and Rr respectively the poles and (Nm ×Nm) residue matrices as modelparameters corresponding to the rth mode of the structure. Although a PFD isnumerically better conditioned than high-order polynomial models (MFDs), thePFD is nonlinear-in-the-parameters. Equivalently, the inverse Laplace transformyields this model in the time-domain

h(t) =

Nm∑

r=1

Rreprt +

Nm∑

r=1

R∗rep∗rt (4.15)

For modal analysis purposes, the residue matrices are further decomposed as [62]

Rr = QrΨrΨTr = ΨrL

Tr (4.16)

with Ψr, Lr respectively the mode shape and modal participation factor vectorand qr the modal scale factor for mode r. Modal participation factors are a resultof multiple reference (input) modal parameter estimation and define how eachmode is excited from each of the input locations.

State Space Formulation

Reformulating Eq. (4.1) as

[

x(t)x(t)

]

=

[

0 I−M−1K −M−1C

] [

x(t)x(t)

]

+

[

0M−1

]

f(t) (4.17)

and defining respectively the state vector y(t) and the output and direct input-output transmission matrices C and D as

y(t) =

[

x(t)x(t)

]

and C = [I 0] , D = [0] (4.18)

a state-space description of the system is given by

y(t) = Ay(t) +Bf(t) + w(t)

x(t) = Cy(t) +Df(t) + v(t) (4.19)

with the matrices A, B respectively the (dynamical) system and input matrix,which together with C and D represent the system’s realization describing thesystem’s response at any time to a known input signal. The eigensolutions ofthe (2Nm × 2Nm) system matrix A yield the system poles, while the mode shapeand modal participation factor vectors are respectively found from the matrices


C and B. The zero-mean white noise vectors w(t) and v(t) can be added tomodel respectively the process noise and output measurement noise. By using anauxiliary state vector, Eq. (4.1) has become a first order differential (or difference)expression, i.e. the state space model. The frequency-domain version of the statespace formulation (4.19) (with the noise vectors w(t) = 0 and v(t) = 0) yields ageneral expression for the transfer function matrix

H(s) = C[

sI −A]−1

B +D (4.20)

4.3 Overview of MPE Algorithms

During the last three decades, extensive effort has been devoted to the develop-ment of methods that aim to produce a reliable identification of the dynamicalbehaviour of a structures by means of the modal parameters. As a result of theintroduction of the Fast Fourier Transform (FFT) and the development of multi-channel acquisition and powerful storage and computation capacity, the methodshave evolved from very simple single degree of freedom (SDOF) techniques tomethods that analyse data from multiple-input excitation and multiple-outputresponses simultaneously in a multiple degree of freedom (MDOF) approach.

SDOF frequency-domain methods

The earliest methods work in the frequency-domain and belong to the class ofSDOF methods. These methods assume that the modes are real and the dynam-ical behaviour within a small frequency band can be described by a single mode.The Peak Amplitude method [17] simply takes the natural frequencies from theobservation of the peaks on the graphs of the magnitude of the frequency responsefunction. The damping ratios are derived from the sharpness of the peaks and themode shapes are calculated from the ratios of the peak amplitudes at various re-sponse locations. Like other SDOF methods, as for instance Quadrature Response[101] or Circle Fitting method [76], the Peak Amplitude method will yield erro-neous results, especially for the damping, when the modes are closely spaced andcoupled, because only one mode can be estimated at the same time. Moreover,the SDOF methods are very sensitive to measurement noise and require exten-sive interaction from the user. Nevertheless, these methods are still used even incommercial analysis software, mainly because of the limited computation time,although during the recent years the multiple degree of freedom methods havecompletely taken over when an accurate MPE is wanted.

MDOF time-domain methods

In the class of MDOF methods, a major grouping is usually done based onthe domain in which the data are treated numerically resulting in time-domainand frequency-domain methods. One of the first MDOF time-domain algorithms

4.3. Overview of MPE Algorithms 91

specifically designed for MIMOMPE was the Ibrahim Time-Domain (ITD) method[66]. By means of the Modal Confidence Factor (MCF) [65], this method has beenadapted for an automated model order selection in order to minimize the requireduser-interaction. The main disadvantage of the ITD is the computational com-plexity that origins from solving the large eigenvalue problems that result fromreformulation the time-domain equivalent of the modal model (4.15) as an eigen-value problem. Since this method starts from the free-response time data, theeigenvectors (mode shapes) cannot be mass-normalized. Taking sparse matrixstructures into account it was possible to reduce the computational load [67], al-though this faster version is still too slow to be used for current MPE practices.

Another well-known time-domain algorithm, which starts from the state spaceformulation (4.19) (with the noise vectors w(t) = 0 and v(t) = 0) to derivethe minimum realization of the system matrices, is the Eigensystem RealizationAlgorithm (ERA) [71, 70]. Although this algorithm has been successfully applied inmany modal analysis applications, it is computationally involved since it requiresthe singular value decomposition of the so-called block Hankel matrix, which isconstructed from the impulse response function (IRF) data and in general a largematrix (NoNs ×NiNs) with Ns the number of time samples. An investigation onthe effects of noise on the estimated parameters using ERA is discussed in [72],while the faster ERA/DC variant, which uses autocorrelation functions as data, isintroduced in [73].

During the recent years the state space formulation (4.19) has also been thebasis for subspace identification methods [141]. Several combined deterministic-stochastic algorithms, such as for instance Balanced Realization (BR), Multivari-able Output-Error State sPace (MOESP) and Canonical Variate Algorithm (CVA)were proposed to identify the state matrices as well as the stochastic variables in(4.19). Subspace identification algorithms are non-iterative, and thus always yielda solution without the risk for convergence problems. However, since the mea-sured force and response time sequences are used without any preprocessing, thedata is in general characterized by high levels measurement noise. Also no re-duction of the amount of data is obtained through an averaging process as is thecase for IRFs, making the computational burden one of the drawbacks. So far inthe domain of modal analysis, subspace identification has been mainly applied forMPE from output-only data (forces are not measured) acquired from civil [100]and mechanical structures [59, 60, 10].

Another class of time-domain system identification methods, having its rootsin the domain of control theory and applications, are the PEM-based methods,such as ARMA, ARMAX,..., using a variant of the ARMAVX model structure.However, since this model is a LMFD, the matrix polynomials have a low orderwith large matrix coefficients, where the product of the model order nj and numberof responses has to be greater than 2Nm in order to find 2Nm system poles. ThePEM approach is less suited for MPE, since an increase of the model order byone results in large increase of the number of system poles, making it difficult to


determine the optimal model order by means of a so-called stabilization chart. In[59], the ARV method has been used for MPE from output-only data.

A much faster time-domain method is the Least Squares Complex Exponential(LSCE) algorithm first introduced by [23] as an extension of Prony’s method [112],which states that the roots of an underdamped system always occur in complexconjugate pairs and therefore can be used to form a characteristic polynomial withreal coefficients. From the IRF data at subsequent discrete time instants, these co-efficients can be computed and the roots then result in the system poles. A MIMOversion of the LSCE is referred to as the Polyreference LSCE [148]. The LSCEmethod is fast and does not require initial estimates for the modal parameters.Moreover, based on auto- and cross-correlations between output measurements,the LSCE algorithm has also been successfully applied for MPE from output-onlymeasurements as demonstrated in [59]. The only unknown is the number of modesthat must be considered in the analysis for which an overspecified number of modesis usually used. However, as for most parametric estimators used for modal analy-sis, this requires the distinction between the physical and computational modes forwhich a so-called stabilization chart was proposed. Displaying the poles (on thefrequency axis) for an increasing model order, indicates the physical poles sincethey in general tend to stabilize for an increasing model order, while the mathe-matical poles scatter around. For low noise levels the PLSCE in combination withthe stabilization chart is suitable for an accurate MPE. However, for high noiselevels the estimation process becomes susceptible to errors and results can differsignificantly depending on the user’s experience. The LSCE only yields the polesand modal participation factors, while the mode shapes have to be estimated ina second step, for which the Least Squares Frequency-Domain (LSFD) algorithm[94] is usually applied. The LSFD is based on the fact that, once the poles (andmodal participation factors) are known, equation (4.14) becomes a linear-in-the-parameters problem that can be solved in least squares sense. Even for a highnumber of outputs (e.g. No = 500 or more) the LSCE-LSFD combination can stillbe executed within a reasonable computation efforts, explaining its frequent usefor modal analysis applications.

The LSCE and ITD methods can be considered as special cases of the Time-domain Direct Parameter Identification (TDPI) algorithm [79]. This MDOFmethod yields global estimates for the system poles, mode shapes and modalparticipation factors (suitable for multiple input data) and directly estimates thecoefficients of a (low order) ARX model (4.10) (in modal analysis literature oftenreferred to as ARMA model) in a least squares sense starting from the measuredforce and response sequences. If the force is an impulse (Dirac pulse), the LSCEis derived.

Within the domain of modal analysis it is a commonly accepted rule to prefertime-domain estimators for systems with a low damping (damping ratio ζ < 1%),while frequency-domain estimators are better for highly damped systems (ζ >2.5%). This is often motivated by the fact that FRF data from low damped systems


contains most of its energy in a very limited number of frequency lines, while theequivalent IRF data has a large magnitude for a large number of time samples.In practice, however, an IRF is computed from a FRF by means of the inverseFFT (IFFT) and since this is an orthogonal transformation exactly the sameinformation will be available in both the frequency and time-domain. As a result,the validity of the previously mentioned rule in practice can be seriously questionedand moreover, the computation of the IRF using the IFFT of the FRF results inaliasing effects, which become important in the case of low frequency resolutionor small damping [8]. The argument that, when using arbitrary excitation signals,time-domain is preferred because of spectral leakage phenomena is also invalidsince the IRFs, used for time-domain identification, are computed from the FRFscontaining this leakage. It should be noticed that, transient effects in general (e.g.leakage) can be modelled in the frequency domain in an analogous fashion as is thecase in the time-domain [111, 56]. Moreover, the use of residual terms to model theeffect of modes outside the analysis frequency band as well as the possibility to usean non-equidistant frequency grid and frequency band selection are also advantagesof frequency-domain identification. Problems of numerical conditioning related toa high model order can be considered as a drawback, although the latter greatlydepends on the choice of parameterization, as will be discussed in Chapter 5.

MDOF frequency-domain methods

A first MDOF frequency-domain method already mentioned is the LSFD esti-mator, which uses the modal model (4.14). In the case that the poles (and modalparticipation factors) are not estimated in a first step, this method is nonlinear-in-the-parameters resulting in a nonlinear least squares problem that has to besolved using an optimization algorithm. Good starting values are required to re-duce the number of iterations. This method as such never became popular, exceptin combination with the LSCE, where it reduces to a linear Least Squares problemyielding global estimates for the mode shapes (i.e. independent of the referencedegree of freedom).

As in the time-domain, a number of frequency-domain methods use a statespace parameterization. A frequency-domain version of the ERA method (ERA-FD), is presented in [73], which formulation is closely related to the time-domainalgorithm. The primary data for the ERA-FD are the FRFs forming a com-plex block matrix from which, through an SVD, a state model is derived andtransformed to the modal space. As a part of this ERA-FD algorithm, accuracyindicators, namely the output modal amplitude coherence and modal spectrumcoherence are developed to distinguish between the physical and mathematicalmodes. A number of important features in frequency-domain analysis, such asoverlap averaging and data windowing and since a frequency band selection ispossible, considerably reduce computation and storage requirements.

As for the ERA method, frequency-domain versions of time-domain subspaceidentification schemes having a reduced computation and storage requirements


were proposed. Starting from MIMO frequency response function measurements,frequency-domain algorithms based on discrete-time deterministic subspace iden-tification [31, 141] have been described in [84, 90]. These methods already use theideas of frequency weighting presented in [12], in order to avoid a distortion ofthe Markov parameter sequences by time aliasing effects when these are computedfrom FRFs using the IFFT. A continuous-time deterministic subspace variant isdiscussed in [140]. The latter approach requires the introduction of orthogonal(Forsythe) polynomials to avoid the problem of badly conditioned data-matricesas for other frequency-domain identification schemes as well [117].

Another method that first identifies the frequency-domain low order completedirect (state space) model (4.20) from MIMO FRF measurements is the Frequency-domain Direct Parameter Identification (FDPI) method [77]. The poles, modeshapes and participation factors are derived from the system matrices of a statemodel. In practice, a data reduction based on a Principal Component Analysis(PCA) of the frequency response function matrix is necessary for computationspeed and memory requirements as well as to construct a proper stabilization dia-gram for proper model order detection . The algorithm assumes that the numberof modes within the analysis band is less than or equal to the number of outputs.

The Complex Mode Indicator Function (CMIF) [102] that uses a SVD of thefrequency response function matrix, has gained some popularity because of itssimple use. The CMIF is a plot of the log-magnitude of the singular values of theFRF matrix as a function of the frequency. Peaks in the CMIF plot indicate thedamped natural frequencies. Based on the assumption that near a resonance theFRF matrix is dominated by a single term in equation (4.14), it follows that themode shape and modal participation factor vectors are found from the left andright singular vectors associated with the largest singular value. Since the CMIFis a multiple input algorithm, it can detect multiple roots resulting in severalsignificant singular values peaking at a specific frequency. It should be noticedthat, for the case of multiple poles, the singular vectors corresponding to thesingular values significantly differing from zero are not the actual mode shapes.Instead, these are given by a (unknown) linear combination of the singular vectors.

A large number of frequency-domain estimators are based on a matrix fractiondescription model and are closely related to the identification techniques discussedin the next two chapters. Similar to the PEM approach [85] in the time-domain,these estimators are based on the minimization of an equation error ε betweenthe measured and the modelled FRF matrix. Considering for example the scalarmatrix fraction description (4.5), this equation error is defined as

ε(ωf ) = H(ωf )−B(ωf , θ)

A(ωf , θ)(4.21)

where ωf is the angular frequency (f = 1, . . . , Nf ) and the polynomial coefficients θthe parameters to estimate. The quadratic-like cost function is then defined as thesum of the Frobenius norm of the squared error function for each angular frequency


ωf , i.e `(θ) =∑Nf

f=1

∥

∥ε(ωf )∥

∥

2

F. The most straightforward method to minimize `(θ)

with respect to the parameters θ using a Least Squares approach. However, sincethis cost function is nonlinear-in-the-parameters, it requires an iterative algorithmbased on for instance a Gauss Newton approach [11]. Alternatively, a two stepapproach can be used to obtain a suboptimal solution in analogy with the LSCE-LSFD procedure [92]. Due to the iterative character of estimators based on thenonlinear cost function, starting values are required and to achieve this, Levi [81]proposed a way to linearize the cost function resulting from the multiplication ofthe equation error (4.21) with A(ωf , θ)

`(θ)LS =

Nf∑

f=1

∥

∥A(ωf , θ)H(ωf )−B(ωf , θ)∥

∥

2

F(4.22)

Based on this idea, a large number of linear least squares estimators have beendescribe in literature during the last two decades. The so-called Rational Frac-tion (Orthogonal) Polynomial (RFOP) method was presented in 1982 [113] forSISO systems. Since the formulation of the LS problem in the continuous-timefrequency-domain resulted in ill-conditioned matrices for high-order systems, theScalar MFD model was expressed in terms of Forsythe polynomials. This methodwas extended to a global method (GRFOP) for SIMO identification [114], whileVan der Auweraer [135] formulated the multiple reference version (OPOL).

According to the two major classes, introduced in [105, 51], all the previouslydiscussed estimators belong to the class of so-called deterministic algorithms, whichin essence are curve fitting techniques. Using however a stochastic approach basedon an errors-in-variables model, as defined in Chapter 2, yields a more realisticdescription of real-life experiments. By taking knowledge about the noise on themeasured data into account in the cost function, it is possible to derive estimatorswith significant higher accuracy compared to the ones developed in the determin-istic framework. Most of the described stochastic estimators start from the Fouriertransform of the input (force) and response time sequences measured using a pe-riodic excitation signal derived in an errors-in-variables framework, from which asample mean and sample variance can be derived [106] (cf. §2.5.4). Based on thisinformation, the model parameters can be derived using the so-called frequency-domain Maximum Likelihood Estimator (MLE) developed by Schoukens and Pin-telon [122] and extended to multivariable systems by Guillaume [41]. In the samestochastic framework, Weighted Total Least Squares algorithms with nearly maxi-mum likelihood properties were developed for both SISO [137, 46, 107] and MIMO[52, 108] system identification. Although these stochastic algorithms feature nicestatistical properties, the computational burden related to the iterative characterand the initial formulation of the algorithms has been a bottleneck to apply thesealgorithms for MPE. Modal data is typically characterized by a large number ofspectral lines and response locations as well as a high modal density. Furthermore,the initial implementations are based on Fourier data acquired under periodic ex-citation, which is also different from the FRF data that is most often acquired


using an arbitrary excitation signal. As a result until a few years ago, these al-gorithms were not well-known in the modal analysis community except for a fewspecific applications in the domain of modal analysis, such as for instance fluttertesting [110, 47, 134]. Flutter analysis requires a stochastic approach since themeasured data is characterized by significant measurement noise. In [55] a ’dedi-cated’ multivariable implementation for frequency-domain estimators, based on acommon denominator transfer function model, has been given. Typical for modalanalysis is the large number of output measurements. This extensive amount ofdata requires optimized algorithms that balance between accuracy and memoryand computation efficiency. This work has been the basis for the further devel-opment of frequency-domain system identification for modal analysis. Recently,the ML approach has also been adapted for modal parameter estimation fromoutput-only measurements by using the spectral density matrix between outputsinstead of FRF matrix [45, 58]. Furthermore, the accurate identification of thefrequency components (sinusoids) present in the vibration behaviour of rotatingmachinery, based on a pole/residue model, is possible as well by means of thefrequency-domain MLE [44].

4.4 Conclusions

This chapter has given an introduction of the concepts of parametric system iden-tification for modal parameter estimation. Different parametric models to describethe vibration behaviour of mechanical systems have been discussed, where threemajor groups of parameterizations can be considered: matrix fraction description,pole-residue or state space.

The interconnection of the different models is illustrated by means of a matrixpolynomial approach. This provided a proper basis for understanding the relationbetween many of the parametric identification methods that originate from controltheory and different engineering areas. An extensive overview of the parametricidentification methods with their potential and drawbacks from the point of viewfor modal parameter has been given. Besides the methods specifically developedfor modal parameter estimation, such as for instance the LSCE, ITD and ERA,recent research in the field of system identification has resulted in more advancedsubspace and frequency-domain maximum likelihood methods.

This overview also introduced the so-called stochastic estimators, such as thefrequency-domain Maximum Likelihood or Weighted Total Least Squares methodsthat take the noise information of the data into account. This thesis focusses onthe applicability of these estimators for modal analysis, where this is motivated byseveral reasons. New applications in the domain of modal analysis, require a highaccuracy of the estimated parameters, with important examples such as damagedetection and structural health monitoring. Other applications, such as flight

4.4. Conclusions 97

flutter analysis, typically result in very noisy data from which accurate dampingestimates should be derived. Since the stochastic estimators do consider the noiseon the data, accurate models can be estimated and confidence bounds for theparameters can be obtained.

A methodology for determining the noise on the FRF data in an EV frameworkwas already developed in Chapter 2. The next two chapters will study the numer-ical efficiency of the parametric estimators using FRF or Input/Output Fourierdata. The initial formulation of the algorithms will be adapted in order to processlarge data sets characterized by a high modal density and dynamical range.

Chapter 5

Frequency-domain MPEfrom FRF Data

This chapter presents the optimization of the (stochastic) frequency-domain meth-ods, such as the Maximum Likelihood estimator, for modal parameter estimation.Large modal data sets require optimized algorithms that balance between accuracyand computation efficiency. A multivariable implementation, based on a common-denominator transfer function model, is given. First, improvements of the classicalimplementation of the frequency-domain linear Least Squares (LS) approach willbe presented in Section 5.3. Aspects, such as robustness for high modal densityand numerical performance (computation time, memory usage) of the different al-gorithm implementations and parameterizations are studied. The final result is afast LS solver, i.e. the so-called Least Squares Complex Frequency (LSCF) method.This estimator offers the user, in analogy with the well-known LSCE method, astabilization chart. The fast implementation is also extended to the class of thestochastic estimators based on a Total Least Squares (TLS) and the MaximumLikelihood (ML) framework in Sections 5.4 and 5.5.

99

100 Chapter 5. Frequency-domain MPE from FRF Data

5.1 Introduction

Since modal analysis usually starts from FRF measurements, the optimization ofthe stochastic frequency-domain methods for modal parameter estimation fromFRF data is one of the key issues of this thesis. The case for I/O will be consid-ered in Chapter 6 for specific applications where only short data records can bemeasured.

Typical for modal analysis is the large number of output measurements. Anextensive amount of data requires optimized algorithms that balance between ac-curacy and memory/computation efficiency. A multivariable implementation forfrequency-domain estimators, based on a common-denominator transfer functionmodel, will be derived.

Improvements of the classical implementation of the frequency-domain linearLeast Squares (LS) approach will be presented for extracting the modal parametersfrom typical modal FRF data. Aspects, such as robustness for high modal densityand numerical performance (computation time, memory usage) of the differentalgorithm implementations and parameterizations are studied and validated bymeans of experimental modal data obtained from a safety critical aircraft compo-nent. The final result is a fast LS solver, i.e. the so-called Least Squares ComplexFrequency (LSCF) method. In analogy with the well-known LSCE method, a fastconstruction of a stabilization chart is studied as well, where different representa-tions are possible. A second step LS solver is proposed for the estimation of theresidues based on the poles selected by the user from this diagram.

Furthermore, it is shown in this chapter how the fast implementation can alsobe extended to the class of stochastic Total Least Squares (TLS) estimators and theMaximum Likelihood (ML) method. The FRF data and required noise informationcan be obtained by means of the Hiv FRF approach presented in Chapter 2. Usingexperimental data, the robustness of the algorithms is demonstrated for modelorders up to 200.

5.2 Parametric model

Given the global character of the poles of a mechanical system, a scalar matrix-fraction description (cf. §4.2) – better known as a common-denominator model(CDM) – will be used for the development of the frequency-domain estimators.The measured Frequency Response Function (FRF) at DFT frequency f , betweenoutput o and input i, is then modelled as

Hk(Ωf , θ) =Bk(Ωf , θ)

A(Ωf , θ)(5.1)

5.2. Parametric model 101

for k = 1, 2, . . . , NoNi with

Bk(Ωf , θ) =

n∑

j=0

bkjΩjf (5.2)

the numerator polynomial between the output/input DOF combination k and

A(Ωf , θ) =

n∑

j=0

ajΩjf (5.3)

the common-denominator polynomial. The coefficients aj and bkj are the unknownparameters θ to be estimated. In general, the order of the denominator polynomialand numerator polynomials can differ.

For a discrete-time domain model, the generalized transform variable Ωf , eval-uated at DFT frequency f , is given by Ωf = e(−iωfTs) (Z-domain) with Ts thesampling period. For continuous-time domain models, other choices for Ωf arepossible such as for instance

Ωf =

(iωf ) lumped continuous-time systems (Laplace domain)

orthogonal polynomials, e.g. Forsythe, Chebyshev

(√

(iωf )) diffusion phenomena

(tanh τR(iωf )) microwaves (Richardson domain) [116]

To obtain an identifiable parameterization (5.1), it is needed to impose a(scalar) constraint. This is readily verified by considering the following expres-sion

Hk(Ωf , θ, θ) =Bk(Ωf , θ)

A(Ωf , θ)=αBk(Ωf , θ)

αA(Ωf , θ)(5.4)

Clearly, for every non-zero scalar α another equivalent scalar matrix-fraction de-scription is obtained. The parameter redundancy can be removed by fixing onecoefficient of the denominator, such as for instance the highest order coefficient ofthe denominator, i.e. an = 1 or by imposing a norm-1 constraint, i.e. θHθ = 1with θ containing all polynomial coefficients aj and bkj , the parameters to be es-timated. Other constraints are possible such as e.g. θHA θA = 1 with θA containingonly the denominator coefficients.

A linear Least Squares approach requires model equations that are linear-in-the-parameters. An often used approximation, first presented by Levi [81] forSISO systems, consists of replacing the model Hk in (5.1) by the measured FRFHk and multiplying with the denominator polynomial

Wk(ωf )

n∑

j=0

bkjΩjf −

n∑

j=0

ajΩjfHk(ωf )

≈ 0 (5.5)


for k = 1, . . . , NoNi and f = 1, . . . , Nf the number of spectral lines. By introduc-ing an adequate weighting function Wk(ωf ) in the equations (5.5), the quality ofthe LS estimate can often be improved as discussed in § 5.6.

5.3 (Weighted) Linear Least Squares

The first part of this chapter presents a linear Least Squares solver for the mini-mization of the LS cost function (4.22), which using the model (5.1) becomes

`(θ)LS =

NoNi∑

k=1

Nf∑

f=1

∣

∣Bk(Ωf , θ)−A(Ωf , θ)Hk(ωf )∣

∣

2(5.6)

Starting from the LS formulation based on both Jacobian and normal equationsseveral algorithms are discussed and optimized for deriving the modal parametersfrom complex high order systems in a time-efficient way.

5.3.1 LS Formulation based on Jacobian Matrix

Since Eqs. (5.5) are linear-in-the-parameters and because a common-denominatormodel is used, they can be reformulated as

Γ1 0 · · · 0 Φ1

0 Γ2 · · · 0 Φ2

.... . .

...0 0 · · · ΓNoNi ΦNoNi

θB1

θB2

...θBNoNiθA

≈ 0 (5.7)

with

θBk =

bk0bk1...bkn

, θA =

a0a1...an

(5.8)

and

Γk =

Γk(ω1)Γk(ω2)

...Γk(ωNf )

, Φk =

Φk(ω1)Φk(ω2)

...Φk(ωNf )

(5.9)

where

Γk(ωf ) = Wk(ωf )[

Ω0f ,Ω

1f , . . . ,Ω

nf

]

, Φk(ωf ) = −Γk(ωf )Hk(ωf )

5.3. (Weighted) Linear Least Squares 103

The Jacobian matrix J of the least-squares problem

J =

Γ1 0 · · · 0 Φ1

0 Γ2 · · · 0 Φ2

.... . .

...0 0 · · · ΓNoNi ΦNoNi

(5.10)

has NfNoNi rows and (n + 1)(NoNi + 1) columns (with Nf À n, where n is theorder of the polynomials). Because every equation in (5.5) has been weighted withWk(ωf ), the matrix entries Γk in (5.10) generally differ.

Contrary to the Φk matrices, the Γk matrices occurring in (5.10) do not containmeasured data and thus are not subject to errors. Because of this, it is possibleto apply the ”mixed LS-TLS”algorithm presented in [139].

The matrix J in (5.10) can be partitioned as J = [Γ,Φ] with the matrix Γexactly known and Φ subject to errors, where Γ is NfNoNi× (n+1)(NoNi) and Φis NfNoNi × (n+ 1). θ is the parameter vector with (n+ 1)(NoNi + 1) elements.

The mixed LS-TLS algorithm determines the denominator coefficients θA suchthat [Γ,Φ−∆Φ]θ = 0 with ∆Φ an perturbation matrix with a minimal Frobeniusnorm and θ = [θTB , θ

TA]

T with θB a vector of length (n+1)(NoNi) and θA a vectorwith n+ 1 elements. The algorithm consists of the following steps

Mixed LS-TLS algorithm using full QR decomposition

• step 1 : Compute the QR-decomposition of J

[Γ,Φ] = [Q1, Q2]

[

R11 R12

0 R22

]

(5.11)

with R22 a (n+ 1× n+ 1) upper triangular matrix.

• step 2 : Compute the SVD R22 = USV H with S = diag(σ1, . . . , σn+1) andV = [ν1, . . . , νn+1] an orthonormal matrix.

• step 3 : The denominator coefficients are found as the eigenvector corre-sponding to the smallest singular value σn+1, i.e. θA = νn+1.

The prove of this algorithm is given in [139] (pp.92–95). The number of singlefloating point operations (flops) gives a first insight in the computation speedof the algorithm by counting each sum or product as one flop [40]. The totalnumber of flops involved in steps 1–3 is approximately O(2(NoNi)

3n2Nf ). Basedon the knowledge of the denominator coefficients θA, this algorithm also allows tocompute the numerator coefficients θB by means of a back-substitution approach,O((NoNi)

2n) [40].


In the first step, a QR-decomposition of the large matrix J must be com-puted. This can be done in a numerical efficient way by exploiting the blockstructure of this matrix. Bayard proposed an algorithm for a so-called sparse QR-decomposition [11]. However, this algorithm assumes that the submatrices Γk inJ are all equal for k = 1, . . . , NoNi. In general, this is not the case for the LSproblem (5.7) since the weighting functions Wk(ωf ) can differ for each k. There-fore, the Sparse QR algorithm, first presented in [11], was generalized solving theLS problem using the following algorithm.

Mixed LS-TLS algorithm using sparse QR decomposition

• step 1 : For each k, with k = 1, . . . , NoNi, compute the QR-decompositionof the submatrices Γk yielding Γk = QkRk.

• step 2 : Compute the orthogonal projection of the submatrices Φk on thespace defined by the orthonormal column vectors of the matrix Qk by meansof the projection operator defined as Q⊥

k = I −QkQHk (with QH

k Qk = I) inorder to form the following matrix (with α = NoNi)

V =

Q⊥1 Φ1

Q⊥2 Φ2

...Q⊥αΦα

(5.12)

• step 3 : Perform the QR decomposition of V yielding V = QVRV .

• step 4 : The matrix R22 (cf. step 2 of LS-TLS algorithm) is given by RV .

Proof : Given the QR-decomposition of the sparse matrix J (5.11) (cf. step 1 ofLS-TLS algorithm)

J =

Q1 0 · · · 0 Q1β

0 Q2 · · · 0 Q2β

.... . .

...0 0 · · · Qα Qαβ

R1 0 · · · 0 R1β

0 R2 · · · 0 R2β

.... . .

...0 0 · · · Rα Rαβ

0 0 · · · 0 Rβ

(5.13)

with β = α + 1 and where Rβ = R22. For the computation of the denominatorcoefficients θA using the mixed LS-TLS algorithm, it is sufficient to know the small(n+ 1× n+ 1) upper triangular matrix Rβ . Indeed, from (5.13) it follows that

Γk = QkRk

Φk = QkRkβ +QkβRβ


The orthogonal projection of the submatrices Φk on the space defined by theorthonormal column vectors of the matrix Qk is obtained by multiplying with theprojection operator Q⊥

k = I −QkQHk

Q⊥k Φk = Q⊥

k QkRkβ +Q⊥k QkβRβ = 0 +Q⊥

k QkβRβ = QkβRβ (5.14)

Since Qk and Qkβ are orthogonal matrices, [I − QkQHk ]Qkβ = Qkβ + 0. Conse-

quently, the matrix V (5.12) can be written as

V =

Q1β

Q2β

...Qαβ

Rβ = QVRV (5.15)

where [QH1β , Q

H2β ,

..., QHαβ ]

H is an (NfNoNi×n) orthogonal matrix and Rβ an (n×n)upper triangular matrix, representing the QR-decomposition of the large matrixV . Hence, the required matrix R22 (cf. step 2 of the mixed LS-TLS algorithm),is given by Rβ = RV .

Similar as for the mixed LS-TLS algorithm [139], it is possible, based on thequantities computed in steps 1–3 of the sparse matrix QR algorithm, to computealso the numerator coefficients based on a back-substitution approach as discussedin [11].

The most important drawback of Bayard’s sparse matrix QR algorithm, requir-ing approximately Nf

2(5nNoNi+Nf ) flops, stems from the orthogonal projectionsQ⊥k Φk derived by the explicit matrix multiplications, O(Nf

3). This also explainsthe assumption of equal matrices Γk in order to form the projection matrix onlyonce to limit the computation time, since in the case of differing weights, a total ofNoNiNf

2(5n+Nf ) flops is required. Another drawback is the QR-decompositionof the large matrix V . However, this algorithm can be further improved by com-puting these orthogonal projections by means of QR decompositions (a techniquewhich is commonly used in subspace identification algorithms [141]). This im-provement was necessary to handle the problem with different matrix entries Γkefficiently.

Mixed LS-TLS algorithm using sparse QR via orthogonal projections

The orthogonal projections Q⊥k Φk (k = 1, . . . , NoNi) forming the matrix V , can be

computed avoiding the explicit matrix products as well as the QR decompositionof the large matrix V , as follows

• step 1 : Compute the QR-decomposition

[Γk,Φk] = [(Qk)a, (Qk)b]

[

(Rk)a (Rk)ab0 (Rk)b

]

(5.16)


for k = 1, . . . , NoNi

• step 2 : Compute the QR decomposition of the matrix Rb

Rb =[

(R1)Hb (R2)

Hb . . . (Rα)

Hb

]H= QVRV (5.17)

with RV the matrix R22 in the mixed LS-TLS algorithm.

Proof : The QR-decomposition of [Γk,Φk] yields

Γk = (Qk)a(Rk)a

Φk = (Qk)a(Rk)ab + (Qk)b(Rk)b

As the QR-decomposition of Γk is unique, Qk = (Qk)a and Rk = (Rk)a. Left-multiplication of Φk with the orthogonal complement of (Qk)a, i.e. (Qk)

⊥a =

[I − (Qk)a(Qk)Ta ], results in

(Qk)⊥a Φk = (Qk)

⊥a (Qk)a(Rk)ab + (Qk)

⊥a (Qk)b(Rk)b

= [I − (Qk)a(Qk)Ta ](Qk)a(Rk)ab + [I − (Qk)a(Qk)

Ta ](Qk)b(Rk)b

= (Qk)a(Rk)ab − (Qk)a(Rk)ab + (Qk)b(Rk)b − 0

= (Qk)b(Rk)b

since (Qk)Ta (Qk)a = I and (Qk)

Ta (Qk)b = 0. As a result, Q⊥

k Φk = (Qk)⊥a Φk =

(Qk)b(Rk)b. This proves that the (Nf × n) submatrices Q⊥k Φk of the matrix V

(5.12) are found by means of a (thin) QR-decomposition of [Γk,Φk] instead ofexplicitly computing the matrix products Q⊥

k Φk.

The explicit computation of matrix product (Qk)⊥a Φk requires Nf

2(n + 1)flops without taking the formulation of the matrix (Qk)

⊥a into account. The QR-

decomposition only requires Nf (2n+ 2)2 flops, with Nf À (2n+ 2).

Using the orthogonal projection algorithm for a fast formulation of the matrixV then results in

V =

(Q1)b(R1)b(Q2)b(R2)b

...(Qα)b(Rα)b

=

(Q1)b 0 · · · 00 (Q2)b · · · 0...

... · · ·...

0 0 · · · (Qα)b

(R1)b(R2)b

...(Rα)b

(5.18)

where the block diagonal matrix with submatrices (Qk)b is orthogonal again (byconstruction).

Finally, the QR-decomposition of the (αn × n) matrix Rb, defined as Rb =[

(R1)Hb (R2)

Hb . . . (Rα)

Hb

]H, is given by Rb = QRRR and hence this corresponds to


the QR-decomposition of the large matrix V = QVRV (5.15), where

RV = RR and QV =

(Q1)b 0 · · · 00 (Q2)b · · · 0...

... · · ·...

0 0 · · · (Qα)b

QR (5.19)

In practice, only the matrix RR is used. Hence the total number of flops requiredfor steps 1–2 is now approximately O(8NoNiNfn

2). As a result, by optimizingthe Sparse QR algorithm, an important reduction of the computation time as wellas memory usage can be obtained. Nevertheless, since the number of measuredfrequencies Nf is typically large for modal testing, the LS formulation based onthe so-called normal equations is commonly-used by many of the modal parameterestimators discussed in Chapter 4.

5.3.2 LS Formulation based on Normal Matrix

Instead of deriving the parameter estimates directly from the Jacobian matrix,many estimators used in modal analysis form the normal equations explicitly bycomputing

JHJ =

ΓH1 Γ1 0 · · · ΓH1 Φ1

0 ΓH2 Γ2 · · · ΓH2 Φ2

......

. . ....

ΦH1 Γ1 ΦH

2 Γ2 · · · ∑NoNik=1 ΦH

k Φk

(5.20)

with the entries of the submatrices given by

[ΓHk Γk]rs =

Nf∑

f=1

|Wk(ωf )|2Ωr−1H

f Ωs−1f

[ΦHk Φk]rs =

Nf∑

f=1

|Wk(ωf )Hk(ωf )|2Ωr−1H

f Ωs−1f

[ΓHk Φk]rs = −Nf∑

f=1

|Wk(ωf )|2Hk(ωf )Ωr−1H

f Ωs−1f (5.21)

In § 5.3.3 it is shown that the submatrices of the normal matrix are structuredmatrices. By exploiting the specific matrix structure, an important reduction ofthe computation time and memory requirements is obtained.

Defining the following submatrices

Rk = ΓHk Γk , Sk = ΓHk Φk , Tk = ΦHk Φk (5.22)


the normal equations are rewritten as

R1 0 · · · S10 R2 · · · S2...

.... . .

...

SH1 SH2 · · · ∑NoNik=1 Tk

θB1

θB2

...θBNoNiθA

≈ 0 (5.23)

Since the submatrices Rk = ΓHk Γk in Eq. (5.23) do not contain any measurementdata (i.e. they are not subjected to errors), it is possible to apply again the mixedLS-TLS approach presented in [139] as discussed in § 5.3.1.

Although the number of rows of the normal matrix (5.20) is much smaller thanthe number of rows of the Jacobian matrix (5.10), its size (i.e. (n+1)(NoNi+1))is often still of importance for the computation time in the case of typical modaltest data (No > 100) since solving the Eqs. (5.23) for θ still requires (NoNin)

3

flops.

Under the condition that, the parameter constraint only applies to the denomi-nator coefficients θA, the numerator coefficients can be eliminated from the normalequations

θBk = −R−1k .Sk.θA (5.24)

by substitution in the last (n+ 1) equations of (5.23)

NoNi∑

k=1

SHk θBk +

NoNi∑

k=1

TkθA ≈ 0 (5.25)

yielding a very compact problem[

NoNi∑

k=1

Tk − SHk R−1k Sk

]

θA = DθA ≈ 0 (5.26)

The square matrix D has a size (n+1) and thus is much smaller than the originalnormal matrix (5.20) with size (n + 1)(NoNi + 1). Notice, that in the case ofthe Jacobian matrix the mixed LS-TLS algorithm uses QR decompositions for theelimination of the numerator coefficients θBk .

A LS solution of θA is found by fixing for instance the highest order coefficientan to 1

θALS =

−[D(1 : n, 1 : n)]−1D(1 : n, n+ 1)1

(5.27)

The mixed LS-TLS solution is given by the eigenvector ve corresponding to thesmallest eigenvalue λe found by solving an eigenvalue problem Dve = λeve.

The LS or mixed LS-TLS solutions for θA, obtained by solving the compactlinear LS problem (5.26) is the same as obtained by solving the full LS problem


(5.23) (with the same constraint). It is proven in [74] (matrix inversion lemma)that the inverse of the normal matrix in Eq. (5.20) is as well a positive definiteHermitian symmetric matrix given as

R | S−− −−SH | T

−1

=

E | F−− −−FH | G

(5.28)

with the submatrices E = (R − ST−1SH)−1, F = −R−1S(T − SHR−1S)−1 andG = (T − SHR−1S)−1 where E and G are both Hermitian matrices. As shown in[30], the jth (j = 1 , . . . , n+1) column of (T −SHR−1S)−1 gives the LS solutionfor the denominator coefficients θA under the constraint aj = 1. Notice that, inthe case that none of the entries is subject to errors (i.e. no noise is present on thedata), the matrix is not of full rank and so the inverse does not exist. However, inthat case, (5.23) and (5.26) are exactly equal to zero and as a result the solutionθA is uniquely defined.

Once the θA coefficients are known, back-substitution based on (5.24) can beused to derive the numerator coefficients θB . The total number of flops requiredfor the elimination of the numerator coefficients, solving D for θA and the back-substitution is O(2NoNin

3). This approach is more time efficient than solving(5.23) directly, i.e approximately No

2.Ni2 times faster.

In Appendix A it is proven that this LS algorithm based on the compactnormal equation yields the same solution as the sparse QR algorithm presented in§ 5.3.1 (under the assumption that both problems are numerically well conditionedcf. §5.3.3). Nevertheless, the LS algorithm, based on the compact normal equationshas the important advantage to be faster.

5.3.3 Optimal LS Implementation for MPE

Based on the results of the previous sections, different implementations of afrequency-domain LS estimator can be derived by varying following algorithmcharacteristics:

• Jacobian or Normal matrix based LS formulation. (The Jacobian implemen-tation is based the fast sparse QR decomposition via projections)

• real or complex valued coefficients θ

• continuous or discrete-time model, defined by the generalized transform vari-able Ω

• parameter constraint, i.e. a LS or mixed LS-TLS constraint


LEADINGEDGEOF WING

SLAT TRACK

SLAT

Figure 5.1: A Slat track (top) mounted in the wing of an Airbus320 aircraft (bottom).

In order to determine an optimal implementation for the problem of modal pa-rameter estimation, the performance of the various implementations is assessedusing modal test data obtained from on a slattrack of an Airbus A320 commercialaircraft.

Slat tracks are located at the leading edge of an aircraft wing and make partof a gliding mechanism that is used to enlarge the wing surface (see Figure 5.1).The enlargement of the wing surface is needed in order to increase the lift force atreduced velocity during landing and take off. The A320 airplane has 5 slats perwing. The first slat (i.e. the inboard slat between fuselage and engine) contains 4slat tracks. The other 4 slats have 2 slat tracks each.

Using a Scanning Laser Doppler Vibrometer setup, shown in Figure 5.2, theresponse (velocities) to a single input excitation was measured in 352 scan-points.An electrodynamic shaker was used to apply a random noise excitation in a fre-quency band of 0-4kHz with a resolution of 1.25Hz. Frequency response functions


SHAKER

STINGER

FORCE SENSOR

SLAT TRACK

SHAKER

STINGER

FORCE SENSOR

SLAT TRACK

SH AK ER

SC ANN ING LASER DO PPLER V IBROM ETER

SH AK ER

SC ANN ING LASER DO PPLER V IBROM ETER

Figure 5.2: Force (input) measurement with shaker, stinger and force sensor attachedto the slat track (left). Velocity (output) measurement with scanning laser Dopplervibrometer (right).

were estimated using the H1 estimator with 5 averages. Using the coherence func-tions, the variances of the FRFs were obtained as well. This data set is a goodrepresentation of a typical modal data set.

Unless otherwise stated, a common-denominator model of 50 modes and afrequency band of 1000–3725Hz is used for the parametric identification by meansof the various LS implementations in this section. Given the variances, obtained bymeans of the knowledge of the coherence functions based on Eq. (2.22), a WeightedLS formulation was used for the various implementations. This was done by meansof the nonparametric weighting function (5.76) that will discussed later in § 5.6.

Real or Complex Valued Coefficients

The equations derived in the previous two sections implicitly assume that thecoefficients θ are complex-valued. Hence, since a common-denominator model(5.1) is used, the denominator polynomial A(Ωf , θ) has scalar coefficients and anorder n equal to the number of modes to be estimated Nm (cf. § 4.2).

To obtain real-valued coefficients, the Jacobian matrix has to be transformedinto a real valued matrix. This can be done by reformulating the Jacobian matrixas follows

JRE =

Γre1 0 · · · 0 Φre1

0 Γre2 · · · 0 Φre2

.... . .

...0 0 · · · ΓreNoNi Φre

NoNi

(5.29)


with

Γrek (ωf ) =

[

Re(Γk)Im(Γk)

]

and Φrek (ωf ) =

[

Re(Φk)Im(Φk)

]

The normal matrix is transformed to a real-valued matrix simply by taking thereal part of JHJ since JTREJRE = Re(JHJ).

In the case that real-valued coefficients are estimated, the order of the denomi-nator polynomial A(Ωf , θ) has to be doubled in order to identify Nm modes again.This, however, is not in favor of the numerical conditioning of the equation matri-ces, especially when the normal equations are formulated in the Laplace domain(Ωf = iωf ) as is will be shown now.

Continuous-time domain – Laplace variable

The classical implementation of the frequency-domain linear LS estimator usesa continuous-time model with real-valued coefficients. A linear, time-invariant(LTI) continuous-time system of order n is modelled by the common-denominatormodel (5.1) by taking the generalized transform variable Ω = iω. As a result, thesubmatrices ΓK and ΦK of the Jacobian matrix of the LS problem (5.7) containentries of the form of a power basis

Γk(ωf ) = Wk(ωf )[

(iωf )0, (iωf )

1, (iωf )2, . . . , (iωf )

n]

Φk(ωf ) = −Γk(ωf )Hk(ωf ) (5.30)

It can be noticed that these submatrices have a ”graded structure”strongly relatedto a so-called structured Vandermonde matrix. The formulation of the Jacobianmatrix (cf. Eq. 5.7) is approximately O(12nNoNiNf ) for the case of complexcoefficients.

From this, it follows that the submatrices ΓHk Γk, ΓHk Φk and ΦH

k Φk appearingin the normal equations (5.23) convert into structured matrices as well

Σω0 iΣω1 −Σω2 · · · i(c−1)Σωn

−iΣω1 Σω2 iΣω3 · · · −icΣω(n+1)

−Σω2 −iΣω3 Σω4 · · · i(c+1)Σω(n+2)

iΣω3 −Σω4 −iΣω5 · · · −i(c+2)Σω(n+3)

......

.... . .

...

−i(r−1)Σωn irΣω(n+1) −i(r+1)Σω(n+2) · · · (−i)(r−1)i(c−1)Σω2n

with r, c = 1, . . . , n+ 1 and where the operator Σωn is defined as

Σωn =

Nf∑

f=1

G(ωf )ωnf (5.31)


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Figure 5.3: Jacobian WLS using a frequency-scaled continuous-time model with real(top) and complex (bottom) coefficients. Measurements (dotted line) and estimatedtransfer function model (solid line).

with G(ωf ) = |Wk(ωf )|2 for the matrices ΓHk Γk, G(ωf ) = |Wk(ωf )Hk(ωf )|2 forthe matrices ΦH

k Φk and G(ωf ) = |Wk(ωf )|2Hk(ωf ) for the matrices ΓHk Φk. Basedon the Hermitian symmetry of ΓHk Γk and ΦH

k Φk, it is sufficient to compute andstore the elements of the first row and last column (i.e. 2n+1 instead of (n+1)2

summations) in order to reconstructed the complete matrix. The same is truefor the submatrices ΓHk Φk, which however are not Hermitian symmetric but onlyfor the reason of the sign of the elements in the lower triangular part. Taking the


matrix structure into account, the normal matrix formulation requires 48nNoNiNf

flops, i.e. only 4 times the number of flops for the Jacobian formulation, whereasthe explicit product JHJ is O((NoNi)

3n2Nf ).

However, for a continuous-time model, Eqs. (5.23) become numerically ill-conditioned, especially for high-order systems. This is certainly the case for com-plex mechanical systems with a high modal density in the studied frequency band.Normalization (scaling) of the frequency axis by a factor ωs = (ωNf − ω1)/2 canimprove the numerical conditioning to a certain extend. Nevertheless, in practice,a model order of n = 20 (i.e. 10 modes in the case of real coefficients) is oftenappears to be a the limit for preserving an acceptable numerical conditioning.

Figure 5.3 shows a synthesized FRF for the Jacobian LS implementation usinga frequency-scaled continuous-time model with real (top) and complex (bottom)coefficients. As can be seen a bad numerical conditioning, i.e. κ = 6E75 for realand κ = 6E45 for complex case (cf. Table 5.2), results in important estimationerrors. It can be seen that the higher frequencies are over-emphasized, typical forthis type of model. Nevertheless, the model obtained is completely wrong for bothcases. For the case of the slattrack, reasonable results could only be obtained withthis implementation by analyzing small frequency bands with Nm not higher than10.

Continuous-time domain – Orthogonal Polynomials

The numerical conditioning of the jacobian or normal matrix can be significantlyimproved by decomposing the numerator and denominator polynomials of eachtransfer function model Hk(Ωf , θ) for k = 1, . . . , NoNi into a well-chosen basis oforthogonal polynomials

Hk(Ωf , θ) =

∑nj=0 ukjpkj(iωf )∑n

j=0 vjqj(iωf )(5.32)

where pkj(iωf ) and qj(iωf ) are the sets of orthogonal polynomials evaluated at theangular frequency ωf (f = 1, . . . , Nf ) and where θ are the new (real) coefficientsto be estimated

θ = [u10, u11, u12, . . . , u1n, u20, . . . , uNoNin, v0, . . . , vn]T

(5.33)

In the domain of modal analysis, Richardson and Formenti [113, 114] used this ap-proach in order to improve the numerical conditioning of their so-called Global Ra-tional Fraction Polynomial (GRFP) method which was also extended to a MIMOmethod (OPOL) by Van der Auweraer [133]. To obtain a better conditioning ofthe normal matrix in Eq. (5.23), Forsythe polynomials can be used. OrthogonalForsythe polynomials can be generated through a recursive scheme presented in[38, 113]. The OPOL estimator, for example, uses the Forsythe polynomials to


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Figure 5.4: Normal-based WLS implementation for continuous-time model with realcoefficients using Forsythe polynomials.

obtain unity matrices for the block matrices on the diagonal of the normal matriximproving the numerical conditioning of this matrix as well as allowing a reductionof the computation time and required memory.

For the practical implementation of the LS approach using Forsythe polyno-mials, the same polynomial basis was used for each transfer function model usingreal coefficients, where the polynomials were orthogonal with respect to the weightT (ωf ) = 1

Nf∑

f=1

T (ωf )pi(ωf )pj(ωf ) = δij (5.34)

and the result for the normal-based implementation is shown in Figure 5.4. Giventhe good numerical conditioning (κ = 2E3) of the normal matrix, the results forthe Jacobian and normal equations yield a very similar result.

However, an important drawback of this parameter estimation approach is thenecessity to transform the estimated coefficients from the orthogonal basis backto the original power polynomial basis in order to solve for the modal parameters(cf. § 5.3.4). Even if frequency scaling is applied, this again represents a numer-ically ill-conditioned problem. A solution to this problem is discussed in [117],i.e. the extension of the use of orthogonal polynomials to the complete identifica-tion process. This includes the extraction of the roots of each numerator and thedenominator polynomial and hence the modal parameters by rewriting each char-acteristic equation into a low order state space model and solving the eigenvalue


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Figure 5.5: Jacobian-based (top) and Normal-based (bottom) WLS implementation fora continuous-time model with complex coefficients and using Chebyshev polynomials.

problem of the so-called orthogonal companion matrix of this model. This is anumerically well-conditioned matrix since it has been formulated in the orthogo-nal basis as well. By doing so, it is possible to identify the modal parameters ofhigh order systems in a numerically stable way as demonstrated for SISO systemsin [117]. Nevertheless, in the case of MIMO systems, a different set of Forsythepolynomials pkj(iωf ) has to be generated for each transfer function model Hk ex-plaining the high computational involvement of this approach. This is a majordrawback for using such estimation schemes for modal parameter estimation.


The use of Chebyshev polynomials instead of Forsythe polynomials can reducethe computation time and memory usage significantly. Using a similar approach asdiscussed in [1] for SISO systems, the construction of the square (n+1) submatricesRk, Sk and Tk (cf. Eq. 5.21) boils down to the computation of 2n+1 entries (i.e.the entries of the first row and last column). Indeed, the other entries can becomputed as a sum from two of these 2n + 1 entries, since the product of twoChebyshev polynomials Ci and Cj is given as CiCj =

12 (Ci+j + Ci−j).

The result for slattrack data obtained by this approach is shown in Figure 5.5for both a Jacobian and Normal based implementation. While results for a Jaco-bian Chebyshev implementation are good, it can be seen that, since a continuous-time model is used and Chebyshev polynomials are only approximately orthogo-nal, the numerical conditioning (κ = 9E8) becomes a problem for Nm = 50 inthe case of the normal equations. Comparison with Figure 5.4, shows that theresults obtained by the normal-based implementation using Forsythe polynomialsis certainly better in the band of 1500–3000Hz, however at the price of a highercomputation time.

Discrete-time domain – Z variable

Considering the common-denominator model (5.1) in the discrete-time domain, thegeneralized transform variable Ωf is defined as Ωf = e(−iωfTs). Since these complexpolynomial basis functions are implicitly orthogonal with respect to the unitycircle, a well-conditioned Jacobian matrix J is usually obtained, which also justifiesthe explicit calculation of the normal equations. It turns out from past experiencein modal analysis with discrete-time domain estimators such as the LSCE that thenumerical conditioning of the normal matrix (or so-called covariance matrix) is nota major problem. The computation of the poles from the estimated denominatorcoefficients is also well-conditioned (cf. § 5.3.4).

The submatrices ΓHk Γk, ΓHk Φk and ΦH

k Φk appearing in the normal equations(5.23) are structured matrices of the following form

Σz0 Σz1 Σz2 Σz3 · · · Σzn

Σz−1 Σz0 Σz1 Σz2 · · · Σz(n−1)

Σz−2 Σz−1 Σz0 Σz1 · · · Σz(n−2)

Σz−3 Σz−2 Σz−1 Σz0 · · · Σz(n−3)

......

......

. . ....

Σz−n Σz−(n−1) Σz−(n−2) Σz−(n−3) · · · Σz0

where the Σzn operator is defined as

Σzn =

Nf∑

f=1

G(ωf )e−iωfTsn (5.35)


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Figure 5.6: Jacobian-based (top) and Normal-based (bottom) WLS implementation fora discrete-time model with complex coefficients.

with G(ωf ) = |Wk(ωf )|2 for the matrices ΓHk Γk, G(ωf ) = |Wk(ωf )Hk(ωf )|2 forthe matrices ΦH

k Φk andG(ωf ) = |Wk(ωf )|2Hk(ωf ) for the matrices ΓHk Φk. Havingentries that are constant along each diagonal, this matrix has a so-called Toeplitzstructure. Toeplitz matrices belong to the larger class of persymmetric matricesand the inverse of a nonsingular Toeplitz matrix is persymmetric as well. Basedon the Hermitian symmetric character of the structured matrices ΓHk Γk and ΦH

k Φk

(since Σz−n = (Σzn)∗ if G(ωf ) is real), it is sufficient to compute and store theelements of the first row (i.e. n + 1 instead of (n + 1)2 summations) in order to


reconstructed the complete matrix. For the matrices ΓHk Φk also the first columnis required, since these are not Hermitian symmetric and hence 2n + 1 elementsmust be stored. Taking the matrix structure into account, the normal matrixformulation now requires approximately 32nNoNiNf flops, i.e. only 2.66 timesthe number of flops for the Jacobian formulation.

In Appendix B it is shown that, if the frequencies are uniformly distributed(e.g., ωf = f.∆ω with ∆ω = 2π

NsTs), a fast computation of the matrix entries (5.21)

can be done using the Fast Fourier Transform [53, 125], in this case requiring15NoNiNf log2(Nf ) flops for the normal matrix formulation. This results in afurther reduction of the computation time if 15 log2(Nf ) < 32n, thus dependingon the model order n and the number of DFT frequencies Nf . In practice, thistypically results in a further reduction of a factor 2–10.

Figure 5.6 shows the comparison for the Jacobian and normal-matrix basedWLS implementation for a discrete-time model of 50 modes with complex coef-ficients. As can be seen both results are the same, as is theoretically expected,since the normal matrix still has a good conditioning (κ = 7E4) for this high num-ber of modes, indicating the important advantage of using a discrete-time model.The benefit from using complex coefficients can be seen by comparing the normal-based implementations using a discrete model with complex and real coefficients,shown in Figure 5.7, where in the case of the real-valued coefficients, the numericalconditioning (κ = 6.5E5) has some affect on the accuracy of the model for the 3closely-spaced modes around 1700Hz (since the order of the polynomial mode isdoubled).

As discussed in [123] the assumption made with respect to the discrete charac-ter of the measured data and the model used to represent the LTI system shouldbe the same. The two most commonly-used assumptions are Zero-Order Hold(ZOH) and Band Limited (BL). In practice, since anti-aliasing filters are appliedduring the measurements, the measured signals have a limited bandwidth (BL).A discrete-time model, implicitly assumes that the measured signals remain con-stant between two consecutive samples (ZOH). Mixing the use of both assumptionsintroduces modelling errors, which, however, by sufficient over-modelling remainsmall in practice.

Using a bilinear transformation [91], it would be possible to model continuous-time systems exactly by means of a discrete-time model wherefore a pre-distortionof the frequency-axis ωDT = 2

Tsarctan (ωCT ) is required. However, due to this

distortion, the complex exponential functions Ωf = e(−iωfTs) are no longer or-thogonal resulting again in a worse numerical conditioning. Moreover, the FFTalgorithm (cf. Appendix B) is not applicable anymore since this requires a uniformfrequency grid, resulting in a increase of the computation time.


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Figure 5.7: Normal-based WLS implementation for a discrete-time model with realcoefficients.

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Figure 5.8: Normal-based Weighted mixed LS-TLS implementation for a discrete-timemodel with complex coefficients.

LS or mixed LS-TLS Approach

By choosing another constraint for the parameters a different solution is found.All the previous LS solutions were found by fixing the highest order denominator


coefficient an to 1. The mixed LS-TLS solution is found by using a norm-1 con-straint on the denominator coefficients, i.e. θHA θA = 1, resulting in Figure 5.8. Itis clear that the choice of the constraint has an important effect on the quality ofthe estimated model since in the case of the norm-1 constraint more mathematicalpoles appear in the transfer function estimate compared to the LS estimate in Fig-ure 5.6 . This will be illustrated as well in § 5.3.5, where a stabilization diagramis constructed by varying the LS constraint by fixing the lowest to highest orderdenominator coefficient.

Comparison of Numerical Properties

Concerning the numerical properties for the different possible implementations andparameterizations two important numerical aspects were considered:

• number of flops: gives a first indication of the computational speed of thedifferent possible algorithm implementations when using the Jacobian orNormal equations. The number of flops is found by counting each sum orproduct as one single floating point operation (flop) according to the “new”definition of flops given in [40].

• matrix condition number and rank : gives an indication for the numericalrobustness of the implementation for a high model order. Depending bothon the choice of parameterization and type of LS equations, the accuracy ofthe transfer function estimate will deteriorate, once the condition numberof the Jacobian or Normal matrix becomes too high. At the same time,the rank of this matrix indicates the maximum number of modes that canbe identified by the number of linear independent rows or columns in thematrix.

These quantities are summarized in Tables 5.1 and 5.2 for the practical case of theslat track data, where No = 352, Ni = 1, Nf = 1075 and Nm = 50. In terms ofcomputation speed, the important gain of the optimized algorithms can be seenfrom the normalized gain. For the case of the slat track example, the Jacobian-based implementation using the mixed LS-TLS algorithm based on the sparse QRvia projections is approximately 10000 times faster (in terms of flops) than solvingthe problem by a full QR decomposition. Compared to the sparse QR algorithmproposed by Bayard, using different weights for the Weighted LS version, there isstill a gain of approximately a factor 100. However, the Normal-based implemen-tation based on the use of the Toeplitz structure, FFT and elimination approachachieves an additional gain of approximately a factor 100 compared to the fast QRapproach for the Jacobian approach. Furthermore, it can be seen from these re-sults that the number of flops required for the formulation of the Normal equationsby exploiting the Toeplitz structure is comparable to the Jacobian matrix.


Flops Slat trackexample

Normalizedgain (×)

Jacobian-based methods

Form. J 12nNoNiNf 2.3E8 –full QR(J) 8(NoNiNf )

3n2Nf 9.4E14 1

sparse QR(J) = W 8Nf2(nNoNi +Nf ) 1.7E11 5.5E3

sparse QR(J) 6= W 12NoNiNf2(2n+Nf ) 5.7E12 1.6E2

sparse QR(J) via proj. 32NoNiNfn2 3.0E10 3.1E4

Normal-based methods

Form. Nsum 32nNoNiNf 6.1E8 –Form. NFFT 15NoNiNf log2(Nf ) 5.7E7 –full EIG(Nsum) 4(NoNin)

3 2.2E13 4.3E1

compact EIG(Dsum) 16NoNin(n2 + 2Nf ) 1.3E9 7.2E5

compact EIG(DFFT) NoNi(16n3 + 15Nf log2(Nf )) 7.6E8 2.1E6

Table 5.1: Comparison of number of flops for different mixed LS-TLS algorithms us-ing a discrete-time model with complex coefficients, including the flops count for theformulation of the Jacobian or Normal matrix.

Table 5.2 compares the condition number and rank of the Jacobian matrix Rv

(cf. Eq. 5.17) and the compact Normal matrix D (cf. Eq. 5.26) for the differentstudied parameterizations, where the last row and column are omitted for theLS problem with an = 1. These results obviously indicate the numerical problemsrelated to the use of the Laplace variable. Besides the very high condition numbers,the rank of the matrices is not higher than 10, which explains that only a single orfew modes are more or less found for the high frequencies in Figure 5.3. The useof Forsythe polynomials clearly improves the numerical conditioning, where theJacobian and Normal matrices have a comparable condition number and are bothof full rank. On the contrary, the normal-based implementation using Chebyshevpolynomials already suffers from the fact that these polynomials are not perfectlyorthogonal resulting in a poor overall synthesis, although most of the physicalmodes are still identified since the normal matrix is still of full rank. Finally, theuse of a discrete-time model also yields a good numerical behaviour. Although,the condition number for the normal matrix using real coefficients is somehowhigher, one can conclude that an implementation based on discrete-time modelis characterized by good numerical properties, combining both the benefits ofaccuracy and computation speed.

Conclusions of Comparative Study

The conclusions drawn from this comparative study of various possible parameter-izations for the frequency-domain least squares implementations for experimentalmodal data can be summarized as follows:


Cond κ Rank

Continuous-time Laplace

Rv(1 : n, 1 : n) – R 5.94E75 10

Rv(1 : n, 1 : n) – C 1.67E45 7

D(1 : n, 1 : n) – R 2.30E143 5

D(1 : n, 1 : n) – C 3.10E75 3

Continuous-time Forsythe

Rv(1 : n, 1 : n) – R 1.04E3 100

D(1 : n, 1 : n) – R 1.92E3 100

Continuous-time Chebyshev

Rv(1 : n, 1 : n) – C 3.55E3 50

D(1 : n, 1 : n) – C 9.27E8 50

Discrete-time

Rv(1 : n, 1 : n) – R 8.65E2 100

Rv(1 : n, 1 : n) – C 8.58E2 50

D(1 : n, 1 : n) – R 6.50E5 100

D(1 : n, 1 : n) – C 7.35E4 50

Table 5.2: Comparison of condition number and rank of Jacobian matrix Rv and Normalmatrix D for different parameterizations and a model with 50 modes (Nm = 50). Thelast row and column are omitted for the LS problem with an = 1, where n = 2Nm orn = Nm for respectively real and complex coefficients.

• The use of the normal equations has the advantage, compared to the Jacobian-based LS formulation, that these equations can be constructed in a computa-tional efficient way, while the size of the normal matrix is also smaller. Thisnormal matrix has a block structure with the submatrices having a prede-fined Vandermonde-like or Toeplitz structure. When using a discrete-timedomain model, these submatrices can be computed using the FFT algorithm,while a good numerical conditioning of the normal matrix is preserved. AJacobian-based implementation is preferred in the case of continuous-timemodel, although numerical conditioning is often a major problem. Bothapproaches allow the elimination of the numerator coefficients in order tofirst identify the system poles. This results in an important gain of com-putation efficiency. The numerator coefficients can be found by means ofback-substitution.

• Although classically real-valued coefficients are estimated, the use of complexcoefficients is preferable with respect to both the numerical conditioning andthe computational performance. For complex coefficients the order of thedenominator polynomial equals the number of modes that can be presentedby the common-denominator model, while this model order is doubled in thecase of real-valued coefficients. For modal analysis applications, this numberof modes can be high (a typical number is 50 modes).


• The choice of the generalized transform variable Ω is an important factorwith respect to the numerical properties of the LS implementation. Using adiscrete-time model with complex-valued coefficients, models with 100 modesor more can be estimated without any numerical problems (cf. § 5.7). Al-though, modelling errors will be introduced by this type of model, a sufficientover-modelling allows to minimize these errors. Given the robustness for highmodel orders, this over-modelling does not introduce numerical problems. Inthe case that the modelling errors become too large or when a very accuratemodel is required, a continuous-time model might be preferred. As discussedbefore, the use of Chebyshev polynomials offers a possible alternative for thefast formulation of the structured matrices and hence it is still possible toimplement a fairly fast version of the linear Least Squares estimator basedon the normal equations. Although, since the Chebyshev polynomials areonly orthogonal in approximation, a normal-based formulation tends to suf-fer again from a deteriorating numerical condition, once the model ordersbecome high. On the contrary, Forsythe polynomials yield a very robustformulation for a continuous-time model identification, however at the priceof an important increase of the computation time. Hence, a MIMO imple-mentation is considered too slow for practical use in modal analysis.

• The parameter constraint has an important effect on the quality of the esti-mated model as well. Fixing the highest order coefficient to 1, has the ten-dency to yield better results than the mixed LS-TLS (norm-1 constraint),although some caution is needed as will be shown in § 5.3.5.

5.3.4 Transformation to Modal Model

Since the goal is to determine the structural dynamics by means of the modalmodel, the modal frequencies, damping ratios and modal residues have to be de-rived from the estimates of the polynomial coefficients θ. This is done by trans-forming the common-denominator model (5.1) into a pole-residue parameterization(4.14), which is done as follows

• Poles: the poles pr (r = 1, . . . , Nm) are found as the roots of the commondenominator polynomial A(Ω, θA) with coefficients θA. From the poles themodal frequency fdr and damping ratio ζr are readily obtained as

fdr =Im(pr)

2πand ζr =

Re(pr)

|pr|(5.36)

• Residues: the residue matrices Rr (No × Ni) can be calculated from thecoefficients θ as follows (k = 1, . . . , NoNi)

Rkr = limΩ→pr

Hk(Ω, θ)(Ω− pr) (5.37)


If a discrete-time pole-residue model (Z-domain) is used, the poles pr and residuesRr have to be transformed to the Laplace domain by means of the impulse-invariant transformation (z = esTs), where the damped natural frequency anddamping ratio are subsequently obtained from the poles as (5.36).

5.3.5 Constructing Stabilization Charts

The presence of noise (measurement noise, computation noise, ...) and possiblemodelling errors (discrete-time domain model) require the model order n to bechosen high enough in order to find all physical modes. Over-modelling, however,introduces many computational poles, which complicates the modal parameter es-timation process. In order to assist the user in distinguishing the physical (struc-tural) from the computational poles, a so-called stabilization chart was proposed.By displaying the poles (on the frequency axis) for an increasing model order (i.e.number of modes in the model), indicates the physical poles since in general theytend to stabilize for an increasing model order, while the computational polesscatter around. First presented in the same period as the LSCE estimator, it hasbecome a common tool in modal analysis today. As a result a fast construction ofthe stabilization chart is one of the requirements of a MPE algorithm.

To construct a stabilization chart, the poles have to be estimated for increasingmodel orders. Based on the knowledge of the square matrix D with size (n+ 1)

D =

[

NoNi∑

k=1


]

(5.38)

this can be done in a time efficient way, by solving the eigenvalue problem ofsubmatrices of D for an increasing size. By doing so, a set of LS or mixed LS-TLSsolutions (and thus the poles) are obtained for a varying order of the denominatorpolynomial, while the order of the numerator polynomial is kept constant andequal to the maximum specified order. Notice that the matrix D plays a similarrole as the so-called covariance matrix of the LSCE estimator.

Figure 5.9 shows the relation between the compact and full normal LS equa-tions for a varying order of the denominator polynomial, while the order of thenumerator polynomials is fixed. As can be seen from Eq. (5.38), omitting for ex-ample n− j rows and columns of the compact matrix D is equivalent to omittingn−j columns in the submatrices Sk and n−j rows and columns in the submatricesTk of the full problem. The submatrices Rk remain the same and consequently,the order of the numerator polynomials is fixed to the maximum order n. As aresult, according Eq. (5.38), the solution found by solving the compact eigenvalueproblem, where D is now a square (n − j + 1) matrix, is the same as found fromsolving the full problem, since the submatrices that are used for both formulationsare still the same.


10

11

1n

20

21

2n

0

1

n

bb

b

bb

b

aa

a

00

000

0

00

0

≈

2R

1R1S

2S

H1S H

2So iN N

kk

T

0

1

n

a

a

a

00

0

≈ !"#D

Figure 5.9: Relation between compact and full normal LS equations for varying orderof denominator polynomial, while order of numerator polynomials is fixed.

Remark : A similar approach is possible for the Jacobian-based implementation.Based on the knowledge of the square matrix RV returned by the first step of themixed LS-TLS algorithm (cf. § 5.3.1) for a given maximum model order, the SVDof submatrices of RV can be computed for an increasing size of this matrix. Giventhe equivalency between the compact and full normal LS problem as well as theresults of Appendix A proving that the Jacobian and Normal equations yield thesame solution, this also implies that same equivalency exists for the results foundby omitting the last n− j rows and columns in Rv and last n− j columns the fullJacobian J .

Figure 5.10 (top) shows stabilization diagram for the case of the slattrack,


1000 1500 2000 25000

5

10

15

20

25

30

35

40

Freq. (Hz)

Num

ber o

f mod

es

1000 1500 2000 25000

5

10

15

20

25

30

35

40

Freq. (Hz)

Num

ber o

f mod

es

Figure 5.10: Stabilization diagram for normal discrete LS estimator with complex coef-ficients with the highest order coefficient an = 1 (top) and a norm-1 constraint (bottom)with (+ = stable), (· = unstable) (i.e. positive real part) and (solid line = averaged sumFRFs).

when using the normal-based discrete estimator with complex coefficients (withthe highest order coefficient an = 1). The poles are plotted for an increasing modelorder, with (+) indicating stable poles, (·) the unstable poles (i.e. positive realpart) and the line is the averaged sum of all FRF measurements.

The effect of the parameter constraint is clearly noticed by comparing withFigure 5.10 (bottom), where a norm-1 constraint is used to yield the mixed LS-


1000 1500 2000 25000

5

10

15

20

25

30

35

40

45

Freq. (Hz)

Den

omin

ator

coe

ffici

ent

Figure 5.11: Stabilization diagram for normal discrete mixed LS estimator with complexcoefficients for a constant model order and variable constraints.

TLS solution. However as can be noticed from both results, the LS implementationindicates the pole at 1870Hz as unstable, while a fairly consistent behaviour ofthis solution can be seen for the increasing model order. The mixed LS-TLSdiagram, shows this pole as a stable solution. This learns us that, although theLS implementation leads to clear stabilization diagram, it can sometimes happenthat a physical pole is estimated as unstable, however with a real part very closeto zero.

Instead of computing the LS or mixed LS-TLS solutions for an increasingmodel order, a variant for the classical stabilization chart can be constructed byconsidering the n + 1 LS solutions (for the maximum model order), derived byfixing one by one all coefficients of the denominator to 1. In practice, this canbe done in an efficient way by computing the inverse matrix of D. As shown in[32], the jth (j = 1, . . . , n + 1) column of D−1 = (T − SHR−1S)−1 gives the LSsolution for the denominator coefficients θA under the constraint aj = 1. Plottingthe n poles obtained for each fixed coefficient, results in Figure 5.11. The physicaland computational poles can be clearly distinguished since the physical poles arefound as stable poles for each parameter constraint, while the computational onesare more scattered and evolve from stable to unstable poles. This result alsodemonstrates that the constraint applied on the denominator coefficients has animportant influence on the clarity of the stabilization chart since the computationalpoles tend to shift into the unstable part of the Nyquist plane when fixing thehigher order coefficients to 1. This approach can also be applied for the Jacobian-based LS.


1000 1500 2000 25000

5

10

15

20

25

30

35

40

Freq (Hz)

Num

ber o

f mod

es

Figure 5.12: Stabilization diagram for normal discrete mixed LS estimator with complexcoefficients with the highest order coefficient an = 1, where relative criteria of 1% for thefrequency and 5% for the damping are used. The markers stand for (? = stable pole),(+ = stable frequency), (x = stable damping), (o = unstable pole), where ’stable’ heremeans within the criterium.

Remark : It should be noticed, however, that in practice the evaluation of thepoles in order to determine their stabilization behavior is often based on the useof relative criteria, e.g frequency variation over 2 model orders ≤ 1% and damp-ing ≤ 5%, as shown in Figure 5.12. However, such criteria should be used withcaution. Especially in the case of lightly damped modes at lower frequencies (0–100Hz) for which the damping ratios are small, the percentage variation will belarge, making that relative criteria are more severe for poles that are close to theorigin of the Nyquist plane. Moreover, such criteria do not take into account theuncertainty on the measurement data, which will introduce variations on the sub-sequent estimates. For this reason, other mode validation criteria are presented inChapter 7 in order to automate the physical poles selection process.

5.3.6 LSCF Estimator

The final practical implementation of the frequency-domain Linear Least Squaresestimator optimized for modal parameter estimation is called the Least SquaresComplex Frequency (LSCF) algorithm. The choice of the most important algo-rithm characteristics was based on the results obtained in § 5.3.3 and can besummarized as:


• Formulation: the normal equations (5.26) are constructed for the commondenominator discrete-time model in the Z-domain. Consequently, by loopingover the outputs and inputs, the submatrices Rk, Sk and Tk are formulatedthrough the use of the FFT algorithm as Toeplitz structured (n+ 1) squarematrices. Using complex coefficients, the FRF data within the frequencyband of interest (FRF-zoom) is projected in the Z-domain in the intervalof [0, 2π[ in order to improve the numerical conditioning. (In the case thatreal coefficients are used, the data is projected in the interval of [0, π[.) Theeffects of the discontinuities at the edges of the interval, can be reduced byprojecting on an interval that does not completely describes the unity circle,say [0, α2π[ where α is typically 0.9–0.95. Deliberately over-modelling isbest applied to cope with the discontinuities. This is justified by the use of adiscrete-time model in the Z-domain, which is much more robust for a highorder of the transfer function polynomials.

• Solver : the normal equations can be solved for the denominator coefficientsθA by computing the LS or mixed LS-TLS solution. The inverse of thesquare matrix D for the LS solution is computed by means of a pseudoinverse operation for reasons of numerically stability, while the mixed LS-TLS solution is computed using an SVD.

• Stabilization Chart : The stabilization chart is computed in a time-efficientway as explained in § 5.3.5. As discussed in § 5.3.5, fixing the highest ordercoefficient (an) generally results in the most clear stabilization charts.

• Residue Estimation: Depending on the purpose of the LSCF estimator, thenumerator coefficients and residues are derived differently. In the case thatthe LSCF is used as a starting value generator for the ML estimator, thenumerator coefficients are computed by back-substitution through Eq. (5.24).On the other hand, if only the physical poles are selected by means of thestabilization chart, the residues are estimated by a second-step LS solver aspresented in the next section.

5.3.7 LS Residue Estimator

An alternative for deriving the estimates of the residues consists of using a 2-stepapproach analogously to the traditional LSCE-LSFD approach. As discussed in§ 5.3.5, the LSCF estimator can be used to estimate the poles for an increasingmodel order in a time-efficient manner where the results are summarized in astabilization chart. In a second step, the selection of the physical poles is passed toa second-step solver, which estimates both the mode shape and modal participationvalues at the same time. For this purpose, the so-called Least Squares Frequency-domain Residue estimator was developed. This estimator uses the frequency-domain formulation of the pole-residue parameterization which is given (for the


case that displacements and forces are considered) as

H(iω) =

Nm∑

j=1

Rj

iω − pj+

N∑

j=1

R∗j

iω − p∗j+ UR− LR

ω2(5.39)

where pj and Rj are respectively the poles and (No×Ni) residue matrices. UR andLR are the (No×Ni) upper and lower residual terms that approximate the effectsof modes below and above the frequency band of interest. Since the poles pj areknown from the first step, the model (5.39) becomes linear in the parameters, i.e.the residues Rj , UR and LR, and so a linear least squares problem has to be solved.The least squares estimate of the complete residue matrices Rj , j = 1, . . . , Nm,can be further decomposed in a mode shape Ψj and modal participation factor Ljvector by means of the (thin) SVD algorithm [40]

Rj = UΣV T (5.40)

Assuming that rank(

Rj

)

= 1 (i.e. only one singular different from zero), the firstcolumn of U represents the mode shape vector Ψ and the first row of V T representsthe modal participation factor vector LTj , while the singular value σ1 can be usedto scale these vectors.

Similar to the CMIF method, for the case of multiple poles, the singular vectorscorresponding to the singular values significantly differing from zero are not theactual mode shapes. Instead, these are given by a (unknown) linear combinationof the left singular vectors.

As an example, Figure 5.13, shows the mode shapes of 2 modes in the lower partof the considered frequency band estimated by means of the LS Residue estimator.

Remark : It should be noticed that, when using a discrete-time model, it is notpossible to take the effects of modes below and above the frequency band of interestinto account by means of the so-called upper and lower residual terms, as is thecase for the continuous-time modal model (5.39). Nevertheless, over-modelling forthe effects of the discontinuities, generally results in a good FRF synthesis, evenat the edges of the analysis band. Moreover, since the LSCF estimator allows todetermine both the poles and residues of the physical modes within the band ina single step, the unknown UR and LR can still be derived from a linear leastsquares problem based on

H(iω)−Nm∑

j=1

Rj

iω − pj−

N∑

j=1

R∗j

iω − p∗j

= +UR− LR

ω2(5.41)

5.3.8 Comparison of LSCF with LSCE

A comparison of the LSCF estimator with the well-known and commonly-usedLSCE estimator learns more about their similarities and differences


(a) (b)

Figure 5.13: Mode shapes of slattrack for modes at 1156.6Hz (a) and 1348.6Hz (b)estimated by the LS Residue estimator.

• Data: both estimators start from FRF data for which a frequency band ofinterest can be selected. However, since the LSCE is a time-domain method,the selected data is transformed into IRFs by means of an IFFT. Conse-quently, possible leakage errors in the FRFs are also present in the IRFdata. Furthermore, the practical implementation of the LSCE only uses thefirst, say 100 samples of the IRFs (containing the most information aboutthe system) in order to reduce the computation time.

• Model : the LSCF uses a discrete-time scalar matrix fraction descriptionmodel (SMFD) in the Z domain with complex polynomial coefficients. Thepoles are determined as the roots of the common denominator polynomial.On the other hand, in the algorithm of the Polyreference LSCE, the polesand participation factors are derived as the eigenvalue solutions of a problemthat is formulated using an AR model with real (Ni×Ni) matrix coefficients[62], where this model corresponds to a right matrix fraction description(RMFD) starting from the transpose of the IRF matrix.

• Solver : in both cases, the least squares solution is found based on the formu-lation of the normal equations. In the case of the LSCE, the normal matrixis called the covariance matrix and plays the same role as the normal matrixD in (5.26).

• Stabilization Chart : As discussed in § 5.3.5, both estimators can constructthe stabilization chart in a time-efficient manner by solving the LS problemfor an increasing size of the normal matrix. However, since the PLSCE uses

5.4. Weighted Generalized Total Least Squares 133

an AR model with real matrix (Ni × Ni) coefficients, the relation betweenthe number of modes Nm that can be described by the model and the orderof the matrix polynomial n is given as nNi ≥ 2Nm. Hence the increment ofthe model order in the stabilization chart is not necessarily equal to one. Inthe case of the LSCF estimator this increment is always equal to one.

• Modal Parameters: since the LSCF estimator uses a SMFD model, boththe poles and residues, and hence the mode shape and modal participationfactor vectors can be estimated in one step. Alternatively, similar to theLSCE–LSFD approach, the LSCF can be used to yield only the poles ina first step from which the user can make a selection using a stabilizationchart. Next, the residues are then obtained by means of the LS Residueestimator. On the other hand, an AR model is used by the LSCE implyingthat besides the poles, also the modal participation factors can be estimatedsimultaneously during a first step, while the LSFD estimator yields the modeshape vectors in a second step. In the case of inconsistent data, the LSCE ingenerally performs worse [136], which partially can be explained by the factthat both the poles and the participation factors are inconsistent hamperingthe estimation process.

Figure 5.14, shows the poles obtained by means of the LSCE estimator (Nm =50). Comparing this result with Figure 5.10 (top) illustrates that the LSCE es-timator does not properly identify the closely-spaced poles around 1650Hz. Al-though, the LSCE estimates real-valued coefficients, the use of a discrete-timedomain model avoids a bad numerical conditioning. Nevertheless, since the LSCEis a time-domain method, no frequency weighting could be applied, which alsoexplains the better results obtained by the frequency-domain LS.

5.4 Weighted Generalized Total Least Squares

By taking also the uncertainty on the estimated FRF data into account duringthe parametric identification, more accurate results can be obtained compared tothe LS approach. This can be done by the so-called (Weighted) Generalized TotalLeast Squares (WGTLS) and Maximum Likelihood estimators. These approachesare discussed for modal parameter estimation from FRF data in this and thefollowing section.

5.4.1 Cost Function Formulation

Given the linearized LS problem formulated as (cf. Eq. 5.7) and shortly noted asJθ ≈ 0, the (Weighted) Generalized Total Least Squares (WGTLS) solution for


1000 1500 2000 25000

5

10

15

20

25

30

35

40

Mod

el o

rder

Freq (Hz)

Figure 5.14: Stabilization diagram for the LSCE estimator using real coefficients withthe highest order coefficient a2n = 1 (Ni = 1).

this estimation problem is given as [139]

argminJ,θ

‖(J − J)C−1J ‖2F subject to Jθ = 0 and θHθ = I (5.42)

A frequency-dependent weighting can be introduced by means of the scalar weight-ingWk(ωf ) in the linear LS equations (5.5). The weighting matrix CJ is the square

root of the covariance matrix of J , i.e. C = CHJ CJ = E

δJHδJ

, with δJ = J− Ja measure for the noise contribution on the Jacobian matrix. These disturbancesare due to the uncertainty on the FRF data.

In [107] it is proven that the elimination of J in (5.42) results in the equivalentcost function minimized by the WGTLS estimator

argminθ

θHJHJθ

θHCHJ CJθ

subject to θHθ = 1 (5.43)

where this cost function is equivalent to the Rayleigh quotient of the parametervector θ. In practice, the solution is found by computing the Generalized SingularValue Decomposition (GSVD) of the matrices J and CJ .

With the matrix JHJ equivalent to the normal matrix (5.20), denoted as Mand C = CH

J CJ the covariance matrix of this normal matrix, it implicitly followsfrom (5.43) that the equivalent cost function minimized by the WGTLS estimator


for the estimation problem (5.23) is given as

argminθ

θHMθ

θHCθsubject to θHθ = 1 (5.44)

In practice, the solution is now found by means of an Eigenvalue Decomposi-tion (EVD) for the generalized eigenvalue problem that directly follows from thisequivalent cost function

Mθ = λCθ (5.45)

where the solution θWGTLS is given by eigenvector corresponding the smallesteigenvalue λ.

Based on the equivalency between (5.43) and (5.44) the WGTLS solution forthis estimation problem is also found as

argminM,θ

‖(M − M)C−1‖F subject to Mθ = 0 and θHθ = 1 (5.46)

where the WGTLS solutions given by (5.42) and (5.46) are the same (cf. Ap-pendix C).

5.4.2 Generalized Eigenvalue Problem Formulation

Referring to the generalized eigenvalue problem (5.45)

JHJθ = λCθ (5.47)

the covariance matrix C contains a measure for the errors on the normal matrixM and hence has the same block structure

C = E

δJHδJ

=

CR10 · · · CS1

0 CR2· · · CS2

......

. . ....

CHS1

CHS2· · · ∑NoNi

k=1 CTk

(5.48)

In the case of FRF data, only the submatrices Φk contain this data and henceonly the last n+1 columns of the Jacobian matrix (5.10) are subjected to errors onthe measurements, i.e. J = J0 + δJ with δJ =

[

[0] . . . [0] | δΦ]

. Consequently, thecovariance matrix C = EδJHδJ is then given as (5.48), where the submatricesCRk = CSk = [0] for k = 1, . . . , NoNi, while only the square (n + 1) matrix

CT =∑NoNi

k=1 CTk is different from zero.

Notice that, by using SMFD model, the FRFs Hk(ωf ) are implicitly assumedto be uncorrelated. Referring to Chapter 2, the extent of correlations between


the FRFs strongly depends on the modal test setup used as well on the noisesources interfering with the setup. In the case that electrodynamic shakers are usedto excite the structure, important correlations can be introduced near resonancefrequencies. However, in order to obtain a fast implementation of the frequency-domain algorithms for MPE possible correlations have to be omitted. Nevertheless,the results obtained by taking only the variances into account are in general moreaccurate that the traditional or so-called deterministic LS algorithm discussed in§ 5.3. Furthermore, by using a SMFD, the WGTLS is consistent without takingany possible FRF correlations into consideration.

With the entries of the submatrices Tk given by

[Tk]rs = [ΦHk Φk]rs = −

Nf∑

f=1

|Wk(ωf )Hk(ωf )|2Ωr−1H

f Ωs−1f (5.49)

the entries of the submatrices CTk are computed as

[CTk ]rs = E

δΦHk δΦk

rs=

Nf∑

f=1

|Wk(ωf )|2var(

Hk(ωf ))

Ωr−1H

f Ωs−1f (5.50)

with var

Hk(ωf )

= E

δHHk (ωf )δHk(ωf )

the variance of the measured FRF.When a discrete-time model in the Z-domain is used with uniformly distributedfrequencies, the entries (5.50) can be computed again in a time-efficient mannerby means of the FFT algorithm (cf. Appendix B) with the matrices CTk having aToeplitz structure.

From this it follows that the mixed LS-TLS solution corresponds to the GTLSsolution of the Generalized Total Least Squares problem (5.45) under the condi-

tions that∑NiNo

k=1 CTk ∝ [I]n+1 (i.e. proportional to identity matrix) and thatCRk , CSk are all zero for k = 1, . . . , NoNi. Since the submatrices Γk in Eq. (5.10)do not contain any measurement data, CRk = CSk = [0] is always satisfied. Themixed LS-TLS approach has two advantages compared to the classical TLS ap-proach:

• the assumptions with respect to the measurement noise are more realisticsince the classical TLS assumes CRk ∝ [I]n+1 with the same scaling factor

as for∑NiNo

k=1 CTk ∝ [I]n+1

• the elimination of the numerator coefficients (5.24) is only possible whenCRk = CSk = [0], which is the case for the the mixed LS-TLS approach.

Based on this, the generalized total least squares problem (5.45) is explicitly writ-

5.5. Maximum Likelihood 137

ten as

R1 0 · · · S10 R2 0 S2... 0

. . ....

SH1 SH2 · · · T

θB1

θB2

...θBNoNiθA

= λ

0 0 · · · 00 0 · · · 0...

.... . .

...0 0 · · · CT

θB1

θB2

...θBNoNiθA

(5.51)From this it can be easily seen that, by using a common denominator model,the WGTLS estimator is consistent without requiring any possible correlationsbetween the FRFs to be taken into account. As for the LS approach, the numeratorcoefficients θB can be eliminated from (5.51) by

θBk = −R−1k .Sk.θA (5.52)

while substitution in the last n+1 equations of (5.51) now yields a compact GTLSproblem

[

NoNi∑

k=1


]

θA = λCT θA (5.53)

where the compact normal matrix is the same as the matrix D found for the LSestimator (cf. Eq 5.26).

Again, once the θA coefficients are known, back-substitution based on (5.52)can be used to derive all θB coefficients. This approach is more time efficient thansolving (5.51) directly, i.e approximately No

2.Ni2 times faster.

For the practical implementation of the fast WGTLS estimator, a similar ap-proach as for the LSCF estimator (cf. § 5.3.6) was followed. Again a fast con-struction of the stabilization chart is possible by means of computing the EVD foran increasing order n, i.e. an increasing size of the D and CT matrices.

5.5 Maximum Likelihood

5.5.1 ML Equations

Under the assumptions that the FRFs are complex normally distributed and themeasured FRFs are uncorrelated, the maximum likelihood cost function reducesto

`ML(θ) =

NoNi∑

k=1

Nf∑

f=1

|Hk(Ωf , θ)−Hk(ωf )|2var(

Hk(ωf )) (5.54)

where Hk(Ωf , θ) is the FRF for k = 1, . . . , NoNi, modelled using the commondenominator model (5.1). Omitting the FRF covariances in the cost function


results in a loss of statistical efficiency of the estimates, while the estimator is stillconsistent (cf. § 5.8).

The ML estimate θML of the polynomial coefficients is obtained by minimizingthe cost function (5.54) with respect to the parameters θ, where (5.54) is nonlinear-in-the-parameters. This can be done for instance by means of a Gauss-Newtonoptimization algorithm, which takes advantage of the quadratic form of the costfunction. The Gauss-Newton iterations are given by

1. solve the normal equations

JHmJmδm = −JHm em for δm (5.55)

2. compute an update of the previous solution θm

θm+1 = θm + δm (5.56)

with em = e(θm) the equation error and Jm = ∂e(θ)/∂θ |θm the Jacobian matrixwhere the equation error or so-called residual vector is given as

e(θ) =

H1(ω1,θ)−H1(ω1)√var(H1(ω1))

...H1(ωNf ,θ)−H1(ωNf )√

var(H1(ωNf ))

H2(ω1,θ)−H2(ωf )√var(H2(ωf ))

...HNoNi

(ωNf ,θ)−HNoNi(ωNf )√

var(HNoNi(ωNf ))

(5.57)

The explicit computation of the Jacobian matrix Jm yields a matrix with the sameblock structure as the Jacobian matrix (5.10) where the submatrices Γk and Φk

are now given by

Γk =

Ω0(ω1)√var(Hk(ω1))A(ω1,θ)

· · · Ωn(ω1)√var(Hk(ω1))A(ω1,θ)

... · · ·...

Ω0(ωNf )√var(Hk(ωNf ))A(ωNf ,θ)

· · · Ωn(ωNf )√var(Hk(ωNf ))A(ωNf ,θ)

(5.58)

and

Φk =

− Ω0(ω1)Bk(ω1,θ)√var(Hk(ω1))|A(ω1,θ)|2

· · · − Ωn(ω1)Bk(ω1,θ)√var(Hk(ω1))|A(ω1,θ)|2

... · · ·...

− Ω0(ωNf )Bk(ωNf ,θ)√var(Hk(ωNf ))|A(ωNf ,θ)|

2· · · − Ωn(ωNf )Bk(ωNf ,θ)√

var(Hk(ωNf ))|A(ωNf ,θ)|2

(5.59)


Given the block structure of the jacobian matrix Jm, the equations (5.55) areexplicitly given (for iteration m) as

R1 0 · · · S10 R2 0 S2... 0

. . ....

SH1 SH2 · · ·∑NoNi

k=1 Tk

δθB1

δθB2

...δθBNoNiδθA

= −

ΓH1 e1ΓH2 e2

...ΓHNoNieNoNi∑NoNi

k=1 ΦHk ek

(5.60)

with the submatrices now defined as Rk = ΓHk Γk, Sk = ΓHk Φk and Tk = ΦHk Φk

using (5.58) and (5.59). Given the block structure of the Eq. (5.60), a time-efficientformulation of the square matrix JHmJm with (NoNi+1)(n+1) rows and columnsand the vector JHm em with (NoNi+1)(n+1) rows is possible in an analog manner aspresented in § 5.3.2. In the case that a discrete-time model is used, these square(n + 1) submatrices again have a Toeplitz structure. In the case of uniformlydistributed frequencies, the entries can be computed using the FFT algorithm.

Furthermore, because of the block structure, the first NoNi − 1 blocks of canbe rewritten as

δθBk = −R−1k

(

ΓHk ek + SkδθA

)

(5.61)

where δθBk and δθA are the perturbations on the numerator and denominatorcoefficients computed in Eq. (5.55) and ek the part of the residual vector (cf.Eq. 5.57) corresponding to the output/input location k. The numerator coefficientscan be eliminated from the last n + 1 equations of (5.55) by means of (5.61),resulting in

NoNi∑

k=1

(


)

δθA = −NoNi∑

k=1

(

ΦHk − SHk R−1

k ΓHk

)

ek (5.62)

This elimination approach decreases the required storage space by a factor NoNi

and the computational operations by a factor No2Ni

2. As a result, the computa-tion time for one iteration is comparable with the LSCF estimator. A time-efficientimplementation of the frequency-domain maximum likelihood algorithm is solelypossible when only the variances of the FRFs are considered in the ML cost func-tion (5.54). Hence this implementation is merely a compromise between accuracyand computation time. As a result, it is significantly faster than the exact MLalgorithm, while the results are in general more accurate that the traditional or so-called deterministic algorithms for modal parameter estimation. The covariancescan optionally be considered for a more accurate computation of the uncertaintybounds for the parameter estimates, as shown in the next section.


5.5.2 Parameter Uncertainty Bounds

Given the uncertainty on the measurements as the covariance matrix of the FRFmatrix (cf. Ch. 2), the covariance matrix of the ML estimate θML can be computedaccording the following approximation [109] (p.287)

Cov(θML) ≈ [JHmJm]−1 (5.63)

with Jm the Jacobian matrix evaluated in the last iteration step of the Gauss-Newton algorithm. As one is mainly interested in the uncertainty on the modalfrequencies and damping ratios, only the covariance matrix of the denominatorcoefficients is required. Starting from (5.62), the covariance matrix of the denomi-nator coefficients θA can be computed without loss of generality but an importantreduction of the computation time as

Cov(θA) = E

δθAδθHA

= E

(

NoNi∑

k=1

(

Tk − SHk R

−1k Sk

)

)−1(NoNi∑

k=1

(

ΦHk − S

Hk R

−1k ΓH

k

)

ek

)

(

NoNi∑

l=1

eHl

(

Φl − ΓlR−1l Sl

)

)(

NoNi∑

l=1

(

Tl − SHl R

−1l Sl

)

)−H

(5.64)

where δθA are perturbations of the estimated coefficients of the denominator poly-nomial, for which EδθA is taken equal to zero, which is justified by Eq. (5.62)if the estimate of θA equals the exact denominator coefficient vector. Under thenoise assumption that no correlation exists between any of the FRFs, the sameholds for the residuals, i.e. E

eHk ek

= INf and E

eHk el

= 0Nf for k 6= l, andhence Eq. (5.64) reduces to

Cov(θA) ≈[(

NoNi∑

k=1


)]−1

(5.65)

Hence it is not necessary to invert the full matrix defined by (5.63). Moreover,this covariance matrix is the inverse of the matrix that is also computed to solvethe normal equations (5.62). From (5.65), the uncertainty on the poles (modalfrequencies and damping ratios) can be computed using the approach given in[54].

Remark : In the case that the FRFs are correlated, it is necessary to computethe covariance matrix according the full expression (5.64). In [143], the influenceof correlations between FRFs measured in near points is studied by means ofmonte carlo simulations demonstrating a systematic overestimation of the bounds(typically up to a factor 2) when only the variances are taken into account.


1000 1200 1400 1600 1800 2000 2200 2400 26000

5

10

15

20

25

30

35

40

Freq. (Hz)

Itera

tion

num

ber

Figure 5.15: Stabilization diagram for the ML estimator.

5.5.3 ML Stabilization Chart

Given the poles derived from each iteration step, a variant of the stabilization chart(cf. § 5.3.5) can be constructed by plotting the poles for an increasing numberof iterations as a function of the frequency. Since the model order is fixed, thesame number of solutions (poles) is found for each iteration step. However, it isobserved that the physical poles will have a more consistent behaviour over theiterations resulting in clear stable lines, as shown in Figure 5.15 for the case of theslattrack data.

5.5.4 Logarithmic ML

The logarithmic equation error is given by

εLOGk (ωf ) = log

(

Hk(ωf )/

Hk(ωf ))

(5.66)

The variance of this equation error is given by

var(

εLOGk (ωf )

)

= var(

log(Hk(ωf )))

≈ var(

Hk(ωf ))

∣

∣Hk(ωf )∣

∣

2 (5.67)

i.e. a measure for the noise-to-signal (power) ratio of the measured FRFs. Hence, ifa nonparametric estimate of var

(

Hk(ωf ))

is available (cf. Chapter 2), the following


weighted complex logarithmic cost function is obtained

`LOG(θ) =

NoNi∑

k=1

Nf∑

f=1

∣

∣ log(

Hk(ωf )/

Hk(ωf ))∣

∣

2

var(

Hk(ωf ))/∣

∣Hk(ωf )∣

∣

2 (5.68)

This estimator is in strict sense ”ML-like”with an asymptotic efficiency close tothe ML estimator while being practically consistent [49]. The logarithmic MLestimator is more appropriate for the analysis of FRF measurements with a veryhigh dynamical range (i.e. 100dB and more) and is more robust to outliers in thedata.

The same fast algorithm as used for the minimization of (5.54) can be appliedto (5.68) to derive the parameter estimates with their uncertainty bounds. Theentries of the Jacobian matrix Jm (cf. Eq. 5.55) are now found to be

Γk =

Ω0(ω1)|Hk(ω1)|√var(Hk(ω1))Bk(ω1,θ)

· · · Ωn(ω1)|Hk(ω1)|√var(Hk(ω1))Bk(ω1,θ)

... · · ·...

Ω0(ωNf )|Hk(ωNf )|√var(Hk(ωNf ))Bk(ωNf ,θ)

· · · Ωn(ωNf )|Hk(ωNf )|√var(Hk(ωNf ))Bk(ωNf ,θ)

(5.69)

and

Φk =

− Ω0(ω1)|Hk(ω1)|√var(Hk(ω1))A(ω1,θ)

· · · − Ωn(ω1)|Hk(ω1)|√var(Hk(ω1))A(ω1,θ)

... · · ·...

− Ω0(ωNf )|Hk(ωNf )|√var(Hk(ωNf ))A(ωNf ,θ)

· · · − Ωn(ωNf )|Hk(ωNf )|√var(Hk(ωNf ))A(ωNf ,θ)

(5.70)

Remark : The idea of fitting the logarithm (of the amplitude) of the FRF data, wasfirst proposed by [128] for the nonlinear least squares estimator. The logarithmicLS cost function tends to be smoother than the traditional nonlinear least squarescost function resulting in a larger convergence region [9]. A generalization forMIMO systems has been given in [68]. In [49], the asymptotical properties arestudied in an EV framework.

5.6 Choice of Frequency Weighting

Depending on the solver used, i.e. a WLS (cf. § 5.3) or WGTLS (cf. § 5.4) and thechoice of the weighting Wk(ωf ) a number of other frequency-domain estimatorscan be derived using the same fast implementation.

5.6. Choice of Frequency Weighting 143

Weighted Least Squares Solver

A drawback of the linearized equation error εk(ωf , θ) = A(Ωf , θ)H(ωf )−Bk(Ωf , θ),in the case of a continuous-time model (a polynomial in Ωf = iωf ), is the over-emphasizing of high-frequency measurements in the cost function. This may resultin a poor modelling at low frequencies and ill-conditioned normal equations in thecase of data with a high modal density and a large dynamic frequency range. By in-troducing a frequency-dependent weighting in the cost function

∑NoNik=1 |εk(ωf , θ)|2,

this problem can be reduced. In the case that a common-denominator model isused, this is done by weighting the linearized equation error εk(ωf , θ) by means ofthe scalar weight Wk(ωf )

εwk (ωf , θ) = Wk(ωf )(

A(Ωf , θ)Hk(ωf )−Bk(Ωf , θ))

(5.71)

for k = 1, . . . , NoNi the output/input DOFs and f = 1, . . . , Nf the number ofspectral lines.

The basic idea was introduced by Sanathanan and Koerner [119], where thefollowing (deterministic) weighting function evaluated using the denominator co-efficients known from a previous estimation

W2k(ωf ) =

1∣

∣A(ωf , θm−1)∣

∣

2 (5.72)

results in an iteratively weighted linear least squares problem minimizing the fol-lowing cost function

`IWLS(θm) =

NoNi∑

k=1

Nf∑

f=1

∣

∣A(ωf , θm)Hk(ωf )−Bk(ωf , θm)∣

∣

2

∣

∣A(ωf , θm−1)∣

∣

2 (5.73)

As a result, a lower weight is given to the data at the higher frequencies. Severalother weighting functions (Stahl, Strobel, Gyurki, ’t Mannetje, Whitfield, etc) aresummarized in [105]. This iterative approach, however, requires adequate startingvalues [145].

The optimal weighting function equals the maximum likelihood weighting inthe cost function (5.54) which, for the linearized least squares equations, can beintroduced by the following (stochastic) weighting

W2k(ωf ) =

1∣

∣A(ωf , θm−1)∣

∣

2var(

Hk(ωf ))

(5.74)

evaluated using the denominator coefficients known from a previous estimation.This results in an iteratively weighted estimator with nearly ML properties, calledthe Iterative Quadratic Maximum Likelihood (IQML) estimator minimizing the


1000 1500 2000 2500 3000 3500−120

−100

−80

−60

−40

−20

0

Freq. (Hz)

Am

pl. (

dB)

1000 1500 2000 2500 3000 3500

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

Freq. (Hz)

Am

pl. (

dB)

Figure 5.16: Unweighted (top) and weighted (bottom) normal-based LS implementationfor a discrete-time model with complex coefficients.

following cost function

`IQML(θm) =

NoNi∑

k=1

Nf∑

f=1

∣

∣A(ωf , θm)Hk(ωf )−Bk(ωf , θm)∣

∣

2

∣

∣A(ωf , θm−1)∣

∣

2var(

Hk(ωf ))

(5.75)

In the case of a discrete-time model, the problem over-emphasizing the high-frequency measurements in the cost function is not experienced, which is one ofthe advantages of using this type of model for modelling modal data with a high


modal density and a large dynamic frequency range. Nevertheless, the idea ofusing a frequency-dependent stochastic weighting can still offer advantages sincethe importance of each measurement in the LS estimation process is weightedaccording the quality of each measurement. Analogously to the nonparametricweighting function as proposed in [107] for Input/Output Fourier data, the fol-lowing frequency-dependent weighting function can be used for FRF data in thecase of a discrete-time model, where possible correlations between the FRFs areneglected,

W2k(ωf ) =

∣

∣Hk(ωf )∣

∣

2

var(

Hk(ωf )) (5.76)

Given the nonparametric character of the weighting function an iterative process isnot needed. In order to illustrate the effect of this frequency weighting, Figure 5.16shows the results obtained for the slattrack data using an unweighted (top) andweighted (bottom) version of the Normal-based LS for a discrete-time model withcomplex coefficients (Nm = 50). As can be noticed, the 3 closely-spaced modesaround 1650Hz are not properly identified by the unweighted LS, while a correctmodel is found for the analyzed frequency band in the case of the weighted LS.

Weighted Generalized Total Least Squares Solver

The WGTLS cost function (5.44) in terms of the linearized weighted equationerror (5.71) is given as

`WGTLS(θ) =

∑NoNik=1

∑Nff=1

∣

∣Wk(ωf )∣

∣

2∣∣A(ωf , θ)Hk(ωf )−Bk(ωf , θ)

∣

∣

2

∑NoNik=1

∑Nff=1

∣

∣Wk(ωf )∣

∣

2∣∣A(ωf , θ)

∣

∣

2var(

Hk(ωf ))

(5.77)

By choosing the weighting function Wk(ωf ) as the nonparametric weighting(5.76), the accuracy can be improved since the importance of each measurement isweighted according the quality of each measurement without the need for startingvalues.

By using Wk(ωf ) the ML-like weighting function (5.74) evaluated using thedenominator coefficients known from a previous estimation, the asymptotical ef-ficiency of the GTLS estimator can be improved. This results in an iterativelyWGTLS estimator with nearly maximum likelihood properties, also called the Boot-strapped Total Least Squares (BTLS) estimator.


1000 2000 3000 4000 5000 6000 7000−130

−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

Freq. (Hz)

Am

pl. (

dB)

1000 2000 3000 4000 5000 6000 7000−130

−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

Am

pl. (

dB)

Freq. (Hz)

Figure 5.17: Results for WLSCF (top) WGTLS (bottom) estimators for Nm = 80.


1000 2000 3000 4000 5000 6000 7000−130

−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

Freq. (Hz)

Am

pl. (

dB)

1000 2000 3000 4000 5000 6000 7000−130

−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

Am

pl. (

dB)

Freq. (Hz)

Figure 5.18: Results for ML (top) and BTLS (bottom) estimator for Nm = 80.


5.7 Validation of Stochastic Estimators for Ex-perimental Data

In order to validate the accuracy of the stochastic algorithms for a high modelorder, the slattrack data was analyzed by estimating a model of 80 modes infrequency band of 200–7800Hz using the WLS, WGTLS shown in Figures 5.17and ML (50 iterations) and BTLS (50 iterations) estimator shown in Figure 5.18.

As can be seen, the and WLSCF and WGTLS yield very similar results, al-though the WLSCF tends to estimate a more accurate model for the very highfrequencies, which can be explained by the difference in parameter constraint. TheML estimator clearly yields the most accurate model, where all the modes in theconsidered frequency band are correctly identified. Interesting modes for compar-ison are the closely-spaced and highly damped modes at 2600Hz and the closelyspaced mode close to the dominant resonance peak at 4000Hz.

5.8 Asymptotical Properties of Stochastic Esti-mators

Based on the cost function formulation it is possible to evaluate the asymptoti-cal properties of the different frequency-domain estimators. However, since thepossible correlations between the FRFs are neglected in the covariance matrix ofthe FRFs, none of the presented estimators will be asymptotically efficient. Theasymptotical consistency of the estimators can be examined using the quick toolsproposed in [109] (p. 189–190).

A necessary condition for consistency is the minimization of the expected valueof the cost function by the exact modal parameters (assuming that the true modelbelongs to the model set)

θe = argminθ

E`(θ) (5.78)

Remark : However, in order to prove the consistency of an estimator (i.e. a nec-essary and sufficient condition), one should prove that the cost function `(θ) con-verges in means square sense to exact cost function for the number of frequencyspectral lines Nf →∞ [109].

Least Squares

Given that the measured FRF can be written as Hk(ωf ) = Hk0(ωf ) + NHk(ωf ),

where EHk0(ωf )N∗Hk

(ωf ) = 0. The expected value of the cost function of the

5.8. Asymptotical Properties of Stochastic Estimators 149

LS estimator (5.6) is computed as

E`WLS(θ) (5.79)

= E

NoNi∑

k=1

Nf∑

f=1

∣

∣A(Ωf , θ)H(ωf )−Bk(Ωf , θ)∣

∣

2

= E

NoNi∑

k=1

Nf∑

f=1

∣

∣(A(Ωf , θ)Hk0(ωf )−Bk(Ωf , θ) +A(Ωf , θ)NHk(ωf )

∣

∣

2

=

NoNi∑

k=1

Nf∑

f=1

(

∣

∣A(Ωf , θ)Hk0(ωf )−Bk(Ωf , θ)∣

∣

2+∣

∣A(Ωf , θ)∣

∣

2var(

NHk(ωf )

)

)

(5.80)

Hence, in the exact parameters, E`WLS(θe) = 0 +∣

∣A(Ωf , θAe)∣

∣

2var(

NHk(ωf )

)

,is not minimal since this still depends on the parameter and the data.

Generalized Total Least Squares

A similar result as for the LS is found for the Total Least Squares estimator. How-ever, referring to the cost function of the GTLS (5.77), where the measured FRFHk(ωf ) = Hk0(ωf ) + NHk

(ωf ), with EHk0(ωf )N∗Hk

(ωf ) = 0 and the variances

var(

Hk(ωf ))

are assumed to be exactly known, the expected value is given as

E`GTLS(θ) (5.81)

=E

∑NoNik=1

∑Nff=1

(

∣

∣A(ωf , θ)Hk0(ωf )−Bk(ωf , θ)∣

∣

2+∣

∣A(ωf , θ)∣

∣

2NHk

(ωf ))

∑NoNik=1

∑Nff=1

∣

∣A(ωf , θ)∣

∣

2var(

Hk(ωf ))

=

∑NoNik=1

∑Nff=1

∣


∣

2

∑NoNik=1

∑Nff=1

∣

∣A(ωf , θ)∣

∣

2var(

Hk(ωf ))

+ 1

From this it follows that the expected value of the GTLS cost function is minimalin the exact parameters, i.e the necessary condition for consistency.

Maximum Likelihood

Under the assumption that the FRFs are complex normally distributed and thevariances var

(

Hk(ωf ))

are exactly known, computing the expected value of thecost function (5.54) based on the linearized equation error results in

E`ML(θ) =NoNi∑

k=1

Nf∑

f=1

∣


∣

2

∣

∣A(ωf , θ)∣

∣

2var(

Hk(ωf ))

+ 1 (5.82)


where the measured FRF Hk(ωf ) = Hk0+NHk, with EHk0N

∗Hk = 0. From this

it follows that the expected value of the ML cost function is minimal in the exactparameters, i.e the necessary condition for consistency.

5.9 Conclusions

In this chapter, the LS problem formulation using the Jacobian or normal equationshas been numerically optimized in terms of the required memory and computationtime by exploiting the block structure resulting from using a common-denominatormodel. For the Jacobian-based implementation, this was achieved by developinga fast sparse QR algorithm for the elimination of the numerator coefficients, whilethis elimination follows directly from the block structure of the normal equations,where this latter approach is still faster than using the Jacobian formulation.

A study of different types of parameterization for experimental modal datahas demonstrated that the use of the normal-based formulation is justified whenusing a discrete-time model with complex coefficients, resulting in a very robustapproach in terms of numerical conditioning and accuracy, where models of 100modes or more could be estimated without any numerical problems. The memoryusage and computation time were further optimized by exploiting the Toeplitzstructure of the block matrices and the use of the FFT algorithm in the case ofthe discrete-time model.

For a continuous-time model, the use of orthogonal polynomials is required inorder to preserve a good numerical conditioning, as shown by the use of Forsytheor Chebyshev polynomials. In the case of Chebyshev polynomials, a time-efficientimplementation could be implemented. However, a model order of typically 50 ormore still results in numerical problems for the normal matrix since the Chebyshevpolynomials are only orthogonal in approximation.

Furthermore, it is shown that the parameter constraint has an important effecton the obtained solution and a method for the fast construction of the stabilizationdiagram has been derived. Likewise the well-known LSCE estimator, the finalimplementation of the LS algorithm (LSCF) can be used in a 2 step approach incombination with the proposed LS Residue estimator.

The fast normal-based implementation was extended to the class stochastic(frequency-domain) estimators such as the Maximum Likelihood and the WeightedGeneralized Total Least Squares. Notice that the time-efficient implementation isa balance between accuracy and memory/computation efficiency since it does notallow to take possible correlations between the FRFs into account. Although thisimplies that the stochastic estimators are not asymptotically efficient anymore, theconsistency is preserved for the ML andWGTLS estimator. It has been shown thata fast implementation of the WLS and WGTLS variants, such as IQML and BTLS,

5.9. Conclusions 151

can be readily obtained. The same features, such as Toeplitz structured matrices,use of FFT algorithm and a fast construction of the stabilization diagram wereapplicable. The robustness and accuracy of the stochastic estimators has beenillustrated for experimental data obtained from a safety critical component of anaircraft.


5.10 Appendix A – Jacobian and Normal Matrixbased LS Algorithms

Given the very compact LS problem, based on the (n+1) last equations of (5.23),[

NoNi∑

k=1


]

θA ≈ 0 (5.83)

Rewriting this as a function of the submatrices Γk and Φk of the Jacobian matrix(5.10), based on the definition of the submatrices Rk = ΓHk Γk, Sk = ΓHk Φk,Tk = ΦH

k Φk (cf. Eq 5.23), yields

[

NoNi∑

k=1

ΦHk Φk − ΦHΓk(Γ

Hk Γk)

−1ΓHk Φk

]

θA ≈ 0 (5.84)

orNoNi∑

k=1

ΦHk

[

I − ΓkΓ+k

]

ΦkθA ≈ 0 (5.85)

with Γ+k the pseudo-inverse of the matrix Γk. Furthermore, the matrix

[

I − ΓkΓ+k

]

is idempotent since[

I − ΓkΓ+k

] [

I − ΓkΓ+k

]

= I − 2ΓkΓ+k + ΓkΓ

+k ΓkΓ

+k =

[

I − ΓkΓ+k

]

(5.86)

with Γ+k Γk = I. The QR-decomposition of the (Nf × n + 1) matrix Γk = QkRk,

from which it follows that

ΓkΓ+k = QkRkR

+k Q

+k = QkQ

Hk (5.87)

since Q+k =

(

QHk Qk

)−1QHk = IQH

k (Qk is orthogonal). As a result,[

I − ΓkΓ+k

]

=[

I −QkQHk

]

, which is the (Hermitian) orthogonal projection operator Q⊥k as de-

fined in step 2 of the Sparse matrix QR algorithm (cf. § 5.3.1).

As a result, given Eq. (5.86), Eq. (5.85) can be written as

[

ΦH1 Q

⊥1 ,Φ

H2 Q

⊥2 , . . . ,Φ

HNoNiQ

⊥NoNi

]

Q⊥1 Φ1

Q⊥2 Φ2

...Q⊥NoNi

ΦNoNi

θA ≈ 0 (5.88)

which based on the definition of the matrix V in step 3 of the mixed LS-TLSalgorithm using Sparse QR decomposition boils down to

V HV θA = RHV Q

HV QVRV θA = RH

V RV θA ≈ 0 (5.89)

proving that normal equations based on the matrix RV will yield the same solutionfor θA as the normal equations (5.26).

5.11. Appendix B – Fast Calculation of Normal Matrix Entries 153

5.11 Appendix B – Fast Calculation of NormalMatrix Entries

If a discrete time-domain model is used, i.e. Ωjf = e(−iωfTs) and if the frequencies

are uniformly distributed (e.g., ωf = f.∆ω with ∆ω = 2πNsTs

), then the summations(5.21) can be rewritten as

[ΓHk Γk]rs =

Nf∑

f=1

|Wk(ωf )|2ei2π(r−s)f/Ns

[ΦHk Φk]rs =

Nf∑

f=1

|Wk(ωf )Hk(ωf )|2ei2π(r−s)f/Ns

[ΓHk Φk]rs = −Nf∑

f=1

|Wk(ωf )|2Hk(ωf )ei2π(r−s)f/Ns (5.90)

Defining the sequences

ΓΓk(n) =

|Wk(ωf )|2 n = 1, . . . , Nf

0 n = 0, Nf + 1, . . . , Ns − 1(5.91)

ΦΦk(n) =

|Wk(ωf )Hk(ωf )|2 n = 1, . . . , Nf

0 n = 0, Nf + 1, . . . , Ns − 1(5.92)

ΓΦk(n) =

−|Wk(ωf )|2Hk n = 1, . . . , Nf

0 n = 0, Nf + 1, . . . , Ns − 1(5.93)

it is observed that

[ΓHk Γk]rs = FΓΓk(r − s)[ΦH

k Φk]rs = FΦΦk(r − s)[ΓHk Φk]rs = FΓΦk(r − s) (5.94)

with

FΓΓk(m) =

Ns−1∑

n=0

ΓΓk(n) expi2πmn/Ns

FΦΦk(m) =

Ns−1∑

n=0

ΦΦk(n) expi2πmn/Ns

FΓΦk(m) =

Ns−1∑

n=0

ΓΦk(n) expi2πmn/Ns

the discrete Fourier transform (DFT) of the above defined sequences. Using anFFT to form the normal equations [53, 125] results in a further reduction of thecomputation time (typically a factor 2–10).


5.12 Appendix C – Equivalency of WGTLS Solu-tions

It can be assumed that C = I without any loss of generality. Given the singularvalue decomposition of the Jacobian matrix J

J = UΣV H (5.95)

the Frobenius norm of the matrix J is then given as the sum of the singular values(n is the rank of the matrix J)

‖J‖F =√

trace(

JHJ)

=

√

√

√

√

n∑

j=1

σ2j (5.96)

The Frobenius norm of the matrix JHJ results in

‖JHJ‖F =

√

√

√

√

n∑

j=1

σ4j (5.97)

which, in general, is not equal to ‖J‖F . However, the TLS solution is only foundin the case that the rank of the perturbed matrix J is equal to n− 1 in order that‖(J − J)‖2F → 0. The SVD of the matrix J of rank n− 1 is given as

J =

n−1∑

j=1

ujσjvHj (5.98)

Hence, based on equations (5.96) and (5.97) it follows that

‖(J − J)‖2F = σ2n (5.99)

‖(JHJ − JH J)‖F =√

σ4n = σ2n

where σn is the smallest singular value. This proves that both the (WG)TLSproblems based on the Jacobian (5.42) and normal matrix (5.46) yield the sameresult.

Chapter 6

Frequency-domain MPEfrom Input/Ouput Data

For the case of an arbitrary excitation and that only a very limited amount of datasamples can be acquired, the nonparametric FRF identification based on averagingtechniques becomes difficult. Sufficient time samples must be available to havean acceptable frequency resolution to detect closely spaced modes. An alternativeapproach then consists of estimating the modal parameters based on the Inputand Output (I/O) Fourier sequences. Based on the results of Chapter 5, a fastimplementation of the frequency-domain (Total) LS and ML methods is derived forI/O data in Sections 6.4, 6.5 and 6.7. Moreover, when using the I/O data insteadof the FRFs, effects of leakage (or transients in general) can be compensated byestimating the system and transient parameters simultaneously as will be discussedin Section 6.8. This approach can also be generalized to estimate the parametersfrom auto and cross power spectra, which can be preferable when longer time recordsare available and a parametric compensation for possible leakage effects is desired(cf. § 6.9).

155

156 Chapter 6. Frequency-domain MPE from Input/Ouput Data

6.1 Introduction

As explained before, the common approach for experimental modal identificationstarts from frequency response functions (FRF), which are derived using a non-parametric FRF estimator such as H1 or Hv. However, FRF-based identificationcan be seriously complicated by one or more of the following problems

• First of all, too restrictive assumptions about the noise model result in incon-sistent FRF estimators. As discussed in Chapter 2, accurate FRF estimationis possible based on errors-in-variables nonparametric estimators, such as theHiv (arbitrary excitation) or Hev (periodic excitation).

• However, since FRF estimators are based on averaging techniques, sufficienttime samples must be available in order to have an acceptable frequencyresolution to detect all the dynamical behaviour, including closely-spacedmodes. When only short data sequences are acquired under a random noiseexcitation, it is preferable to estimate the parameters based on the raw (I/O)Fourier series.

• Another problem relates to the use of arbitrary excitation signals (e.g ran-dom noise) which introduces leakage effects, especially in the case of shorttime sequences obtained from lightly damped structures. Although, periodicsignals, such as multisine or periodic chirp signals, avoid effects of leakage,random noise excitation is still most often used in modal testing requiringspecial attention as was also discussed in Chapter 3.

Several applications are confronted with one or more of these problems. In thecase of ’flutter testing’, the time to perform measurements is very limited due tosafety and cost reasons. Only very short time sequences can be measured at eachflutter speed and the measured data typically contains high noise levels [47, 134].When using a SLDV setup to measure in a high number of response DOFs and theoverall testing time is limited, the measurements can also be restricted to shortdata sequences.

In [110, 47, 42] a frequency-domain Maximum Likelihood algorithm and [108,42] frequency-domain Total Least Squares-based algorithms are presented to esti-mate modal parameters for both SISO (Single Input Single Output) and MIMO(Multiple Input Multiple Output) models from Input/Output data. However thenumerical (computation time, memory usage and numerical conditioning) perfor-mance was not optimized to handle extensive data-sets characterized by a highmodal density and corrupted by leakage.

In this chapter, the frequency-domain estimators for FRF-based modal param-eter estimation presented in Chapter 5, are adapted for the case of I/O data. Asdiscussed in the beginning of this section, the use of I/O data is certainly pref-ered when only short data sequences corrupted with leakage errors are available.

6.2. Parametric Model 157

The formulation of the I/O estimators is based on an errors-in-variables set-up(cf. § 6.3), where both the noise on the input and output signals are considered.However, as for the FRF-based estimators where possible correlations between theFRFs are ommited, the fast implementation of the I/O estimators requires thatonly the output variances are considered. Nevertheless, in the case of a SLDVmeasurement setup, this noise model is still exact.

In § 6.4, both the Jacobian and Normal-based formulations of the LS estimatorare derived for the I/O data. Using the same approach as in § 5.3.2, a fast im-plementation of the LS estimator is possible. However, as will be shown in § 6.5,the fast formulation of the WGTLS I/O estimator was less straightforward sincetwo approximations are required. Besides ommiting possible output correlations,the derivation the WGTLS requires a linear approximation in the formulation ofa compact WGTLS problem, which has been validated by means of Monte Carlosimulations. The I/O-based ML estimator is discussed in § 6.7.

By estimating simultaneously the initial conditions and the system model pa-rameters, it is possible to deal with arbitrary signals in the frequency-domainwithout any approximation and under the same assumptions as in the time-domain[111, 56]. Using this approach, the proposed I/O estimators can be made robustfor errors due to leakage effects (cf. § 6.8).

In § 6.9, discusses the generalization of the I/O estimators to the case of Autoand Cross Power Spectral Density functions. The use of PSD functions has theadvantage, compared to using the raw I/O data, that the noise on the data isreduced by means of the averaging process, however, this requires sufficient data.On the other hand, compared to using FRF data, the effects of leakage can nowbe compensated for as shown in § 6.8.

6.2 Parametric Model

Analogous to Chapter 5, a common denominator model is used to model theFrequency Response Function between output o and input i (for o = 1, 2, . . . , No

and i = 1, 2, . . . , Ni) at angular frequency ωf with f = 1, . . . , Nf (Nf the numberof spectral lines)

Hoi(ωf ) =Boi(Ωf , θ)

A(Ωf , θ)(6.1)

with Boi(Ωf , θ) =∑n

j=0 boijΩjf the numerator and A(Ωf , θ) =

∑nj=0 ajΩ

jf the

common-denominator polynomial, where n is the order of the polynomials and Ωf

the generalized transform variable (cf. § 5.2). The polynomial coefficients aj andboij are the parameters θ to be estimated.


6.3 Errors-in-Variables Noise Model

In practice, the measured responses (velocities and/or accelerations) as well asforces are affected by errors. As discussed in § 2.2, the measured I/O Fourier dataZ = [FH , XH ]H can be represented using a errors-in-variables stochastic noisemodel

Z(ωf ) = Z0(ωf ) + EZ(ωf )[H0(ωf ),−INo ]Z0(ωf ) = 0

f = 1, . . . , Nf (6.2)

with Z0 = [FH0 , X

H0 ]H the ”true” input-output (I/O) Fourier data and EZ =

[EHF , E

HX ]H some random perturbations and Z(ωf ), Z0(ωf ), EZ(ωf ) ∈ C (Ni+No×1).

The errors EZ(ωf ) are assumed to be complex normally distributed with thefollowing a-priori known covariance matrix

CEZ (ωf ) = EEZ(ωf )EZ(ωf )H =(

CEF (ωf ) CEFEX (ωf )CEXEF (ωf ) CEX (ωf )

)

(6.3)

where the non-diagonal elements of the output noise covariance matrix CEX arenot considered throughout this chapter. This follows from the practical issuesrelated to the derivation of a full rank covariance matrix CEZ (ωf ) for a largenumber of responses (No) by means of measurements (cf. § 2.4). Furthermore, atime-efficient implementation of the I/O estimators requires that CEX is diagonalas will turn out in this chapter.

For the particular case of scanning laser Doppler vibrometer (SLDV) measure-ments, the errors on the (subsequent) output measurement setup are uncorrelatedsince each response location is measured separately resulting in zero non-diagonalelements of CEX (zk). On the other hand, since the number of inputs is typicallysmall (Ni ≤ 5), input correlations as well as input-output correlations can be deter-mined, resulting in CEF (ωf ) and CEXEF (ωf ) being full matrices for each measuredoutput. Taking, besides the noise variances, also the cross-correlations of the er-rors on the I/O Fourier data into account, can further decrease the uncertainty ofthe estimates.

6.4 (Weighted) Linear Least Squares

Using the model (6.1) and given the input and output Fourier data (2.1) as definedin § 6.3, the linearized (Levi) weighted equation error for output o and spectralline f is then written as

Wo(ωf )(

bBo(Ωf , θ)c F (ωf ) −A(Ωf , θ)Xo(ωf ))

≈ 0 (6.4)

where F (ωf ) represents the input (Ni × 1) Fourier vector and Xo the response(output) Fourier coefficient measured at DOF o for spectral line f as a result of


the Ni inputs. The (1 × Ni) row vector bBo(Ωf , θ)c is the oth row of the (No ×Ni) numerator polynomial matrix B(Ωf , θ). An adequate (frequency-dependent)weighting function Wo(ωf ) in the equations (6.4), can improve the quality of theparameter estimates. Notice that in the specific case of SLDV measurements, adifferent input Fourier vector Fo(ωf ) is measured for each output DOF.

Since the equations (6.4) are linear in the parameters and in the Fourier data,they can be formulated as Jθ ≈ 0

Γ1 0 0 · · · Φ1

0 Γ2 0 · · · Φ2

......

. . ....

...0 0 · · · ΓNo ΦNo

θB1

θB2

...θBNoθA

≈ 0 (6.5)

with J the Jacobian matrix now having NfNoNi rows and (n+ 1)(NoNi + 1)columns. A similar formulation can be obtained for real coefficients for which thenumber of equations and coefficients is doubled. The entries of the submatricesΓo = [Γo1, . . . ,ΓoNi ] (Nf ×Ni.(n+ 1)) and Φo (Nf × (n+ 1)) are given by

Γoi(ωf ) = Wo(ωf )[Ω0f ,Ω

1f , . . . ,Ω

nf ]Fi(ωf )

Φo(ωf ) = −Wo(ωf )[Ω0f ,Ω

1f , . . . ,Ω

nf ]Xo(ωf ) (6.6)

while the parameter vector entries contain the (unknown) coefficients

θBo = [bo10, bo11, . . . , boNin]T (6.7)

θA = [a0, a1, . . . , an]T (6.8)

Since an analogous formulation is obtained as in § 5.3.1, the Least Squaressolution can again be found by means of the sparse QR algorithm. However, forthe same reason as in § 5.3.2, (i.e. Nf À n), the normal equations JHJθ ≈ 0results in a more compact formulation of the identification problem (JHJ having(n+ 1)(No.Ni + 1) rows and columns)

R1 0 · · · S10 R2 0 S2... 0

. . ....

SH1 SH2 · · · ∑Noo=1 To

θB1

θB2

...θBNoθA

≈ 0 (6.9)

with Ro having (n+ 1).Ni rows and columns,

Ro =

R11o · · · R1Ni

o

R21o · · · R2Ni

o...

. . ....

RNi1o · · · RNiNi

o


So having (n+ 1).Ni rows and (n+ 1) columns,

So = [S1o , S

2o . . . , S

Nio ]T

and To having (n+1) rows and columns. The parameter vector entries containingthe (unknown) coefficients are given by

θBo =

bo10bo11...

boNin

, θA =

a0a1...an

The entries of the submatrices are given by

[Ri1,i2o ]rs =

Nf∑

f=1

|Wk(ωf )|2F ∗i1(ωf )Fi2(ωf )Ω

r−1H

f Ωs−1f

[To]rs =

Nf∑

f=1

|Wk(ωf )Xo(ωf )|2Ωr−1H

f Ωs−1f

[Si1o ]rs = −Nf∑

f=1

|Wk(ωf )|2F ∗i1(ωf )Xo(ωf )Ω

r−1H

f Ωs−1f (6.10)

The same time efficient approach as presented in § 5.3.2 can be used to derive anestimate of the coefficients θ. The elimination of the numerator coefficients θBo

θBo = −R−1o SoθA , o = 1, . . . , No (6.11)

yields[

No∑

o=1

To − SHo R−1o So

]

θA ≈ 0 (6.12)

Once the denominator coefficients θA are known, a back-substitution based onEq. (6.11) can be used to find θB . Based on (6.12), a stabilization chart can beconstructed again in a time efficient manner as was explained in § 5.3.5.

From Eqs. (6.9) and (6.10) it clearly follows that the submatrices of the normalmatrix have a predefined structure. By exploiting the specific matrix structurein a similar way as discussed in § 5.3.2 and § 5.3.3, a further reduction of thecomputation time and memory requirements is obtained. More specific, in thecase that a discrete-time model is used (i.e. Ωf = e(−iωfTs)) the submatrices Ro,So and To have a Toeplitz structure while the entries can be computed by meansof the FFT algorithm if the frequencies are uniformly distributed. Using a discretetime-domain model leads to both a well-conditioned Jacobian and normal matrix.In the case of a continuous time-domain model, orthogonal polynomials have tobe used to significantly improve the numerical conditioning (cf. § 5.3.3).


6.5 Weighted Generalized Total Least Squares

6.5.1 Cost Function Formulation

The same general formulation of the Weigthed Generalized Total Least Squares,as derived in § 5.4.1 for FRF-based identification, is applicable for the I/O formu-lation of this estimator. With the Jacobian matrix J and normal matrix M nowformulated as in Eqs. (6.5) and (6.9), the WGTLS solutions are then respectivelygiven by Eqs. (5.42) and (5.46).

Hence, given the normal equations (6.9), the equivalent cost function minimizedby the WGTLS estimator for this estimation problem is given as (whereM = JHJ)

argminθ

θHMθ

θHCθsubject to θHθ = 1 (6.13)

In practice, the solution is found by means of an Eigenvalue Decomposition (EVD)for the generalized eigenvalue problem that directly follows from this equivalentcost function

Mθ = λCθ (6.14)

where the solution θWGTLS is given by the eigenvector corresponding to the small-est eigenvalue λ. The covariance matrix C is a measure for the errors on thematrix M resulting from the measurement noise on the I/O Fourier data. Thiscovariance matrix C is defined as C = CH

J CJ = E

δJHδJ

, with δJ = J − J ameasure for the noise contribution on the Jacobian matrix (6.5).

6.5.2 Generalized Eigenvalue Problem Formulation

Considering the errors-in-variables model with the noise assumptions made in § 6.3,the generalized eigenvalue problem (6.14) is explicitly given as

R1 0 · · · S10 R2 · · · S2...

.... . .

...SH1 SH2 · · · T

θB1

θB2

...θA

= λ

CR10 · · · CS1

0 CR2· · · CS2

......

. . ....

CHS1

CHS2· · · CT

θB1

θB2

...θA

(6.15)


where T =∑No

o=1 To and CT =∑No

o=1 CTo . Given the Jacobian entries (6.6), theentries of the covariance matrix C are computed as (i1, i2 = 1, . . . , Ni)

[CRi1i2o

]rs = E

δΓHoi1δΓoi2

rs=

Nf∑

f=1

|Wo(ωf )|2covar

(

Fi2(ωf ), Fi1(ωf ))

Ωr−1H

f Ωs−1f

[CSi1o]rs = −E

δΓHoi1δΦo

rs= −

Nf∑

f=1

|Wk(ωf )|2covar

(

Xo(ωf ), Fi1(ωf ))

Ωr−1H

f Ωs−1f

[CTo ]rs = E

δΦHo δΦo

rs=

Nf∑

f=1

|Wk(ωf )|2var

(

Xo(ωf ))

Ωr−1H

f Ωs−1f (6.16)

where var(

Xo(ωf ))

are the diagonal elements of the (diagonal) output noise covari-

ance matrix CEX and covar(

Fi2(ωf ), Fi1(ωf ))

and covar(

Xo(ωf ), Fi1(ωf ))

, respec-tively the elements of the input noise CEF (ωf ) and input/ouput noise covariancematrices CEFEX (ωf ) (cf. Eq. 6.3). When a discrete-time model in the Z-domainis used with uniformly distributed frequencies, the entries (6.16) can be computedin a time-efficient manner by means of the FFT algorithm with the matrices CRo ,CSk and CTk having a Toeplitz structure.

As for the I/O WLS estimator (cf. § 6.4), the computation time can be sig-nificantly reduced by eliminating the numerator coefficients θBo , since only thedenominator coefficients θA are required to compute the system poles. In the caseof I/O Fourier data, elimination of the numerator coefficients θBo in (6.15) yields(o = 1, . . . , No)

θBo =[

I − λR−1o CRo

]−1 [λR−1

o CSo −R−1o So

]

θA (6.17)

from which it can be seen that additional terms in λ (with λ unknown) appear inthe case that also the input noise sources are taken into account (i.e. CRo 6= 0and CSo 6= 0). Notice that for FRF-based GTLS identification this problem doesnot occur since errors are only present on the FRF data appearing only in thelast (n + 1) columns/rows of the normal equations (cf. § 5.4.2). Consequently,a straightforward formulation of a compact generalized eigenvalue problem in thedenominator coefficients θA is not possible by eliminating θBo from the last nequations of (6.15), i.e

No∑

o=1

SHo θBo + TθA = λ

No∑

o=1

CHSoθBo + λCT θA (6.18)

and as a result, the exact solution of the GTLS identification problem (6.14)is only given by solving the generalized eigenvalue problem (6.15). However, thecomputational load for solving this eigenvalue problem, in the case of typical modalanalysis applications (i.e. No ≥ 500 , Ni = 3 , n ≥ 50), is not acceptable inpractice.


One obvious way to benefit from the elimination approach is by consideringonly errors on the measured output sequences (i.e. H1 noise-model assumptionscf. § 2.5.2), with all matrices CRo and CSo equal to zero in (6.15) and (6.17). Thisresults, after eliminating the numerator coefficients, in the following generalizedeigenvalue problem in θA

[

T −No∑

o=1

SHo R−1o So

]

θA = λCT θA (6.19)

Once the θA coefficients are known, Eq. (6.17) (with CRo = CSo = 0) can be usedto derive all numerator coefficients. However, ignoring the input noise sources hasthe important disadvantage of loosing the errors-in-variables noise-model charac-teristics together with the asymptotical consistency of the GTLS approach.

Therefore, another option was considered in order to combine the time-efficientelimination process with the advantages of EV (GTLS) identification. Using a

Taylor expansion, the inverse matrix[

I − λ.R−1o .CRo

]−1in Eq. (6.17) can be ap-

proximated as

[

I − λR−1o CRo

]−1 ∼= I + λR−1o CRo + (λR−1

o CRo)2 + . . . (6.20)

Introducing this approximation for the first order in (6.17) and substitution in(6.18) results in, after retaining the first order terms in λ

θBo∼= −R−1

o SoθA − λR−1o CRoR

−1o SoθA + λR−1

o CSoθA (6.21)

which now again yields a compact generalized eigenvalue problem in θA

No∑

o=1

[

To − SHo R−1o So

]

θA (6.22)

= λ

No∑

o=1

[

CTo + SHo R−1o CRoR

−1o So − 2herm

(

No∑

o=1

SHo R−1o CSo

)]

θA

Compared to (6.19), two additional terms in λ appear after linearizing, containingthe noise characteristics of the input and between the input and output Fourierdata, i.e. CRo and CSo . Notice that, the matrix in the right part of (6.22) is thecovariance matrix of the compact normal matrix

D =

[

T −No∑

o=1

SHo R−1o So

]

(6.23)

Indeed, the covariance matrix CD = EδD where

δD =

No∑

o=1

δTo −No∑

o=1

(

δSHo R−1o So − SHo R−1

o δRoR−1o So + SHo R

−1o δSo

)

(6.24)


mode pole residue1 -1.5234E-3 + 0.7500E+1i 3.9282E-8 + 2.5015E-3i2 -1.3441E-3 + 1.2167E+1i 1.2874E-7 + 1.4997E-3i3 -0.7553E-3 + 2.2164E+1i -4.9174E-6 + 1.1966E-3i4 -1.2464E-3 + 2.3837E+1i 1.0541E-5 + 1.3087E-3i5 -0.5506E-3 + 2.5336E+1i -7.9402E-7 + 1.7064E-3i6 -1.1167E-3 + 4.4167E+1i 1.2672E-9 + 1.9000E-3i

Table 6.1: Poles and residues of theoretical SISO transfer function.

and hence taking the expected value results in

CD =

[

No∑

o=1

CTo +

No∑

o=1

SHo R−1o CRoR

−1o So − 2herm

(

No∑

o=1

SHo R−1o CSo

)]

(6.25)

since CTo = EδTo, CRo = EδRo and CSo = EδSo.

Instead of solving (6.15) directly, this elimination approach results again in animportant reduction of the computation time since the GSVD algorithm can beapplied to much smaller matrices (dimensions reduced by No.Ni). From equation(6.20) it follows that the linear approximation will be valid in the case that theeigenvalues of (R−1

o CRo)¿ λ, where R−1o CRo is a measure for the signal-to-noise

ratio (SNR) of the input Fourier data.

Similar to the approach used in § 5.3.5, a time efficient construction of thestabilization chart is possible again. Since the model equations in (6.12) and(6.19) are formulated for a given maximum model order, the generalized eigenvalueproblem can be re-solved for all smaller model orders by simply taking sub-matricesof the full matrices. By doing so, a set of solutions is obtained for an increasingorder of the denominator polynomial, while the order of the numerator polynomialis kept constant and equal to the maximum specified (polynomial) model order.

6.5.3 Validation of Linear Approximation

The validity of this linear approximation is now studied by means of monte carlosimulations. A Single Input Single Output system having 6 modes with polesand residues, given in Table 6.1 and shown in Figure 6.1, was used to generatean I/O data set of 1200 equally distributed frequencies in the band 0.6–7.9Hzi.e. F0(ωf ) = 1 , X0(ωf ) = H0(ωf ) (f = 1, . . . , 1200). Independent zero-mean Gaussian noise was added to both the input and output data. In order toevaluate the validity of approximation (6.20), the variance of the input noise wasincreased over 21 logarithmically distributed steps in the range of [1E-8,1E+0]while the output noise level remained constant with a variance of 1E-6. For eachI/O noise combination, 1000 disturbed data sets were generated and for each


0 2 4 6 8−70

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10

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Am

plitu

de (d

B)

0 2 4 6 80

20

40

60

80

100

120

140

160

180

f (Hz)

Pha

se (d

egre

es)

Figure 6.1: Theoretical transfer function used for SISO Monte Carlo simulations.

set the modal parameters of 20 modes (poles, residues) were estimated using theestimators summarized in Table 6.2. The frequency weighting function Wo(ωf )was equal to 1 for all estimators. The accuracy of the different estimators can be

abbreviation descriptionGTLS eqn. (6.14)FGTLS eqn. (6.22)FGTLSX eqn. (6.19) with CR = 0

TLS eqn. (6.22) with C = diag(1, 1)FTLS eqn. (6.19) with CT = 1

Table 6.2: Different estimators studied during Monte Carlo simulations. (CS = 0 forall considered cases.)

compared with respect to their squared bias, total variance and mean squared error(MSE) for both the estimated poles and residues. These quantities are related asfollows

MSE(P ) = E

(P − P0)H(P − P0)

= E

(P − EP)H(P − EP)

+ (EP − P0)H(EP − P0)

= total variance(P ) + squared bias(P ) (6.26)

with P and P0 the estimated and theoretical values of the poles (residues). TheMonte Carlo simulation results can be compared for the different estimators as


10−8

10−6

10−4

10−2

100

10−9

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noisevar input

MS

E

GTLSFGTLSFGTLSXTLSFTLS

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E

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squa

red

bias


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red

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tota

l var

ianc

e


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100

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10−8

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10−5

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noisevar input

tota

l var

ianc

e

GTLSFGTLSFGTLSX TLSFTLS

Figure 6.2: Monte Carlo simulation results: mean squared error (MSE), squared biasand total variance for poles (left) and residues (right).

6.6. Validation of (WG)TLS for Experimental Data 167

shown in Figure 6.2. The important errors in case of both TLS estimators are dueto the high bias errors, originating from the inconsistency properties of the TLSapproach, since no noise covariance information is taken into account. Consideringonly output noise results in the FGTLSX, improving the accuracy as long as theinput variance is small. However, once the effect of the input noise outweighs theoutput noise, the errors for the FGTLSX increase significantly. Since input noiseis disregarded, the FGTLSX is still inconsistent, explaining the increase in biasand the agreement with the TLS results for high input noise variances. The gainin accuracy by taking input noise into account is clearly illustrated by comparingthe FGTLSX and the GTLS where the MSE is 5dB to 10dB smaller for the GTLS.The good agreement of the errors for the GTLS and FGTLS indicates the validityof the approximation (6.20). Only for very high variances of the input noise (SNRaround 0dB), the errors are slightly higher for the FGTLS. This result provesthat no loss in accuracy is encountered when using the approximation (withinpractical noise levels), making the fast implementation of the fast algorithm of the(W)GTLS suitable for an accurate analysis of large modal data sets as shown inthe next section.

6.6 Validation of (WG)TLS for Experimental Data

Consider again the experiments performed on the slat track presented in Chapter 5,using the scanning Laser Doppler Vibrometer setup as shown in Figure 5.2. Noticethat this setup completely satisfies the errors-in-variables noise model defined insection 6.3 since no correlations exist between the output signals. Since a periodicchirp excitation was used, the noise information could be obtained by computingthe sample variances of the measured signals during 5 experiments, while leakagewas avoided.

Using the fast implementations of the I/O TLS and GTLS estimator, a SMFDmodel with 22 modes was estimated in a frequency band of 250–1500Hz containing1000 spectral lines. No frequency weighting was used in this case. Figure 6.3 showsthe ratio of the FFT of a measured velocity and the corresponding force (dots) aswell as the estimated model (solid line). The FGTLS has a better performance,explained by the use of noise covariance matrix information during the estimationprocess, making the FGTLS consistent and less sensitive to measurement noise.

Comparing the stabilization diagram for both estimators, shown in Figure 6.4,it is clear that the use of a proper noise (EV) model results in an improved stabi-lization behavior of the physical modes. In addition, the computational poles areestimated as unstable in the case of the FGTLS, improving the user-friendlinessof the chart, since in general, only the stable poles are plotted.

Another important part of the modal model is the spatial information presentin the estimated mode shapes. Figure 6.5 shows the mode shape for the mode


400 600 800 1000 1200 1400−50

0

50

f (Hz)

Am

pl.(d

B)

400 600 800 1000 1200 1400−200

−100

0

100

200

Freq. (Hz)

Pha

se (d

egre

es)

400 600 800 1000 1200 1400−50

0

50

f (Hz)

Am

pl. (

dB)

400 600 800 1000 1200 1400−200

−100

0

100

200

Freq. (Hz)

Pha

se (d

egre

es)

Figure 6.3: Parametric results for the slat track using the FTLS (top) and FGTLS(bottom). Amplitude and phase of the measurements (dots) and the estimated transferfunction model (solid line).

6.6. Validation of (WG)TLS for Experimental Data 169

400 600 800 1000 1200 14000

2

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es

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f (Hz)

Num

ber o

f mod

es

Figure 6.4: Stabilization chart with the stable (+) and unstable (×) poles (i.e. positivereal part) for the FTLS (top) and FGTLS (bottom) method.


Figure 6.5: Mode shape for mode of 1234Hz, obtained by FTLS (left) and FGTLS(right).

at 1234Hz. Again the results obtained by the FGTLS are better since an accu-rate noise model is taken into account. The mode shape for the FTLS is clearlycorrupted by estimation errors due to the measurement noise.

0 20 40 60 80 100106

107

108

109

1010

1011

1012

1013

Flop

s

Outputs

107

108

109

1010

1011

1012

Flop

s

0 20 40 60 80 100model order

Figure 6.6: Flops count for GTLS (dashed line) and FGTLS (solid line) for varyingnumber of outputs and Nm = 20 (left) and varying model order and No = 10 (right).

6.7. Maximum Likelihood Estimator 171

Finally, the numerical efficiency is illustrated in Figure 6.6 by comparing theGTLS and FGTLS algorithms in terms of (Matlab) flops count for a varyingnumber of outputs No (with n=20, Ni=1) and a varying model order n (withNo=10, Ni=1). Considering the output dimension No, the results indicate thatthe GTLS problem is solved in O(N 3

o ) flops, whereas O(No) flops for the FGTLS,resulting in a gain of O(No)

2 flops, explained by the elimination approach. On theother hand, for an increasing model order, the number of flops used by the GTLSand FGTLS is similar, i.e. O(n3).

6.7 Maximum Likelihood Estimator

Based on Eq.(6.4), the following equation error can be defined

εo(Ωf , θ) = bBo(Ωf , θ)c F (ωf ) −A(Ωf , θ)Xo(ωf ) (6.27)

The maximum likelihood (ML) cost function based on the weighted equation error

εso(Ωf , θ), i.e. εo(Ωf , θ)var(

εo(Ωf , θ))−1/2

, is given by

`ML(θ) =

No∑

o=1

Nf∑

f=1

|εso(Ωf , θ)εs∗o (Ωf , θ)|2

=

No∑

o=1

Nf∑

f=1

|bBo(Ωf , θ)c Fo(ωf ) −A(Ωf , θ)Xo(ωf )|2

var(

εo(Ωf , θ)) (6.28)

where the variance of the (scalar) equation error (6.27)

var(

εo(Ωf , θ))

= bBo(Ωf , θ)cCF (ωf ) bBo(Ωf , θ)cH + |A(Ωf , θ)|

2var(

Xo(ωf ))

−2herm(

bBo(Ωf , θ)cCFXo(ωf )A(Ωf , θ)∗)

(6.29)

with CF (ωf ) = CEF (ωf ) and CFXo(ωf ) = CEFEX [:,o](ωf ) (cf. Eq. 6.3). The ML

estimates of the polynomial coefficients θML = [θTB1, . . . , θTBNo , θ

TA]

T are obtained

by minimizing (6.28) with respect to these parameters θ.

This can be done by means of a Gauss-Newton optimization algorithm, whichtakes advantage of the quadratic form of the cost function (6.28). The Gauss-Newton iterations are now given by

1. solve the normal equations

JHmJmδm = −JHm em for δm (6.30)

2. compute an update of the previous solution θm

θm+1 = θm + δm (6.31)


with em = e(θm), Jm = ∂e(θ)/∂θ |θm and e(θ) = [εs1, . . . , εsNo

]T .

However, in the case that complex-valued coefficients are used, the equationerror εso(Ωf , θ) is not analytical in the (complex) parameters θ and hence, theentries of the Jacobian matrix Jm can only be computed by considering the realand imaginary part of the coefficients as

δm = [Re(δB1)T , Im(δB1

)T , . . . ,Re(δA)T , Im(δA)

T ]T (6.32)

JRem = ∂e(θRe,Im)/∂θRe |θRem,Imm

JImm= ∂e(θRe,Im)/∂θIm |θRem,Imm

(6.33)

The entries (6.33) of the Jacobian matrix Jm, for the weighted equation errorat spectral line f , are found to be given as

bJRem,Immco =

1

var(εo)12

∂εo∂θRe,Im

∣

∣

∣

θRem,Imm

− 1

2

εo

var(εo)32

∂var(εo)

∂θRe,Im

∣

∣

∣

θRem,Imm

(6.34)

where εo(Ω, θRem,Imm) is denoted as εo.

Since the correlations between the outputs are ommited and a SMFD modelis used, the Jacobian matrix Jm has a similar block structure as the Jacobianmatrix of the Least Squares formulation (6.5). As a result, it is again possible toform the normal equations (i.e., JHmJm and JHm em) in a time-efficient way usingthe Toeplitz structure and FFT algorithm in the case that a discrete-time modelis used. Furthermore, given the block structure of Jm, the first No blocks can beeliminated from the normal equations analogously to the MLE algorithm presentedin § 5.5.

The practical implementation of the ML algorithm for MPE from I/O data hasyet been done only for the case that the variances of the outputs are taken intoaccount (i.e. CF = 0 and CFXo

= 0 in Eq. 6.29). While the cost function (6.28)is uniquely defined, this is not the case for the equation error εso(Ωf , θ) and hencethe equation error εo(Ωf , θ) can as well be formulated as

εo(Ωf , θ) =⌊

Ho(Ωf , θ)⌋

F (ωf ) −Xo(ωf ) (6.35)

with⌊

Ho(Ωf , θ)⌋

=bBo(Ωf ,θ)cA(Ωf ,θ)

and the following expression for the variance

var(

εo(Ωf , θ))

= var(

Xo(ωf ))

(6.36)

Weighting the equation error (6.35) by var(

εo(Ωf , θ))−1/2

results in an expressionthat is analytical in the complex polynomial coefficients θ. The submatrices Γo

6.8. MPE in the Presence of Leakage Phenomena 173

and Φo of the Jacobian matrix Jm, have the same structure as the Jacobian matrixin (6.5)

Γo =

Ω0(ω1)Fi(ω1)√var(Xo(ω1))A(ω1,θ)

· · · Ωn(ω1)Fi(ω1)√var(Xo(ω1))A(ω1,θ)

... · · ·...

Ω0(ωNf )Fi(ωNf )√var(Xo(ωNf ))A(ωNf ,θ)

· · · Ωn(ωNf )Fi(ωNf )√var(Xo(ωNf ))A(ωNf ,θ)

(6.37)

and

Φo =

−Ω0(ω1)bBo(ω1,θ)cF (ω1)√var(Xo(ω1))|A(ω1,θ)|2

· · · −Ωn(ω1)bBo(ω1,θ)cF (ω1)√var(Xo(ω1))|A(ω1,θ)|2

... · · ·...

−Ω0(ωNf )bBo(ωNf ,θ)cF (ωNf )√var(Xo(ωNf ))|A(ωNf ,θ)|

2· · · −Ωn(ωNf )bBo(ωNf ,θ)cF (ωNf )√

var(Xo(ωNf ))|A(ωNf ,θ)|2

(6.38)

The uncertainy bounds on the estimated parameters θML can be derived using thesame approach as discussed in § 5.5.2.

6.8 MPE in the Presence of Leakage Phenomena

A typical problem for frequency-domain estimators in general arises from the pres-ence of transient phenomena in the data, of which spectral leakage is an importantexample. Until recently, frequency-domain system identification assumed that theinput and output signals are periodic or time-limited within the observation win-dow as in the case of e.g. an impulse or burst random excitation. In [111] theapplicability of frequency-domain system identification techniques has been gen-eralized to arbitrary signals and these results can be applied for modal parameterestimation from I/O data.

When the modal parameters are derived directly from the I/O data there is ingeneral a difference between time and frequency-domain methods. This differenceis mainly due to the manner how the initial conditions of the system are takeninto account. This can be understood by considering the Laplace transform of thej-th derivative of the force time signal f(t)

L

djf(t)

dtj

= sjF (s)−j−1∑

r=0

sj−1−r drf(t)

dtr

∣

∣

∣

∣

∣

t=0

(6.39)

where the sum is a polynomial in the Laplace variable of order j − 1 which onlyexists if the initial conditions differ from zero. Hence, when taking the initialconditions into account, the Laplace transformed SISO input-output differentialequation is given as

A(s)X(s) = B(s)F (s) + T (s) (6.40)


where the order of T (s) equals the maximum order of B(s) and A(s) minus 1 withunknown coefficients θT = [c0, c1, . . . , cn−1]

T .

From this it follows that a transient polynomial T (Ωf , θ) can be used to modelthe initial conditions (e.g. the effects of leakage) present in the I/O data. Forexample in the case of the ML estimator presented in § 6.7, adding the polynomialT (Ωf , θ) to the equation error εo results in the following generalized cost function

`ML(θ) =

No∑

o=1

Nf∑

f=1

∣

∣

∣

⌊

Bo(Ωf , θ)⌋

F (ωf ) −A(Ωf , θ)Xo(ωf )∣

∣

∣

2

var(

εo(Ωf , θ)) (6.41)

with⌊

Bo(Ωf , θ)⌋

= bBo(Ωf , θ), T (Ωf , θ)c

F (ωf ) =

F (ωf )1

The (unknown) coefficients of T (Ωf , θ) are included in θ and are estimated si-multaneously with the coefficients Bo(Ωf , θ) and A(Ωf , θ). This result indicatesthat taking a ’transient’ polynomial into account is equivalent with having oneadditional input with a constant Fourier spectrum equal to 1, and as a result,the same I/O algorithms can still be applied. This result shows that parametricfrequency-domain estimators can be made robust for leakage by estimating theinitial (and final) conditions of the system together with the system parameters.

The use of a transient polynomial is now illustrated for the case of the slattrackstructure. Using a similar setup as discussed in § 5.3.3, the time histories of theapplied force and response (acceleration) were now measured using an impedancehead. In this case the structure was excited using both a multisine and a randomnoise excitation in the frequency band of 0-4000Hz with 16384 frequency lines.

Using the time records obtained by the multisine, containing 32768 samples,the empirical transfer function estimate (ETFE) was computed. Based on thisETFE, a SMFD discrete-time model was estimated using the FRF ML estimator(discussed in § 5.5). Since no leakage effects appear when using multisine exci-tation, this model will be used as a reference for the comparison of the differentpossible approaches for analyzing the random noise time histories.

As a first case, the complete random noise time records (Ns = 32768 = 32K)were considered. Figure 6.7 (top) compares the model obtained by the I/O MLestimator using the FFT of the time records, when using a transient polynomialor not. The dashed-dotted line corresponds to the reference model obtained fromthe multisine data. The ETFE for the random noise data (dots) is a very noisefunction as can be expected. From this data, the I/O ML estimator derived themodel without (dashed line) and with using a transient polynomial (solid line).

6.9. MPE from Auto and Cross Power Spectra 175

A model with 30 modes was estimated for a frequency band of 200-1500Hz. Byzooming in, the effects of leakage can already be noticed in the case that notransient polynomial is used, as for instance for the mode at 1156Hz. In this casethe zoomed frequency band has 921 spectral lines (Nf = 921).

Decreasing the size of the records clearly introduces more leakage effects in thecase of random noise data, as is shown in Figure 6.7 for Ns = 16K (Nf = 461)(middle) and Ns = 8K (Nf = 231) (bottom) and Figure 6.8 for the cases Ns = 4K(Nf = 116 in zoom) (top), Ns = 2K (Nf = 59) (middle) and Ns = 1K (Nf = 30)(bottom). As can be seen for this experimental case, the approach using theI/O ML with transient polynomial is very robust for short data records. Evenin the case of 1K time samples corresponding to a frequency resolution of 8Hz,this approach can still estimate a fairly accurate model compared to the approachwithout using this transient polynomial.

The I/O ML approach can also be compared with the FRF-based ML. TheFRF data can be obtained by either computing a H1 estimate, which howeverrequires the averaging of a number of subrecords, or by computing the ETFE alsousing a Hanning window. Since, in the general case of short data records, the firstpossibility is not applicable, a comparison was done using the ETFE. Figure 6.9,shows a zoom on the results found when using the FRF ML for the same analysisparameters as for the I/O estimation for a decreasing size of the records. As can beseen, the estimated model is acceptable in the case of the 32K samples, however,due to imporant leakage errors the results detoriate fast for a decreasing blocksizewhen compared to the I/O ML approach that also estimates the transients. Theuse of a Hanning window is not sufficient once the records become short, in thiscase Ns = 2K or less.

6.9 MPE from Auto and Cross Power Spectra

In the case that sufficient data is measured to perform an average based non-parametric processing, the presented I/O estimators can also be used for MPEfrom the Auto and Cross Power Spectra functions. Indeed, assuming that theinput F [m](ωf ) and output X [m](ωf ) Fourier sequences, as defined in § 6.3, areobserved during M records (m = 1, . . . ,M), the Auto and Cross Power Spectracan be estimated. Using for instance the Periodogram estimator (cf. § 3.3), thePower Spectra matrix is obtained as

GZ(ωf ) =1

M

M∑

m=1

Z [m](ωf )Z[m]H(ωf ) (6.42)

where Z [m](ωf ) =[

F [m](ωf )H , X [m](ωf )

H]H ∈ C (Ni+No)×1 is the Fourier data

observed during the mth measurement.


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Freq (Hz)

Am

pl. (

dB)

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pl. (

dB)

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10

20

30

40

50

60

70

Freq (Hz)

Am

pl. (

dB)

Figure 6.7: Comparison of I/O ML results without (dashed) and with (solid) tran-sient polynomial with Reference model (dashed-dotted) and the ETFE for random noiseFourier data (dots). Cases: Ns = 32K (Nf = 921 in zoom) (top), Ns = 16K (Nf = 461)(middle) and Ns = 8K (Nf = 231) (bottom).

6.9. MPE from Auto and Cross Power Spectra 177

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60

70

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Am

pl. (

dB)

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70

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pl. (

dB)

1000 1050 1100 1150 1200−10

0

10

20

30

40

50

60

70

Freq (Hz)

Am

pl. (

dB)

Figure 6.8: Comparison of I/O ML results without (dashed) and with (solid) tran-sient polynomial with Reference model (dashed-dotted) and the ETFE for random noiseFourier data (dots). Cases: Ns = 4K (Nf = 116 in zoom) (top), Ns = 2K (Nf = 59)(middle) and Ns = 1K (Nf = 30) (bottom).


1000 1050 1100 1150 1200−10

0

10

20

30

40

50

60

70

Freq (Hz)

Am

pl. (

dB)

Figure 6.9: Comparison of FRF (ETFE with Hanning window) ML results with Ref-erence model (dashed-dotted) for the cases: Ns = 32K (Nf = 921 in zoom) (solid),Ns = 8K (Nf = 231) (dashed) and Ns = 2K (Nf = 59) (dotted).

Referring to Eq. (6.9), the normal equations[

JHJ][m]

θ can be computed for

each set of the measured records Z(ωf )[m], which can be averaged over the number

of experimentsM . The entries of the submatrices of the obtained normal equationsare then given as

[Ri1,i2o ]rs =

Nf∑

f=1

|Wk(ωf )|2(

1

M

M∑

m=1

F[m]i1

(ωf )F[m]∗i2

(ωf )

)

Ωr−1H

f Ωs−1f

[To]rs =

Nf∑

f=1

|Wk(ωf )|2(

1

M

M∑

m=1

X [m]o (ωf )X

[m]∗o (ωf )

)

Ωr−1H

f Ωs−1f

[Si1o ]rs = −Nf∑

f=1

|Wk(ωf )|2(

1

M

M∑

m=1

F[m]∗i1

(ωf )X[m]o (ωf )

)

Ωr−1H

f Ωs−1f

where the expressions between the parentheses represent the Auto and Cross PowerSpectra as defined in (6.42). This shows that the generalization of the I/O esti-mators, as developed in this chapter, is straightforward. The use of the PowerSpectra has the advantage, compared to using the raw I/O data, that the noiseon the data is reduced by means of the averaging process. This, however, requiresthat sufficient data samples could be measured. On the other hand, compared tousing FRF data, the effects of leakage can still be compensated for as shown in§ 6.8.


6.10 Conclusions

The use of an I/O estimator is preferred in the case of short data sequences, sincethen the nonparametric processing for FRF estimation is not possible. This canbe the case for flight flutter testing and structural health monitoring applications.

The formulation of the I/O estimators is based on an errors-in-variables set-up,where both the noise on the input and output signals are considered. Similar tothe FRF-based estimators, where possible correlations between the FRFs are ne-glected, the fast implementation of the I/O estimators omits possible correlationsamong the noise on the output measurements, balancing between computationalefficiency and accuracy. Moreover, in the case of a SLDV measurement setup, thissimplified noise model is still exact.

Based on the results of Chapter 5, a fast implementation of the frequency-domain LS method has been derived for the case of I/O data, where both theJacobian and Normal equations-based formulations can be used, while again afast construction of a stabilization diagram is possible. It has been shown thatin the case of the WGTLS estimator, a reduction to a compact formulation wasnot straightforward in the case that input noise is considered. Nevertheless, theinput noise can be taken into account at the price of using a linear approximationfor which, however, the validity has been demonstrated for a wide range of inputsignal-to-noise ratios by means of Monte Carlo simulations. Furthermore, neglect-ing the output correlations only results in a loss of asymptotical efficiency whilethe WGTLS remains consistent. Next, it has been shown that a fast implementa-tion of the I/O ML estimator is possible for the case that also the input noise isconsidered.

Furthermore, by estimating simultaneously the initial conditions and the sys-tem model parameters, it is possible to deal with arbitrary signals in the frequency-domain without any approximation and under the same assumptions as in thetime-domain. Using this approach, the proposed I/O estimators can be made ro-bust for errors due to leakage effects, as is demonstrated for experimental dataobtained from a safety critical component of an airplane.

The fast I/O estimators could also be generalized to estimate the parametersstarting from auto and cross power spectra, which is preferable when longer timerecords are available. Compared to using the raw I/O data, the noise on the datais reduced by means of the averaging process. On the other hand, compared tousing FRF data, the effects of leakage can now be compensated for during theparametric identification.

Chapter 7

Automated ModalParameter Estimation andTracking

In this chapter it is demonstrated that, by using a frequency-domain MaximumLikelihood estimator, features such as high accuracy and confidence bounds for theestimated parameters and robustness for high measurement noise levels can signif-icantly contribute to the automation of the modal identification process. For thevalidation of the model, criteria based on a statistical approach were developed inaddition to well-known criteria such as modal phase colinearity and mode complex-ity. Two different approaches for the final mode selection are based on a weightedvalidation criterium or a fuzzy clustering algorithm. In addition, an accuratemode-tracking algorithm is presented. It is shown that the problem of coincidingpoles, due to shifting or crossing modes under changing structural dynamics, canhamper the mode tracking and a robust solution is proposed. The possibilities andlimitations of the proposed automated modal parameter estimation approaches areinvestigated and tested for two cases. The first application concerns the automatedanalysis of data obtained from a slat track of an Airbus A320 commercial air-plane. Serving as a preliminary study for future monitoring practices during lifecycle tests on slat tracks, the method for mode tracking is tested on this same data.The second application focusses on the automated processing of data from a bridge.

181

182 Chapter 7. Automated Modal Parameter Estimation and Tracking

7.1 Overview of “State-of-the-Art”

During the last few years a lot of effort has been put in the development of vi-bration based damage detection techniques [34]. Since the dynamic behaviourof structures is influenced by damage, one is able to detect the occurrence andevolution of damage by monitoring the modal parameters (such as resonance fre-quencies) [61, 132]. However, varying environmental and operational conditionscan seriously complicate in situ structural health monitoring affecting the modalparameters as well [3, 100]. Nevertheless, an important element for the appli-cability of damage detection techniques as a part of monitoring practices is anautomated identification and tracking procedure since traditional modal identifi-cation still requires extensive interaction from an experienced user. In the case ofmodal analysis, the model order is typically over-specified in order to capture allphysical modes present in the frequency range of interest. However, this requiresthe distinguishing between the physical and mathematical modes, which is moreinvolved than merely determining a proper model order.

Most of the classical model order selection tools (Akaike, F-test, MDL,...) thatare used in time-domain identification were developed in the framework of controltheory [2, 129, 115]. The shortcomings of these methods have been extensivelystudied in [78]. They only allow to verify if the model order used is appropriateor not (detection of under-modelling or over-modelling).

Order selection techniques, used for frequency-domain identification, have beenproposed in the literature, based on a statistical analysis of the residual errors(such as a whiteness test) [121]. This approach can also detect the presence ofnon-linear distortions. A more recent model order selection approach is basedon a statistical analysis of the global minimum of a weighted non-linear least-squares cost function [118]. Another possible order selection criterion consists indetermining the rank of the Jacobian matrix or the sampled covariance matrix[118, 29]. These techniques can be used to verify if the model order is appropriateor not, or to see if there are non-linear distortions present in the measurements,but they do not allow to separate physical from computational modes. Moreover,the model orders considered in modal analysis are usually much higher and alsothe amount of data to be processed is much larger than for control applications(typically systems with a very limited number of inputs and outputs).

Today, in modal analysis, a stabilization diagram is still an important tool toassist the user in separating physical from mathematical modes. Since this is donemanually, selecting the physical poles can be very difficult and time-consumingdepending on the type and quality of the measurement data, the performance ofthe estimator and the experience of the user. Moreover, this “manual” approachis certainly not suitable for monitoring purposes.

Recently, several approaches for automated modal identification for diagnostic

7.2. Autonomous Modal Parameter Identification 183

purposes were proposed, most of them based on the use of Neural Networks [83],Fuzzy Logic [150] or Genetic Algorithms [25]. Unfortunately the computationalload for these methods can be quite high. Classical time-domain identificationalgorithms (e.g. ERA/DC) were used, not taking any information about the mea-surement noise into account. The main application focus of the research conductedby the NASA [96, 97, 150] and Zimmerman and James [25, 69] is to reduce theuser-time to perform a detailed ground vibration test/analysis of the Space ShuttleOrbiter after each flight. Essentially, they follow a rule-based approach to buildup a system model pole table by consecutively adding or updating 1 pole. Thefollowed approaches are very heuristic, and as a result dedicated for their specificapplication.

In [146, 144] automated modal identification approaches are presented basedon the use of a frequency-domain Maximum Likelihood estimator (MLE) andstochastic validation criteria, yielding promising results. In particular the avail-ability of accurate parameter estimates with their uncertainty information con-tributes greatly to a more simplified validation and selection of the physical poles.This statistical information is furthermore very important to make a distinctionbetween changes in the modal parameter caused by damage and changes causedby measurement noise. While, the computational load of the MLE algorithm hasbeen significantly reduced such that one iteration of the algorithm is as fast asthe well-known LSCE algorithm (cf. Chapter 5), the use of this estimator is stillcurrently not applicable for the on-line identification of “fast” time-varying sys-tems. Examples are a rocket consuming fuel or dangerous and expensive flightflutter tests, requiring fast online data processing techniques for which some pos-sible approaches are discussed in [28, 19, 63]. On the other hand, numerous othervibration-based condition monitoring applications are conducted on a totally dif-ferent time scale, resulting in sufficient time for online data analysis using moreaccurate algorithms such as the MLE. Examples are the monitoring of in-operationlarge structures such as bridges [100, 34] or offshore platforms [22, 34], or groundinspections of airplanes and the space shuttle orbiter [64, 34]. Another applicationwithin the same category is the condition monitoring of structures during life cycletests, which will be studied in this chapter as well.

7.2 Autonomous Modal Parameter Identification

The approach used to automate the modal parameter identification process con-sists of 3 main steps, discussed in detail in the following paragraphs:

1. Data Analysis: Identification of a model with sufficiently high order using afrequency-domain Maximum Likelihood estimator

2. Mode Validation: Computation of the mode validation criteria


3. Physical Mode Selection: Based on these criteria, the structural (physical)modes can be pinpointed by means of amode quality index or by classificationusing a fuzzy clustering approach

7.2.1 Data Analysis

The FRF-based frequency-domain Maximum Likelihood estimator (MLE), as pre-sented in § 5.5, is used as “data analysis engine”. In this case, the MPE algorithmstarts from the LSCF estimators to obtain starting values. The MLE algorithmminimizes the log-likelihood cost function based on the FRF and correspondingvariance data. Important algorithm parameters are the model order, the numberof iterations and the frequency band of interest. A maximum of 50 iterations ischosen since, for structural dynamics analysis, this typically leads to convergence.For each iteration, the modal parameters and poles and zeros are calculated.

In order to capture all physical modes in the model, a model with a high orderis estimated. Due to this over-modelling, a large number of computational modesoccur. A first straightforward elimination of such modes is done by removing allmodes having poles with a frequency within 1% of the edges of the analysis bandor a damping ratio smaller than 0% (unstable pole) or larger than say 10% (appli-cation dependent). The frequency cutoff is applied to eliminate the computationalpoles on the edges of the analysis bandwidth that result from the fact that theMLE uses a discrete-time model. This first mode elimination step is called thephysical thresholding step.

7.2.2 Mode Validation Criteria

Based on the knowledge of the uncertainty on the measured data, it is possible,besides the classical deterministic validation criteria, to define stochastic criteriathat give more information regarding the physicalness of a mode.

Stochastic Mode Validation Criteria

Given the scalar matrix fraction description used as model for the MLE implemen-tation (cf. § 5.5), the polynomial coefficients are related to a poles/zeros model andas discussed in [130, 118] for SISO systems, the computational or mathematicalmodes are characterized by so-called cancelling pole-zero pairs. This is explainedby the fact that the estimator tries to minimize the contribution of a computa-tional pole in the model by placing a cancelling zero very close to that pole. Suchpole-zero pairs will occur for two possible reasons:


• Computational mode: a computational mode should not affect the transferfunction and as a result the pole should be located far from the imaginaryaxis (high damping) or close to a (cancelling) zero.

• Poorly excited/measured physical mode: when a mode is poorly excited ormeasured, e.g. in a nodal point, it has only a small contribution to thetransfer function, and hence the physical pole will be cancelled by a zero.

Based on this property and given the uncertainty information for the poles andzero, obtained from the MLE algorithm, the following stochastic mode validationcriteria can be defined:

• Pole-zero pairs: a large number of zeros within a uncertainty circle aroundpole pr indicates a computational mode. The radius of the circle is calculatedas

R =√

−2log(1− pb)σpr (7.1)

where σpr = std(pr) the uncertainty on pole pr, for r = 1, . . . , Nm thenumber of modes and with pb the probability for the event that the truepole falls inside the circle. As discussed in [146], the uncertainty levels ofcomputational poles are often higher (i.e. typically order 10 to 100) than forthe physical poles.

• Pole-zero correlation: the correlation between poles and zeros gives moreinsight for the detection of cancelling pole-zero pairs, since these are alwaysstrongly mutually correlated. In the case of a high modal density or a highuncertainty on the measurement data, it is possible that other zeros fallwithin the uncertainty circle and as a result the correlation matrix is thenrequired to determine which pole and zeros are cancelling pairs. Based onthe uncertainty on the measurement data, the MLE can calculate the covari-ance matrix of the ML estimates (cf. Eq. 5.63), from which the correlationcoefficient between each pole and zero is derived. In order to limit the com-putational load, only the nearest zero is used in practice resulting in thefollowing correlation coefficient for pole pr

corr(pr, z) =covar(pr, zc)

std(pr) std(zc)(7.2)

where zc is the nearest cancelling zero of each estimated transfer function.

However, in the case of weakly excited modes, it is possible that both stochasticmode validation criteria falsely indicate physical modes as computational. Theautonomous procedure can be made robust for this type of failure by performingin addition an analysis for each FRF separately (SISO analysis) using again theMLE or for example the frequency-domain Least Squares (LSCF) estimator (cf.§ 5.3.6) in order to reduce analysis time. In the case of a weakly excited mode, the


pole should be identified for most of the FRFs, while computational poles do varyfor the different FRF analyses. The probability of having poorly excited modescan be reduced by using a multi-input excitation setup. However, in the case ofsingle-input excitation, an optimal experimental set-up can tackle the problem ofweakly excited modes for monitoring practices, where only the modes that varymost during the deterioration of the structure must be tracked over time andshould be well excited. On the other hand, when a robust autonomous estimationprocess is required for analyzing data without a priori knowledge of the structure,it is best to incorporate this check for weakly excited poles at the expense ofadditional analysis time.

Deterministic Mode Validation Criteria

Besides the validation approach based on cancelling pole-zero pairs, well-knownvalidation criteria specifically developed for modal analysis purposes can be consid-ered as well. Most of them give an indication based on the mode shape properties,such as the Modal Phase Collinearity (MPC), Mean Phase Deviation (MPD) andthe Mode Overcomplexity (MOV)

• The ”Modal Phase Collinearity”as defined in [71] is a measure for the degreeof complexity of a mode shape by evaluating the functional linear relation-ship between the real and imaginary parts of the mode shape coefficients.For proportionally or lightly damped structures, physical modes behave as’normal’ (real) modes and the MPC index approaches unity. A mode witha low index is rather complex, indicating a computational or noisy mode.Caution is needed in the case of complex physical modes, which are typicalfor highly damped systems.

• The ”Mode Overcomplexity”as defined in [62], is based on the mode complex-ity check, which evaluates the sensitivity of the damped natural frequencyfor a mass change at each response degree of freedom (DOF). The percentageof the response DOFs for which a mass addition yields a negative frequencysensitivity is the MOV for a mode. Physical modes yield a MOV close tounity. The MOV criterion requires the availability of a driving point mea-surement, which, in practice, is not always available.

One of the major drawbacks of both criteria is their deterministic character (no useof uncertainty information) and as a result they can be expected to be less robustfor poor quality data. Especially in the case of medium values (30−70%) they canindicate a computational or a noisy mode, where the latter can still be a structuralmode. Nevertheless, given the limited computational load, their integration in thisautonomous process can increase the robustness for mode evaluation.


7.2.3 Physical Mode Selection

Based on the results of each validation criterium, the modes remaining after thedata analysis step (cf. § 7.2.1) have to be grouped in 2 classes, i.e. the physicaland computational modes. Two possible approaches are now proposed.

Fuzzy Clustering Approach

An iterative Fuzzy C-means clustering algorithm [15], proven to be useful for theautomation of the modal mode extraction [144], is used in this case. The algorithmis implemented in the MATLAB Fuzzy Logic Toolbox [151]. The input variables forclustering are based on the validation criteria discussed above. An input variableXk consists of Nm objects, with Nm the total number of modes (physical andmathematical) after the data analysis step. The following input variables wereconsidered:

• Variable X1: standard deviation of the estimated pole pr (r = 1, . . . , Nm).

• Variable X2: the number of transfer functions having a zero within e.g. the99% uncertainty circle (pb=99%) of the estimated pole pr divided by thetotal number of transfer functions (o = 1, . . . , No number of responses, i =1, . . . , Ni number of forces)

ρpairs(pr) =#

|pr − zc| < R

NoNi(7.3)

• Variable X3: the number of cases where the correlation between pole pr andthe nearest zero zc in each transfer function model is higher than e.g. 90%divided by the total number of transfer functions

ρcorr(pr) =#

corr(pr, zc) > 90

NoNi(7.4)

• Variable X4: the inverse of the modal phase colinearity (MPC).

• Variable X5: the inverse of the RMS magnitude of the mode shape ψr .

• Variable X6: the number of SISO analyses that could find a pole withina certain interval around the global pole estimate pr divided by the totalnumber of transfer functions.

ρsiso(pr) =#∣

∣pr − pk∣

∣ ≤ δ∣

∣pr∣

∣

NoNi(7.5)


where pk is a pole estimate for each SISO analysis (k = 1, . . . , NoNi). Theinterval δ can be user-defined or related on the uncertainty of global poleestimate. Thus, X6 =1 when all NoNi SISO estimations return a pole closeto the global pole.

An interval scaling of the input variables transforms them into the same interval(e.g. [0;1]). In a first step of the interval scaling procedure, the variables with alarge range (i.e. variable X1) are transformed using a logarithmic scale. Next, eachvariable is subtracted by its minimum and divided by its range (i.e. the maximumminus the minimum). Finally, all variables, except X6, are subtracted from unity.Values larger than 0.5 indicate physical modes. The output of the algorithm is amembership function for the 2 classes, i.e. physical and computational modes. Ifthe membership function for an object with respect to a class is larger than 0.5(or 50%) the object belongs to that class.

Weighted Mode Quality Index

Another straightforward method to synthesize the results of the mode valida-tion criteria is to compute a (weighted) index, defined as the “Mode Quality In-dex”(MQI), which is a measure for the physical behavior of a mode. Using thevariables X1 – X6, scaled between [0;1], this MQI can be defined as a weightedratio

MQIr = αstd(1−X1r ) + αpairs(1−X2r ) + αcorr(1−X3r )

+αmpc(1−X4r ) + αrms(1−X5r ) + αsisoX6r (7.6)

with∑

αi = 1. A weighted index has the advantage that, since the stochasticcriteria (pole-zero pairs and correlation) are more robust for noisy measurementdata, they can have a larger contribution to the overall MQI than the deterministiccriteria (MPC and MOV). A high MQI, typically above 70%, indicates a structuralmode, while lower values are found for computational. The final set of modes isselected from the results of the last iteration, by identifying those modes having a”Mode Quality Index”above or equal to the threshold of physicalness with a typicalvalue of 50–70%. The mode quality index is certainly appropriate for automatedmonitoring practices were the proper weights are chosen during the setup of themonitoring system.

7.3. Case Study I: Airbus A320 Slat Track 189

7.3 Case Study I: Airbus A320 Slat Track

7.3.1 Test Structure and Experiments

The autonomous approach is illustrated for the application of structural healthmonitoring of a slat track, which was already discussed in § 5.3.3. Safety criticalcomponents such as slat tracks are rigorously tested by means of rig endurancetests to prove their ability to withstand all safety regulations. It is commonlyaccepted that the track should outlive the plane by five times, i.e. 240000 flights.Until today, commonly used damage detection techniques in the aircraft industryare mostly visual and/or local (ultrasound techniques, analysis of magnetic fields,radiology, thermal methods). All these techniques require that the sensors be closeto the damaged area of the investigated part. Vibration-based damage detectionhowever allows a more global analysis of the structure with the potential of beingdone in situ. Besides the development of shortened endurance tests without losingtheir ability to represent actual in flight loads, the availability of vibration-basedcondition monitoring system can result in an improved damage assessment of thecrack initiation and propagation in the slat track throughout the life cycle tests.In this section, the results of a preliminary study of an approach for autonomousmodal parameter identification and tracking, applied to the slat track monitoringproblem, are presented. Structural changes were introduced in a controlled man-ner and forced-vibration tests resulted in frequency response function data. Thisallowed ’simulating’ the effect of crossing modes, which can significantly hamperthe mode tracking process as well as the damage assessment, which is discussedin [147, 99]. Nevertheless, it is important to notice that the approach presentedcan be applied to a large number of other applications, including in-operationcondition monitoring of bridges, and (off-line) flight flutter analysis.

Given the scope of the research presented, i.e. a feasibility study of an au-tomated identification and tracking process, the measurements were performedunder controlled free-free conditions using a shaker and accelerometers. As shownin Figure 7.1, seven accelerometers were attached to the slat track on the boarderof the inner-side of the surface. This part of the slat track was also measured bymeans of a scanning laser Doppler vibrometer in 500 points, serving as a refer-ence model for optimal sensor location and a better understanding of the dynamicbehaviour.

A broadband periodic excitation was applied by means of a multisine signal[57] in the frequency band 0–4096Hz with 16384 spectral lines. The frequencyresponse and coherence functions were calculated using the H1 FRF estimatorwith 5 averages. Figure 7.2 shows some of the seven FRFs indicating a high modaldensity, while Figure 7.3 shows 9 of the mode shapes obtained from data measuredby the scanning laser vibrometer. In order to ’simulate’ varying structural modesin a controlled manner, an increasing mass was added (by means of magnets) in


SLATTRACK

FORCE SENSOR

STINGER

SHAKER

S1

S2

S3

S4

S5

S6S7

Figure 7.1: Measurements using accelerometers. Excitation with a shaker and forcemeasurement by means of a force sensor. Response measurements by means of 7 ac-celerometers.

400 600 800 1000 1200 1400−60

−40

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(m/s2 /N

− d

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Figure 7.2: FRFs measured between the force transducer and the accelerometers 1(dots), 2 (solid) and 5 (dashed).

between accelerometers 2 and 3, during 14 steps with an increment of 2–2.5 10-3kg.The total mass added was 30.5 10-3kg, i.e. 0.77% of the total weight of the slattrack (3.96kg). This approach was preferred to a difficult controllable initiationand propagation of cracks by means of saw cuts.


578.8Hz 614.7.Hz 622.5Hz

764.3Hz 926.6Hz 936.6Hz

1136.9Hz 1203.0Hz 1310.6Hz

Figure 7.3: Mode shapes obtained by SLDV measurements.


7.3.2 Autonomous Estimation Results for Slat Track

Data Analysis

A model order of 50 modes, 50 iterations and an analysis frequency band of 300–1375Hz were used during the ML identification. Analysis of the reference data set(i.e. no mass added) yields 17 modes after removing the modes with unstable andhighly damped poles from the model (cf. §7.2.1).

Mode Validation Criteria

Based on the stochastic mode validation criteria presented in § 7.2.2 the followingresults were derived for the slat track:

• Pole-zero pairs: Figure 7.4, shows the real and imaginary part of the poles(+) and zeros (o) as well as the corresponding (dashed) 99% uncertaintycircle in a band 550–630Hz (both the axes are scaled by a factor 2π). One canclearly recognize the clustering pattern of zeros around the computationalpoles within their uncertainty circles, while no or only a few zeros are presentin the small uncertainty circles around the physical poles. (The horizontaldashed lines indicate the physical poles.)

• Pole-zero correlation: Figure 7.5 gives the correlation coefficient between thepole and nearest zero in each transfer function for some of the modes in theband 550–630Hz. High correlation coefficients for the modes at 554.3Hz and564.4Hz indicate a computational character for these modes.

Notice however that, in the case of the weakly excited mode at 554.3Hz, bothstochastic mode validation criteria falsely indicate a physical mode as computa-tional. This is easily understood by the fact that, since a weakly excited modebarely appears in the measured data (Figure 7.2), a zero in each FRF approxi-mately cancels the physical pole. The autonomous algorithm is made robust byperforming in addition a SISO analysis for each FRF separately using e.g theLSCF estimator. Figure 7.6 (left) shows the absolute value of the difference be-tween the global pole at 554.3Hz and the closest poles found by each separateFRF analysis. The dashed line indicates the product of the absolute value of theglobal pole and its uncertainty. The same was done for the computational pole at564.4Hz in Figure 7.6 (right). This result demonstrates that the weakly excitedphysical mode can be identified for most of the FRFs


−50 −25 0 25 50550

560

570

580

590

600

610

620

630

real/2/pi

imag

/2/p

i

Figure 7.4: Nyquist plot of the real and imaginary part (scaled by a factor 2π) of poles(+) and zeros (o) after physical thresholding. The computational poles are characterizedby a typically large uncertainty circle (dashed line) containing a cluster of zeros. Thehorizontal dashed lines indicate the physical poles.


Fuzzy Clustering

The various input variables that can be used for the fuzzy clustering are shownin Figure 7.7. Considering the threshold of 0.5%, it can be noticed that thedeterministic criteria (X4 and X5) have a significant higher error ratio than thestochastic criteria. Indeed, variable X4 indicates all 4 computational poles asphysical, while 2 physical poles are below 0.5%. Variable X5 indicates the 2computational poles as physical. On the other hand, the stochastic criteria X1

and X2 only indicate the pole of the weakly excited mode falsely as computational,while X3 and X6 give no erroneous indications.

In order to study the influence of the choice of input variables on the classifi-cation result, Figure 7.8 shows the results for 4 possible combinations of variables.Considering the threshold of 0.5, where all physical poles should be above in orderto be correctly classified, where the best result is obtained by using the variablesX2, X3 and X6 (at least in the case of the slattrack data). Figure 7.8, also demon-


1 2 3 4 5 6 70

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Cor

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coef

f.

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Figure 7.5: Correlation coefficients between the pole and nearest zero in each transferfunction indicating computational modes at 554.3Hz and 564.4Hz.

strates that considering all the possible input variables does not necessarily yielda better clustering result. This is explained by the use of the deterministic cri-teria (i.e. (MPC)−1 and (rms(|ψr|)−1), which have a higher error ratio than theother (stochastic) criteria. It is clear that the variable X6 makes the automatedprocedure more robust when weakly excited modes are present. In the analyzedfrequency band, 13 poles were correctly classified as physical. The natural fre-quency and damping ratio of the classified poles are summarized in Table 7.1 forthe case of the reference model.

It is important to notice that for structural health monitoring applications onlya few or even a single sensor will be used. As a result, care should be taken forsome of the proposed input variables. In the case of a single sensor, the modalphase colinearity has no physical meaning, while the phase difference between theforce and response (acceleration) should still be close to 90 degrees for physicalmodes. Similarly, the variable X2 should be used with care when the sensor is ina node of a mode while variable X6 is not applicable anymore. For this reason, itis important to first select the modes to be tracked and then choose an optimalsensor position, i.e. not in any node of the selected set of modes. The othervariables remain applicable without any constraints.


1 2 3 4 5 6 70

1

2

3

4

5

6

7

FRF number

|∆(p

)|

554.3Hz

1 2 3 4 5 6 70

10

20

30

40

50

60

70

|∆(p

)|FRF number

564.4Hz

Figure 7.6: The absolute value of the difference between the global pole and the closestpoles found by each separate FRF analysis. The dashed line indicates the product of theabsolute value of the global pole and its uncertainty.

mode fMLE (Hz) ζMLE (%) fLSCE (Hz) ζLSCE (%)1 362.24 0.103 362.17 0.1022 554.19 0.439 554.08 0.4283 578.84 0.657 578.77 0.5674 603.91 1.032 603.90 0.6965 614.66 0.611 614.91 0.6626 622.48 0.192 623.81 0.2417 667.89 0.109 668.03 0.0938 764.27 0.174 764.72 0.1749 926.63 0.218 926.56 0.21310 936.56 0.127 936.55 0.12511 1136.88 0.177 1136.47 0.17212 1203.02 0.099 1203.79 0.09813 1310.62 0.186 1311.74 0.191

Table 7.1: Autonomous identification results for the slat track (reference data set):natural frequency and damping ratio and comparison with LSCE results.

Mode Quality Index

Alternatively, the Mode Quality Index can be used to classify the physical andmathematical modes. Based on the results in Figure 7.7, proper weights can be


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Figure 7.7: Input variables X1–X6 that can be used for classifying the poles for thereference data. The X-axis represents the frequency in Hz. The dotted lines indicate thephysical poles (in the upper part > 50%).

chosen for the MQI (7.6). Figure 7.9 shows the result for the reference data andlast ML iteration, for two different choices:

• case 1: αstd = 0.20, αpairs = 0.20, αcorr = 0.30, αsiso = 0.30

• case 2: αpairs = 0.30, αcorr = 0.35, αsiso = 0.35


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Figure 7.8: Classification result for the slat track (reference data set) by using severalcombinations of the input variables X1–X6 for fuzzy clustering. The X-axis representsthe frequency in Hz. The horizontal dashed line indicates a threshold of 0.5, whereall physical poles should be above in order to be correctly classified. The dotted linesindicate the physical poles.

As can be expected, these results are in line with those of the fuzzy clusteringapproach. Nevertheless, the use of the clustering approach has the advantage thatit avoids the user-defined choice of the weights for the different input variables,making this approach more automated and in general resulting in an improvedseparation of the physical and computational poles.


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MQ

I (%

)

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I (%

)

Figure 7.9: Mode Quality Index for slat track (reference data set) for the 2 cases of theweights, case 1 (left), case 2 (right). This result is in line with the results in the bottomfigures of Figure 7.8

Validation

In order to validate the final model extracted by the autonomous identificationprocess, a conventional modal analysis on the reference data set was performedas well. The results of the LSCE estimator (in LMS CADA-X software) [86] aresummarized in the stabilization diagram presented in Figure 7.10 for the band490–805Hz. Analyzing in several sub-bands with extensive over-modelling wasnecessary in order to identify all physical modes. Based on the pole indicators,(s, f and o), an experienced user should be able to select all physical poles, eventhe weakly excited ones at 554.3Hz and 620Hz, although a large number of math-ematical poles complicate this selection. All physical modes identified by theLSCE algorithm where also returned by the autonomous process. However, it isin general, even for an experienced user, a difficult task to select all the physicalmodes using the classical approach. The use of the MLE algorithm as well as themode validation tools is an important step towards an automated estimation pro-cess. Table 7.1 demonstrates a good agreement between the natural frequenciesand damping ratios of the selected poles using the LSCE and those returned bythe autonomous process. Around 600Hz and 925Hz two groups of closely spacedmodes can be noticed, which will be considered to illustrate the robustness of theproposed tracking algorithm in § 7.5.

7.4. Case Study II: I40 Bridge 199

Figure 7.10: LSCE Stabilization diagram for the reference data set.

7.4 Case Study II: I40 Bridge

7.4.1 Test Structure and Experiments

A prototype of the proposed autonomous approach has been successfully appliedfor a number of experimental test cases [146] under laboratory conditions result-ing in data-sets with relatively good signal-to-noise ratios. However, data-setscorrupted with high noise levels are not uncommon in practice. Typical examplesare modal test data obtained from large structures such as civil structures (bridgesand buildings), automotive structures (fully assembled cars) and aerospace struc-tures (aircrafts, rockets). For this reason, the automated approach is also validatedby means of forced vibration test data obtained on a highway bridge.

The bridge was located along Interstate Highway 40 across the Rio GrandeRiver in Albuquerque, New Mexico. A series of modal tests was performed on thisbridge after it had been closed for traffic prior to demolition in 1993. The sectionof the bridge that was instrumented for this series of modal tests consisted of 3spans with a combined length of about 130 m (see Figure 7.11).


Figure 7.11: Geometry of the portion of the I40 bridge (sideview) that was tested.

Figure 7.12: Measurement point geometry of the I-40 highway bridge. The measure-ment points, numbered 1–26, are indicated by ‘•’. The shaker position is indicated by’SE’.

Furthermore 13 accelerometers were placed on each of the 2 plate girders alongthe length of the 3 spans, resulting in a total of 26 DOFs and the structure wasexcited in DOF 3 (Figure 7.12). A typically measured FRF together with its coher-ence function is depicted in Figure 7.13. The coherence function indicates the poorquality of the measurements, mainly due to the resistance of the large structure tothe forced excitation, especially at resonance frequencies [37] and lower signal tonoise ratios around the anti-resonances. More details on the bridge and performedvibration experiments can be found in [37]. The data can be downloaded at the LosAlamos National Laboratory website (http://ext.lanl.gov/projects/damage id/).

7.4.2 Model Order Reduction

In general, the number of structural modes, within the frequency band of analysis,is a priori not known. Since the optimal model order depends on the type ofstructure and frequency band studied, an optional step can be considered to reducethe initial arbitrary model order to a more appropriate size before the actual modalparameter estimation based on the MLE algorithm is performed. Two possibleapproaches for model order reduction are illustrated for the case of the I40 bridge.


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Figure 7.13: Typical FRF (top) and corresponding coherence function (bottom) ob-tained on the I-40 Highway Bridge.

Pole Uncertainty Analysis

Running the Weighted LSCF estimator with an arbitrary chosen (high) modelorder, e.g. 50 modes, results in a first set of poles. Due to the extensive over-modelling, a large number of computational modes are present. A first approachto reduce this initial model order, is the analysis of the uncertainties on the esti-mated poles. Availability of the variances on the FRFs allows to do a sensitivityanalysis of the parameters for noise on the measured data resulting in an uncer-tainty level on the estimated poles. Figure 7.14 shows the uncertainties of thepoles in case of the bridge data. Ranking the uncertainty (standard deviation) ofall estimated poles from low to high and calculating the derivative of this func-tion results in Figure 7.15. Since the uncertainties of physical poles are generallysignificantly smaller compared to the computational ones, the maximum of thisderivative function should appear close to the number of physical poles. As shownin Figure 7.14 for the lower frequency part (2–5Hz) of the analysis band, a cleargap of 10–15dB separates the uncertainties for the physical (indicated by dashedlines) and computational poles. However, between 5–9Hz the measurement data ischaracterized by poor signal-to-noise ratios and as a result the uncertainties of thephysical poles increase, making a clear distinction between physical and compu-tational poles impossible. Although pole uncertainty analysis can be successfullyapplied in numerous cases [146], this illustrates clearly the shortcomings of thisapproach for measurement data of poor quality.


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dB

Figure 7.14: Uncertainties (standard deviation) for poles estimated by WLSCF for 50modes (dotted lines: physical poles).

Pole-Zero Pairs Elimination

The second approach tested for model order reduction, is based on the detectionof cancelling pole-zero pairs as presented in § 7.2.2, a typical characteristic forcomputational modes. Eliminating modes corresponding to these pole-zero pairs,often results already in a fairly good indication for the number of structural modes.

Figure 7.16 shows most of the poles (×) and zeros () estimated by the weightedLSCF estimator for an initial model order of 50 modes. The resonance frequenciesare indicated on the scaled imaginary axis. Again, it can be noticed that a largenumber of poles are surrounded by dense clusters of zeros, which is typical anindication of cancelling pole-zero pairs. Furthermore, most of these poles areunstable (in right part of Nyquist plane).

The probability parameter in Eq. (7.1) is chosen as pb = 68% in order toidentify only the most obvious cancelling pole-zero pairs since the goal is only toreduce the model to a more appropriate order without the risk of excluding anyof the physical poles. As shown in figure Figure 7.16 by the ellipses (due to scaledimaginary axis) all computational poles clearly have dense clusters of zeros fallenwithin the ”threshold circle”. By eliminating these poles, the model order can be


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pl.

f(stdp)=sort(db(stdp))

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number of modes

diff(f(stdp))

Figure 7.15: Analysis of pole uncertainties. Ranking of the uncertainty (standard devi-ation) of all estimated poles from low to high (top) and calculating the derivative function(bottom).

reduced and this is done in an recursive way until no more ”pole-zero pairs”canbe detected for the given threshold parameters. In the case of the bridge data, theinitial model order was reduced from 50 to 28 modes during 5 WLSCF runs.

7.4.3 Autonomous Estimation Results for I40 Bridge

For the analysis of the I40 bridge data, a model order of 28 modes, 50 iterationsand an analysis frequency band of 2–9Hz were used. The averaged sum of allFRFs and the ML poles over 50 iterations for a fixed model order of 28 are givenby the pole-diagram in Figure 7.17. Next, the physical thresholding step consistsof a first elimination of unfeasible modes generated by the ML estimator. Afterapplying the heuristic criteria discussed in § 7.2.1, the poles in Figure 7.17 wereobtained.

Although the physical thresholding eliminated a large number of non-physicalmodes, this is often not sufficient for eliminating all computational modes. Thisis also the case for the bridge data. As can be seen in Figure 7.17, several closely


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RE

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/PI

Figure 7.16: Poles (×) and zeros () in the Nyquist plane. The ellipses (circles re-scaledby 2π) indicate the 2 dimensional 68% uncertainty bounds around each pole.

spaced lines appear, whereas in the frequency range 5–7Hz no structural modesare expected [37].


Although, all lines exhibit a “stable” behavior, it is most likely that not all ofthese lines correspond to structural modes and a manual selection of the physicalpoles based in this diagram is likely to yield erroneous results. For this reason,the physical pole selection was done again by using a fuzzy clustering approach aspresented in § 7.2.2.

The input variables X1 – X4 were available for the Fuzzy clustering approach,in the case of the bridge data, and are shown in Figure 7.18. Again in this case,the Modal Phase Colinearity has a higher error ratio compared to the stochasticcriteria (X2 and X3). From the classification result shown in Figure 7.19, it followsthat 10 modes are classified as physical, where there is a very clear distinctionbetween the physical and computational poles in the case that only the stochastic

7.5. Mode Tracking 205

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Figure 7.17: Pole-diagram: poles (×) remaining after physical thresholding.

criteria (i.e. X1 – X3) are considered. The first 6 physical modes indicated inFigure 7.19 are in good correspondence with those reported in [37], while theother 3 modes were out of the analysis band considered in that report.

7.5 Mode Tracking

Since the natural frequencies and damping ratios of the modes change due tochanges in the structure, it is not always straightforward to determine which modesfrom two subsequent models are corresponding. Furthermore, it can happen thattwo modes cross each other in terms of natural frequency or damping ratio or eventhat the estimation algorithm returns a single mode at the moment of crossing,masking the other mode and seriously hampering the tracking process. In thissection, a robust mode tracking approach is presented for the case of the slattrack.


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Figure 7.18: Input variables X1–X4 that can be used for classifying the poles. TheX-axis represents the frequency in Hz. The dotted lines indicate the physical poles.

7.5.1 Tracking Approach

One obvious way for mode tracking is to monitor the changes of the estimatednatural frequencies. However in the case of closely spaced or crossing modes moreinformation is needed to determine which modes are related over the subsequentmonitoring instances. The damping ratio could help in some cases to separateclose modes. In [33] the use of mode shape information in the form of the ModalAssurance Criterion (MAC) is successfully applied for mode tracking for flight flut-ter test data. For tracking purposes, the MAC evaluates the degree of correlation

between two mode shapes over subsequent instants, ψ[t−1]k and ψ

[t]l , resulting in a


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Figure 7.19: Classification result for the bridge data by using all (top) and only thestochastic input variables X1–X3 (bottom) for fuzzy clustering. The X-axis representsthe frequency in Hz. The horizontal dashed line indicates a threshold of 0.5, whereall physical poles should be above in order to be correctly classified. The dotted linesindicate the physical poles.

value close to one for corresponding modes

MAC(

ψ[t−1]k , ψ

[t]l

)

=

∣

∣ψ[t−1]H

k .ψ[t]l

∣

∣

2

(

ψ[t−1]H

k .ψ[t−1]k

)(

ψ[t]H

l .ψ[t]l

)

(7.7)

with k = 1, . . . , Nmt−1 the number of modes at instant t − 1, l = 1, . . . , Nmt thenumber of modes at instant t.

However, structural health monitoring in normal operating conditions is oftendone by means of a very limited number of sensors, and as a result, the MACcriterion is then not always applicable. This can be overcome by using the distance

(or resemblance) between two residue matrices over subsequent instants, R[t−1]k

and R[t]l (with Rr = QrΨrΨ

Tr = ΨrL

Tr , with Lr the row vector of participation

factors). The Frobenius norm of their difference, results in a value close to zero


for corresponding modes

∥

∥(R[t−1]k , R

[t]l )∥

∥

F=

√

trace(

(

R[t−1]k −R[t]

l

)H(R[t−1]k −R[t]

l

)

)

(7.8)

In the case of a single response sensor, the use of the distance measure will onlyfail when two closely spaced modes have a similar residue values at the sensorlocation. The use of high spatial resolution (laser vibrometer) measurements canthen contribute to an optimal selection for the sensor location for the modes to betracked.

Notice that in general the distance criterion cannot be used in the case ofoperational data (output-only data), since in that case the participation factorscannot be estimated. An example is the monitoring of a bridge by means of roadtraffic excitation, where the level of excitation varies. However, bridge inspectionsystems integrate sufficient sensors such that the MAC criterion can then be usedto perform the in-operation mode tracking. This is different for monitoring ap-plications having only a limited number of sensors, such as in the case of routineairplane ground inspections or life cycle tests. In these cases however, output-onlymeasurements over the subsequent instances are then performed in similar con-ditions by means of a patch actuator or acoustic excitation. Although the MACcriterion is not applicable (not enough sensors), the distance criterion can thenbe used since the source of excitation is identical during the periodic inspectionsand the scaling factor remains similar. However, in the case of general operationalconditions, the input forces vary as a function of the amplitude, frequency coloringand location where the forces are applied.

7.5.2 Tracking Algorithm – Problem of Missing Modes

Initially, the modes of interest for the tracking are selected from the referencemodel obtained from an analysis of the first data set. Mode tracking over theconsecutive instants is done based on a model comparison between the new andprevious modal model. Considering the poles in the Nyquist plane, the distance

between pole p[t−1]k of mode m

[t−1]k (k = 1, . . . , Nmt−1 number of modes at t− 1),

and all poles of the modes at the next instant t can be determined. Depending onthe amount of a structural change and the sensitivity of the mode for that change,the influence on a certain pole (frequency and/or damping) can be large and as a

result the closest pole at instant t does not necessarily correspond to the pole p[t−1]k .

For this reason the y closest poles at instant t are considered (y being an algorithmparameter which is typically chosen between 5–Nmt). Next the residue distancemeasure (7.8) is used to determine which of the y modes at instant t corresponds

to the mode m[t−1]k . After having determined the “closest mode” at instant t for

each mode at instant t−1, an additional step is necessary to check if each mode isuniquely matched over the subsequent instants. It can happen that a mode from


the model at instant t is used more than once for different modes at t− 1, due tothe problem of missing modes at instant t. This problem can appear in the caseof crossing modes with very close or coinciding poles, with the possibility that thebroadband MLE identification only returns a single mode. In this case, a model re-estimation by means of the autonomous identification process in a small frequencyband is initiated. Given the performance of the MLE approach, this quasi alwaysresults again in a set of very close but different modes. Using the same modelcomparison approach based on the distance measure (7.8) for the small band, thedifferent modes can now still be tracked over the subsequent time instants. In thecase that the re-estimation process is not successful, the modes still missing oninstant t are ultimately replaced by the modes at the previous instant ensuring thecontinuity of tracking all modes once they are estimated again for the followinginstants.

7.5.3 Tracking Results for Slat Track

To illustrate the robustness of the tracking approach for missing modes, the slattrack modes at 926.6Hz and 936.6Hz in the reference model are followed over thesubsequent measurements for an increase of the mass between sensors 2 and 3(see Figure 7.1). Applying the tracking algorithm without checking for missingmodes results in Figure 7.20 (top), where it is clearly shown that for instant 7only a single mode is estimated during the broadband MLE analysis, used twicefor two, very closely spaced but different modes at that instant. As a result, forthe subsequent instants only the initial mode of 936.6Hz is further tracked, missingthe initial mode of 926.6Hz. The type of structural changes, i.e. a mass changebetween sensors 2 and 3 allowed for ’simulating’ the crossing effect of the modes,since this mass is located in a node for the mode of 926.6Hz while in a point ofhigh amplitude for the mode of 936.6Hz (see Figs. 3 and 5). Re-estimation ofthe model for instant 7 in a small band around 926Hz (±25Hz) results in twoclosely spaced but different modes at 926.16Hz and 926.41Hz, with a MAC valueof 33.36%. The re-estimation step makes the tracking algorithm robust to missingor coinciding modes as is also shown in Figure 7.20 (bottom).

In order to illustrate that the tracking algorithm also works for larger changesin the structure, different amounts of mass changes were considered. Besides theincrement of about 2 10-3kg over 14 steps, tracking results were obtained foran increment of 10 10-3kg over 3 steps and 30 10-3kg over one single step. Aspresented by Figure 7.21, the tracking algorithm is perfectly capable of trackingthe crossing modes at 926.6Hz and 936.6Hz even if the structural changes becomelarges. For example, the natural frequency of the mode at 936.6Hz shifted by2.63% for a mass increment of 30 10-3kg.

The robustness of the algorithm for closely spaced modes is also illustrated inFigure 7.22 for the modes around 610Hz, for which tracking, based on the distance


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Figure 7.20: Mode tracking results (modal frequency and damping ratio) for the modesat 926.6Hz (*) and 936.6Hz (o) using the tracking algorithm without (top) and with(bottom) checking for missing modes.

criterion, is successful again. The mass change between sensors 2 and 3 clearlyaffects some of the modes while others remain unchanged since they have a nodeat that location as can also be seen in Figure 7.3. A crossing of the damping ratioscan be noticed now.

Figure 7.23 compares the use of the MAC (7.7) and the distance (7.8) criterionfor 3 different measurement setups. In the case that the 7 response sensors areused (dash-dotted line), there is no difference in the tracking results obtainedby using the MAC or the distance criterion. However, in the case that a single


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Figure 7.21: Mode tracking results (modal frequency and damping ratio) for the modesat 926.6Hz and 936.6Hz using different mass changes for the subsequent instants. Track-ing of mode at 926.6Hz over 14 steps (dotted *), 3 steps (o) and 1 step (♦). Tracking ofmode at 936.6Hz over 14 steps (dotted +), 3 steps (2) and 1 step (×).

sensor is used, the MAC criterion always returns one, and thus is not applicablefor tracking purposes. On the other hand, the distance criterion still gives goodresults in the case that a single response sensor is located in position 1 (dashed line)(see Figure 7.1 for sensor positions) since it uses the amplitudes of the differencein residues which is still be meaningful in the case of a single sensor. Only inthe case that the residues of two closely spaced modes have similar amplitudesfor that specific sensor position, also this criterion fails. This is the case whenthe single sensor is located in position 7 (dotted line). This can also be seen inFigure 7.3, where the bottom part of the slat track (i.e. sensor 7) has similarvibration behaviour for both modes at 926.6Hz and 936.6Hz. Prior knowledgeof the dynamic behaviour by means of a scanning laser vibrometer allows for anoptimal sensor positioning depending on the modes selected for tracking.

The availability of the confidence levels on the estimated parameters givesadditional information to determine if a small change of a modal parameter is dueto a structural change or due to the uncertainty on the measured data. Figure 7.24shows the natural frequency and damping ratio for the mode of 926.6Hz with the95% confidence interval. Since the mass changes are applied in a quasi-nodalpoint for this mode, the change of model parameters remains small. Nevertheless,


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Figure 7.22: Mode tracking results for the closely spaced modes around 610Hz.

the changes are significantly larger than the sum of the confidence levels over twosubsequent instants clearly still indicating that in the structural changes appeared.Only for instant 7, the uncertainty levels are higher explained by the difficultestimation problem of the two coinciding modes.

7.5.4 Damage Assessment during Tracking

Mode tracking can serve autonomous monitoring systems by detecting structuralchanges, and threshold levels for different modal parameters can be set for notifyingan operator. However, this is not sufficient in the case that one wants to knowthe location and severity of the damage. As a result a damage assessment hasto be done based on the results obtained from the autonomous identification andtracking process. Recently, a novel sensitivity-based damage assessment techniquewas proposed in [98]. However, since a driving point is required, the applicabilityof this method was restricted to forced vibration tests only, while most monitoringapplications only have access to output-only measurement data. The applicabilityof the sensitivity-based approach has been extended to output-only measurementsand illustrated based on the results discussed in this chapter [99].


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Figure 7.23: Comparison of modal frequency and damping ratio for the two modesat 926.6Hz (*) and 936.6Hz (o) obtained from the tracking algorithm using the MAC(top) and distance criterion (bottom) for different measurement setups using 7 sensors(dash-dotted), one sensor at position 1 (dashed), one sensor at position 7 (dotted).


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Figure 7.24: MLE uncertainty information during tracking process. Horizontal barsindicate the 95% confidence intervals for the natural frequency and damping ratio ateach instant.

7.6 Conclusions

This chapter presented a novel approach for an autonomous modal parameteridentification and mode tracking illustrated by means of two practical case studies.The results are very promising since the approach is capable of identifying andtracking all physical modes of interest even in difficult cases of closely spaced oreven crossing modes due to changes (damage) in the structure. The proposedapproach is applicable for both forced-vibration and output-only measurementsand provides a tool for structural health monitoring.

It is shown that accurate parameter estimates with confidence intervals are es-timated by means of the frequency-domain Maximum Likelihood algorithm, whilestochastic mode validation criteria, make the validation process more robust tomeasurement noise. The final model is selected using a mode quality index or afuzzy clustering approach, which separate physical from computational modes.

The mode tracking process is based on model comparison between subsequentmeasurements. Using the minimal distance between mode residue matrices, corre-sponding modes can be successfully determined even without the need of extensive


spatial information, which is important for the implementation of many monitoringsystems. The tracking approach provides a small band re-estimation in the caseof missing or wrongly estimated modes due to closely spaced or crossing modesnot captured by the model during the broadband analysis. Eventually, it has beendemonstrated that the proposed tracking approach still works for larger structuralchanges.

Chapter 8

Conclusions

8.1 Thesis Summary

In this thesis, the applicability of frequency-domain identification methods in thedomain of modal analysis has been studied. Special attention has been paid to theproblems and requirements related the different steps in the overall identificationprocess. It is now worthwhile to review the most important results achieved in thedifferent chapters.

Chapter 2 is devoted to the first step of modal analysis. The nonparametricidentification of FRFs has been revised in a maximum likelihood framework basedon an errors-in-variables stochastic noise model. A nonparametric approach forthe determination of both the FRF matrix and its covariance matrix from the samemeasurement data set has been presented for arbitrary excitations. It is shownthat the so-called nonparametric instrumental variables (Hiv) estimator has quasimaximum likelihood properties, without requiring any a priori information aboutthe measurement noise. A study of the applicability of the proposed estimator forthe specific case of modal analysis has shown that the nonparametric Hiv estimatorremains consistent when a MISO approach is used to process the extensive amountof data, while the practical limitations with respect to the determination of thenoise information have been pinpointed.

The problem of leakage, encountered when using arbitrary excitation for modaltesting, has been addressed in Chapter 3. Several approaches for reducing theleakage on the estimated FRFs by means of exponential windowing have beenstudied and compared with techniques proposed in literature. This has resultedin a method that is based on the use of an exponential window in combinationwith the classical FRF estimators. Although this results in biased nonparametric

217

218 Chapter 8. Conclusions

results, the additional damping is known and its effect on the modal parameterscan be compensated, which is not the case for e.g. a Hanning window. Moreover,the proposed approach is simple to apply in practice, using commercial measure-ment equipment for modal analysis that often only stores the measured FRF data.Furthermore, the noise information can still be determined when using this ap-proach. Based on the results of Chapter 2, this method has been generalized toan errors-in-variables framework.

An introduction of the concepts of parametric system identification for modalparameter estimation is given in Chapter 4. Different parametric models to de-scribe the vibration behaviour of mechanical systems have been discussed, and anoverview of parametric identification methods with their potentials and drawbacksfrom the point of view of modal parameter estimation has been given. This chap-ter motivated the application of so-called stochastic frequency-domain estimators,such as the Maximum Likelihood estimator.

Chapter 5 has been devoted to the optimization of the (stochastic) frequency-domain modal parameter estimators using FRF measurements. An extensiveamount of data, typical for modal analysis, requires optimized algorithms thatbalance between accuracy and memory/computation efficiency. A multivariableimplementation of frequency-domain estimators, based on a common denominatortransfer function model, was derived omitting however possible correlations be-tween the FRFs. A study of different types of parameterization for experimentalmodal data has demonstrated that the use of normal equations is justified whenusing a discrete-time transfer function model with complex coefficients. The finalLS implementation resulted in the so-called Least Squares Complex Frequency(LSCF) method. This estimator offers the user, in analogy with the well-knownLSCE method, a stabilization chart. The fast implementation could also be ex-tended to the class of stochastic Total Least Squares (mixed LS-TLS) estimatorsand the Maximum Likelihood (ML) methods, making them applicable for modalanalysis. Using experimental data, the robustness of the algorithms was demon-strated for model orders up to 200.

Based on the results of Chapter 5, a fast implementation of the frequency-domain methods has been derived for I/O data. The use of these estimatorsis preferred in the case of short data sequences, since then the nonparametricprocessing for FRF estimation is not possible. This can be the case for flightflutter testing and structural health monitoring applications. The formulationof the I/O estimators is based on an errors-in-variables set-up, where both thenoise on the input and output signals are considered. Similar to the FRF-basedestimators, where possible correlations between the FRFs are neglected, the fastimplementation of the I/O estimators omits possible correlations among the noiseon the output measurements. It has been shown that a compact formulation forthe WGTLS estimator was only possible by using a linear approximation for thecontributions of input noise. Furthermore, by estimating simultaneously the initialconditions and the system model parameters, the proposed I/O estimators could

8.2. Ideas for Further Research 219

be made robust for errors due to leakage effects. Next, a generalization has beengiven of the fast I/O estimators to estimate the parameters starting from auto andcross power spectral density functions. This is preferable when longer time recordsare available while a parametric compensation of the leakage is still possible.

Finally, Chapter 7 presents an approach for an autonomous modal parameteridentification and mode tracking . The approach is capable of identifying andtracking all physical modes of interest even in difficult cases of closely spaced oreven crossing modes due to changes (damage) in the structure. It is shown thataccurate parameter estimates with confidence intervals are estimated by meansof the frequency-domain Maximum Likelihood algorithm, while stochastic modevalidation criteria make the validation process more robust to measurement noise.The mode tracking process is based on model comparison between subsequent mea-surements. It was demonstrated that modes could be successfully tracked withoutthe need of extensive spatial information. This is important for the implementa-tion of many monitoring systems. The tracking approach also checks for missingor wrongly estimated modes due to closely spaced or crossing modes. The trackingapproach remains applicable even for larger structural changes. The approach hasbeen validated for two experimental cases from the domain of mechanical and civilengineering.

8.2 Ideas for Further Research

In chapters 5 and 6, it has been shown that the use of a common-denominatormodel results in structured matrices, which has been exploited to improve thespeed and memory usage of the algorithms. Nevertheless, further optimizationcan be considered in order to broaden the possible applications that can benefitfrom the accurate stochastic WGTLS and ML estimators. Possible ideas for furtherimprovement in terms of computation speed and memory usage are related to:

• the exploitation of the structured Toeplitz matrices for internal operationssuch as e.g. the matrix product, inversion and factorizations (QR, SVD, ...).

• the implementation of the algorithms using parallel processing. The compactnormal matrix

D =

[

NoNi∑

k=1


]

(8.1)

is perfectly suitable for a parallel implementation that divides the compu-tation of the submatrices Rk, Sk and Tk over multiple processors, since thematrix D can be composed as the sum of multiple submatrices correspond-ing to different batches of input/output DOFs. This will allow to furtheroptimize the stochastic estimators for applications such as online monitoringof the flutter behaviour of an aircraft during flight testing.

220 Chapter 8. Conclusions

In chapter 2, the noise covariance matrix for the FRFs or Input/Output signalscan be computed using the Hiv methodology. Since in practice M < No, thismatrix is not of full rank and hence cannot be inverted. During the parametricestimation, the correlations between the FRFs or output signals are not takeninto account, which results in some loss of asymptotically efficiency while theconsistency is remained. Nevertheless, the uncertainty bounds obtained for theparameters using the approach discussed in § 5.5.2 will not be correct, which canbe a problem for applications such as e.g. damage detection and structural healthmonitoring and hence further research should be devoted to this problem. Afirst idea consists of computing the uncertainty on the modal parameters directlyfrom the nonparametric noise covariance matrix as Cθ = SHCHiv

S, where S is asensitivity matrix. This is possible since no inversion of the matrix CHiv

is required,although the computational burden is too high to be applied in practice requiringa numerical optimization of this approach. Another idea for deriving uncertaintybounds for the parameters obtained by the LSCF estimator is as follows. Based onthe stabilization chart using variable LS constraints, clusters of poles are found fora specific model order for which a statistical analysis yields the mean and varianceof each pole. How are these “variances” related to the stochastic variances (orconfidence bounds) introduced by the noise on the measurements is the questionto be assessed. Comparing this with Monte Carlo simulations will indicate if thisapproach can yield acceptable confidence bounds for the LSCF estimator.

Other types of matrix fraction description models can be considered. The useof a RMFD will lead to a polyreference implementation of the LSCF, where thepoles and modal participation factors can be estimated together in a first step.The computation time will be limited since the RMFD model still results in struc-tured matrices by uncoupling the output DOFs. At first sight, this polyreferenceapproach would avoid the use of the SVD of the residue matrices, where infor-mation is lost due to the rank reduction. Nevertheless, the performance in termsof accuracy and stabilization chart should be compared by the SMFD implemen-tation, especially in the case of inconsistent data due to the effects of e.g. massloading.

Furthermore, extending the applicability of the algorithms to operational modalanalysis is an important topic for research. This requires first the revision of thenonparametric processing in order to obtain suitable data, i.e. Auto and CrossPower Spectra between the output signals as well as the uncertainty on this data.Since the operational conditions often correspond to random noise excitation, theapplicability of windowing techniques should be studied in order to reduce effectsof leakage. Moreover, if the excitation consists of a part that is measured and apart that is unknown due to the presence of process noise, the possibility of using acombined approach forced/operational is an interesting topic for further research.

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