APPLIED FREQUENCY-DOMAIN SYSTEM IDENTIFICATION IN THE …mech.vub.ac.be/avrg/PhD/PhDBCauberghe.pdf · 2015-03-20 · APPLIED FREQUENCY-DOMAIN SYSTEM IDENTIFICATION IN THE FIELD OF

SREVIN

U

ITEIT

EJI

RV

BR

US

SE

L

ECNIV

RET

EN

EB

RA

S

AI

TN

EI

CS

VRIJE UNIVERSITEIT BRUSSEL

FACULTEIT TOEGEPASTE WETENSCHAPPEN

VAKGROEP WERKTUIGKUNDE

Pleinlaan 2, B-1050 Brussels, Belgium

APPLIED FREQUENCY-DOMAIN SYSTEM

IDENTIFICATION IN THE FIELD OF EXPERIMENTAL

AND OPERATIONAL MODAL ANALYSIS

Bart CAUBERGHE

Promotor:

Prof. dr. ir. P. Guillaume

Proefschrift ingediend tot

het behalen van de academische

graad van doctor in de

toegepaste wetenschappen

May 2004

Bart CAUBERGHE – APPLIED FREQUENCY-DOMAIN SYSTEM IDENTIFICATION

IN THE FIELD OF EXPERIMENTAL AND OPERATIONAL MODAL ANALYSIS – May 2004

SREVIN

U

ITEIT

EJI

RV

BR

US

SE

L

ECNIV

RET

EN

EB

RA

S

AI

TN

EI

CS

VRIJE UNIVERSITEIT BRUSSEL

FACULTEIT TOEGEPASTE WETENSCHAPPEN

VAKGROEP WERKTUIGKUNDE

Pleinlaan 2, B-1050 Brussels, Belgium

APPLIED FREQUENCY-DOMAIN SYSTEM

IDENTIFICATION IN THE FIELD OF EXPERIMENTAL

AND OPERATIONAL MODAL ANALYSIS

Bart CAUBERGHE

Jury:

Prof. dr. ir. G. Maggetto, voorzitter (ETEC, VUB)

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

VUB)

Prof. dr. ir. P. Guillaume, promotor (WERK, VUB)

Prof. dr. ir. R. Pintelon, secretaris (ELEC, VUB)

Prof. dr. ir. W. Heylen (PMA, KULeuven)

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

Dr. ir. B. Peeters (LMS International)

Prof. dr. ir. M. Van Overmeire (WERK, VUB),

Dr. ir. P. Verboven (WERK, VUB)

Proefschrift ingediend tot

het behalen van de academische

graad van doctor in de

toegepaste wetenschappen

May 2004

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

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Acknowledgements

It is very difficult to express in words the great feelings of gratitude I feel forthe people, who helped me making this work possible. First of all, I’m speciallygrateful to my supervisor, prof. dr. ir. Patrick Guillaume, for his support, en-couragement and guidance throughout the course of this work. I thank him forgiving me his time and advices, for sharing his scientific knowledge and for givingme all resources needed to accomplish this work.

I would also like to thank all members of the jury for their time and interestin my work. I’m specially grateful to prof. dr. ir. Rik Pintelon for the severalinteresting discussions and to dr. ir. Peter Verboven for the careful reading of mytext.

I thank all me colleagues of the Acoustics and Vibration Research Group, Pe-ter Verboven, Steve Vanlanduit, Eli Parloo, Gert De Sitter, Joris Vanherzeele andGunther Steenackers for the stimulating work environment, for their excellent co-operation and many joint publications. Many colleagues became close friends andtogether we spent nice times besides the daily work.

Furthermore I would like to thank the different partners from the FLITE project,especially dr. ir. Bart Peeters and dr. ir. Herman Van der Auweraer from

i

ii

LMS Intl., the engineers from Airbus, Dassault and PZL, prof. Albert Benvenistefrom INRIA, Ivan Goethals from K.U.Leuven and finally ir. Carlos Refinetti fromEmbraer to give me insight in real-life applications of modal parameter estimationmethods and for the enthusiasm they have shown for my research work. Many ofour discussions at different meetings and conferences lead to new innovative ideas,which contributed greatly to the successful accomplishment of this thesis.

I would like to express my gratitude to Prof. Francesco Benedettini and his re-search team from the University of L’Aguilla for inviting my to join them in a testcampaign on bridges.

The Fund for Scientific Research Flanders (FWO) is gratefully acknowledged formy research associate grant and the Research Council (OZR) of the VUB for theirfinancial support.

I would also like to thank Thierry Lenoir for his help with computer problemsover the years and the secretaries of our department and prof. dr. ir. DirkLefeber, head of the mechanical engineering department, for all the given support.

Last but not least, I am greatly indebted to my family, especially my parents,who have always supported and motivated me in an outstanding manner through-out my studies and life. I will always be grateful for the inexhaustible faith theyhad in me. And finally I would like to thank my girlfriend Sofie, who has morecontributed than she realizes, for her patience and valuable support. Thanks mydarling for all your warm love.

Brussels, May 2004

Bart Cauberghe

Nomenclature

List of operators

i i2 = −1

(·)Tmatrix transpose

(·)−1matrix inverse

(·)∗ complex conjugate

(·)HHermitian transpose:complex conjugate transpose of matrix

(·)† Moore-Penrose pseudo-inverse

(·)−TTranspose of the inverse matrix

(·)−HHermitian transpose of the inverse matrix

Re (·) real part of

Im (·) imaginary part of

|x| absolute value of a complex number x

E (X) mathematical expectationof a stochastic variable X

var(x) variance of a scalar x

σ2x = var(x)

cov(x) covariance of a vector x

tr(x) trace of a matrix x

iii

iv

List of symbols

Ni number of inputs

No number of outputs

Nm number of modes

N number of frequencies

number of time samples

Nref number of references

Nb number of blocks

n model order

∆t sampling period

ω angular frequency

y(t) time-domain vibration response

x(t) time-domain state sequence

f(t) time-domain force signal

x(i), xi sample at time i∆t of a signal x(t)

Y (ω) response spectra, Fourier spectra of x(t)

F (ω) input force spectra, Fourier spectra of f(t)

X(ω) state spectra, Fourier spectra of y(t)

X(ωk),Xk spectral line k of the Fourier spectra X(ω)

H(ω) frequency Response Function

λr system pole r

ωr natural frequency of mode r

σr damping of mode r

φr mode shape vector of mode r

Lr modal participation vector of mode r

M , K, C1 mass, stiffness and damping matrix

v

List of abbreviations

CMC Changes in Mode shape Curvature

CMF Changes in Modal Flexibility

ABS Averaged Based Spectral

AR Auto Regressive

ARMA Auto Regressive Moving Average

BTLS Bootstrapped Total Least Squares

CLSF Combined Least Squares Frequency-domain

DOF Degree Of Freedom

EMA Experimental Modal Analysis

ERA Eigenvalue Realization Algorithm

EVD Eigen Value Decomposition

DFT Discrete Fourier Transform

FDPI Frequency Domain Direct Parameter identification

FEM Finite Element Model

FFT Fast Fourier Transform

FRF Frequency Response Function

GEVD Generalized Eigen Value Decomposition

GTLS General Total Least Squares

IDFT Inverse Discrete Fourier Transform

ITD Ibrahim Time-Domain algorithm

IQML Iterative Quadratic Maximum Likelihood

LMFD Left Matrix Fraction Description

LS Least Squares

LSCE Least Squares Complex Exponential algorithm

LSCF Least Squares Complex Frequency-domain

LSFD Least Squares Frequency Domain

MDOF Multiple Degree of Freedom

ML Maximum Likelihood

MIMO Multiple Input-Multiple Output

MPE Modal Parameter EstimationMSE Mean Square Error

vi

MSRE Mean Square Relative Error

OMA Operational Modal Analysis

OMAX Operational Modal Analysis with eXogenous inputs

PEM Prediction Error Method

RMS Root Mean Square

SDOF Single Degree of Freedom

SISO Single Input-Single Output

RMFD Right Matrix Fraction Description

SNR Signal to Noise Ratio

SVD Singular Value Decomposition

TLS Total Least Squares

UMPA Unified Matrix Polynomial Approach

WGTLS Weighted Generalized Total Least Squares

WNLLS Weighted Non-Linear Least Squares

XP Auto and Cross Power Spectra

Contents

Acknowledgements i

Nomenclature iii

Contents vii

1 Introduction 1

1.1 Research context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Focus, outline and organization of the thesis . . . . . . . . . . . . . 5

1.3 Original contributions in this work . . . . . . . . . . . . . . . . . . 8

2 Frequency-domain models for dynamical structures 11

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

2.2 The Modal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Common-denominator models . . . . . . . . . . . . . . . . . . . . . 13

2.4 State-space models . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Matrix fraction polynomial models . . . . . . . . . . . . . . . . . . 15

2.5.1 Left-Matrix Fraction Description (LMFD) models . . . . . 15

2.5.2 Right-Matrix Fraction Description (RMFD) models . . . . 17

2.6 Acceleration, Displacement, Velocity . . . . . . . . . . . . . . . . . 18

2.7 Continuous-time models and discrete-time models . . . . . . . . . . 18

2.8 Real versus complex models . . . . . . . . . . . . . . . . . . . . . . 20

2.9 Primary identification data . . . . . . . . . . . . . . . . . . . . . . 20

2.9.1 Input-Output measurements: a deterministic approach . . . 20

vii

viii Contents

2.9.2 Output-Only measurements: a stochastic approach . . . . . 23

2.9.3 Input-Output measurements: a combined deterministic-stochasticapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Non-Parametric preprocessing steps 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Modal testing for EMA . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 FRF estimation for EMA . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.1 Arbitrary input signals . . . . . . . . . . . . . . . . . . . . . 35

3.3.2 Periodic input signals . . . . . . . . . . . . . . . . . . . . . 38

3.3.3 Decreasing the noise levels on FRFs . . . . . . . . . . . . . 38

3.4 Estimation of the power spectra for OMA . . . . . . . . . . . . . . 42

3.4.1 The correlogram approach . . . . . . . . . . . . . . . . . . . 42

3.4.2 The periodogram approach . . . . . . . . . . . . . . . . . . 44

3.4.3 The ’positive’ power spectra approach . . . . . . . . . . . . 45

3.5 Combined FRF and XP estimation for OMAX . . . . . . . . . . . 47

3.5.1 No structure-exciter interaction . . . . . . . . . . . . . . . . 48

3.5.2 Structure-Exciter interaction . . . . . . . . . . . . . . . . . 50

3.6 Simulations and measurement examples . . . . . . . . . . . . . . . 52

3.6.1 Experimental Modal Analysis . . . . . . . . . . . . . . . . . 53

3.6.2 Operational Modal Analysis . . . . . . . . . . . . . . . . . . 54

3.6.3 Combined Operational-Experimental Modal Analysis . . . . 60

3.6.4 Measurement examples . . . . . . . . . . . . . . . . . . . . 61

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.8 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.9 Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Identification of common-denominator models 71

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 An extended parametric model for the H1 estimator . . . . . . . . 73

Contents ix

4.3 Weighted Least Squares Complex Frequency-domain (LSCF) esti-mator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.1 FRF driven identification . . . . . . . . . . . . . . . . . . . 75

4.3.2 Remarks on the extended LSCF estimator . . . . . . . . . . 81

4.3.3 IO data driven identification . . . . . . . . . . . . . . . . . 83

4.4 Maximum Likelihood identification . . . . . . . . . . . . . . . . . . 85



4.4.3 Noise intervals . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5 Combined deterministic-stochastic identification . . . . . . . . . . . 89



4.6 Comparison between common denominator based algorithms . . . 92

4.7 Mixed LS-TLS, SK, IQML, WGTLS and BTLS estimators . . . . . 93

4.8 Simulation and Measurement examples . . . . . . . . . . . . . . . . 93

4.8.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.8.2 Measurement on an aluminium plate . . . . . . . . . . . . . 101

4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Identification of right and left matrix fraction polynomial models107

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2 Frequency-domain identification of RMFDmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.1 Poly-reference Weighted Least Squares ComplexFrequency-domain estimator . . . . . . . . . . . . . . . . . 109

5.2.2 Poly-reference Maximum Likelihood Estimator . . . . . . . 111

5.2.3 Fast Poly-reference Maximum Likelihood Estimator . . . . 114

5.2.4 Fast Poly-reference IQML . . . . . . . . . . . . . . . . . . . 117

5.2.5 Poly-reference WGTLS and fast BTLS Estimator . . . . . . 117

5.2.6 RMF description for IO data . . . . . . . . . . . . . . . . . 119

5.2.7 From matrix coefficients to modal parameters . . . . . . . . 119

5.3 Left Matrix Fraction Description . . . . . . . . . . . . . . . . . . . 121

x Contents

5.3.1 Linear Least Squares estimator for IO data . . . . . . . . . 121

5.3.2 Linear Least Squares estimator for FRF data . . . . . . . . 123

5.3.3 From matrix coefficients to modal parameters . . . . . . . . 123

5.3.4 Data condensation . . . . . . . . . . . . . . . . . . . . . . . 124

5.4 Output-Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.5 Illustrating examples . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.5.1 Body-in-white . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.5.2 Fully trimmed car . . . . . . . . . . . . . . . . . . . . . . . 126

5.5.3 Villa Paso Bridge . . . . . . . . . . . . . . . . . . . . . . . . 128

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.7 Appendix: Confidence intervals on the estimated poles from thep-ML estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Deterministic Frequency-domain Subspace Identification 135

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2 Basic Frequency-Domain Projection Algorithm . . . . . . . . . . . 137

6.3 Starting from FRFs or power spectra . . . . . . . . . . . . . . . . . 139

6.4 A weighted frequency-domain projection algorithm . . . . . . . . . 139

6.5 Extended state-space model for initial and final conditions . . . . . 141

6.5.1 State-Space Model for Arbitrary Signals . . . . . . . . . . 141

6.5.2 Remarks on the extended state-space model . . . . . . . . . 142

6.5.3 A mixed non-parametric/parametric FRF estimator for val-idation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.5.4 State-Space Model for FRF data . . . . . . . . . . . . . . . 144

6.6 Simulation and Measurement examples . . . . . . . . . . . . . . . . 145

6.6.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.6.2 Measurements on a subframe of a car . . . . . . . . . . . . 148

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7 Stochastic frequency-domain subspace identification 153

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Contents xi

7.2 A first approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.3 A second approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.4 Connection to time domain stochastic subspace identification . . . 163

7.5 Geometrical Interpretation . . . . . . . . . . . . . . . . . . . . . . . 164

7.6 Final Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.8 Simulation and Measurement example . . . . . . . . . . . . . . . . 168

7.8.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.8.2 Measurement example: Subframe of an car . . . . . . . . . 170

7.8.3 Flight flutter testing . . . . . . . . . . . . . . . . . . . . . . 171

7.8.4 Villa Paso bridge . . . . . . . . . . . . . . . . . . . . . . . . 173

7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8 Combined frequency-domain subspace identification 177

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.2 Theoretical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.2.1 System description . . . . . . . . . . . . . . . . . . . . . . . 179

8.2.2 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.2.3 From states to system matrices . . . . . . . . . . . . . . . . 182

8.2.4 Taking into account effects of transients and leakage . . . . 183

8.3 Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . 183

8.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.5 Simulations and Real-life measurement examples . . . . . . . . . . 186

8.5.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . 186

8.5.2 Flight flutter simulation . . . . . . . . . . . . . . . . . . . . 188

8.5.3 Flight flutter measurements . . . . . . . . . . . . . . . . . . 190

8.5.4 ABS-function driven identification . . . . . . . . . . . . . . 192

8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

8.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

xii Contents

9 The secrets behind clear stabilization charts 199

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.2 Theoretical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.2.1 LS Identification . . . . . . . . . . . . . . . . . . . . . . . . 201

9.2.2 Influence of the constraint on an AR model . . . . . . . . . 201

9.2.3 Stochastic State-Space models . . . . . . . . . . . . . . . . 205

9.2.4 Extrapolation to ARX models . . . . . . . . . . . . . . . . 206

9.2.5 Combined deterministic-stochastic frequency domain sub-space identification . . . . . . . . . . . . . . . . . . . . . . . 208

9.3 Application for Modal Parameter Estimation methods . . . . . . . 209

9.3.1 Least Squares Frequency-domain (LSCF) algorithm . . . . 210

9.3.2 Poly-reference Least Squares Frequency-domain (p-LSCF,PolyMAX) algorithm . . . . . . . . . . . . . . . . . . . . . . 210

9.3.3 Least Squares Complex Exponential (LSCE) method . . . . 211

9.3.4 Frequency-domain subspace algorithms . . . . . . . . . . . 212

9.3.5 Coupled stochastic-deterministic dynamics . . . . . . . . . . 212

9.3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.4 Application of experimental structural testing . . . . . . . . . . . . 214

9.4.1 Measurements on a car door . . . . . . . . . . . . . . . . . . 214

9.4.2 Measurements on a fully trimmed car . . . . . . . . . . . . 216

9.4.3 In-flight aircraft measurements . . . . . . . . . . . . . . . . 216

9.4.4 Villa Paso bridge . . . . . . . . . . . . . . . . . . . . . . . . 219

9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

9.6 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

9.7 Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

10 Conclusions 227

10.1 Summary and main contributions . . . . . . . . . . . . . . . . . . . 227

10.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Bibliography 232

Chapter 1

Introduction

This chapter contains the general introduction and motivation of the research thatwas conducted in the frame work of this thesis. The research context is described inparagraph 1.1. The focus of the thesis and the organization of the text are discussedin paragraph. 1.2, while paragraph 1.3 gives an overview of the new contributionsin this work.

1

2 Chapter 1. Introduction

1.1 Research context

During the last decade modal analysis has become a key technology in structuraldynamics analysis. Starting from simple techniques for trouble shooting, it hasevolved to a ”standard” approach in mechanical product development. Beginningfrom the modal model, design improvements can be predicted and the structurecan be optimized. Based on the academic principles of system identification, ex-perimental modal analysis helps the engineers to get more physical insight fromthe identified models. Continuously expanding its application base, modal anal-ysis is today successfully applied in automotive engineering (engine, suspension,body-in-white, fully trimmed cars, ...), aircraft engineering (ground vibration test,landing gear, control surfaces, in-flight tests), spacecraft engineering (launchers,antennas, solid panels, satellites,...), industrial machinery (pumps, compressors,turbines, ...) and civil engineering (bridges, off-shore platforms, dams, ...).

Experimental modal analysis (EMA) identifies a modal model from the measuredforces applied to the test structure and the measured vibration responses. Themodal model expresses the dynamical behavior of the structure as a linear combi-nation of different resonant modes. Each resonance mode is defined by a resonancefrequency, damping ratio, mode shape and participation vector. These modal pa-rameters depend on the geometry, material properties and boundary conditionsof the structure. Vibrations of the structure originate from its resonance modesthat are inherent properties of the structure. Forces exciting the structure at oneor more of these resonance frequencies cause large vibration responses resulting inpossible damage, discomfort and malfunctioning.

More recently, system identification techniques were developed to identify themodal model from the structure under its operational conditions from vibrationresponses only. These techniques, referred as operational modal analysis (OMA)or output-only modal analysis, take advantage of the ambient excitation as e.g.wind, traffic and turbulence. During an EMA, the structure is often removed fromits operating environment and tested in laboratory conditions. The latter exper-imental situation can differ significantly from the real-life operating conditions.An important advantage of OMA is that the structure can remain in its normaloperating condition. This allows the identification of more realistic modal modelsfor in-operation structures.

In practice, performing a modal analysis typically consists of three basic steps:

1. Design of the test setup and conducting the experiments: concerns the ex-perimental setup (e.g., placement of sensors (and actuators), boundary con-ditions, . . . ) and the data acquisition parameters, (e.g., measurement time,sampling frequency, . . . ). During an EMA, the test structure is excited bymeans of artificial excitation devices (shakers, impact hammers). For OMAtesting, freely available ambient excitation sources, are usually considered as

1.1. Research context 3

excitation and eliminate the need for artificial excitation equipment.

2. Processing of the measured data and identification of a modal model : de-pending on the length of measured time signals, the time signals can firstbe converted into Averaged Based Spectral (ABS) functions, which serveas primary data for the system identification algorithms. In the case offrequency-domain system identification, the raw time domain data can betransformed into Frequency Response Functions (FRFs) (EMA case) or autoand cross power spectra (OMA case). Next, starting from the primary data,the modal model can be estimated by using an appropriate parametric iden-tification algorithm.

3. Validating the model : the extracted modal model must be assessed for itsphysical representation of the dynamical behavior of the structure in thestudied frequency band. Therefore, the modal model must satisfy severalcriteria, based on physical properties of the modal model.

For engineers, the modal model is often not the final goal, but only an inter-mediate result that can be used for a wide range of applications:

• Response prediction: modal models are often used for the purpose of predict-ing the response of the structure to a given dynamic loading. These loadingsusually corresponds to the forcing sequences encountered during the real-lifeoperating conditions. This way, designers can check the robustness of thedeveloped product under a variety of working conditions.

• Sensitivity analysis and structural modification: the modal model can beused in order to predict the effect of structural modifications to a test struc-ture [125]. For instance, if a structure (e.g., prototype) suffers from a vibra-tional problem, a variety of structural modifications (that attempt to solvethe problem) can be evaluated without actually having to apply any high-cost changes to the prototype. As long as the structural modifications canbe considered small, a (linear) sensitivity analysis can be used in order topredict the most sensitive areas of the structure for applying a structuralmodification that aims at solving the problem.

• Model updating : the initial values chosen for the material properties, geomet-rical properties and boundary conditions of a FEM often do not guaranty areliable model of the structure under test. For this reason, the use of exper-imental data is required to update (correlation, optimization, verification)the initial model and produce an FEM which can more reliably predict the(dynamical) behavior of the structure [33].

• Sub-structuring : given the modal model of different components of a complexstructure, the dynamical behavior of the complete structure can be computedby using sub-structuring techniques [72].


• Structural health monitoring and damage detection: given a reference modelof a healthy undamaged structure, a decision can be made on the struc-tural integrity of that structure by comparing newly estimated models tothe reference one. The topic of structural health monitoring has receiveda considerable amount of attention during the last few decades [30]. Apartfrom the detection of damage, the information contained in the modal modelcan also allow a localization and assessment of the structural defects. Themain disadvantage of these techniques is that other changes, such as bound-ary conditions and/or environmental changes, can also produce changes inthe modal parameters that are of the same order of magnitude as those re-sulting from the occurrence of damage. In this context, model analysis canalso be a tool for automated model based quality control [13].

• Load identification: is the inverse problem of response prediction. Given amodal model of a structure, the idea consists in the identification of unknownforces that gave rise to a specific measured response [84].

• Vibro-acoustics: the existence of vibro-acoustical coupling between the struc-tural vibration and radiated noise is an important aspect during the designprocess of for instance cars, airplanes, heavy machinery and control cabins,in terms of comfort. Based on the modal parameters it is possible to com-pute the sound intensity radiated from the vibrating structure without theneed of performing expensive acoustic experiments [104].

Nevertheless, the current evolution in mechanical engineering towards the use ofComputer Aided Design (CAD) like Finite Element Models (FEM) results in achanging role for testing [113], [114], [112]. Today the optimization process inproduct development is under strong pressure because of the competitive mar-ket, increasing customers’ demands and by consequence the design cycle becomesshorter in time. This results in an increasing use of simulations based on numericalmodels to reduce the number of prototypes and expensive experiments. Still, test-ing plays an important and evermore critical role, in every step of the developmentprocess for target setting, bench-marking and model updating. The limitations ofFEM approaches lie in the long calculation times due to the need of huge modelsizes, required to properly describe complex structures (models with over 1 mil-lion degrees of freedom) and accuracy limitations related to modelling damping,non-homogeneous materials, structural joints, ... [114]. All this, together with thedecreasing expertise of the users, since EMA/OMA have been transferred from therealm of the research experts to the product development workfloor [89], makesthat the demands for modal parameter estimation (MPE) algorithms still increasein terms of accuracy, speed, automation and physical interpretation.

1.2. Focus, outline and organization of the thesis 5

1.2 Focus, outline and organization of the thesis

In this thesis, several dedicated extensions of existing frequency-domain modalparameters estimation (MPE) algorithms and new frequency-domain algorithmsfor the estimation of modal parameters from experimental measurements are pro-posed. In general the proposed algorithms try to fulfill the requirements for newMPE algorithms [113], i.e.

• Reduction of the test time.

• Allow maximal test data exploitation

• Increase the accuracy of the estimates

• Decrease the complexity of the analysis, allowing less-experienced staff toprocess the data

• Extend the limits and ranges (large number of sensors, high damping, noisydata, short data records, ...)

Furthermore, this thesis extends the EMA and OMA concepts, to a so-called com-bined EMA-OMA framework. In this combined framework, the vibration responseis considered as a result of both measured artificial applied forces and unmea-surable ambient excitation, i.e. an Operational Modal Analysis with eXogenousinputs (OMAX), resulting in a maximum data exploitation. In the OMAX frame-work, the stochastic contribution, i.e. the part of the response which can notbe related to the measurable input forces, is considered as valuable information.Under the assumption that the stochastic contribution in the response is relatedto unmeasurable ambient forces, extra information about the system can be ex-tracted from this contribution i.e. both modes excited by the measurable and/orby the unmeasurable forces can be identified from the data.

The following gives a short overview of the content of the different chapters.A chapter-by-chapter outline is also presented in figure 1.1

Instead of estimating the modal parameters directly from the measurements,the proposed algorithms identify first mathematical polynomial or state-spacemodels. These mathematical models can be related to the modal parametersin a next step. In chapter 2 different mathematical models, the modal model andtheir relation is given. Next, the deterministic approach for the EMA case anda stochastic approach for the OMA case are discussed, to introduce the OMAXframework. Finally, a distinction is made between the data driven and AveragedBased Spectral (ABS) function driven identification algorithms.

In chapter 3, an overview of ABS function (i.e. FRFs for EMA, power spectrafor OMA, and both simultaneously for OMAX) estimators is given. The differ-ent non-parametric FRF and power spectra estimators are discussed and special


Modal Testing

Data driven ABS driven

Mathematical models for EMA, OMA and OMAX

Modal Applications

Construction of clearstabilization charts

Cost function based Subspace Algorithms

Frequency-domain System Identification

Common-denominatormodels

Left and right matrixfraction description

Deterministic identification

Stochastic identification

Combined deterministic-stochastic identification

Chapter 4

Chapter 5

Chapter 7

Chapter 3

Chapter 2

Chapter 6

Chapter 8

Chapter 9

Figure 1.1: Organization of the text.

attention is paid to obtain the noise information on these ABS functions. It isshown, that the use of a rectangular window, can reduce the noise levels anderrors introduced by leakage, if the final estimated parameters are corrected forboth time- and frequency-domain leakage. The idea of ’positive’ power spectra isintroduced to eliminate the 4-quadrant symmetry in case of an OMA. In this way,MPE algorithms can start simultaneously from FRFs and positive power spectrain an OMAX framework to estimate the modal parameters.

1.2. Focus, outline and organization of the thesis 7

During the last years, several MPE algorithms based on a common-denominator(scalar polynomial fraction) model were developed in the Acoustics and VibrationResearch Group of the Vrije Universiteit Brussel for both EMA and OMA applica-tions [131],[80]. Special attention was paid to the accuracy, calculation speed andmemory requirements. In chapter 4, these algorithms were generalized to startfrom FRFs starting from arbitrary signals, without introducing bias errors fromleakage. Furthermore a combined stochastic-deterministic algorithm is proposed,together with a discussion of the analogy between the algorithms starting fromFRFs as primary data and those starting directly from the input/output spectra(data driven).

In chapter 5 a generalization of the fast common-denominator algorithms tothe identification of right and left matrix fraction description models is presented.It is shown that for Multiple Input/Multiple Output (MIMO) measurements onhighly damped structures the new proposed maximum likelihood based algorithmsoutperform their common-denominator counterparts. A least-squares based algo-rithm and both a scalar and matrix implementation of a maximum likelihoodalgorithm are developed, with special attention paid to speed up the algorithmand to reduce the memory requirements.

Chapter 6 starts with the introduction of frequency-domain subspace algo-rithms to identify state-space models from both FRFs and input/output spectra.In this chapter an extension of the frequency-domain state-space model is discussedto take into account the effect of initial and final conditions of the vibrating struc-ture. In this way transient effects in the measured data are considered as anextra input to estimate the modal parameters. Based on this extension a mixednon-parametric/parametric estimation for FRFs is proposed for model validation.Finally, this extension is applied to estimate state-space models from FRFs.

Until recently, frequency-domain subspace identification was limited to esti-mate deterministic models from input/output or FRF data. In chapter 7, astochastic frequency-domain subspace algorithm is proposed starting directly fromoutput spectra. It is shown that is algorithm is closely related to the stochas-tic time-domain subspace algorithms. This algorithm is very useful to estimatestochastic models from a short data sequence in a frequency band of interest as itis the case for operational in-flight aircraft tests.

Next, the extension to a combined deterministic-stochastic frequency domainsubspace algorithms is proposed in chapter 8. This algorithm results in consistentestimates in presence of process and output noise on the primary data. Com-pared to the cost function related algorithms proposed in chapters 4 and 5 thecombined deterministic-stochastic frequency-domain subspace algorithm is capa-ble to estimate consistent parameters in a single step (no need for an optimizationalgorithm), without requiring any a priori known noise information. Similar tothe inconsistent least-squares based estimators, the subspace algorithms can con-struct a stabilization diagram in a fast way. This combined frequency-domain


subspace algorithm is closely related to its time-domain equivalent. In the caseinput/output spectra serve as primary data the combined algorithm fits in theOMAX framework, since the vibration responses are considered as the result ofboth measurable and unmeasurable forces. Therefore, this algorithm gives goodresults to process in-flight flutter measurements, where the airplane is simultane-ously excited by both an artificially applied force and by atmospheric turbulence.Nevertheless, the proposed algorithm also results in consistent estimates startingfrom noisy FRFs and (’positive’) power spectra.

Finally, in chapter 9, the secret behind clear stabilization charts is revealed. Bymany researchers it was noticed that the least-squares frequency-domain identifi-cation algorithms results in very clear and easy-to-interpret stabilization diagrams.It is now shown that this can be explained by the choice of the constraint on theparameters to solve the least-squares problem. A smart choice of this constraintresults in the identification of stable physical poles modelling the deterministic con-tribution and unstable mathematical poles modelling the stochastic contribution.This property results in a simple distinction between the physical and mathemat-ical poles based on the sign of the damping ratios and the start for an automaticinterpretation of the diagram. This key idea for a clear distinction between thephysical and mathematical poles based on the damping, is closely related to thechoice of the basis functions in frequency-domain models and to feed-forward/feed-backward time-domain identification. It is shown that this property also canbe used for the implementations of frequency-domain subspace algorithms andother well-known time-domain algorithms.

1.3 Original contributions in this work

The research presented in this paper is based on several presentations given at in-ternational conferences, published and submitted in international journals. Next,an overview is given of the most important contributions together with their ref-erences:

• The introduction of a combined EMA/OMA framework as Operational ModalAnalysis with eXogenous Inputs (OMAX) which considers the vibration re-sponse as a combination of a contribution caused by measurable and unmea-surable forces [22], [23].

• The use of ’positive’ power spectra to eliminate the 4-quadrant symmetry inthe poles of power spectra in combination with its compensation for time-domain leakage. The noise and data reduction for FRF data based on theuse of a rectangular window without the introduction of bias errors due toleakage [20].

• The extension of FRF based common-denominator models to take into ac-

1.3. Original contributions in this work 9

count the initial/final conditions to prevent the introduction of bias errorscaused by leakage [24], [14]. This work was rewarded with the ’Best paperaward’ of the VIII International Conference on Recent advances in StructuralDynamics (Southampton, UK, 2003) organized by the Institute of Sound andVibration Research.

• The presentation of a combined deterministic-stochastic algorithm based ona common-denominator model, together with the close analogy between in-put/output spectra driven and FRF driven algorithms [22], [23], [132].

• The development of a scalar and matrix implementation of the maximumlikelihood algorithm to estimate right matrix fraction models, resulting inan increased accuracy for MPE from noisy, highly damped, MIMO tests [21],[89].

• The extension of frequency-domain state-space models to include transientand initial/final conditions. This resulted in an increased accuracy and anenhanced mixed non-parametric/parametric validation tool based on FRFs[18], [17].

• The development of a consistent stochastic frequency-domain subspace al-gorithm for operational test data [16].

• The development of frequency-domain counterparts of the combined determi-nistic-stochastic time-domain subspace algorithms. This algorithm has thespecific advantages of frequency-domain identification and results in con-sistent estimates starting from input/output spectra, FRFs or (’positive’)power spectra [19].

• The mathematical explanation behind the construction of clear stabilizationdiagrams. The explanation of the importance of the constraint on the pa-rameters, the basis functions and the time axis in identification algorithmsresulted in the capability of constructing clear stabilization diagrams forseveral well-known algorithms [25], [27].

• Application of all these contributions for several experimental test cases inautomotive, civil and aerospace engineering [26],[132].


Chapter 2

Frequency-domain modelsfor dynamical structures

In this chapter, different mathematical models are introduced to describe the dy-namical behavior of vibrating structures. Their relation with the physical param-eters of the modal model of a structure is established. This chapter discussescommon-denominator models, left- and right fraction description models and state-space models. Finally, the concepts of an experimental modal analysis, operationalmodal analysis and combined experimental-operational modal analysis are intro-duced

11

12 Chapter 2. Frequency-domain models for dynamical structures

2.1 Introduction

This chapter discusses several mathematical models that can be used to describethe vibrational behaviour of a structure with a limited number of parameters.From an engineering point of view the modal model of a structure provides thebest physical understanding. However, since this model is highly non-linear in itsparameters most identification algorithms do not directly identify the model pa-rameters. Instead, the modal parameter estimation (MPE) methods proposed inthe next chapters identify scalar matrix fraction models, also known as common-denominator models, left-and right matrix fraction description models and state-space models from the experimental measurements. In the next sections, the rela-tion between these models and the modal parameters are discussed. Furthermorea distinction between continuous-time and discrete-time models is discussed. Adistinction is made between data driven and spectral function driven identificationalgorithms. Finally, it is shown how these models can be used to identify modelparameters from both input-output measurement and from output-only measure-ments in absence of the input measurements and the concepts of a Experimen-tal Modal Analysis, a Operational Modal Analysis and Combined Operational-Experimental Modal Analysis are introduced.

2.2 The Modal Model

Newton’s equations of motion for a finite-dimensional linear structure are a set ofNm second-order differential equations, where Nm is the number of independentdegrees-of-freedom, given by

My(t) + C1y(t) + Ky(t) = f(t) (2.1)

with M , C1 and K ∈ RNm×Nm respectively the mass, damping and stiffness

matrices, f(t) ∈ RNm×1 the applied force and y(t) ∈ R

Nm×1 the structure’sdisplacement. Using the Laplace transform and neglecting the initial conditionsresults in the frequency-domain equivalent given by

Z(s)Y (s) = F (s) (2.2)

with the dynamical stiffness Z(s) = Ms2 + C1s + K and s = jω. Inverting Eq.2.2 yields

Y (s) = H(s)F (s) (2.3)

with H(s) = Z−1(s) the transfer function matrix. The transfer function matrixcan be formulated in its modal form [52], [72]

H(s) = φ[

sINm

− Λ]−1

LT + φ∗[sINm

− Λ∗]−1LH (2.4)

2.3. Common-denominator models 13

where the modal parameters λr, φr and Lr are respectively the pole, mode shapeand modal participation factor of mode r. The diagonal matrix Λ is given by

Λ = diag(

λ1, λ2, . . . , λNm)

(2.5)

with Nm the number of modes. The poles λr = σr + iωr contain the naturalfrequencies fr = ωr/(2π) and the damping ratios dr = −σr/

√

σ2r + ω2

r . In real-lifeapplications the number of modes Nm differs from the number of measured outputdegrees of freedom No and the number of input forces Ni.

2.3 Common-denominator models

The common-denominator, also called scalar matrix fraction model, considers therelation between output o and input i as a rational fraction of two polynomials, ofwhich the denominator polynomial is common for all input-output relations. Thetransfer function matrix H(s) can be expressed as

H(s) =Zadj(s)

|Z(s)| (2.6)

with Zadj(s) the adjoint matrix, containing polynomials of order 2Nm − 1. Thecommon-denominator is then given by the characteristic equation |Z(s)|, a poly-nomial in s of order 2Nm, which roots are the poles of the structure. In generalthe common-denominator model can be expressed as

H(s) =

B11(s) . . . B1Ni(s)

.... . .

...BNo1(s) . . . BNoNi

(s)

A(s)(2.7)

The relation between the modal model and the common-denominator model isobtained by considering the frequency response function between output o andinput i

Hoi(s) =

Nm∑

r=1

(

φorLir

s − λr+

φ∗orL

∗ir

s − λ∗r

)

=Boi(s)

A(s)(2.8)

From this equality it is clear that the structure poles are given by the roots of thedenominator A(s), while the mode shapes and participation factors are obtainedfrom a singular value decomposition (SVD) of the residue matrix Rr ∈ C

No×Ni ofmode r

Rr = φrLTr (2.9)


with the elements of the residue matrix Rr given by

Roi,r = lims→λr

(s − λr)Hoi(s) (2.10)

= φorLir (2.11)

and φr = [φ1rφ2r . . . φNor]T , Lr = [L1rL2r . . . LNir]

T . From modal analysistheory it follows that this residue matrix is of rank 1. Nevertheless the common-denominator model does not force rank 1 residue matrices on the measurements.The implications on the modal model are discussed in chapter 5. For a common-denominator model each relation between one output and one input can be con-sidered separately, this turns out to be an advantage in terms of the optimizationof the calculation speed of the identification algorithms proposed in chapter 4.

2.4 State-space models

An other type of models are the so-called state-space models, which introduce theconcept of the states of a dynamical system. Reformulating Eq. 2.2 as

[

sY (s)s2Y (s)

]

=

[

0 I−M−1K −M−1C1

] [

Y (s)sY (s)

]

+

[

0M−1F (s)

]

(2.12)

results in a state-space formulation given by

sX(s) = AX(s) + BF (s) (2.13)

Y (s) = CX(s) + DF (s) (2.14)

with

A =

[

0 I−M−1K −M−1C1

]

, C =[

I 0]

(2.15)

B =

[

0M−1

]

and D = [0] (2.16)

By using the auxiliary state vector X, given by

X(s) =

[

Y (s)sY (s)

]

(2.17)

Eq. 2.2 is transformed in a first order differential expression. Identification algo-rithms based on a state-space model identify the system matrices A, B, C and Dfrom the measurement data. The transfer function matrix between the outputsand inputs is then given by

H(s) = C [sI − A]−1

B + D (2.18)

2.5. Matrix fraction polynomial models 15

By considering the righthand eigenvectors V of the system matrix A defined by

AV = V Λ (2.19)

the state-space equations are transformed to their modal form

H(s) = CV [sI − Λs]−1

V −1B (2.20)

=[

φ φ∗ ][

sI −[

Λ 00 Λ∗

]]−1 [LT

LH

]

(2.21)

This last expression is totally equal to the more commonly used expression ofthe modal model given by Eq. 2.4. In the most general case the number ofresponses No differs from the number of modes Nm and consequently A ∈ R

n×n,C ∈ R

No×n, B ∈ Rn×Ni and D ∈ R

No×Ni with the model order n = 2Nm.

2.5 Matrix fraction polynomial models

The common-denominator model (scalar matrix fraction description) can be con-sidered as a special case of multivariable transfer function models described usinga Matrix Fraction Description (MFD), i.e. the ratio of two matrix polynomials[59], [42]. This set of models can be divided in two set of models i.e. a left MFD(LMFD) and a right MFD (RMFD). Based on the relationship between the LMFD,RMFD and the common-denominator model, a so-called Unified Matrix Polyno-mial Approach (UMPA) was proposed in [4] for comparison of different estimationalgorithms using a common mathematical framework.

2.5.1 Left-Matrix Fraction Description (LMFD) models

The LMFD models consider all input-output measurements simultaneously by thefollowing model

H(s) = A−1(s)B(s) (2.22)

with A(s) = Isn + An−1sn−1 + . . . A0 a matrix polynomial with (No ×No) matrix

coefficients and B(s) = Bnsn+Bn−1sn−1+. . . B0 a matrix polynomial with square

(No ×Ni) matrix coefficients. This LMFD model can be obtained from the state-space model. The denominator coefficients are obtained by solving the followinglinear set of equations [2]:

[

An−1 . . . A1 A0

]

CAn−1

...CAC

= −CAn (2.23)


The numerator coefficients are obtained by considering the following set of equa-tions

H(s)sH(s)

...snH(s)

=

CCA...

CAn

X(s) +

D 0 . . . 0CB D 0...

. . ....

CAn−1B CAn−2B . . . D

IsI...

snI

(2.24)

denoted in short as

Hn = OnX + ΓIn (2.25)

Pre-multiplying by [ A0 . . . An−1 I ] yields

[

A0 . . . An−1 I]

Hn =[

A0 . . . An−1 I]

ΓIn (2.26)

The term in X is cancelled because of Eq. 2.23. The left-hand side of Eq. 2.26is nothing else than A(s)H(s) and thus by comparing this equation with Eq. 2.22the right-hand must be an expression for the polynomial B(s). As a result the Bn

coefficients can be expressed as

[

B0 . . . Bn−1 Bn

]

=[

A0 . . . An−1 I]

Γ (2.27)

In the case that the number output observations No equals the number of modesNm, the order n of the polynomials will be 2. Using the expressions for the systemmatrices A, B, C and D defined by Eq. 2.15 and 2.16 and the formulas 2.23 and2.27 respectively for the Aj and Bj polynomial coefficients results in

A2 = I , A1 = M−1C1 and A0 = M−1K

B2 = 0 , B1 = 0 and B0 = M−1 (2.28)

where B2 and B1 are zero because D = 0 and CB = 0. This is not the case ifH(s) is considered as the transfer function matrix between the accelerations andforces, instead of the displacements. Indeed, for the situation where Nm = No therelation with the dynamic stiffness matrix Z(s) is directly given by

H(s) =(

Is2 + M−1C1s + M−1K)−1

M−1 (2.29)

=(

Is2 + A1s + A0

)−1B0 (2.30)

and therefore methods which identify LMFD models with n = 2 are referred toas direct identification methods. The system poles are given by the roots of thecharacteristic equation det(A(s)), which is of order 2Nm. Nevertheless, in the casethat the number of modes Nm exceeds the number of response locations No, thepolynomials A(s) and B(s) must be expanded to higher orders in order to identifyNm modes. The order n of A(s) must be larger or equal to 2Nm/No to captureall dynamics. For No = 1 the limit of this expansion process results in a common

2.5. Matrix fraction polynomial models 17

denominator model with a denominator polynomial of high order n = 2Nm. Forthe case of modal analysis the number of responses No is often larger than thenumber Nm and therefore the number of response locations is often artificiallyreduced by a data reduction steps [63].

Once the coefficients An are known, A(s) = 0 can be reformulated into ageneralized eigenvalue problem, resulting in nNo eigenvalues, yielding estimatesfor the system poles λr and the corresponding left eigenvectors φr, correspondingto the modal mode shapes [63]. The participation factors can be obtained fromthe B coefficients [59] or from a least squares problem in a second step estimationprocedure by the Least Squares Frequency Domain (LSFD) method. Since boththe poles and participation factors are known, the FRFs are a linear function ofthe mode shapes.

2.5.2 Right-Matrix Fraction Description (RMFD) models

The RMFD model is given by

H(s) = B(s)A−1(s) (2.31)

with A(s) = Isn + An−1sn−1 + . . . A0 a matrix polynomial with square (Ni ×Ni)

matrix coefficients and B(s) = Bnsn + Bn−1sn−1 + . . . B0 a matrix polynomial

with (No × Ni) matrix coefficients. This RMFD model can be considered as aLMFD model of the transposed transfer matrix HT (s)

HT (s) = A−T (s)BT (s) (2.32)

and by consequence similar relations between the state-space model and the RMFDmodel can be derived. In practice, since the number of input Ni is typically muchsmaller than the number of modes Nm, RMFD models have a higher model ordern than LMFD models, but the denominator coefficients have smaller dimensions.This fact has some implications on the performance of the system identificationalgorithms based on LMFD and RMFD models as discussed in chapter 5.

Similar as for the LMFD model, the poles λr and modal participation vectorsare obtained from reformulating A(s) = 0 into a generalized eigenvalue problem,resulting in nNi eigenvalues and the corresponding left eigenvectors. The modeshapes can be obtained from the B coefficients or from a second step estimationprocedure by solving a linear least-squares problem, since the modal model is linearin the mode shapes.


2.6 Acceleration, Displacement, Velocity

In a practical modal analysis experiment, the vibration response can be measuredas displacements, but also as velocities (laser vibrometer) or accelerations. Untilnow, only displacement response measurements were considered. Consider thegeneral case were the No response measurements can be either accelerometers,velocity or displacement transducers. In that case the observation equation isgiven by

Y ′(s) = Cas2Y (s) + CvsY (s) + CdY (s) (2.33)

where Y ′(s) are the spectra of the outputs; Ca, Cv, Cd ∈ N0No×No are the selection

matrices for the accelerations, velocities and displacement. These matrices containonly zeros and a few ones and indicate which output is measured as an acceleration,velocity or displacement. E.g. in the case that only accelerometers are usedCa = I, Cv = 0 and Cd = 0. The corresponding state-space model is given by

sX(s) = AX(s) + BF (s) (2.34)

Y ′(s) = CX(s) + DF (s) (2.35)

with A and B defined by Eq. 2.15, 2.16 and C and D given by

C =[

Cd − CaM−1K Cv − CaM−1C1

]

and D =[

CaM−1]

(2.36)

Important to notice is that a modal analysis experiment, based on accelerometers,requires a direct term D. For acceleration based tests, CB differs from the zeromatrix and thus by result the coefficients B2 and B1 in the LMFD model forthe case No = Nm differ from zero. Reducing the size of the matrix coefficientsand increasing the order of the polynomials finally results in a scalar commondenominator model. In the case one starts from accelerations both the numeratorsand denominator in the common-denominator model have the same order.

2.7 Continuous-time models and discrete-time mod-els

For time-domain models a distinction is made between continuous-time and discrete-time models. Until now, all proposed models in this chapter are continuous-timefrequency-domain models. Unlike in time-domain, frequency-domain models allowto identify continuous-time models from samples of the Fourier transforms of themeasured signals, where time-domain identification is restricted to discrete-timemodels. The continuous-time state-space-model is given by

x(t) = Ax(t) + Bf(t) (2.37)

y(t) = Cx(t) + Df(t) (2.38)

2.7. Continuous-time models and discrete-time models 19

and since the continuous-time signals are not available for identification, themodel must be converted to the frequency domain model for identification of acontinuous-time model. By using the discrete Fourier transformation, the frequency-domain model is given by

sX(s) = AX(s) + BF (s) (2.39)

Y (s) = CX(s) + DF (s) (2.40)

where F (s) and Y (s) are only available for discrete values sk of s

Y (sk) =1√N

N−1∑

n=0

y(n∆t)e−2πnk/N and F (sk) =1√N

N−1∑

n=0

f(n∆t)e−2πnk/N (2.41)

with sk = i2πk/(N∆t) and N the number of time samples.

Discrete-time models give the relation between discrete time samples yn =y(n∆t) of continuous-time signals. In this thesis it is assumed that the sampleperiod ∆t is constant during the measurements. Under the Zero Order Hold(ZOH) assumption, i.e. the input is piecewise constant over the sample period, acontinuous-time model converts to a discrete-time model [55]

xn+1 = Adxn + Bdfn

yn = Cdxn + Ddfn (2.42)

with yn and fn the sampled time signals and where the discrete system matriceshave the following relation to the continuous system matrices under the ZOHassumption

Ad = eA∆t , Bd =

∫ ∆t

0

eAζdζB (2.43)

Cd = C , Dd = D (2.44)

For this discrete-time model the frequency domain counterpart is given by

zkXk = AdXk + BdFk

Yk = CdYk + BdFk (2.45)

with zk = esk∆t = ei2πk/N and Yk = Y (sk), Fk = F (sk), Xk = X(sk). Theproposed continuous-time models i.e. the modal model, the common-denominatormodel, the LMFD and RMFD models and state-space models all have their discrete-time equivalent, which is of the same form but the basis functions sn are replacedby zn. In the next chapters most of the proposed identification algorithms identifydiscrete-time models in the frequency-domain.


2.8 Real versus complex models

In the modal model each mode appears twice: once with a positive frequency λr =σr + iωr and once with a negative frequency λ∗

r = σr − iωr. This symmetry in themodal model results in a polynomial model with real coefficients and state-spacemodels with real system matrices. Nevertheless, in modal analysis applications,one is often interested in a modal model in a frequency band of interest. In thisfrequency band of interest the contribution of poles with negative frequencies isoften much smaller than the out of band poles. Therefore this frequency band(which includes only poles with positive frequencies) can be modelled by complexparameters. In this way, the model order can be reduced by a factor 2, resulting ina better numerical conditioning. In practice a small over-modelling (higher modelorder) is required to take care of the systematic errors introduced by modelling afrequency band selection by a discrete-time model. To conclude each model i.e.common-denominator model, state-space model, LMFD and RMFD can be chosento have real or complex parameters.

2.9 Primary identification data

In the previous paragraphs different types of mathematical models were proposedand briefly discussed. In this section attention will be paid to the primary datafrom which one starts to identify a mathematical model. A primary distinctionis made between the availability of input-output measurements or only outputmeasurements. Next, a distinction is made based on the amount of available datasamples resulting in data driven identification or averaged-based-spectral (ABS)function driven identification. Finally, the concept of a combined operational-experimental modal analysis approach is introduced as a generalization of input-output and output-only based approaches.

2.9.1 Input-Output measurements: a deterministic approach

It is a well known fact that linear time invariant systems can be modelled by start-ing from input-output measurements. The identified parametric model essentiallycontains the same information of the studied system as the original non-parametricdata, but is often preferred because of its compact form and possible physical inter-pretation. Several important references on the subject of system identification in-clude [67], [110], [123] for time domain identification and [98] for frequency-domainidentification. Originating from electrical and control engineering, system identifi-cation is today used in many different fields such as e.g. chemical engineering, civilengineering, mechanical engineering, biomedical engineering, econometrics, ... .

2.9. Primary identification data 21

The application of system identification techniques for the identification ofmodal models – natural frequencies, damping ratios, mode shapes and modalparticipation factors – for linear time invariant mechanical structures is known asExperimental Modal Analysis (EMA). Many textbooks give an extensive overviewof EMA and input-output modal parameter estimation method [31], [52], [72].As mentioned in chapter 1, these experimentally determined modal models canbe used in a wide range of structural dynamics applications. Typical for modalanalysis, the system identification techniques must be able to deal with

• multiple inputs and multiple outputs (e.g. Ni = 3 and No = 100)

• high model orders (e.g. Nm = 100)

• very low and high damping (from 0.01% up to 10%)

• high modal density (close-coupled modes)

Experimental modal analysis starts from identifying modal models from themeasured applied forces and vibration responses of the structure, when artificiallyexcited in one or more locations (illustrated by figure. 2.1). These experimentsare performed under laboratory conditions to obtain high quality measurements(typically SNR of 40dB). From these input-output measurements a mathematicalmodel is identified, which can be converted to a modal model of the structure.

Structure

-FNi

-F2

.

.

.

-F1

- YNo

- Y2

.

.

.

- Y1

Figure 2.1: Deterministic Input-Output Model

I/O and FRF based Experimental Modal Analysis

In the field of system identification most identification algorithms start from themeasured input and output time histories or in the case of frequency-domain iden-tification algorithms from the input and output spectra. Nevertheless in the case


of a typical modal analysis experiment a large amount of data is available andsome preprocessing of the data is recommended to reduce both the size of thedata set and the noise levels before starting the parametric identification of themodal parameters.

Therefore in EMA applications it is common practice to reduce the amount ofdata and the noise levels by using the Frequency Response Functions (FRFs) asprimary data instead of the input and output spectra. These FRFs are estimated ina non-parametric preprocessing step. The measured forces and vibration responsesare divided in different data blocks in order to be averaged and estimate the FRFs.In the case that only a limited data is available and averaging reduces the frequencyresolution below a critical value, it is advised to start the identification directlyfrom the raw input spectra and output spectra (IO data driven).

From the estimates of the transfer function matrix i.e. the FRFs or the IOFourier spectra, the discussed mathematical models e.g. common-denominator,RMFD, LMFD or state-space can be identified.

• The common denominator model for FRFs is given by

Hoi(ωk) =B(Ωk)oi

A(Ωk)(2.46)

with Ωk = sk for continuous-time models and Ωk = zk for discrete-timemodels and Hoi(ωk) the FRF between output o and input i. The IO basedversion is given by

Yo(ωk) =

Ni∑

i=1

Boi(Ωk)

A(Ωk)Fi(ωk) (2.47)

• The state-space model for FRFs is given by

ΩkXk = AXk + B (2.48)

H(ωk) = CXk + D (2.49)

with X ∈ Cn×Ni and Hk ∈ C

No×Ni . The IO based version is given by

ΩkXk = AXk + BF (ωk) (2.50)

Y (ωk) = CXk + DF (ωk) (2.51)

with Xk ∈ Cn×1, Y (ωk) ∈ C

No×1 and F (ωk) ∈ CNi×1

• The LMFD and RMFD are similar as the common denominator model fore.g. the LMFD for FRFs is given by

H(ωk) = A−1(Ωk)B(Ωk) (2.52)

while the IO based version is given by

Y (ωk) = A−1(Ωk)B(Ωk)F (ωk) (2.53)


2.9.2 Output-Only measurements: a stochastic approach

In some applications (e.g. civil engineering [88], in-flight testing [10], [60]) one ismore interested to obtain modal models from structures during their operationalconditions to model the interaction between the structure and its environment e.g.wind, traffic, boundary conditions, turbulence, ... on the structure. An other ad-vantage of an in-operational modal analysis is that non-linear effects are linearizedaround the operational working point. For these applications the structures arenaturally excited by ambient excitation forces e.g. wind, traffic, seismic activity(micro-earthquakes) etc. [36, 32], which are difficult or even impossible to mea-sure. Elimination of this ambient excitation is often impossible and applying anartificial measurable force which exceeds the natural excitation is expensive andsometimes difficult. In these cases, one only measures vibration responses (illus-trated by figures. 2.2). From this output-only data only one can again estimate

Structure

-

-unmeasurable forces

.

.

.

-

- YNo

- Y2

.

.

.

- Y1

Figure 2.2: Stochastic Output-Only Model

the natural frequencies, damping values and mode shapes. The knowledge of theinput signal is replaced by the assumption that the response is a realization ofa stochastic process with unknown white noise as an input. Identifying systemparameters from these responses only is referred to as stochastic system identi-fication [123, 85]. More specific to the identification of vibrating structures theterms output-only modal analysis and in-operation or Operational Modal Analysis(OMA) are commonly used.

In the field of stochastic system identification one can generally divide theidentification techniques in two basic subcategories i.e.

• Data- driven stochastic identification algorithms which directly start theidentification from the output time sequences or output spectra.

• Correlation or power spectral density driven stochastic identification algo-rithms which estimate in a first step the power spectra between the outputs


and certain reference sensors. Next, from these functions a deterministicmodel is extracted. This model is now related to the stochastic system.

Data-driven stochastic identification

Data-driven stochastic identification algorithms directly start from the output timesequences and their frequency-domain counterparts use the output spectra as pri-mary data. For the time-domain, Auto-Regressive (AR) and Auto-RegressiveMoving Average (ARMA) models are described in [67]. Both AR and ARMA canbe considered as time-domain counterpart of polynomial models in the frequency-domain [98]. An ARMA model is written as

Iyn + A1yn−1 + . . . Anayn−na

= B0en + B1en−1 + . . . Bnben−nb

(2.54)

with Ai ∈ RNo×No and Bi ∈ R

No×No . The AR model can be considered as aspecial case of an ARMA model with B1 = B2 = . . . = Bnb

= 0. Their frequency-domain counterpart is obtained by taking the Discrete Fourier Transform of Eq.2.54

(Izna

k + A1zna−1k + . . . Ana

)Yk = (B0znb

k + B1znb−1k + . . . Bnb

)Ek (2.55)

Notice that the influence of the initial and final conditions are neglected in thisexpression. However, in chapters 4 and 6 it is shown how these initial/final con-ditions are taken into account for common-denominator and state-space models.A major drawback of the formulation given by Eq. 2.55 is that for the coefficientsBi 6= 0 the identification problem becomes highly non-linear in the system param-eters and for larger number of outputs No, which is typically the case for modaltesting, the algorithms become too slow to be used in practice. Nevertheless, ARmodels and their frequency domain equivalent can be used to model the structurewith a model order na = 2 if No ≥ Nm (referred to as direct models). For thecase where No < Nm it can be shown that an AR model with infinite order isequivalent of a finite-order ARMA model. Unfortunately, the theoretical assump-tion of an infinite order to obtain a reasonable fit, practically means that manymathematical poles are introduced [85].

A second type of stochastic data driven models are stochastic state-spacemodels. Prominent references on stochastic state-space model identification bysubspace methods are [120] and their application to civil engineering [87]. Thestochastic time-domain state-space model is given by

xn+1 = Axn + wn (2.56)

yn = Cxn + vn (2.57)

In chapter 7 of this thesis a frequency-domain counterpart subspace algorithm isdeveloped to identify state-space models from output Fourier spectra only. The


frequency-domain data driven stochastic state-space model is given by

zkXk = AXk + Wk (2.58)

Yk = CXk + Vk (2.59)

with Wk and Vk correlated white noise, representing the unmeasurable ambi-ent forces. All data driven stochastic identification algorithms are developed fordiscrete-time models. In practice this implies that the frequency-domain coun-terparts will only be theoretically consistent if the frequency band from DC toNyquist is processed simultaneously as will be discussed in more detail in chapter7.

Correlation and auto/cross power density driven stochastic identifica-tion

Power spectra driven stochastic identification algorithms can be considered as thestochastic counterpart of the FRF-driven deterministic algorithms, since similaras for FRF-driven identification, ABS functions are used as primary data. At thesame time the correlation driven algorithms can be considered as stochastic coun-terparts of identification methods, which start from Impulse Response Functions(IRF). In fact, both correlation and power spectra based identification methodsfirst estimate respectively correlation functions and power spectra between theresponses and certain reference response. In a next step a deterministic model,related to the stochastic model, is fitted through these functions.

In this paragraph the modal decomposition of power densities and correlations isbriefly discussed and references for a more profound discussion are given. Accord-ing to the modal theory of mechanical systems, the FRF matrix can be decomposedas shown by Eq. 2.4, i.e.

H(ω) = φ[

iωINm

− Λ]−1

LT + φ∗[iωINm

− Λ∗]−1LH

=

Nm∑

r=1

(

φrLTr

iω − λr+

φ∗rL

Hr

iω − λ∗r

)

(2.60)

where λr, φr and Lr are respectively the pole, mode shape and modal participationfactor of mode r.

In [67] it is shown that for stationary stochastic processes the power spectra ofthe outputs Syy(ω) ∈ C

No×No are given by

Syy(ω) = H(ω)Sff (ω)H(ω)H

(2.61)

where Sff (ω) ∈ CNi×Ni contains the cross power spectra of the (unknown) input

forces. Under the assumption that the forces are white noise sequences Sff (ω) can


be considered to be a constant matrix with respect to the frequencies. In [51], [80]it is shown that by substituting Eq. 2.60 in Eq. 2.61 power spectra (XP) of theoutputs Syy(ω) evaluated at frequency ω can be modally decomposed as follows

Syy(ω) =

Nm∑

r=1

(

φrKTr

iω − λr+

φ∗rK

Hr

iω − λ∗r

+Krφ

Tr

−iω − λr+

K∗r φH

r

−iω − λ∗r

)

(2.62)

where φr and Kr are respectively the mode shape and operational reference vec-tor for mode r. This reference vector is a function of the modal parameters andthe cross power spectrum matrix of the unknown random input force(s). Un-fortunately, the modal participation factors and by consequence the modal scalefactors can not be determined from an OMA test. Based on a sensitivity analysisa technique to estimate the modal participation vectors L is proposed in [83]. Touse this technique a second set of measurements is required, where the structureis modified with known modification e.g. adding a known mass. It should benoticed that the modal decomposition of the power densities of the outputs hasa symmetry in the poles i.e. both the positive and negative poles are present inthe model. This symmetry is referred to as a 4-quadrant symmetry. Thanks tothe similarity between the modal decomposition of the Auto and Cross spectraldensities of the outputs and the modal decomposition of the FRFs the modal pa-rameter estimation techniques for FRFs can be used to start from power spectra inthe output-only case. In practice only a limited number of reference sensors Nref

are used, by consequence Syyref∈ C

No×Nref . In the case of multi-patch measure-ments these reference output sensors remain fixed for the different patches.

Taking the Inverse Discrete Fourier Transform (IDFT) of Eq. 2.62 yields thecorrelation functions matrix R(k) for positive and negative time lags k [51]

rk =

Nm∑

r=1φrK

Tr eλrkTs + φ∗

rKHr eλ∗

rkTs for k ≥ 0

Nm∑

r=1Krφ

Tr e−λr|k|Ts + K∗

r φHr e−λ∗

r |k|Ts for k < 0

(2.63)

Interesting to note is that the causal part (positive lags) of the correlation functionscontain the stable poles i.e. λr = σr + iωr and λ∗

r = −σr − iωr, while the non-causal part (negative lags) contains the unstable poles −λr = σr + iωr and −λ∗

r =−σr + iωr. Usually time-domain modal identification methods estimate the modalparameters from the causal part only. In this way the number of modes is reducedby a factor 2. Furthermore the time-domain modal decomposition of the causalpart of the correlations is similar to the modal decomposition of IRFs and henceclassical modal parameter estimators can still be used.

The estimation of the correlation and power spectra can be considered as theestimation of smooth functions (the random character is reduced) by an averaging


process, which also reduces the original amount of data. Since the parametric iden-tification step starts from these smooth functions as primary data, a continuous-time model can be used in contrary to the data driven stochastic identification.Finally for the ABS function driven stochastic identification the stabilization dia-gram turns out to be very useful to distinguish physical poles from mathematicalones, as discussed in more detail in chapter 9.

2.9.3 Input-Output measurements: a combined deterministic-stochastic approach

In the previous two subsections it was shown that Experimental Modal Analy-sis (EMA) and Operational Modal Analysis (OMA) differ in the fact that theyrespectively consider the input forces as known and unknown. Consider a test ex-ample were both measurable and unmeasurable forces are acting on the structureas shown in figure 2.3. In a EMA one is only interested in the deterministic rela-

Structure

-FNi

-F2...

-F1

-unmeasurable forces-

- YNo

- Y2

.

.

.

- Y1

Figure 2.3: Operational Modal Analysis with eXogenous inputs (OMAX), F measurableinputs and E unmeasurable inputs

tion between the measured inputs and outputs and therefore the contribution inthe response resulting from the unmeasurable forces is considered as undesirable,disturbing measurement noise. This is in contradiction with an OMA approach,which considers the output-only of a stochastic process as the primary data toestimate the modal parameters. Therefore the concept of a combined EMA-OMAapproach is now introduced, which considers the response as both a deterministiccontribution from the measurable inputs and a stochastic contribution from theunmeasurable forces. Considering specific situations as a combined EMA-OMAtest results in a maximal data exploitation. For example a mode weakly excitedby the measurable forces and strongly excited by the unmeasurable forces is not orinaccurately identified in a purely EMA framework, while in a combined approachthis mode is still well identified from the stochastic contribution. This concept of


a combined approach, also known as a Operational Modal Analysis with eXoge-nous inputs (OMAX) [45], was already introduced under the form of combineddeterministic-stochastic models in electrical and control engineering [122]. Never-theless, in this engineering field, the noise model (i.e. the model for the stochasticcontribution) is usually not considered as a model describing the studied systemand thus no physical information is extracted from the noise model parameters.Practical examples of OMAX situations are e.g. in-flight aircraft tests, where theairplane is excited by both an artificially measurable excitation an unmeasurableturbulent forces, civil structures excited simultaneously by a measurable drop-mass or impact hammer and by unmeasurable ambient forces caused by traffic,wind and seismic activity.Similar to the purely deterministic and stochastic case combined algorithms canbe divided in data-driven methods and ABS function driven methods.

Data-driven based combined stochastic-deterministic identification

Data driven combined stochastic-deterministic identification can be consideredas data driven deterministic identification of which some inputs are replaced byunknown white inputs. The combined frequency-domain state-space model is givenby

zkXk = AXk + BFk + Wk (2.64)

Yk = CXk + DFk + Vk (2.65)

From Eq. 2.66 the IO relationship is obtained:

Yk =[

C (Izk − A)−1

B + D]

Fk + C (Izk − A)−1

Wk + Vk (2.66)

In chapters 4, 5 and 8 the data driven based combined identification is discussedin more detail respectively for a common denominator model, a LMFD model anda frequency-domain state-space model.

ABS-driven based combined stochastic-deterministic identification

In the case that sufficient data is available, the deterministic and stochastic con-tribution of the responses can be separated in a non-parametric way. This cor-responds with estimating the FRFs or IRF between the outputs and measuredinputs and estimating the power densities or correlation function of the stochasticcontribution in the responses. In a next step, a model with common dynamics(i.e. system poles) is fitted through the FRFs and power spectra. The variancesof the noise on the estimated ABS functions can be taken into account in the al-gorithm to present the relative importance of their contributions in the vibrationresponses. More details about the non-parametric estimation of the deterministicand stochastic ABS functions is given in chapter 3.

2.10. Conclusions 29

2.10 Conclusions

This chapter presented different mathematical models to describe the dynamicbehavior of vibrating structures. Starting from the differential equations basedon the law of Newton, the modal model is introduced. The relationship betweenthe state-space, LMFD and RMFD, common denominator and the modal modelare briefly discussed. These models form the basis of the identification algorithmspresented in the next chapters of this thesis. A first distinction was made be-tween deterministic models for EMA, stochastic models for OMA and combineddeterministic-stochastic models for an OMAX analysis. A second distinction ismade between data driven and ABS function driven identification algorithms. Thedata driven methods for both EMA and OMA are preferred in case only a limitedamount of data is available, while the ABS function based algorithms require alarger amount of data.

MPESmall amount of data

(data driven)

Large amount of data

(ABS driven)

Inpu

ts a

ndou

tput

s

Art

ifici

alfo

rces

Out

puts

onl

y

Am

bien

tfor

ces

I/O data-drivenEMA

FRF-driven EMA

O-driven OMA XP-driven OMA

OM

AX

OM

AX

Figure 2.4: Overview of the EMA, OMA and OMAX approach in function of theamount of data


Chapter 3

Non-Parametricpreprocessing steps

In this chapter, the non-parametric estimation of ABS functions is discussed inmore detail for the EMA, OMA and combined OMAX cases. Several methodsproposed in literature are presented. The use of a rectangular window to reducethe noise levels on FRFs is introduced and the correction for time and frequency-domain leakage is proposed. The concept ’positive’ power spectra is introduced toeliminate the 4-quadrant symmetry in the poles of power spectra. The techniquesare illustrated and validated by a simulation example and real-life applications.

31

32 Chapter 3. Non-Parametric preprocessing steps

3.1 Introduction

In chapter 2, it was already mentioned that, depending on the time length ofthe measured signals, the frequency identification algorithms start directly fromthe spectra of the measured sequences i.e. data driven methods or from Aver-aged Based Spectral (ABS) functions. In this chapter, more detailed attention ispaid to the non-parametric estimation of the ABS functions from sufficiently longmeasured time sequences.

Since a classical modal analysis experiment typically results in long data se-quences, the classical modal parameter estimation (MPE) techniques start fromFRF data to reduce the noise levels and size of the data set. Furthermore, in casethe measurements took place in different patches, starting from FRF data simpli-fies the parametric identification. In literature many attention has been paid tonon-parametric FRF estimators [67], [64], [131], [52]. This chapter starts with apresentation of well-known estimators like the H1, the H2 and Hev in the frame-work of the Total Least Squares Hv estimator. Given the Multiple Input-MultipleOutput (MIMO) character of modal testing, the different estimators are proposedfor a multivariable system. Besides the influence of noise on the measured data,the estimation of the FRFs is complicated by leakage in the case arbitrary in-put signals are used. To deal with leakage and measurement noise, the use of arectangular window is proposed and compared to the classical approach.

Similar as for EMA, many OMA modal parameter identification algorithmsstart from correlations or power density spectra between all the responses and afew reference responses. In this chapter both the correlogram and periodogrambased power spectra estimators are briefly discussed [50], [80]. Next, it is shownhow the 4-quadrant symmetry of the poles in the power density spectra is reducedto a 2-quadrant symmetry by the use of rectangular window on the correlations.

Finally, the non-parametric estimators for the EMA and OMA case are com-bined in the context of an OMAX process. The key idea of this non-parametric es-timator is to estimate both the FRFs, as a result of the deterministic contributionsand the power spectra as a result of the stochastic contributions. Parametric iden-tification starting from both FRFs and XP results in the identification of modesexcited by the artificially applied forces, modes excited by the ambient excitationand modes excited by both simultaneously. For the combined identification, a dis-tinction is made between structures, with exciter-structure (e.g. electrodynamicalshaker) and without any exciter-structure interaction.

This chapter ends by illustrating the techniques by simulations, and two ex-perimental cases. The first experimental case illustrates the applicability for mea-surements on a subframe. In the second case, the applicability for measurementsof a bridge is studied.

3.2. Modal testing for EMA 33

3.2 Modal testing for EMA

In a typically EMA test setup the structure is flexibly mounted to obtain so-calledfree-free conditions. Figure 3.1 illustrates several laboratory test setups.

(a) (b)

(c) (d)

Figure 3.1: Examples of EMA tests. (a) Ground vibration shaker test of an aircraft;(b) Shaker test of a subframe; (c) Hammer test of a frame; (d) scanning laser vibrometertest of a slattrack

Modal analysis experiments are typically characterized by the large amounts ofdata and therefore the use of FRFs as primary data is preferred for reasons ofdata reduction, noise reduction and the combination of measurements in separatedpatches. Furthermore, since the estimation of FRFs is based on an averaging pro-cess the covariance matrix of the stochastic errors on the FRFs can be estimatedand be used as a weighting for the parametric identification. Depending on the ex-citation device the EMA test can be categorized as a shaker or hammer excitation


test.

Shaker-based testing

Shaker testing excites the structure in one or several locations through a fixedshaker-stinger-structure connection. This type of testing is typically character-ized by a large number of outputs and a small number of inputs. The stingerensures that the force from the shaker is solely applied in the longitudinal direc-tion. Typically for this test setup is the interaction between the structure and theshaker, which results in a drop of the signal amplitude of the transmitted force atthe resonant frequencies of the structure. Furthermore, this interaction causes asignificant difference between the resonances in the measured vibration responsesand the FRFs of the structure. Since the force is measured between the stingerand structure, the ratio between the measured response spectra and force spectrastill results in the correct FRF describing the structures dynamical behavior. Onthe contrary, if the spectra of the electrical signals, that are sent to the shakeramplifier are used as the input signals and the vibration responses as the outputsignals, the FRF matrix describes the dynamics of the structure including thestingers, shakers and electrical drive system.

The vibration response is typically measured by accelerometers or by a scan-ning laser Doppler vibrometer. Accelerometers are preferred to measure a three-dimensional structure. Often the number of accelerometers or acquisition channelsis smaller than the number of response locations. In this case the structure is mea-sured in several patches. Since every patch is measured separately, the noise onthe FRFs corresponding with different patches is uncorrelated. An advantage ofFRF-driven identification compared to IO-driven identification, is that the FRFscorresponding to the different patches can easily be processed simultaneously toestimate the modal parameters. Although one must be careful for data inconsis-tencies between the FRFs caused by the mass-loading effect of the accelerometerson the structure [118], [15]. The use of the scanning laser vibrometer avoidsmass loading problems, since this measurement is based on the Doppler effect ofa laser (light) beam and no contact with the structure is required. A scanninglaser vibrometer setup, can be considered as a multi-patch measurement, whereevery patch consists of 1 response sensor. Since all the responses are measuredindependently in time, the noise on the measurements for different responses isuncorrelated. Special modal parameter identification procedures are proposed in[126] to process high resolution scanning laser vibrometer measurements. Scan-ning laser vibrometers are especially interested in the case of panel-like structures,but less suitable to measure three-dimensional geometries.

3.3. FRF estimation for EMA 35

Hammer-based testing

A second class of modal testing is the so-called hammer excitation testing, wherethe structure is excited in different locations by a roving hammer impact, whilethe vibration response is measured in a limited fixed number of reference locations.Different from shaker excitation testing, hammer excitation testing measures onlyvibration responses at a few locations, while the structure is excited separately intime at the different locations of interest. From the reciprocity property of themodal model ( i.e. the transfer function between a force at location 1 and theresponse at location 2 is equal to the transfer function between a force at location2 and the response at location 1), it follows that exciting at Ni locations and mea-suring the vibration response at No locations, results in the transpose of the FRFmatrix obtained by exciting in No locations and measuring in Ni response loca-tions. Therefore, the FRF matrix measured by hammer testing can be consideredas a No ×Ni matrix, where No are the number of excitation locations and Ni thenumber of reference accelerometers. When an impact hammer test is used, thereis no fixed interaction between the structure and the excitation device. Further-more, the noise on the signals for excitation at different locations is uncorrelated,resulting in exact maximum likelihood (ML) estimates for the ML algorithms pro-posed in chapters 4 and 5. Hammer testing is simple in use, does not suffer frommassloading, needs only a limited amount of sensors and acquisition channels andtherefore is still commonly used for modal analysis purpose. Drawbacks are thelimitation of the excitation signals (only impacts or repeated impacts), the factthat exciting at many locations is time-consuming and exciting in all 3 directionsmight be difficult or impossible.

3.3 FRF estimation for EMA

3.3.1 Arbitrary input signals

Arbitrary input signals, such as random noise, are still commonly used for modaltesting because of the their general availability as well as the averaging effect theyhave for possible non-linear effects. Of course the use of arbitrary signals compli-cates the estimation of FRFs due to leakage problems. Therefore window tech-niques, such as a Hanning window, are applied to reduce the influence of leakage.However, the errors introduced by leakage can not be completely eliminated bythe use of a Hanning window, especially for the case of lightly damped structure.In chapter 4 an extension of classical FRF based algorithms is proposed whichmodels the leakage exactly by considering Nb (the number of averaged blocks)extra inputs.

The use of time-limited burst random excitation was introduced for modal


analysis. For this input signal no leakage is introduced since the signals decays tozero within the length of the time window. Similar as for impact hammer testing,an exponential window can be applied in order to amplify this decay by artificiallyincreasing the damping. In [131] it is shown that the effect of the exponential win-dow can be totally compensated on the modal parameters. Disadvantages of burstsignals compared to random noise excitation is that less energy is injected in thestructure within the same time window, resulting in a lower signal-to-noise ratioand in addition a higher crest factor. The use of periodic broadband excitationavoids leakage problems [1]. The FRF estimators discussed in this section startfrom the power spectra, i.e. the Auto Power spectra of the inputs and the outputsand the Cross Power spectra between the inputs and outputs respectively givenby

Syy(ωk) =1

Nb

Nb∑

b=1

Yb,kY Hb,k ∈ C

No×No (3.1)

Sff (ωk) =1

Nb

Nb∑

b=1

Fb,kFHb,k ∈ C

Ni×Ni (3.2)

Sfy(ωk) =1

Nb

Nb∑

b=1

Fb,kY Hb,k ∈ C

Ni×No (3.3)

Syf (ωk) =1

Nb

Nb∑

b=1

Yb,kFHb,k ∈ C

Ni×No (3.4)

= SHfy (3.5)

with Nb the number of blocks by which the time data is divided and the inputFb,k and output spectra Yb,k given by the Discrete Fourier Transform (DFT) incombination with a window wn

Fb,k =1√N

N−1∑

n=0

wnfb,nz−nk (3.6)

Yb,k =1√N

N−1∑

n=0

wnyb,nz−nk (3.7)

with zk = ei2πk/N and fb,n, yb,n the time samples of the input and output signalsfor block b at simple time n∆t (∆t the sample period).

The Hv estimator

Under the assumption that the noise on the measured inputs and outputs is uncor-related and of equal amplitude, the Total Least Squares (TLS) method Hv results


in a consistent estimate of the FRFs. The Hv estimator [64], [52] estimates theFRFs by solving the following eigenvalue problem

[

Sff Sfy

Syf Syy

] [

HHv

−INo

]

=

[

HHv

−INo

]

Λ (3.8)

(the indication for the spectral line k is dropped) In [131] it is shown that the Hv

estimator is not consistent if these noise assumptions are violated. In that case, ageneralized TLS estimator must be used, which uses the noise covariance matrix asa weighting in order to be consistent [131]. However, in practice this is impossiblesince this covariance matrix is not a priori known. Furthermore, it is shown in[131] that the FRF matrix obtained by processing each output separately is onlyequal to the one obtained from considering all outputs simultaneously if HHH isdiagonal. Since there is no reason for this, the MIMO and MISO Hv estimatorgenerally differ.

The H1 estimator

Under the assumption that only noise is present on the outputs (and this noise un-correlated with the input signals), the Hv FRF estimator reduces to the commonly-used H1 FRF estimator given by

H1 = SyfS−1ff (3.9)

where Sff is non-singular if the input forces are not totally correlated and enoughblocks Nb are used to calculate the auto power spectra Sff . Under the specifiednoise assumptions the H1 estimator is consistent (if leakage is neglected). Inthe case no exciter-structure interaction exists e.g. impact hammer excitation,the noise assumptions for the H1 estimator are often realistic, since process noisecaused by unmeasurable forces results as output noise uncorrelated with the inputs.The covariance matrix CH1

of the noise on the estimated FRFs is obtained froma sensitivity analysis given in appendix 3.8.

The H2 estimator

Similar to the H1 estimator, the assumption of only noise on the input signals andnoise-free output signals results in the H2 estimator given by

H2 = SyyS−1fy (3.10)

In order to compute the inversion S−1fy the H2 estimator can only be used for

Ni = No. In case the FRFs are estimated output by output, the H2 estimatoris only applicable in the single input case. In [131] is it shown that both the H1

and H2 estimator belong to the class of maximum likelihood estimators and thecovariance matrix on the estimated FRFs results in the Cramer-Rao lower boundfor their specific noise assumptions.


3.3.2 Periodic input signals

Although the use of periodic input signals (deterministic input signals) avoidsproblems of leakage and results in better signal-to-noise ratios (SNR) [1], [98], [47]they are not always available in signal generators of commercial measurement de-vices and therefore one might have to stick with random input excitation signals.Special design of the deterministic input signals e.g. multisines and Schroeder sig-nals result in optimal SNR and crest factors [43], [37]. Depending on the dampingratios of the structure and the length of the period, the first periods of the mea-sured outputs must be omitted to leave out the transient response. In [106], [107]special designed input signals i.e. odd-multisines, odd-odd multisines and specialodd-multisines are proposed to characterize the possible non-linear behavior of thestructure [140].

In the case a synchronized multi-excitation measurement setup is used i.e. theexternal generator is triggered with the data-acquisition system, the so-called non-parametric errors-in-variables estimator [38] Hev, based on cyclic averaging, is pro-posed. Consider Ni linear independent inputs stacked as F = [F (1), . . . , F (Ni)] ∈C

Ni×Ni and the resulting responses Y = [Y (1), . . . , Y (Ni)] ∈ CNi×Ni , for a Nb

number of periods. In practice, these signal assumptions can easily be realized byapplying Ni times, at Ni input locations periodic broadband excitation sequences,e.g. multisines, by measuring Nb periods. The Hev FRF estimator is than givenby

Hev =

(

Nb∑

b=1

Xb

)(

Nb∑

b=1

Fb

)−1

(3.11)

where b indicates the Nb different periods. Applying the Hev output by outputresults in the same FRF estimates as considering all outputs simultaneously. Itcan easily be proven that this FRF estimator is both asymptotically unbiased andconsistent in presence of both input and output noise. In [131] an expression isgiven for the covariance matrix of the noise on the estimated FRFs. In the caseof asynchronous periodic measurements several other FRF estimators based onnon-linear averaging techniques in an error-in-variables framework are proposedin [39].

3.3.3 Decreasing the noise levels on FRFs

In many practical cases the engineer responsible for the modal parameter estima-tion only has the FRFs and in the best case also their covariances to start from.The measurements itself are often carried out by the test engineers and they havealready done the data reduction step by estimating the FRFs. In the case thathigh frequency resolution measurements are given with large noise levels, the noise


levels can be reduced by a rectangular window, but the price paid is the lower fre-quency resolution. Nevertheless, for several practical reasons, one is often willingto pay this price:

• Since modal testing is characterized by a large data sets typically 100-1000FRFs and large model orders the processing speed of the parametric identi-fication is still an important issue in practice. A reduced data set certainlyresults in a gain of process time for the parametric identification.

• A modal model is often validated by comparing the synthesized FRFs withmeasured FRFs. Therefore, reducing the noise levels results in an easiervisual validation.

• Many well-known modal parameter estimators are not consistent and effi-cient, e.g Least Squares Complex Exponential (LSCE), Least Squares Com-plex Frequency (LSCF), PolyMAX and Eigenvalue Realization Algorithm(ERA). Therefore, reducing the noise levels on the primary data is highlydesired in order to reduce the bias errors on the estimated parameters. Moreadvanced maximum likelihood identification algorithms use the covariancematrix of the noise on the FRFs as a weighting in their cost function. Sincethese algorithms require starting values, better convergence properties areobtained if the initial values are estimated from FRFs with reduced noiselevels.

The reduced FRFs

Consider the given FRF data H(ωk) and the covariance matrix CH(ωk) with thediscrete-time domain model given by

H(ωk) =

Nm∑

r=1

(

φrLTr

1 − λrz−1k

+φ∗

rLH[:,r]

1 − λ∗rz

−1k

)

(3.12)

notice that the discrete modal model has discrete poles λr and discrete participa-tion factors Lr which are related to their continuous-time equivalents. The InverseDiscrete Fourier Transform (IDFT) of the FRF matrix is the Impulse ResponseFunction matrix (IRF), for time-lag n is given by

hn =N∑

k=1

H(ωk)znk (3.13)

=

Nm∑

r=1

(

φrLTr

λnr

(1 − λNr )

+ φ∗rL

Hr

λ∗n

r

(1 − λ∗Nr )

)

(3.14)

the proof is given in the appendix (N the number of spectral lines). This meansthat the IRF can be considered as a sum of exponentially damped sines. The term


(1 − λNr ) is introduced by so-called time-domain aliasing. Under the assumption

that H(ωk) is obtained from periodic excitation and the use of the Hev estimator,the FRF will be leakage free. In this case, identification methods formulated inthe time-domain (by starting from IRF, obtained by the IDFT) need to correctthe identified mode shapes or participation factors with this term 1−λN

r to avoidthe influence of time-domain aliasing.Depending on the damping ratio and the frequency resolution of the FRFs, theIRF contains low signal levels and high noise levels in its tail. Therefore, the DFTtaken from the first Nw samples of the IRF results in

Hr(ωk) =

Nw−1∑

n=0

hnz−nk (3.15)

=

Nm∑

r=1

(

φrLTr (1 − λNw

r )

(1 − λrz−1k )(1 − λN

r )+

φ∗rL

Hr (1 − λ∗Nw

r )

(1 − λ∗rz

−1k )(1 − λ∗N

r )

)

(3.16)

=

Nm∑

r=1

(

φrL′Tr

1 − λrz−1k

+φ∗

rL′Hr

1 − λ∗rz

−1k

)

(3.17)

with L′r = Lr

(1−λNwr )

(1−λNr )

. The proof follows directly from

N−1∑

n=0

xn =1 − xN

1 − x(3.18)

The FRFs given by Hr(ωk) are considered as the reduced FRFs. The poles λr

of Hr(ωk) are not effected by considering only the first Nw samples of the IRF.On the contrary the participation vectors Lr are multiplied by a correction factor,

which depends on the specific pole λr. Notice that this correction term(1−λNw

r )(1−λN

r )is

equal to 1 for Nw = N or for both N and Nw → ∞. In fact, the correction termis caused by both time-domain aliasing and frequency-domain leakage. This dataand noise reduction process is visually presented by figure 3.2.

Another approach to reduce the noise levels and leakage for the estimationof FRFs is based on exponential windows [139]. This method weights the IRFwith an exponential decaying window to reduce the noise levels. The exponentialwindow artificially increases the damping in such a way that 1− λNw

r ≈ 1. In thisway the modal participation factors should not be corrected, but the poles shouldbe instead.

Comparing the exponential and rectangular window it must be noticed thatthe use of the exponential window requires a correction of the damping values,while the rectangular window requires a correction of the participation factors.


FRF

FRF

1

( ) ...1

mNr r

kr r k

LH

z

φωλ=

= +−

1

(1 )

(( ) .

1.

1 ).

wm NNr r

kr r k

rNr

LH

z

φωλ

λλ=

=−

+−

−∑

IRF

Rectangular window

X

Dat

a an

dno

ise

redu

ctio

n

IDFT

DFT

Figure 3.2: Procedure to reduce the data and noise levels.

The covariances of the reduced FRFs

Consider the time window

wn = 1 for 0 ≤ n ≤ Nw − 1

wn = 0 for Nw ≤ n ≤ N (3.19)

and h′

n defined by h′

n = hnwn. The DFT of h′n is given by

H′(ωk) =

N−1∑

n=0

h′

nz−nk (3.20)

with zk = ei2πk/N , while the reduced FRF Hr(ωk) is given by

Hr(ωk) =

Nw−1∑

n=0

hnznk (3.21)

with zk = ei2πk/Nw . For m = N/Nw ∈ N0 it follows that

Hr(ωk) = H′(ωmk) (3.22)


Since multiplying hn with wn corresponds with convoluting H(ωk) with W (ωk) =∑N−1

n=0 wnznk , the covariance on H

′is given by

CH′ (ωk) = |W (ωk)|2 ∗ CH(ωk) (3.23)

with ∗ the convolution symbol. The validity of this expression can easily beproved since the convolution is equivalent with a sum in the frequency-domainand cov(

∑

i aiYi) =∑

i |ai|2cov(Yi). As a result the covariance of the reducedFRFs Hr(ωk) is given by

CHr (ωk) = CH′ (mk) (3.24)

3.4 Estimation of the power spectra for OMA

Section 3.2 discussed different FRF estimators for the EMA case. In this para-graph, the estimation of the power spectra between the outputs and the referenceoutputs is discussed. The auto and cross power spectra Syyr

are the primary datafor model parameter identification based on Eq. 2.62. Similar as for the EMA case,this ABS-function technique is in the case of long-in-time available data sequencespreferred to the data driven approach.

Basically, two classical approaches exist for the estimation of auto and crosspower spectra. The periodogram [102] estimator operates directly on the spectraof different time blocks resulting from a division of the time sequences. The cor-relogram [9] approach first estimates the correlation functions in the time-domainand next the power spectra are obtained by transferring the correlations to thefrequency-domain.

A procedure is proposed to eliminate the unstable poles from the power spectraby considering so-called ’positive’ power spectra. In this way a stable model canbe fitted in the frequency domain based on ABS-functions for OMA applications.

3.4.1 The correlogram approach

The correlogram approach starts by estimating the correlation functions. In thenext step, the correlation functions are transferred to the frequency-domain bytaking the DFT to obtain the power spectra. The unbiased discrete-time correla-tion estimate between the response signals yn ∈ R

No×1 for (n = 0, . . . , N −1) and

3.4. Estimation of the power spectra for OMA 43

the reference output signals yrefn ∈ R

Nref×1 is given by

rn = 1N−n

N−n−1∑

k=0

yk+nyref Tk for 0 ≤ n ≤ N − 1

rn = 1N−|n|

N−1∑

k=|n|yk+|n|y

ref Tn for − (N − 1) ≤ n < 0

(3.25)

The biased correlation estimate uses 1/N rather than 1/(N − n). The powerspectra Syy(ωk) are given by Fourier transforming the correlation functions

Syy(ωk) =

N∑

n=−N

wnrnz−nk (3.26)

with zk = ei2πk/N . To reduce the effect of leakage the use of an adequate (2N +1)-point time window wn (e.g. Hanning, Hamming, ...) symmetric around theorigin is advised. This window reduces the effect of leakage and thus the biaserror in the power spectra. For instance, applying a Hanning window to thecorrelation functions will force the correlation to zero at the higher lags. Moreover,the application of such a window reduces the stochastic uncertainty on the crosspower estimate due to the presence of a higher stochastic uncertainty near thehigher lags of the correlation function estimate.

Nevertheless, the application of a time window introduces bias errors on thefinal modal parameters [24]. In the case an exponential window is used, this bias onthe estimated parameters can be corrected which is not the case for other windowse.g. Hanning, Hamming. The introduction of additional artificial damping by thedouble sided exponential window given by

wn = e−β|n|∆t with − N ≤ n ≤ N (3.27)

reduces both the influence of leakage and the influence of the stochastic uncertain-ties in tails. The factor β is typically chosen such that the amplitude at time lagN of the window is 1% of its initial amplitude [52]. The poles are finally correctedby removing the artificially added damping β.

Another approach to estimate the correlation functions is based on dividingthe time sequences of the outputs in Nb equally sized adjacent blocks. In a nextstep the correlation function can estimated for each block and finally averagedover the different blocks. This has the advantage that the covariances of the noiselevels on the correlations can be obtained in a straightforward way.

In [7] it is proven that a fast calculation of the linear correlation can be obtainedby the use of the discrete Fourier transform by a technique called zero-padding.Each of the Nb adjacent data blocks, each N output samples, is extended by an


additional block of N zeros. In other words, new blocks qb,n of length 2N arecreated such that

qb,n = yb,n for 0 ≤ n ≤ N − 1 (3.28)

qb,n = 0 for N − 1 < n ≤ 2N − 1 (3.29)

Consider the DFT without applying a window

Qb(ωk) =

2N−1∑

n=0

qb,nz−nk (3.30)

and calculating the corresponding power spectra by

Srefqq (ωk) =

1

Nb

Nb∑

b=1

Qb(ωk)Qref,Hb (ωk) ∈ C

No×Nref (3.31)

then it is shown in [7] that the IDFT of this power spectra followed by a correction,given by Eq. 3.33, results in the linear correlation function rn:

rsn =

1

N

2N−1∑

k=0

Srefqq (ωk)zn

k (3.32)

rn =N

N − nrsn (3.33)

In fact this correlation rn is exactly the same correlation function as the one byaveraging the correlation functions from the Nb adjacent blocks, but the use of theDFT and IFT functions speeds up the calculation time. This procedure is shownin figure 3.8.

3.4.2 The periodogram approach

A second widely-used method is the so-called periodogram power density estima-tor, also known as the Welch estimator [102], [143]. The basic idea of the peri-odogram estimator is to divide the response signals in Nb blocks of equal length.Next, the Fourier spectra, for each block weighted with a time window wn, arecomputed as

Yb(ωk) =

N−1∑

n=0

wnyb,nz−nk (3.34)

with N the number of time samples within a block. The time window wn (e.g.Hanning, Hamming) typically reduces the influence of leakage. Since the windowreduces the contribution of the data at the begin and end of the record, introducing

3.4. Estimation of the power spectra for OMA 45

an overlap between the adjacent blocks in order to obtain a better contribution ofeach raw time data sample is advisable. The power density spectra are given by

Srefyy (ωk) =

1

Nb

Nb∑

b=1

Yb,kY ref Hb,k ∈ C

No×Nref (3.35)

which is a similar expression as for the power spectra used by the Hv, H1 andH2 FRF estimators. The choice of the number of samples N within a block andthus the number of blocks Nb is a trade-off between variance and bias on the esti-mated power spectra. Choosing a higher amount of data samples N and a loweramount of blocks Nb reduces the effect of leakage but increases the stochasticuncertainty i.e. the variance on the estimated power spectra. The periodogrammethod is a well-established technique, which is available as a tool in most com-mercial software packages. In [7] it shown that the inverse Fourier transformationof the periodogram estimates of the power functions results in the so-called circularcorrelation.

Variances on the Auto and Cross Power densities

Similar as for the FRFs, the variances on the estimated auto and cross powerdensities can be used as a weighting in the parametric identification process inorder to improve the consistency and efficiency properties of the algorithms. In [7]the following formulas for the variances on the estimated power spectra are given(for each reference considered separately)

var(Syref yref) =

2

NbS2

yref yref(3.36)

cov(Syoyref) ≤ 2

NbSyoyo

Syref yref(3.37)

Of course the variance can also be calculated from the sample mean variance i.e.

cov(Syoyref) =

1

Nb − 1

Nb∑

b=1

|Yo,bYref ∗b − Syoyref

|2 (3.38)

When the stochastic process is not stationary, a technique to process operationaldata is proposed in [83].

3.4.3 The ’positive’ power spectra approach

For the identification of modal parameters from output-only measurements severalfrequency-domain identification methods start from the power spectra inspired byexpression Eq. 2.60 [50]. Nevertheless this technique has several disadvantages:


• The power spectra have a 4-quadrant symmetry i.e. the OMA modal modelcontains λ, λ∗, −λ and −λ∗ as poles. This results in a model order, which istwice the modal order needed to model FRFs. A higher model order resultsfor all identification methods in an increasing calculation time and in a lessgood numerical conditioning.

• The power spectra contain both stable λ, λ∗ and unstable poles −λ, −λ∗poles in its model. This results in less interesting properties for the interpre-tation of stabilization diagrams, when distinguishing physical from mathe-matical poles. A more detailed discussion is given in chapter 9.

• Power spectra estimated from a limited amount of data are typically char-acterized by high noise levels compared to FRFs. Therefore, an additionalnoise reduction would be preferable.

• When using the periodogram approach to estimate power spectra a trade-off must be made between the stochastic uncertainties and the bias errorsintroduced by leakage.

Some of these disadvantages can be overcome be using time-domain identifica-tion algorithms starting from the first Nw positive lags of the correlation functionrn given by

rn =

Nm∑

r=1

φrKTr λn

r + φ∗rK

Hr λn

r for n ≥ 0 (3.39)

Where rn only contains the stable poles. However, the focus of this thesis lies onfrequency-domain identification for modal analysis and this for reasons like of e.g.simple frequency band selection and the use of frequency weighting functions inthe identification procedure.

Since Eq. 3.39 is analog to the expression for the IRF given by Eq. 3.14, asimilar approach as for the reduction of the noise levels on the FRFs (paragraph3.3.3), can be applied by taking the Fourier transform of the first Nw samples.The ’positive’ power spectra S+

yy are defined by the DFT of the first Nw samplesof the positive lags of the correlation function:

S+yy(ωk) =

Nw−1∑

n=0

rnz−nk (3.40)

=

Nm∑

r=1

(

φrKTr (1 − λNw

r )

1 − λrz−1k

+φ∗

rKHr (1 − (λ∗

r)Nw)

1 − λ∗rz

−1k

)

(3.41)

=

Nm∑

r=1

(

φrK′Tr

1 − λrz−1k

+φ∗

rK′Hr

1 − λ∗rz

−1k

)

(3.42)

3.5. Combined FRF and XP estimation for OMAX 47

with K ′r = Kr(1 − λNw

r ) and zk = ei2πk/Nw . Since Kr has no direct physicalinterpretation and the mode shapes φr and poles λr are not affected, no correctionis needed. Similar as for the FRF Hr(ωk), the variance is now given by

var(S+yy(ωk)) = var(S+′

yy (ωmk)) (3.43)

with

var(S+′

yy (ωk)) = |W (ωk)|2 ∗ var(Syy(ωk)) (3.44)

for W (ωk) defined as the Fourier spectra of the time window given by Eq. 3.19and m = N/Nw. Similar as for the IFT, the side lobe of the positive samplescontains most of the stochastic uncertainty. Since only the first Nw positive lags areconsidered, the stochastic uncertainty on the positive power spectra will decrease.Finally the modal parameters can be estimated starting from these positive powerspectra and their variances as a frequency domain weighting.

The estimation of the positive power spectra S+yy can be summarized as follows

and is visually given by figure 3.8:

1. Divide the time records in Nb adjacent blocks of N data samples. Extendeach block with an additional N zeros.

2. Based on these extended blocks, calculate the power spectra with the pe-riodogram method. The inverse Fourier transform of these power spectraresults in the linear correlation functions.

3. The positive power spectra S+yy and its variance is calculated by considering

only the first Nw positive lags of the correlation function.

3.5 Combined FRF and XP estimation for OMAX

An Operational Modal Analysis with eXogeneous (OMAX) inputs considers thevibration response as a combination of a deterministic contribution caused by themeasurable forces and a stochastic contribution caused by unmeasurable forces.As a result both the deterministic and stochastic contribution contain informationabout the system.

In the classical EMA framework the stochastic contribution of the unmeasur-able forces is considered as measurement noise. In fact no information from thestochastic contribution in the vibration response is taken into account to estimatethe modal parameters. This is in contradiction with the approach followed inthe OMA framework, where all modal parameters are estimated from the purelystochastic vibration responses. Therefore, in this paragraph both the contributionof the deterministic part (i.e. the FRFs) and the stochastic part (i.e. the power


densities) are taken into account resulting in an optimal data exploitation i.e. bothmodes excited by the measurable and unmeasurable forces can be identified.

Since an interaction between the structure and excitation device results in acorrelation between the measurable forces and unmeasurable forces a distinction ismade between experiments without structure-exciter interaction and experimentswith structure-exciter interaction.

For a set of measurable input forces F (ωk) ∈ CNm

i ×1 (Nmi number of mea-

surable input forces) and a set of unmeasurable ambient random white forces

E(ωk) ∈ CNu

i ×1 (Nui number of unmeasurable input forces) the vibration response

spectra Y (ωk) ∈ CNo×1 are given by

Y (ωk) = H(ωk)

[

F (ωk)E(ωk)

]

(3.45)

= Hm(ωk)F (ωk) + Hu(ωk)E(ωk) (3.46)

= Y d(ωk) + Y s(ωk) (3.47)

with Hm(ωk) ∈ CNo×Nm

i the FRF matrix between the measurable inputs F (ωk)

and the outputs and Hu(ωk) ∈ CNo×Nu

i the FRF matrix between the unmeasur-able forces E(ωk) and the outputs. The deterministic and stochastic contributionof the vibration response is respectively noted as Y d(ωk) and Y s(ωk). The mea-surable forces F (ωk) are given by

F (ωk) = G(ωk)U(ωk) + T (ωk)E(ωk) (3.48)

where U(ωk) are the spectra of the electrical signals, that are sent to the excitation

devices, G(ωk) ∈ CNm

i ×Nmi the transfer functions characterizing the excitation

devices and the interaction with the structure. The transfer path between theunmeasurable forces and the forces measured at the exciter-structure connectionis given by T (ωk) ∈ C

Nmi ×Nu

i . The multivariable frequency-domain OMAX modelis illustrated in figure 3.3.

3.5.1 No structure-exciter interaction

Impact hammer and acoustic excitation are all examples were no fixed connectionexists between the structure and excitation device and thus the transfer pathT (ωk) is equal to zero T (ωk) = 0. For some other applications, the influenceof the exciter is negligible and thus no interaction should be taken into accounte.g. exciter on a large bridge. For these types of test setups, the measured forcesF (ωk) are uncorrelated with the unmeasurable forces E(ωk) and thus the H1

estimate is consistent under the assumption that the measurable forces contain nomeasurement noise. The FRFs and the covariances of the noise on the FRFs are

3.5. Combined FRF and XP estimation for OMAX 49

G Hm

Hu

U -

E -

F

- l+ -

6

T

6

?l+ -

6

Y

Figure 3.3: Frequency-domain OMAX model with F the measurable input forces, E

the unmeasurable random forces and Y the vibration responses.

then given by

Hm(ωk) = Syf (ωk)S−1ff (ωk) (3.49)

CHm(ωk) =1

NbS−T

ff (ωk) ⊗ CY (ωk) (3.50)

The power densities of the stochastic contribution Y s(ωk) = Hu(ωk)E(ωk) be-tween the outputs and the reference outputs Y ref,s are given by

Sysyref,s(ωk) = Syyref (ωk) − HmSfyref (3.51)

Interesting to notice is that Sysyref,s(ωk) = CY (ωk)[:,ref ] (i.e. the columns corre-sponding to the reference outputs) with CY , defined by Eq. 3.65, the covariancematrix of the stochastic contribution in the output signals. In an EMA approachthis covariance matrix CY is considered only in the expression for the covariancematrix of the noise on the estimated FRFs. In practice, this means that the largerthe stochastic contribution, the larger the uncertainties on the deterministic con-tribution (i.e. on the FRFs) will be. In contrast with the EMA approach, theOMAX approach considers the variance on the responses CY as the power spectraof the contribution of the ambient unmeasurable forces and these now serve asprimary data for the modal parameter identification.


Since the power spectra given by Eq. 2.61 have a 4-quadrant symmetry, whilethe FRFs given by Eq. 2.4 have a 2-quadrant symmetry, identification based onboth spectral functions simultaneously would result in a highly non-linear problem.In order to remove the unstable poles from the power spectra and to reduce thenoise levels, only the first Nw samples of the inverse Fourier transform of Hm(ωk)and Sysyr,s(ωk) are used, resulting in Eq. 3.17 and Eq. 3.42 given by

[

Hm(ωk) S+yy(ωk)

]

=

Nm∑

r=1

(

φrQTr

1 − λrz−1k

+φ∗

rQHr

1 − λ∗rz

−1k

)

(3.52)

with Qr =[

L′r K ′

r

]

. Notice that Eq. 3.52 has exactly the same formulationas the modal model and thus all classical EMA identification algorithms basedon state-space models, common-denominator models and matrix fraction modelscan be used in the OMAX framework, when starting from both FRFs and the’positive’ power spectra of the stochastic contribution. It should be remarkedthat the FRFs in the EMA, the positive power spectra in the OMA and bothsimultaneously FRFs and positive power spectra in the OMAX case are describedby the same model structure and only the participation vectors in the models differin terms of the physical interpretation.

3.5.2 Structure-Exciter interaction

In the case that a fixed connection exist between exciter and structure e.g. shaker-structure connection by a stinger, the transmission path T (ωk) differs from zero.The auto power Sff (ωk) and cross power spectra Syf (ωk) are then given by

Sff = GSuuGH + TSeeTH (3.53)

Syf =(

HmGSuuGH + (HmT + Hu) SeeTH)

(3.54)

under the assumption that U(ωk) and E(ωk) are uncorrelated (the notation for thespectral lines ωk is dropped for simplicity of the expressions). For T 6= 0 the H1

estimator is biased and not consistent. The H1 FRF estimator fails because theexciter-structure connection introduces a correlation between F (ωk) and E(ωk).A consistent estimate for the FRF matrix Hm(ωk) is obtained by measuring addi-tionally the electrical generator signals U(ωk) as additional signals or by the useof periodic electrical generator signals.

An Instrumental Variable approach

Instead of starting from the auto and cross power spectra Sff (ωk) and Syf (ωk)lets consider the cross powers between the force spectra, response spectra andthe electrical generater spectra given by (under the assumption that the electrical

3.6. Simulations and measurement examples 51

generator has a linear behaviour)

Sfu = GSuu (3.55)

Syu = HmGSuu (3.56)

since U(ωk) is uncorrelated with the unmeasurable forces E(ωk). In fact, thiselectrical generator signal U is used as a instrumental variable [139]. An unbiased,consistent estimate of the FRFs Hm(ωk) is then obtained by

Hm(ωk) = Syu(ωk)Sfu(ωk)−1 (3.57)

which is known as the instrumental variables FRF estimator. Analogous as for theH1 FRF estimator the covariance matrix of the noise on the FRF estimates canbe calculated by a sensitivity analysis [131]. The power spectra of the stochasticcontribution can be found by

Sysyref,s =1

Nb

Nb∑

b=1

Y sb Y ref,s

b (3.58)

substituting Y sb = Yb − HmFb results in

Sysyref,s = Syyref − Hm,refSfyref − SyfHm,ref H − HmSffHm,ref H (3.59)

with Hm,ref ∈ CNref×Ni the part of the FRF matrix Hm containing the FRFs

between the reference outputs and the measurable inputs. To eliminate the un-stable poles from Sysyref,s the positive power spectra S+

ysyref,s are calculated andthe same model as Eq. 3.52 can be identified.

Periodic input signals

In the case that periodic excitation signals are used, the instrumental variablesapproach reduces to the Hev FRF estimator discussed in paragraph 3.3.2. ThusFRFs between the outputs and the measurable inputs are given by Eq. 3.60, i.e.

Hmev =

(

Nb∑

b=1

Xb

)(

Nb∑

b=1

Fb

)−1

(3.60)

The use of periodic signals avoids the need for the electrical generator signalsto obtain consistent estimates for H1. Similar as for the instrumental variableapproach the power spectra of the stochastic contribution are obtained by Eq.3.59. This approach has the advantage that no extra signal must be measured, noerrors are introduced by leakage and no linear behaviour of the electrical generatormust be assumed.


Table 3.1: Mass, damping and stiffness characteristics of the 7 DOF system and theexciter

exciter structure

me = 15 m1 = m2 = m5 = m6 = m7 =10ke = 3000 m4 = 26, m3 = 9ce = 75 k1 = . . . = k8 = 150000

c1 = . . . = c8 = 10

3.6 Simulations and measurement examples

In order to validate the results presented in this chapter a 7 DOF system, shown infigure 3.4, is used in order to simulate several types of modal analysis experiments.All three cases i.e. an EMA, OMA and OMAX are investigated by means ofsimulations. In some simulations the system is excited by an electrodynamic shakerwith its body grounded. The exciter coil has a mass me and is supported on aflexure of stiffness ke and damping ce. The exciter is fixed to the system by arigid stinger. The system itself consists of 7 masses connected to each other bysprings and dampers. The system and exciter characteristics are given in table3.1. The first and last masses are connected to the ground. This model of anelectrodynamic shaker is considered as a good approximation of a real shakerused in a modal analysis experiment. The electrodynamic shaker results in aninteraction between the structure and the exciter leading to force signal drop at theresonance frequencies. Obviously, the larger the shaker compared to the structure,the larger the interaction and influence of the exciter on the measurement setup.Given the M , C and K matrices, respectively the mass, damping and stiffnessmatrices both the vibration response caused by a force f(t) and the exact modalparameters of the structure can be calculated. The vibration response is obtainedby solving Newton’s differential equations in the time-domain, which also allowsto include the effects of initial and final conditions. This makes it possible toinvestigate the effect of leakage on the ABS function estimators. The goal of thesimulations is to shown the effectiveness of a rectangular window to reduce thenoise levels on the FRFs and to eliminate the unstable system poles from theauto and cross power density functions. Finally, the advantage of a combinedoperational-experimental modal analysis i.e OMAX-analysis is illustrated.


ms1ms2 ms7

me

…

y1 y1 y7

f1f2 f7

fe

Figure 3.4: Simulated 7 DOF system excited by an electrodynamical shaker

3.6.1 Experimental Modal Analysis

The system is excited at the 5-th mass with a white gaussian noise (N(0,1)) signalduring 128s with a sample frequency of 1024 Hz. A first simulation was done tostudy the effect of leakage on the estimation of the FRFs and therefore no noise wasadded to the force and vibration response signals. All 7 modes of the structure areexcited and the modes are within the frequency band 5-40Hz. To illustrate the useof a rectangular time window to reduce the noise levels of the FRFs, a comparisonis made between the classical H1 FRF estimator with a Hanning window and theH1 estimator with Hanning window combined with a noise reduction procedure asexplained in paragraph 3.3.3. The FRF and its standard deviation estimated from2k blocks is compared to the FRF and its standard deviation from 2 blocks followedby rectangular time window with m = N/Nw for m = 1, 4, 16, 32, resulting in arespectively frequency resolution of 0.0156, 0.0625, 0.25 and 0.50Hz. Figure 3.5clearly shows the initial variance on the FRFs for Nb = 1 due to leakage errors(even with the use of a Hanning window on the relative long records of 64s) andsmall simulation errors. Increasing the number of blocks Nb averaged for the classicH1 estimator results in low SNR ratios in the resonances and biased damping ratioscaused by the use of the Hanning window, which is necessary to reduce the leakage.On the other hand, the approach where only 2 blocks are averaged followed by anoise reduction by the use of a rectangular window clearly results in a increaseof the SNR in the resonance frequencies. Since in modal analysis often leastsquares estimators are used for the reason of their process speed if they are properimplemented, the quality of the FRFs in the resonances is of prior importance.Comparing the FRF estimated for Nb = 64 and Nb = 2,m = 32 with the initialFRFs for Nb = 2 it is clear that the modal parameter estimators starting fromthe FRF H1 estimate for Nb = 64 results in overestimated damping ratios anda tendency to identify a double pole around 6.5Hz. Finally, it should be notedthat although the SNR ratios on the estimated FRFs by the use of a rectangularwindow are not always that high, the FRFs are very smooth. This is explained


by the high correlation of the noise on the FRFs over the different spectral lines.This can be observed in figure 3.7 showing the FRFs, their mean and standarddeviation from 10 independent simulation runs.

The primary goal of an FRF estimation is to reduce the data set for bothmemory requirements and process time of the parametric identification and toincrease the SNR. A second set of simulations were done with 30% relative noiseon the FRFs and the results are illustrated by figure 3.6. Increasing the numberof blocks Nb results in a better SNR for the classic H1 estimates, but comparedto the use of a rectangular window the SNR is still low in the natural frequenciesand the damping values seem to be biased.

From a comparison between the exact FRF and the estimated FRF it is difficultto make statements about the quality of both procedures, since the rectangularwindow introduces an error on the participation factors, which does not harmthe quality of the estimates, since it can be compensated. Therefore a MonteCarlo simulation of 30 runs was done with 10% relative (in the frequency do-main) colored noise added on the responses and the quality of the estimates of thenatural frequencies and damping ratios from the combined deterministic-stochasticfrequency-domain subspace algorithm (proposed in chapter 8) is compared to eval-uate the quality of the primary FRF data. Tables 3.2 and 3.3 respectively comparethe mean estimated natural frequencies and damping ratios with their standarddeviation. Both the natural frequency and damping ratio estimates are unbiasedfor the rectangular window approach, while the classic H1 approach results inlargely biased damping ratios and a much higher variance on both the estimatednatural frequencies and damping ratios. Especially the damping ratios of the lowfrequency modes suffer from the bias introduced by the leakage and the use of theHanning window. From this, it is clear that the use of a rectangular window to re-duce the noise and data sets outperforms the classically-used of the H1 estimator.Finally, it should be noticed that the noise and data reduction by the rectangularwindow can be used in combination with any FRF estimator.

3.6.2 Operational Modal Analysis

Traditional frequency-domain estimators for an OMA experiment start from autoand cross power spectra. These power spectra are obtained by averaging overdifferent blocks and contain a 4-quadrant symmetry in the structures poles e.g.both the stable and unstable poles appear in Eq. 2.63. This classic approach hasthe disadvantage that it requires a double order. Furthermore, the periodogramapproach suffers from a bias introduced by leakage similar as for the H1 estimatorin the EMA case. During a simulation of 512s the structure was excited by twouncorrelated gaussian distributed white noise sequences. The first force (normallydistributed noise N(0,1)) was directly applied to the 4th mass, while the secondforce (N(0,0.4)) was applied directly to the 5th mass. Figure 3.8 gives a schematic


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Nb=2

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pl (

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Nb=2, m=1

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Nb=8

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Nb=2, m=4

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pl (

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Nb=32

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pl (

dB)

Nb=2, m=16

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Freq (Hz)

Am

pl (

dB)

Nb=64

0 10 20 30 40−180

−160

−140

−120

−100

−80

−60

Freq (Hz)

Am

pl (

dB)

Nb=2, m=32

Figure 3.5: No noise added, left: classic H1 FRF estimate from Nb blocks; right: H1

estimate from 2 blocks, followed by a rectangular window (full line: FRF, dotted line:standard deviation)


0 10 20 30 40−180

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pl (

dB)

Nb=2

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pl (

dB)

Nb=2, m=1

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pl (

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Nb=8

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pl (

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Nb=2, m=4

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Am

pl (

dB)

Nb=32

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Freq (Hz)

Am

pl (

dB)

Nb=2, m=16

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Freq (Hz)

Am

pl (

dB)

Nb=64

0 10 20 30 40−180

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−120

−100

−80

−60

Freq (Hz)

Am

pl (

dB)

Nb=2, m=32

Figure 3.6: 30% relative noise, left: classic H1 FRF estimate from Nb blocks; right: H1



0 20 40−140

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Freq (Hz)A

mpl

(dB

)

Nb=4, m=4

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−110

−100

−90

−80

−70

−60

Freq (Hz)

Am

pl (

dB)

Nb=4, m=4

Figure 3.7: left: FRFs estimated for 10 independent runs (no noise added); right: meanestimated FRF (full line) and the standard deviation (dotted line)

Table 3.2: Monte Carlo Simulation (30 runs). Mean value and standard deviation ofthe estimated natural frequencies from primary FRF data obtained by the classic H1

technique (Nb = 16) and from the H1 followed by a rectangular window (RW) (Nb =4, m = 4).

fexact (Hz) fh1(Hz) σfh1

(Hz) frw (Hz) σfrw(Hz)

6.411 6.458 0.134 6.411 0.00115.012 14.921 0.011 15.011 0.00118.672 18.656 0.002 18.673 0.00127.858 27.846 0.002 27.860 0.00129.593 29.585 0.003 29.595 0.00136.141 36.002 0.038 36.147 0.00436.913 36.936 0.101 36.929 0.005

overview for the estimation of the ’positive’ power spectra. Figure 3.9 comparesthe power spectra estimated by the periodogram technique with the estimation ofthe ’positive’ power spectra. Comparing the resonance peaks, it is clear that theperiodogram approach suffers from leakage, which will result in an overestimationof the damping ratios.


10 20 30 40 50 60

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time

10 20 30 40-20

-10

0

10

20

30

40

50

freq

10 20 30 40 50 60

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time

10 20 30 40-20

-10

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50

freq

10 20 30 40 50 60

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time

10 20 30 40-20

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freq

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freq

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-1000

0

1000

2000

0 10 20 30 4040

50

60

70

80

90

100

freq

Ampl (dB)

…

…

Zero

padding

Block 1 Block 2 Block Nb

Response

spectra

DFT DFT DFT

Power spectra

4-quadrant symmetry

Positive and negative

correlation

Window (first Nw

samples)

‘Positive’ power spectra

2-quadrant symmetry

DFT

IDFT

rectangular window

Figure 3.8: Procedure to determine the ’positive’ spectra


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pl (

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Nb=64

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pl (

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Nb=16, m=4

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pl (

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Nb=128

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pl (

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Nb=16, m=8

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Nb=256

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pl (

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Nb=16, m=16

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pl (

dB)

Nb=512

0 10 20 30 4020

40

60

80

100

Freq (Hz)

Am

pl (

dB)

Nb=16, m=32

Figure 3.9: Comparison between the power spectra (left) and the ’positive’ powerspectra (right) (full line) and their standard deviation (dotted line).


Table 3.3: Monte Carlo Simulation (30 runs). Mean value and standard deviation of theestimated damping ratios from primary FRF data obtained by the classic H1 technique(Nb = 16) and from the H1 followed by a rectangular window (RW) (Nb = 4, m = 4).

dexact (%) dh1(%) σdh1

(%) drw (%) σdrw(%)

0.201 1.214 0.154 0.234 0.0070.472 0.924 0.054 0.467 0.0070.587 0.890 0.009 0.593 0.0060.875 1.041 0.014 0.877 0.0060.930 1.055 0.012 0.934 0.0061.135 1.325 0.096 1.139 0.0121.160 0.986 0.057 1.155 0.013

3.6.3 Combined Operational-Experimental Modal Analysis

Consider the same simulation as for the output-only case, but with a measurableforce f4(t), while the force f5(t) is unmeasurable. For this case no interactionbetween the structure and exciter is assumed i.e. f4(t) and f5(t) are uncorrelated.The unmeasurable force introduces so-called process noise on the FRFs. A MonteCarlo simulation was done for 30 runs. Tables 3.4 and 3.5 compare the mean andstandard deviation of the natural frequencies and damping ratios estimated by thecombined frequency-domain subspace algorithm for several cases of primary data:

• FRFs estimated by the classic H1 estimator with a Hanning window andNb = 128.

• FRFs estimated by the H1 estimator with Nb = 4 and m = 32.

• both FRFs and positive power spectra of the stochastic contribution withNb = 4 and m = 32.

• FRFs estimated by the H1 estimator with Nb = 8 and m = 16.

• both FRFs and positive power spectra of the stochastic contribution withNb = 8 and m = 16.

It is clear that the classic H1 approach results in inaccurate and biased estimates.Furthermore, by using the combined approach, where the parametric identifica-tion starts from both the FRFs and the positive power spectra of the stochasticcontribution, the accuracy of the estimation of the 2 and 4th mode is improvedby a factor 2 to 3. This is explained by the fact that these modes are less excitedby the measurable force than by the unmeasurable forces. Furthermore, it can beseen that the damping ratios of the first modes are identified with higher accuracyif the initial FRFs are estimated from 4 blocks instead of 8 blocks. This can beexplained by the effect of leakage on the initial respectively 4 and 8 blocks.


Table 3.4: Monte Carlo Simulation (30 runs). Mean value and standard deviation ofthe estimated natural frequencies for the OMAX case. Comparison between the classicH1 technique (fh1) with Nb = 128, the H1 combined with rectangular window (fhrw ) forboth Nb = 4, m = 32 and Nb = 8, m = 16 and the combined FRF-positive power densityapproach (fc) for both Nb = 4, m = 32 and Nb = 8, m = 16

fexact fh1σfh1

fhrwσfhrw

fc σfcfhrw

σfhrwfc σfc

Nb = 32 Nb = 4 m = 32 Nb = 4 m = 32 Nb = 8 m = 16 Nb = 8 m = 16

6.41 6.40 0.02 6.41 0 6.41 0 6.41 0 6.41 015.01 15.10 0.11 15.03 0.05 15.02 0.02 15.03 0.05 15.02 0.0118.67 18.68 0.02 18.67 0 18.67 0 18.67 0 18.67 027.86 27.86 0.08 27.86 0.07 27.87 0.04 27.87 0.06 27.87 0.0229.59 29.60 0.01 29.60 0.01 29.59 0.01 29.60 0.01 29.60 0.0136.14 36.16 0.15 36.16 0.11 36.17 0.11 36.18 0.16 36.19 0.1536.91 36.91 0.02 36.91 0.03 36.91 0.03 36.91 0.04 36.91 0.04

Table 3.5: Monte Carlo Simulation (30 runs). Mean value and standard deviation ofthe estimated damping ratios for the OMAX case. Comparison between the classic H1

technique (σh1) with Nb = 128, the H1 combined with rectangular window (σhrw ) forboth Nb = 4, m = 32 and Nb = 8, m = 16 and the combined FRF-positive power densityapproach (σc) for both Nb = 4, m = 32 and Nb = 8, m = 16

dexact dh1σdh1

dhrwσdhrw

dc σdcdhrw

σdhrwdc σdc

Nb = 32 Nb = 4 m = 32 Nb = 4 m = 32 Nb = 8 m = 16 Nb = 8 m = 16

0.201 1.601 0.236 0.209 0.015 0.207 0.015 0.222 0.027 0.216 0.0260.472 0.916 0.365 0.501 0.319 0.435 0.109 0.588 0.269 0.451 0.0910.587 1.032 0.126 0.587 0.004 0.587 0.004 0.588 0.004 0.588 0.0040.875 1.011 0.237 0.937 0.241 0.860 0.109 0.984 0.274 0.886 0.1510.930 1.100 0.042 0.938 0.047 0.937 0.046 0.933 0.036 0.931 0.0341.135 1.059 0.352 1.102 0.277 1.092 0.256 1.157 0.378 1.154 0.3661.160 1.196 0.088 1.111 0.066 1.112 0.063 1.111 0.065 1.115 0.065

3.6.4 Measurement examples

Subframe of car

The subframe shown in figure 3.1(b) was excited by an electrodynamic shakerand the vibration responses are measured by 23 accelerometers during 64s with asample frequency of 2048Hz. Similar as for the simulation study the classic H1

approach is compared to the noise reduction obtained by the use of rectangularwindow in figure 3.10. It is clear that the same reduction in data by the classicalapproach results in biased FRF estimates and much higher uncertainty levels thanthe FRF obtained from a rectangular window.

Villa Paso bridge

The Villa Paso bridge, situated in the southeast of Italy, is an arch bridge, recon-structed after the second world war. Measurements were performed on the deckof the bridge in cooperation with the university of L’Aguilla in the context of a


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Nb=64

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pl (

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Figure 3.10: Subframe. left: classic H1 FRF estimate from Nb blocks; right: H1



structural health monitoring program for the local authorities. Figure 3.11 (a),(b)shows the bridge and the hammer used for the force excitation. The span was di-

(a) (b)

1 2 3 4 5 6 7

8 9 10 11 12 13 14

(c)

Figure 3.11: (a) Villa Paso arch bridge; (b) Hammer exciter (c); Schematic overviewof the measured grid

vided in 28 measurement locations shown by figure 3.11 (c), which were measuredin both horizontal and vertical direction. During the modal analysis experiment


the bridge was open for traffic and a forced vibration experiment was performedby excitation by a calibrated hammer in node 2.

(a)

0 100 200 300 400 500 600 700−6

−4

−2

0

2

4

Time (s)

Acc

el. m

/s2

495 500 505 510 515 520

−2

0

2

Time (s)

Acc

el. m

/s2

(b)

Figure 3.12: a) Autopower of the Hammer excitation force spectra b) Vertical vibrationresponse in node 2

The hammer excitation generated low force levels for the low frequencies andhigher force levels for the higher frequencies, which is illustrated by Figure 3.12(a). The traffic typically excites the structure well for the lower frequencies. Fig-ure 3.12 (b) illustrates the vibration response in the vertical direction in node2 from the repeated hammer excitation and the traffic. An OMAX approach isapplied to estimate the FRFs and the positive power spectra of the stochasticcontribution together with their uncertainty levels. From Figure 3.13 it is clearthat the SNR for the positive power spectra is better for the low frequencies thanthe SNR of the FRFs for the low frequencies, while the FRFs have better SNRfor the higher frequencies. This can be explained by the bad excitation of thelow frequencies by the hammer impacts and the low frequent excitation of thetraffic. Since the hammer excitation only excited the vertical modes by hittingthe bridge perpendicular to the deck and the traffic excited both the vertical andhorizontal modes, the positive power spectra contain several modes at the lowerfrequencies which are not visible in the FRFs. Finally, it must be noticed thatthe amplitudes of the positive power spectra of the horizontal measurement areclearly higher than the vertical measurement, since the most important low fre-quent modes are horizontal modes. Figure 3.14 shows 6 modes obtained from thecombined operational-experimental parametric analysis. The first two modes (a)and (b) of respectively 1.6Hz and 2.6Hz are horizontal bending modes, the nexttwo modes (c) and (d) of respectively 3.6Hz and 5.7Hz are vertical bending modes,while the last three modes are coupled horizontal-vertical torsion modes of respec-tively 9.3Hz, 14.9Hz and 17.5Hz. The first 4 modes could only be extracted withhigh quality from the stochastic contribution from the traffic excitation, while thelast three modes are mainly excited by the hammer excitation. In fact, even forthe case where the bridge was closed for any traffic and thus the FRFs are less

3.7. Conclusions 65

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Positive power spectra Nb=8, m=8

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FRF Nb=8, m=8

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Positive power spectra Nb=8, m=8

(b)

Figure 3.13: FRF versus Positive power spectrum for node 4. (a) vertical direction, (b)horizontal direction (full line: FRF or Positive power spectrum, dotted line: standarddeviation)

disturbed by ambient vibration, the first 2 modes could not be extracted from theFRF data and the 3th and 4th only with a very poor quality. Another exampleof operational vibration test on a bridge can be found in [81], where sensitivitybased mode shape normalization is applied.

3.7 Conclusions

In this chapter the estimation of Averaged Based Spectral functions like FRFsand power spectra has been discussed. Several FRF estimators from literature areproposed and discussed together with the periodogram and correlogram estima-tors for power spectra. The use of a rectangular window for noise reduction on theFRFs was introduced and the corrections for time and frequency-domain leakageare presented. Furthermore, the concept of ’positive’ power spectra is introducedby the use of a rectangular window and its benefits are given. Finally, the esti-mation of both the FRFs from the deterministic contribution and positive powerspectra from the stochastic contribution is proposed in the OMAX framework.Simulations are performed to show the reduction in leakage errors (especially on


the damping) by the use of the rectangular window for both experimental andoperational modal analysis. OMAX simulations illustrated the gain in accuracyif both the measurable and unmeasurable forces are considered in the vibrationresponse. Several techniques are applied to real-life vibration experiments on asubframe of a car and an arch bridge.

3.8 Appendix 1

In this appendix an expression for the covariance matrix CH1on the FRFs esti-

mated by the H1 estimator is derived from a sensitivity analysis. A perturbationof the outputs ∆Yb results in a perturbation ∆H1 given by

∆H1 =1

Nb

(

Nb∑

b=1

∆YbFHb

)

S−1ff (3.61)

Applying the Vector operator and eliminating the perturbation of the perturbation∆Yb by the use of the Kronecker product results in

vec(∆H1) =1

Nb

Nb∑

b=1

(

S−Tff F ∗

b ⊗ INo

)

∆Yb (3.62)

Assuming the output noise to be stationarity and uncorrelated over the differentblocks the covariance matrix is given by

CH1= E

(

vec(∆H1)(vec(∆H1)H))

(3.63)

=1

N2b

Nb∑

b=1

(

S−Tff F ∗

b FTb S−1

ff

)

⊗ CY (3.64)

(E the expected value) with the covariance matrix of the output noise ∆Y givenby

CY = E (∆Y ∆Y H) = Syy − H1Sfy (3.65)

since Y = HF + ∆Y . Applying the property (FFH)T = F ∗FT and substituting3.2 in 3.64 results in

CH1=

1

NbS−T

ff ⊗ CY (3.66)

The multiple input coherence function [52] mγ2o describes the amount of energy

of output o correlated with the Ni inputs and is given by

mγ2o =

SyofS−1ff Sfyo

Syyo

(3.67)

=H1,oSffHH

1,o

Syyo

(3.68)

3.9. Appendix 2 67

with Syyo, Syof , Sfyothe auto and cross power for output o and H1,o ∈ C

1×Ni

the FRFs of the H1 FRF matrix estimate between output o and the inputs. Thecovariance matrix of Ho is given in function of the multiple coherence function as

CH1,o=

1

Nb(1 − mγ2

o)SyyoS−1

ff (3.69)

In the single input case the expressions for the coherence function mγ2o and co-

variance matrix CH1,oreduces to

mγ2o = γ2

o =SyofSfyo

SffSyy,o(3.70)

(3.71)

and

CH1,o= varH1,o

=(1 − γ2

o)

Nbγ2o

|H1,o|2 (3.72)

In [105] it is shown that for parametric identification algorithms that use thecovariances on the FRFs as a frequency-domain weighting, at minimum of inde-pendent blocks must be averaged to result in consistent estimates of the systemparameters.

3.9 Appendix 2

In many cases frequency-domain data, i.e. FRFs, are given as primary data. Inthe case that periodic input signals such as multisines are used, these measuredFRF are often of high quality and contain no errors caused by leakage. Thediscrete-time domain modal model is given by

H(ωk) =

Nm∑

r=1

(

φrLr

1 − λrz−1k

+φ∗

rLHr

1 − λ∗rz

−1k

)

(3.73)

Since many identification methods start from IRF as primary data, the FRFs areconverted to IRF by considering the IDFT

hn =1

N

N∑

k=1

H(ωk)znk (3.74)

=1

N

Nm∑

r=1

(

φrLr

N∑

k=1

znk

1 − λrz−1k

+ φ∗rL

Hr

znk

1 − λ∗rz

−1k

)

(3.75)

Consider the sequence xn given by

xn = λn (3.76)


the DFT X(ωk) is defined by

X(ωk) =

N−1∑

n=0

xnz−nk (3.77)

=

N−1∑

n=0

(λz−1k )n (3.78)

=1 − λNz−N

k

1 − λz−1k

(3.79)

=1 − λN

1 − λz−1k

(3.80)

with zk = ei2πk/N and thus zNk = 1. Since the IDFT and DFT operations cancel

each other it holds that

xn = λn (3.81)

=1

N

N−1∑

k=0

X(ωk)znk (3.82)

=1

N

N−1∑

k=0

(1 − λN )znk

1 − λz−1k

(3.83)

and thus it is proven that

λn

1 − λN=

1

N

N−1∑

k=0

znk

1 − λz−1k

(3.84)

Substituting Eq. 3.84 in 3.75 results in

hn =

Nm∑

r=1

(

φrLrλn

r

(1 − λNr )

+ φ∗rL

Hr

λ∗n

r

(1 − λ∗Nr )

)

(3.85)

=

Nm∑

r=1

(

φrL′Tr λn

r + φ∗rL

′Hr λ∗n

r

)

(3.86)

with L′r = Lr

(1−λNr )

. Several discrete time-domain algorithm as e.g. the Least

Squares Complex Exponential (LSCE), Eigenvalue Realization Algorithm (ERA),time-domain subspace methods estimate a modal model based on Eq. 3.86. Oftenthese IRF are obtained from the IDFT of the FRFs, but yet often the correction ofthe estimated participation vectors L

′

r is neglected. Although this correction canbe essential in the case of lightly damped structures. In [77] a similar correctionfor models obtained from IRF calculated from the DFT of the FRFs is proposedfor discrete-time state-space models.

3.9. Appendix 2 69

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 3.14: Vibration mode shapes of the deck of the Villa Paso bridge. (a)-(b)horizontal bending modes, (c)-(d) vertical bending modes, (e)-(f)-(g) torsion modes


Chapter 4

Identification ofcommon-denominatormodels

This chapter starts with a short overview of modal parameter identification algo-rithms. Next, the attention is focussed to the identification of common denom-inator models starting from both ABS-functions and IO data. The models areextended for dealing with leakage when starting from FRFs obtained by the H1

estimator. It is shown how the identification starting from IO data from differ-ent experiments can be combined without suffering from transients and leakage.The analogy between the FRF-driven and IO data-driven algorithms is illustrated.Finally, a combined deterministic-stochastic identification algorithm is proposed,which estimates a common-denominator model from the measurements. Finallyan introduction to in-flight tests is given, together with a comparison of differentmethods based on in-flight simulations of an aircraft.

71

72 Chapter 4. Identification of common-denominator models

4.1 Introduction

During the last decades many researchers have devoted an extensive effort in thedevelopment of reliable identification algorithms for the characterization of dy-namical structures by its modal parameters. Starting from Single Degree Of Free-dom (SDOF) SISO methods e.g. the Peak Picking method [8] and Circle Fittingmethod [61], the algorithms have evolved to Multiple Degree Of Freedom (MDOF)MIMO methods as a result of the introduction of multi-channel acquisition, pow-erful computer memory and processors. Furthermore, since the introduction of theFast Fourier Transform (FFT) frequency-domain algorithms have been developedas the time-domain counterparts for identification for modal analysis. MDOFestimators are able to estimate several modes simultaneously, while MIMO arecapable to handle data from several vibrations responses and force inputs simulta-neously. The first MDOF MIMO time-domain model parameter estimators werethe Ibrahim Time-Domain (ITD) method, the Eigensystem Realization Algorithm(ERA) [56], [57] and the Least Squares Complex Exponential (LSCE) method.Both the ITD and LSCE estimate polynomial coefficients. The LSCE algorithmis closely related to the class of Prediction Error Methods (PEM) [67]. The ERAalgorithm starts from a state-space formulation and requires the singular valuedecomposition (SVD) of the so-called block Hankel matrix. More recently thestate-space formulation has been the basis for subspace identification algorithms[123]. Several MDOF MIMO frequency-domain algorithms are developed, likee.g. the Least Squares Frequency Domain (LSFD), ERA frequency domain andFrequency-domain Direct Parameter Identification algorithm (FDPI). The LSFDdirectly estimates the modal parameters by solving a non-linear optimization. Inthe case that the poles and/or participation factors are already estimated in a firststep by e.g. LSCE, ITD, the LSFD reduces to a linear problem in the unknownmodal parameters (mode shapes) [68]. The ERA frequency domain version startsfrom a complex block matrix with the FRFs as primary data and estimates themodal parameters by the use of SVD and eigenvalue decomposition (EVD) [58].Based on a LMFD in the Laplace domain, the FDPI estimates the modal pa-rameters from the FRFs [63]. More recently several frequency domain algorithmsbased on a common-denominator description are proposed to handle large andnoisy data-sets from EMA [131] and OMA [80] tests. Several implementationssolve the identification problem in a least-squares (LS), total least squares (TLS),generalized TLS (GTLS) and maximum likelihood (ML) sense, each of them cor-responding to different asymptotic properties [93]. A more profound overview ofmodal parameter estimation algorithms can be found in [131], [52].

In this chapter a generalization for common-denominator (see paragraph 2.3)based algorithms is given by taking into account the initial and final conditionsfor every block used by the H1 estimator [24]. Next, the strong analogy betweenthe FRF and IO data driven algorithms is illustrated [132] and finally a combineddeterministic-stochastic algorithm is proposed [22].

4.2. An extended parametric model for the H1 estimator 73

4.2 An extended parametric model for the H1 es-timator

In chapter 3 several non-parametric FRF estimators were discussed with theirrelation to the introduction of leakage errors. In practise special attention mustbe paid to errors introduced by leakage especially for low-damped structures. Itis well-known that the use of a Hanning window reduces the effect of leakage bysuppressing the initial and final samples of each time block to zero. Although,for short time blocks still significant errors are introduced, since no correction onthe final parameters is possible to compensate for the Hanning window. Thereforethe use of exponential and rectangular windows are proposed. These windowsintroduce bias errors on the estimated FRFs, but the final parameters can becorrected for this bias. Although both the use of an exponential and rectangularwindow require an initial leakage free estimate of the FRFs e.g. from a smallnumber of blocks. In this chapter a different approach to deal with leakage isproposed.

In [99] it is shown for SISO systems that the initial and final conditions of theresponse signal can be modelled by an additional transient polynomial. In the casethat the initial and final conditions are equal (i.e. the response is periodic) theextra polynomial disappears. Consider the vibration response yo(t) in location oand the input fi(t) in location i, their relation can be written as

n∑

r=0

an−ryo((n − r)∆t) =

Ni∑

i=1

n∑

r=0

boi,n−rfi((n − r)∆t) (4.1)

with ∆t the sample period. In the appendix it is shown that the discrete Fouriertransformation of equation 4.1 is given by

A(zk)Yo(zk) =

Ni∑

i=1

Boi(zk)Fi(zk) + To(zk) (4.2)

with zk = ej2πfk∆t and

Boi(zk) =

n∑

j=0

boi,jzjk

the numerator polynomial,

A(zk) =n∑

j=0

ajzjk

the common-denominator polynomial and

To(zk) =

n−1∑

j=1

tojzjk


the transient polynomial for output o. In fact, this transient polynomial modelsthe non-steady state response of the structure and in this way errors introducedby leakage are avoided. The use of the additional transient polynomial improvesthe damping estimates and the overall quality of the estimated modal model. Bytaking into account this extra polynomial it is possible to deal with both arbitrarysignals (non-periodic) and periodic signals corrupted by transients under the sameassumptions as time-domain methods [99].

For system identification algorithms starting from IO spectra the transientpolynomial can easily be modelled by considering an extra input. Eq. 4.1 canbe formulated for each data block b. The transient term coefficients tob,j willdepend on the initial and final conditions of the corresponding block b. Multi-plying Eq. 4.2 by the hermitian of the input force spectra at frequency line k[ F1(zk) . . . FNi

(zk) ]T and averaging over all blocks results in

SYoF (ωk) =[Bo(zk)]

A(zk)SFF (ωk) +

Nb∑

b=1

Tob(zk)

A(zk)FH

b (ωk) (4.3)

with Bo(zk) = [ Bo1(zk) . . . BoNi(zk) ]. Multiplication by the inverse of S−1

FF

leads to an extended parametric model for the H1 FRF estimates. Generally

Hoi(ωk) =Boi(zk)

A(zk)+

Nb∑

b=1

Tob(zk)

A(zk)F t

bi(ωk) (4.4)

with F tbi(ωk) the ith element of the row vector F t

b (zk) defined by

F tb (ωk) = Fb(ωk)HS−1

FF (ωk) (4.5)

From these equations it is clear that generally the H1 estimator is not equal to

the exact model B(zk)A(zk) and that leakage and transient phenomena have some noisy

influence, if the responses are not periodic. Since the spectra F tbi(ωk) can be cal-

culated from the force measurements the transient polynomials can be estimated.This extended formulation of the H1 estimator makes FRF based identificationmethods attractive to process shorter data sequences. The major advantage ofall identification methods based on FRF data is that the noise levels decrease byaveraging different data blocks, while this is not the case by identification meth-ods starting from the raw input and output spectra (except if periodic signals areused). It is important to remark that time domain methods e.g. LSCE [52] alsosuffer from leakage, if they start from impulse response functions obtained by theInverse Fourier Transformation (IFT) of the FRFs.

4.3. Weighted Least Squares Complex Frequency-domain (LSCF) estimator 75

4.3 Weighted Least Squares Complex Frequency-domain (LSCF) estimator

4.3.1 FRF driven identification

The model for the FRF between output o and input i estimated by the H1 esti-mator is given by

Hoi(ωk) =Boi(zk)

A(Ωk)+

Nb∑

b=1

Tob(zk)

A(zk)F t

bi(ωk) (4.6)

with Boi(zk), A(zk) and Tob(zk) scalar polynomials in the basis functions zjk given

by

Boi(zk) =n∑

j=0

boi,jzjk (4.7)

A(zk) =n∑

j=0

ajzjk (4.8)

Tob(zk) =n∑

j=1

tob,jzjk (4.9)

The polynomial coefficients can be grouped together in one column vector θ

θ =

η1

...ηNo

α

(4.10)

with

ηo =

βo1

...βoNi

τo1

...τoNb

, α =

a0

a1

...an

, βoi =

boi,0

boi,2

...boi,n

, τob =

tob,0

tob,2

...tob,n

The unknown polynomial coefficients differ in their level of being global parame-ters:

• the denominator coefficients α are global parameters of the structure inde-pendent of the output o, input i and data block b


• the numerator coefficients βoi are local parameters depending on the outputo and input i location

• the transient coefficients τob depend on both the output o location and thedata block b.

Tob(zk)

A(zk)-F e

bi(ωk)

Boi(zk)

A(zk)-1

@@@R h+ -?

Woi(ωk)

A(zk)

?

Eoi(ωk)

Hoi(ωk)

Figure 4.1: LSCF FRF model (Nb = 1)

Equation Error and Cost Function:

By multiplying Eq. 4.6 with A(zk) and by taking the difference between the leftand right hand side an equation error is obtained that is linear in the parameters

ELSoi (ωk) =

1

Woi(ωk)

(

A(zk)Hoi(ωk) − Boi(zk) −Nb∑

b=1

Tob(zk)F tbi(ωk)

)

(4.11)

with Woi(ωk) a frequency-domain weighting function. Figure 4.1 visually repre-sents the different contributions in the weighted linear least-squares model. Theweighted linear least-squares problem is found by minimizing

lLSCFFRF (θ) =

No∑

o=1

Ni∑

b=1

Nf∑

k=1

|ELSoi (ωk)|2

which corresponds to solving

Jθ = 0 (4.12)


with J the Jacobian matrix (which is in this case independent of the parameters)given by

J =

κ1 0 . . . 0 Υ1

0 κ2 . . . 0 Υ2

......

. . ....

...0 0 . . . κNo

ΥNo

(4.13)

with

κo =

Γo1 0 . . . 0 Ξo11 . . . Ξo1Nb

0 Γo2 . . . 0 Ξo21 . . . Ξo2Nb

......

. . ....

.... . .

...0 0 . . . ΓoNi

ΞoNi1 . . . ΞoNiNb

(4.14)

Υo =

ςo1

...ςoNi

(4.15)

and the submatrices Γoi, ςoi and Ξoib

Γoi = −

1Woi(ω1)

[

1 z1 . . . zn1

]

1Woi(ω2)

[

1 z2 . . . zn2

]

...1

Woi(ωN )

[

1 zN . . . znN

]

(4.16)

ςoi =

1Woi(ω1)

Hoi(ω1)[

1 z1 . . . zn1

]

1Woi(ω2)

Hoi(ω2)[

1 z2 . . . zn2

]

...1

Woi(ωN )Hoi(ωN )[

1 zN . . . znN

]

(4.17)

Ξoib = −

1Woi(ω1)

F tbi(ω1)

[

1 z1 . . . zn1

]

1Woi(ω2)

F tbi(ω2)

[

1 z2 . . . zn2

]

...1

Woi(ωN )Ftbi(ωN )

[

1 zN . . . znN

]

(4.18)

Reduced normal equations

A modal analysis experiment is typically characterized by a large number of out-puts and therefore special attention must be paid to reduce memory requirementsand processing time. The use of a common denominator model results in dedicated


algorithms if the structure of the matrices is optimally used. The Jacobian matrixJ of this least squares problem has NNoNi rows and (n + 1)(NoNi + NoNb + 1)columns (with the number of rows larger than the number of columns to makethe problem identifiable). The number of frequencies N can be eliminated byformulating the normal equations, i.e.

JHJθ = 0 (4.19)

K1 · · · 0 L1

.... . .

...0 KNo

LNo

LH1 · · · LH

No

∑No

o=1 To

η1

...ηNo

α

= 0 (4.20)

The specific structure in the normal equations result from the use of a common-denominator model and will allow a fast implementation. All the parameters θ aredivided in output dependent parameters ηo and output independent parametersα. By taking a closer look to the model it can be seen that the parameters ηo

can be split up in a set βoi dependent of the input location i corresponding withthe FRF Hoi(ωk) and a set τob independent of the input location. As a result thesubmatrices of Eq. 4.20 have a structure by itself given by the following equations.

Ko =

Mo1 · · · 0 No1

.... . .

...0 MoNi

NoNi

NHo1 · · · NH

oNiRo

Lo =

Qo1

...QoNi

So

(4.21)

with the submatrices Moi, Noi, Qoi, Ro, So and To given by

Moi = ΓHoiΓoi

Qoi = ΓHoiςoi

Noi = ΓHoi

[

Ξoi1 . . . ΞoiNb

]

Ro =

Ni∑

i=1

[

Ξoi1 . . . ΞoiNb

]H [Ξoi1 . . . ΞoiNb

]

(4.22)

So =

Ni∑

i=1

[

Ξoi1 . . . ΞoiNb

]Hςoi

To =

Ni∑

i=1

ςHoi ςoi


Forming the reduced normal equations

The maximum gain in calculation time is obtained by taking into account thestructure of the normal equations. By simple algebraic elimination procedures thenumerator polynomial coefficients boi can be eliminated from Eq. 4.20 and 4.21yielding

βoi = −M−1oi

[

Noi Qoi

]

[

τo

α

]

(4.23)

Further calculation and elimination of the parameters to leads to

τo = −Ro−1Soα (4.24)

with

Ro = Ro −Ni∑

i=1

NHoi M

−1oi No,i (4.25)

So = So −Ni∑

i=1

NHoi M

−1oi Qo,i (4.26)

From the last n rows of the normal equations Eqs. 4.20 the following equationscan be formed

No∑

o=1

(

LHo ηo + Toα

)

= 0 (4.27)

Substituting Eq. 4.23 and Eq. 4.24 in Eq. 4.27 results in

No∑

o=1

(

−SoHRo

−1So + To

)

α = Dα = 0 (4.28)

with D =∑No

o=1

(

−SoHRo

−1So + To

)

and

To = To −Ni∑

i=1

QHoiM

−1oi Qo,i (4.29)

Eq. 4.28 can be solved by a simple matrix inversion, with dimension n (the orderof the polynomial A(zk)), leading to the denominator coefficients α. Backwardsubstitution in Eq. 4.24 and Eq. 4.23 provides the coefficients τob and βoi. Ex-amining the submatrices Moi, Noi, Qoi, Ro, So and To in more detail reveals that


the entries of these matrices equal

Moi[r,s] =

N∑

k=1

|Woi(ωk)|−2zs−rk (4.30)

Qoi[r,s] =

N∑

k=1

− |Woi(ωk)|−2Hoi(ωk)zs−r

k (4.31)

To[r,s] =∑

i=1

Ni

N∑

k=1

|Woi(ωk)|−2 |Hoi(ωk)|2 zs−rk (4.32)

Since the entry [r, s] of the submatrices only depends on r−s, these matrices havea so-called Toeplitz structure. In practice this means that only one column in caseof an Hermitian matrix e.g. Moi or one column and one row in the general case e.g.Qoi must be calculated to construct the total matrix resulting in both a memoryand time efficient computation. Furthermore, an additional gain in computationtime can be obtained by the using the Fast Fourier Transform (FFT) to computethe entries Eqs. 4.30-4.32 of the matrices. The matrices Nob, Rob and So can bedecomposed in submatrices with similar structures and expressions as the abovediscussed matrices and thus the memory and time efficient computation can alsobe applied in the same way.

Solving the reduced normal equations

To solve the identification problem, the parameter redundancy (and to avoid thetrivial solution with all coefficients equal to zero) of the transfer function modelmust be removed. It can easily be seen that a transfer function model defined bya set of parameters θ, can also exactly be defined by γθ

Hoi(ωk) =Boi(zk)

A(Ωk)+

Nb∑

b=1

Tob(zk)

A(zk)F t

bi(ωk) (4.33)

=γBoi(zk)

γA(Ωk)+

Nb∑

b=1

γTob(zk)

γA(zk)F t

bi(ωk) (4.34)

To remove this parameter redundancy a constrained must be applied to the co-efficients of the polynomials by e.g. fixing a coefficient to a constant value orby fixing the L2 norm of the coefficients to 1. The least squares solution of thereduced normal equations when fixing coefficient aj = 1 is given by

α[1:j−1,j+1:n+1] = D−1[1:j−1,j+1:n+1],[1:j−1,j+1:n+1]D[1:j−1,j+1:n+1],1 (4.35)

The mixed LS-Total Least Squares (TLS) solution given α = V[:,n+1] with V theright singular vectors obtained by an SVD of the matrix D

D = USV H (4.36)


forces a L2 norm equal to 1 on the coefficients α. In chapter 9 the impact of theconstraint on the final solution is discussed in more detail with a special focuson the quality of the stabilization diagram. In [137] it is shown that solving thereduced normal equations by exploiting its predefined structure is much fasterthan solving Eq. 4.12 directly (approximately N2

o N2i ).

4.3.2 Remarks on the extended LSCF estimator

Continuous-time model

Instead of using a discrete-time domain model a continuous-time model can beused by changing the basic functions zj

k into sjk. A similar extended model to

take into account the influence of the initial and final conditions can be formu-lated based on the results given in [96]. Although, special attention must be paidto the numerical conditioning of the identification problem. Therefore the use oforthogonal polynomials such as Forsythe and Chebyshev polynomials is advisedto improve the numerical conditioning [100], which were applied in the domainof modal analysis by [117]. In [100] a numerical well-conditioned solution is dis-cussed to transform the estimated coefficients in the orthogonal basis to the modalparameters. The use of Forsythe polynomials has however the drawback that adifferent set of polynomials must be generated for each individual FRF resulting ina high computation time. This computation can be reduced by the use of Cheby-shev polynomials, but for high model orders Nm = 50 the numerical conditionbecomes still a problem since the polynomials are only approximately orthogonal.A profound study and comparison between the use of continuous- and discrete-time models for MIMO frequency domain identification is given in [137]. The useof discrete-time models and an uniform frequency grid solves the numerical con-ditioning problem, since the basic functions zj

k are orthogonal with respect to theunity circle. The use of the discrete-time model allows a fast and memory efficientimplementation based on the Toeplitz structure of the submatrices and the use ofthe FFT algorithm to compute the matrix entries. Furthermore, it is shown inchapter 9 that discrete-time models identified by a least squares algorithm underthe correct constraint result in a high quality of the stabilization diagram, whichis a major advantage in modal analysis applications. When using of discrete-timemodel the frequencies in the frequency band of interest are mapped between 0Hzand 1Hz. However, this introduces model errors which require a model order tobe chosen higher than the true one to obtain a good model fit. The use of bilineartransformation allows the possibility to model a continuous-time system exactly bya discrete-time model [73]. However, the transformation does not result anymorein a uniform distribution of the zj

k values over the unit circle and as consequencethe complex basic functions or not longer orthogonal. This results in both a worsenumerical conditioning and less clear stabilization charts. Although it should benoted that for lower orders the bilinear transformation improves the quality of the


damping estimates.

With or without transient polynomials

In the case that the transient polynomials are neglected, the extended LSCF es-timator reduces to the original LSCF estimator as proposed in [46]. The originalLSCF estimator models a common denominator model, given by

Hoi(ωk) =Boi(zk)

A(zk)(4.37)

on the FRFs in a least squares sense. The use of a steady state response ofa system excited by periodic excitation signals eliminates this problem since noleakage is introduced. In practice, even in the case of arbitrary excitation, theinfluence of leakage can often be neglected in the case the non-parametric FRFestimation has taken place with caution as shown in chapter 3. However, in thecase of very lightly damped structures and the use of arbitrary excitation signalsa significant improvement of the estimated parameters and stabilization diagramcan be observed by modelling the initial and final conditions of each block. TheLSCF estimator, initially proposed as a starting value generator for the MaximumLikelihood estimator, forms the basis of several other algorithms. They can employthe same procedures to obtain a fast and memory efficient parameter estimationmethod as discussed in this section.

LSCF for operational and combined operational-experimental data

Thanks to the mathematical similarity between the modal model for FRFs andpower spectra in the case of an operational modal analysis, the estimator with-out the estimation of the transient polynomials can also be applied to start frompower densities [40]. In case one is interested to process operational data withtransients one has to stick to data driven methods e.g. the combined deterministic-stochastic frequency-subspace algorithm of chapter 8. In the case one starts frompower spectra the algorithms are not capable to make a distinction between tran-sients (deterministic contribution) and the stochastic contribution (notice that thetransients i.e. intitial/final conditions differ from block to block and thus no dis-tinction can be made based on an averaging procedure). Starting from the positivepower spectra results in more interesting properties concerning the constructionof stabilization diagrams as shown in chapter 9.

Asymptotic properties

Modal parameter estimation often requires that a trade-off must be made betweenaccuracy and computation speed, since the data sets are typically characterized


by a huge amount of data. In the general case the LSCF is not (asymptotic)unbiased, not consistent and not efficient, since the equation error Eoi(ωk) is notwhite uncorrelated noise with bounded second order moments. In practice thismeans that in the presence of noise on the FRFs:

• the mean of the estimated parameters from different experiments with N →∞ differs from the exact parameters (asymptotically biased)

• the expected value of the estimated parameters for N → ∞ differs from theexact one

• the uncertainty on the estimated parameters is larger the Cramr-Rao uncer-tainty bounds

To improve these properties more advance estimators like the Maximum Likelihoodestimator are proposed (discussed in more detail in paragraph 4.4). A detailedstudy and comparison of the different frequency-domain identification approachescan be found in [93].

4.3.3 IO data driven identification

In some cases one is interested to start directly from the measured input and outputspectra to identify the system parameters e.g. if only a limited of time samplesare available. In the most general case, different blocks of data from differentexperiments can be combined in one identification procedure. The output o forblock b can be modelled as

Yob(ωk) =

Ni∑

i=1

Boi(zk)

A(zk)Fib(ωk) +

Tob(zk)

A(zk)(4.38)

which is schematically illustrated by Figure 4.2. The linearized least squaresequation error and cost function are then given by

ELSob (ωk) = A(zk)Yo,b(ωk) −

Ni∑

i=1

Boi(zk)Fi,b(ωk) − Tob(zk) (4.39)

lLSCFIO (θ) =

No∑

o=1

N∑

k=1

Nb∑

b=1

|ELSob (ωk)|2 (4.40)

Classical FRF based estimators do not take into account the influence of thetransient and leakage effects. The extension of the LSCF FRF driven estimatorto its extended formulation takes into account the leakage and transient effects.Comparing these extended FRF estimators with the LSCF IO based estimatorsreveals their strong analogy. Consider both the FRF and IO models respectively


Boi(zk)

A(zk)-Fib(ωk)

Tob(zk)

A(zk)-1

@@@R h+ -?

1A(zk)

?

Eob(ωk)

Yob(ωk)

Figure 4.2: LSCF IO model (Ni = 1)

0246810

0

5

10

15

20

25

30

0

1

2

Z

Nb

No

Ni

Figure 4.3: Data space visualization (e.g. Ni = 2, No = 30 and Nb = 10)

given by Eqs. 4.6 and 4.38. It is clear that the role of the inputs i is swapped withthe blocks b. This strong analogy results in the same mathematical implementationfor the LSCF FRF driven and the LSCF IO data driven estimators. Only theprimary data to the estimator and the interpretation of the estimated parametersare changed. This analogy can be visualized in a 3-dimensional input-output-blockspace shown in figure 4.3. IO data driven estimators work in the No × Nb andNi × Nb planes and need as a result Nb(No + Ni)Nf complex data samples. ForH1 based FRF estimators the data is situated in the No ×Ni and Nb ×Ni planes.Compared to IO estimators the block axis is swapped with the input axis. Fromthis point of view it is clear that in the case Nb > Ni FRF based estimators can

4.4. Maximum Likelihood identification 85

operate from a smaller amount of data. In case no leakage and transient effects arepresent (e.g. periodic excitation) the block axis disappears, since both IO data andFRF data can be averaged over the blocks and no transient polynomials have tobe estimated. Classical FRF driven estimators often neglect the block axis, whilethis condition (no leakage and transient effects) is not fulfilled, resulting in biaserrors. The use of periodic excitation signals simplifies the estimation problem alot, since no transient polynomials have to be identified. The block dimension willdisappear, if cyclic averaging takes place over the different periods. Furthermore,cyclic averaging reduces the noise levels on both the responses and forces and givesthe possibility to obtain noise information.

4.4 Maximum Likelihood identification

Linear least-squares based algorithms like the LSCF algorithm are often used inmodal analysis for their ability to handle large amounts of data in a reasonabletime. However, the LS approach is not consistent and not efficient. This results inbiased estimates and in the cases that a LS approach is not accurate enough moresophisticated estimators must be used. In the following paragraph a consistentalgorithm is proposed, which estimates a common-denominator model startingfrom FRFs or IO data in a Maximum Likelihood (ML) sense. This algorithmwas introduced in [46] and applied to both experimental and operational modalanalysis experiments [40].


Equation Error and Cost Function:

Consider the common-denominator model for a measured FRF, disturbed by col-ored independent circular complex noise with variance σ2

Hoi(ωk), given by

Hoi(ωk) =Boi(zk)

A(zk)+

Nb∑

b=1

Tob(zk)

A(zk)F t

bi(ωk) + σHoi(ωk)Eoi(ωk) (4.41)

Minimizing the quadratic cost function results in the maximum likelihood solutionunder the assumption that Eoi(ωk) is uncorrelated over the different FRFs [37],[98]. The (negative) log-likelihood function is then given by

lMLFRF (θ) =

No∑

o=1

Ni∑

b=1

Nf∑

k=1

|EMLoi (ωk)|2 (4.42)


with the equation error given by

EMLoi (ωk) =

A(zk)Hoi(ωk) − Boi(zk) −∑Nb

b=1 Tob(zk)F tbi(ωk)

Woi(ωk)A(zk)

with Woi(ωk) = σHoi(ωk). Notice that the equation error is non-linear in the sys-

tem parameters and thus a non-linear optimization algorithm such as e.g. Gauss-Newton is required to estimate the polynomial coefficients. The ML equation erroris equal to the LSCF equation error but weighted by its standard deviation result-ing in an equation error, which is white uncorrelated noise over the spectral linesk. Notice that if the transients are taken into account by means of the use of theH1 estimator the inputs are assumed to be free from noise. In [41] it is shown thatthe logarithmic equation error, given by

Elog oi(ωk) =log(Hoi(ωk)) − log

(

Boi(zk)A(zk) +

∑Nb

b=1Tob(zk)A(zk) F t

bi(ωk))

Woi(ωk)|Hoi(ωk)|

is more robust for a large dynamical range, outliers and incorrect noise information.

Weighting function

The ML estimator is equivalent to a weighted non-linear least-squares (WNLLS)with a weighting Woi(ωk) = σHoi

(ωk). It is shown in [98] that independent ofthe weighting Woi(ωk) the WNLLS results in consistent estimates if the primarydata i.e. the FRFs are estimated in a consistent way (so no bias in presence ofnoise). The choice of the weighting function only influences the efficiency of theestimates i.e. the closer the weighting function agrees with the true variance on theprimary data the better the efficiency. To obtain the noise information sufficientdata must be available to average several blocks. However, in many cases theengineer responsible for the modal analysis has only access to the FRFs. In thiscase different alternatives for the weight Woi(ωk) can be proposed:

• If no noise information is available, the assumption of white, additive noiseon the FRFs corresponds with a weight Woi(ωk) = 1.

• The assumption of white additive noise on the responses Yob(ωk) correspondsto a weight

Woi(ωk) =

√

√

√

√

Ni∑

j=1

|S−1Fji(ωk)|2σ2

Fj

4.4. Maximum Likelihood identification 87

with S−1Fji defined by

S−1F11 . . . S−1

F1Ni

.... . .

...S−1

FNi1. . . S−1

FNiNi

= S−1

FF (4.43)

By weighting the equation error with this weighting function, frequency lineswhich contain not much energy over the different blocks, are not taken intoaccount in the cost function.

• The assumption of relative noise to the FRFs corresponds with a weightWoi(ωk) = |Hoi(ωk)|. This assumption gives in general good results in termsof a fit of the FRFs considered in a logarithmic scale, since both resonancesand anti-resonances have the same weight in the cost function. In the caseof an operational modal analysis, the FRFs are replaced by power spectraor ’positive’ power spectra and for these spectral functions as primary datathe assumption of relative noise is often in close agreement with true noiseconditions.

Gauss-Newton optimization

A Gauss-Newton optimization is applied to minimize the cost function given byEq. 4.42 by taking into account the quadratic form. The Gauss-Newton iterationsare given by

(a) solve JHm Jm∆Θm = −JH

m E (4.44)

(b) set Θm+1 = Θm + ∆Θm (4.45)

with the Jacobian matrix J defined as the derivative of the equation errors Eoi(k)with respect to the parameters θ for all spectral lines k, outputs o and inputs i.It can easily be shown that the Jacobian matrix has a similar structure as the one(cf. Eqs. 4.13 and 4.14) in the LSCF case. The submatrices Γoi, ςoi and Ξoib are


now given by

Γoi = −

1A(z1)Woi(ω1)

[

1 z1 . . . zn1

]

1A(z2)Woi(ω2)

[

1 z2 . . . zn2

]

...1

A(zN )Woi(ωN )

[

1 zN . . . znN

]

(4.46)

ςoi =

Boi(z1)+∑Nb

b=1 Tob(z1)Ftbi(ω1)

A(z1)2Woi(ω1)

[

1 z1 . . . zn1

]

Boi(z2)+∑Nb

b=1 Tob(z2)Ftbi(ω2)

A(z2)2Woi(ω2)

[

1 z2 . . . zn2

]

...Boi(zN )+

∑Nbb=1 Tob(zN )F t

bi(ωN )

A(zN )2Woi(ωN )

[

1 zN . . . znN

]

(4.47)

Ξoib = −

1A(z1)Woi(ω1)

F tbi(ω1)

[

1 z1 . . . zn1

]

1A(z2)Woi(ω2)

F tbi(ω2)

[

1 z2 . . . zn2

]

...1

A(zN )Woi(ωN )Ftbi(ωN )

[

1 zN . . . znN

]

(4.48)

The normal equations JHJ∆θ = −JHE have a similar the structure as in theLSCF case. The left hand-side JHJ is given by Eq. 4.20, Eq. 4.21 and Eq. 4.22,while the right hand-side JHE is given by

JHE =

[

PT1 . . . PT

No

No∑

o=1

WTo

]T

(4.49)

Po =[

UTo,1 . . . UT

o,NiV T

o

]T(4.50)

with

Uo,i = ΓHoiEoi (4.51)

Vo =

Ni∑

i=1

[Ξoi1 . . . ΞoiNb]Eoi (4.52)

Wo =

Ni∑

i=1

ςoiEoi (4.53)

and Eoi defined by

Eoi =[

Eoi(ω1) Eoi(ω2) . . . Eoi(ωN )]T

(4.54)

Finally, it can be shown that the submatrices in JHJ have a Toeplitz structureand the entries can be calculated in a time-efficient way using the FFT algorithm

4.5. Combined deterministic-stochastic identification 89

similar as in the LSCF case. The initial parameter estimates θ0 (starting values)to construct the normal equations in the first iteration are estimated by the LSCFestimator. In fact, the ML estimator optimizes the results obtained by the leastsquares procedure. From practical studies, a serious improvement of the accuracyis observed after even 10 iterations, while the use of a Levenberg-Marquard loopforces the algorithm to converge by solving

(

JHJ + λLMdiag(JHJ))

θ = −JHE (4.55)

Increasing the parameter λLM forces the cost function to decrease, but decreasesthe convergence speed. Therefore this factor λLM is adapted in every iterationdepending on the evolution of the cost function.


The same analogy between the FRF driven ML and IO data driven ML is presentas it is the case between the FRF and IO driven LSCF. In [136] it is shown howboth the noise on the input measurements and output measurements can be takeninto account by the use of periodic excitation and cyclic averaging.

4.4.3 Noise intervals

One of the advantages of the ML estimator is the capability to estimate the con-fidence intervals on the estimated parameters by the knowledge of the noise infor-mation on the primary data. In [46] this capability is discussed in detail, whilein [141], [130] these noise intervals are considered as a tool to distinguish physicalfrom mathematical poles. Even in the case that the noise assumptions are violatedthis tool still seems to work well.

4.5 Combined deterministic-stochastic identifica-tion

The combined deterministic-stochastic common denominator estimator estimatesboth the deterministic model and the color of the noise on the primary data i.e.a polynomial describing the color of the noise. For IO data driven identification,the algorithm can be interpreted in OMAX framework by assuming that the noisepresent on the response measurements is caused by unmeasurable stochastic forces.


Bo(zk)

A(zk)-Fb(ωk)

To(zk)

A(zk)-1

@@@R h+ -?

Co(zk)

A(zk)

?

Eo(ωk)

Yob(ωk)

Figure 4.4: CLSF IO model


Consider the situation where Nb measured time blocks yob(t) and fb(t) are givento start from. (All further developed formulas are written for Ni = 1 for reason ofsimplicity). For each output o and block b, with Yo,b and Fb respectively the DFTspectra of yo,b(t) and fb(t), the input-output relation is given

Yo,b(ωk) =Boi(zk)

A(zk)Fb(ωk) +

To,b(zk)

A(zk)+

Co(zk)

A(zk)Eo,b(ωk) (4.56)

with Eo,b(ωk) unknown Gaussian white noise. The combined least-squares frequency-domain (CLSF) estimator will estimate the system parameters A(z), Bo(z) andCo(z) simultaneously with the initial and final conditions Tob(z). This Gaussian

contribution in the vibration response Co(zk)A(zk) Eo,b(ωk) can be physically interpreted

as the contribution from unmeasurable random forces [22], [23]. The CLSF-IOmethod, visualized in Figure 4.4, forces a parametric model on the input-outputmeasurements in such a way that the residues Eo,b(ωk) (in our case the unmeasuredforces) will be white noise. The vibration response consists in three contributionsrespectively from the measurable forces, from the noise and unmeasurable forcesand from the transients. Identification of this model is closely related to the iden-tification process of ARMAX models in the time-domain where the minimizationof the so-called prediction error leads to the model parameters [67]. Minimizingthe cost function

lCLSFIO (θ) =

No∑

o=1

Nb∑

b=1

Nf∑

k=1

|Eo,b(ωk)|2 (4.57)

4.5. Combined deterministic-stochastic identification 91

with

Eo,b(ωk) =A(zk)Yo,b(ωk) − Bo(zk)Fb(ωk) − To,b(zk)

Co(zk)(4.58)

leads to the estimates. In [98] the theoretical background is discussed for SISOsystems. Special attention must be paid to ensure numerical stability during theestimation process, since this CLSF IO cost function can decrease be increasing thepolynomials Co(zk). By rejecting the zeros of Co(zk) into the unity circle duringeach iteration numerical stability can be enforced [22]. Since only the magnitudeof the polynomial Co(ωk) matters in the cost function, only the magnitude of themode shapes can be extracted from the stochastic contribution in the vibrationresponse. As a result mode shapes of modes, only excited by the unmeasurableforces can only by identified in magnitude.

Gauss-Newton optimization

Similar as the MLE cost function, the CLSF cost function is the sum of thequadratic equation errors over the different outputs. A Gauss-Newton optimiza-tion is required, since the equation error is again non-linear in the parameters. Ingeneral the same procedures as explained for the LSCF and ML estimators canbe used to speed up the algorithm and to reduce the memory requirements. Thestructure of the normal equations, the Toeplitz structure of the submatrices andthe fast calculation of the entries by the FFT algorithm can be used to obtain apractical algorithm in term of memory requirements and calculation time. Morespecific details about the implementation of the algorithm and some applicationscan be found in [23], [109]. The starting values for the coefficients of the polynomi-als A(z), Bo(z) and Tob(z) are obtained from the IO data driven LSCF algorithmor in the more noisy case from the IO data driven ML algorithm. The startingvalues for the coefficients of the polynomials are assumed to be equal to the de-nominator coefficients corresponding to the case of white noise on the responses.During the optimization, the model has the freedom to shape the white noise tocolored noise. In [98] it is shown that to be consistent the method requires a uni-form frequency grid covering the halve unit circle in the case of real coefficients i.e.Re(

∑Nk=1 zk) = 0 and the complete unit circle in the case of complex coefficients

i.e.∑N

k=1 zk = 0. Therefore in practice, when processing a frequency band, thefrequencies are re-scaled to fulfill this requirement. Unfortunately this requirementintroduces model errors. Therefore, overmodelling is required and often only 90%of the halve or complete unit circle is covered to make a trade-off between modelorders and consistency. Notice that the ML estimator does not require a uniformfrequency grid and full coverage to be consistent.



The strong analogy between FRF and IO data driven identification makes theimplementation of the IO CLSF algorithm also applicable to process FRF data.In general the CLSF estimator is preferred to process noisy FRFs, if no a prioriknown noise information is available.

4.6 Comparison between common denominator basedalgorithms

Different criteria are evaluated to draw a relative comparison between the differentfrequency-domain estimators based on a common-denominator model.

• How fast are the estimators?

• How well can the estimators deal with process noise?

• How well can the estimators deal with measurement noise?

• How robust are the estimators?

• How do the estimators deal with short data sequences?

• Is the information in the data maximally exploited?

• How much data must be stored?

CRITERIUM LSCF IO MLCF IO CLSF IO LSCF FRF MLCF FRF CLSF FRF

computation speed +++ ++ + +++ ++ +process noise −− ++ ++ − ++ ++

measurement noise −− +++ + − +++ ++robustness (convergence) ++ ++ + ++ ++ ++

short data sequences ++ ++ ++ + + +data exploitation − − ++ − − +

data storage (for Nb > Ni) − − − + + +

The most important conclusions are that LSCF methods are fast, but fail in pres-ence of large noise levels. MLE methods are robust and can deal with noise. Animportant benefit is the availability of uncertainty bounds on the estimates, whichare useful to detect mathematical poles [141]. However, the major drawback isthat ML methods require long data sequences to determine the a priori needednoise information. Nevertheless, even if no noise information is a priori known,the ML algorithm reduces to a NLLS, which is still consistent. However, slowerthan the ML algorithm, since more parameters have to be estimated, the CLSF

4.7. Mixed LS-TLS, SK, IQML, WGTLS and BTLS estimators 93

algorithm is still relative fast. Their major advantage is that they can deal withnoise even in the case that only short data sequences are available and their com-bined interpretation in the OMAX framework. Important to mention is that allmethods can be further speed up in the case that the leakage and transient ef-fects are neglected, although this can introduce small bias errors and increases theuncertainty on the estimates.

4.7 Mixed LS-TLS, SK, IQML, WGTLS and BTLSestimators

Finally, it should be noted that many other variants exist to estimate a common-denominator model starting from both FRF and IO data. The mixed LS-TLSapproach estimates the denominator coefficients from the compact normal equa-tions forcing a L2 norm fixed to 1 as a constraint on the denominator coefficients.The Sanathanan-Koerner approach solves the weighted LS equations iteratively bytaking the weighting function for iteration j Wn(ωk) = Aj−1(zk) with Aj−1(zk)the denominator coefficients from the previous iteration j − 1 [103]. The Itera-tive Quasi Maximum Likelihood (IQML) estimator differs from the Sanathanan-Koerner approach since it takes into account the noise information in its weightingWj,oi(ωk) = Aj−1(zk)σHoi

(ωk). Finally, the consistent Generalized Total LeastSquares (GTLS) and the consistent iterative Bootstrapped Total Least Squares(BTLS) solvers can be applied to estimate the common-denominator model [131].Both the GTLS and BTLS are discussed in more detail in paragraph 5.2.5 for theestimation of a RMFD model. A theoretical overview and comparison of the dif-ferent solvers is given in [93] and more applied to estimate common-denominatorsfor modal analysis in [131].

4.8 Simulation and Measurement examples

Both the LSCF and ML common-denominator estimator have proven their appli-cability for both experimental and operational modal analysis. In [46], [116], [115]both the LSCF and ML estimators starting from FRF data are applied to severalstructures like e.g. noisy flight flutter measurements and automotive parts. TheirIO data driven LSCF and ML counterparts are presented and applied in [136]. ABTLS IO data driven implementation is used to estimate the modal parametersfrom a time-varying system in [134]. In [40], [50], [82] the ML estimator is appliedto operational modal analysis starting from auto and cross power densities. Thesimulations and examples in this chapter are mainly focussed on the extended FRFmodels and the combined stochastic-deterministic estimator.


4.8.1 Simulations

The extended LSCF estimator versus the classical LSCF estimator

The same 7-DOF system as explained in paragraph 3.6 is now used for the simu-lations. All simulations are performed in the time domain in order to be able tomodel the transient effects. The structure is excited by a Gaussian random noisesignal at mass 5. The vibration responses and force signals are divided in 4 blocksto estimate the FRFs. In the next step, the modal parameters are estimated fromthese FRFs. For this process a distinction is made between three different cases:

• The FRFs are estimated by the H1 estimator without the use of a windowand the modal parameters are estimated by the classic LSCF estimator (i.e.no transient polynomials are taken into account). This procedure is denotedas ’LSCF No window’.

• The second estimation procedure, denoted as ’LSCF Hanning’, is the mostoften used approach. FRFs are derived by the H1 estimator with the use ofa Hanning window and the parameters are estimated by the classical LSCF.

• The third procedure, denoted as the ’Extended LSCF’ uses an H1 estimatorwith no window and takes into account the initial and final conditions ofevery block by considering the signals F t

bi(ωk).

Figure 4.5 clearly shows the quality of the estimated damping ratios for the threeapproaches for a decreasing block size from 40s to 2s corresponding with a fre-quency resolution from 0.025Hz to 0.5Hz. It is clear that the first approach resultsin the worst results, while the second approach for long blocks in combination witha Hanning window still ends up with acceptable estimates. Nevertheless, takinginto account the contribution of the initial and final conditions by modelling anextra transient polynomial for each block results in accurate estimates for thedamping ratios. Figure 4.6 (a) and (b) show the FRF estimated from 4 blocks ofeach 8s when using a Hanning and rectangular window, while (c) and (d) illustratethe synthesized FRF and error (difference) with the exact FRF for respectivelythe classic LSCF and the extended LSCF algorithm. It is clear that the extendedLSCF estimator results in more accurate results, although the primary FRF datashown in figure 4.6 (b) for the extended LSCF looks more noisy than the primaryFRF data shown in figure 4.6 (a) for the classic LSCF. Nevertheless, the noisycontribution on the FRFs can perfectly be described by taking into account thecontribution of the initial and final conditions.

4.8. Simulation and Measurement examples 95

40 32 24 16 8 4 2

−7

−6

−5

−4

−3

−2

−1

0

1

Block length (s)

Dam

ping

rat

io (

%)

(a)

40 32 24 16 8 4 2

0

0.5

1

1.5

2

Block length (s)

Dam

ping

rat

io (

%)

(b)

40 32 24 16 8 4 2

0.5

1

1.5

Block length (s)

Dam

ping

rat

io (

%)

(c)

Figure 4.5: Damping ratios estimated by (a) the classic LSCF (rectangular window),(b) by the classic LSCF (Hanning window) and (c) the extended LSCF as a function ofthe block length. (∗: estimated damping ratios, dotted line: exact damping ratios)

The combined stochastic-deterministic common denominator estimator:Flight flutter testing

The benefits of using of combined stochastic-deterministic approach is shown in theframework of the EUREKA-FLITE project, where new methods were developedand applied to the analysis, validation and interpretation of aircraft structuraldynamics [132]. The main focus of the project is on the identification of modalparameters from flight test data. Aircraft and winged-launch vehicles must be freefrom aerodynamic instabilities such as flutter to ensure safe operation. Flutter isa dynamic instability that results from the coupling of aerodynamic, elastic andinertial forces acting on the structure. During flight the structure extracts energyfrom the airstream. At speeds larger than the critical airspeed, the energy dissi-pated by the available structural damping is less than the energy injected by theairstream and the motion becomes divergent. Airworthiness regulations requirethat a full-scale aircraft is demonstrated free from flutter by a flight flutter test.In these tests, natural frequencies and modal damping ratios are estimated fordifferent flight conditions. Most common approaches track the damping ratios of


5 10 15 20 25 30 35 40−140

−130

−120

−110

−100

−90

−80

−70

−60

Freq. (Hz)

Am

pl. (

dB)

(a)

5 10 15 20 25 30 35 40−140

−130

−120

−110

−100

−90

−80

−70

−60

Freq. (Hz)

Am

pl. (

dB)

(b)

5 10 15 20 25 30 35 40

−240

−220

−200

−180

−160

−140

−120

−100

−80

−60

Freq. (Hz)

Am

pl. (

dB)

(c)

5 10 15 20 25 30 35 40

−240

−220

−200

−180

−160

−140

−120

−100

−80

−60

Freq. (Hz)

Am

pl. (

dB)

(d)

Figure 4.6: FRF estimated by (a) the H1 estimator and Hanning window and (b) bythe H1 estimator and rectangular window. Synthesized FRF and error (difference) by(c) the classic LSCF (d) by the extended LSCF. (full line: FRF, dotted line: error

the different flight conditions, which are then extrapolated in order to determinewhether it is save to proceed to the next flight point as shown in figure 4.7. Flut-ter can occur when one of the damping values tends to become negative. Thespeed at which such an instability takes place is called the flutter speed and isone of the most important design parameters for an aircraft wing. Before start-ing the flight tests, ground vibration tests as well as numerical simulations andwind tunnel tests are used to get some prior insight in the problem. Flight fluttertesting continues to be a challenging research area because of the concerns withcosts, time and safety in expanding the envelope of new or modified aircrafts. Theaerospace industry desires to decrease the flight flutter testing time for practical,economical and safety reasons partially by improving the accuracy and reliabilityof the parameter estimation process. The costs of test-flights are enormous andhence as many flight conditions as possible should be verified in one single flight.Moreover flight flutter tests can be highly dangerous even when approached withcaution. An aircraft represents a huge investment in terms of time and money anda flutter occurence can be radically destructive. Several fatal cases are reportedin literature. An important goal of the FLITE project is to reduce the required


Figure 4.7: Overview of modal analysis during flight flutter testing

test time and to improve the accuracy of the damping ratio estimates by means ofimproved system identification techniques. There are numerous system identifica-tion methods that allow the estimation of the modal parameters from a vibratingsystem, but not all of them can deal with flight flutter data. In-flight test datais typically characterized by short time records and noisy measurements due tothe unmeasured atmospheric turbulent forces. Many traditional methods used inmodal testing work well with clean data, but, as the noise increases and measure-ment time decreases, the accuracy of the flutter parameter determination rapidlydegrades, especially for the damping ratio estimates. Increasing the level of theartificial applied force to increase the signal-to-noise ratio is no option, since theresponse level is limited to structural integrity or comfort reasons. Furthermore,non-linear effects will appear and the measured working point will differ fromoperational flight conditions. However, the use of specially designed broadbandsignals as multisines and chirp signals can improve the quality of the measurements[12], [47]. Many traditional flight flutter techniques start from averaged data likefrequency response functions (FRFs) or impulse response functions (IRFs), sinceusually there is an attempt to reduce the influence of the noise level on the outputsby collecting many data blocks. However, this increases the required flight time ata fixed flight condition, which is adversely to the desired procedures. The state-of-the-art of flight flutter testing techniques in aircraft industry has been extensivelyreviewed in [29], [65] and [10].


Today many research is focussed on operational modal analysis, where thestructure is excited by ambient noise excitation such as e.g. atmospheric turbu-lence [90], [50] and [5]. Although these output-only methods have been used withsome success, there are several disadvantages. The turbulence is often not intenseenough and excites usually only the lower modal modes. Furthermore, output-onlymethods need long data records to obtain accurate parameter estimates and flighttime is lost during the search for sufficient turbulence levels. [60].

Figure 4.8: Aircraft and the grid used for the GVT

To compare the CLSF IO estimator with other approaches simulations weredone to check for bias errors and to compare the efficiency. The simulations arebased on a modal model extracted from a ground vibration test (GVT). Figure4.8 illustrates the actual airplane and the grid used for the GVT. The simulationswere done in the continuous-time with a state space model based on the first sixmodes, visualized in Figure 4.9. In all the simulations one random noise forcef(t) is applied on a discrete point perpendicular to the surface of the left wing. Asample frequency of 256Hz was used and 16K samples, corresponding to a totalmeasurement time of 64s, were simulated for each test.

To simulate the influence of the atmospheric turbulence, spatially correlatedwhite noise sources were acting on the nose, wings and tail of the aircraft. The levelof atmospheric turbulence is measured as the ratio of the RMS values between the


(a) (b)

(c) (d)

(e) (f)

Figure 4.9: Different mode shapes (from a to f: increasing natural frequency)

mean stochastic contribution in the vibration responses over the different outputsand the mean deterministic contribution over all outputs. Simulation flight testswere carried for levels of atmospheric turbulence of 7, 14, 21, 28, 35 and 42%(10 different runs for each turbulence level). The ML FRF driven, ML IO drivenand CLSF IO driven algorithm were compared for the different simulations. Figure4.10 shows the mean estimated value and 68% confidence bars of the damping ratioestimates (for the 10 runs) of each mode for an increasing level of the turbulence.Since only a limited amount of data is considered the FRFs estimates are very noisyand biased due to leakage effects. Since a classical LSCF FRF estimator failed, the


1 2 3 4 5 61.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

turbulence level

dam

ping

rat

io (

%)

Damping ratio: mode 1

1 2 3 4 5 61.4

1.6

1.8

2

2.2

2.4

2.6

2.8

turbulence level

dam

ping

rat

io (

%)


1 2 3 4 5 6−10

−8

−6

−4

−2

0

2

4

6

8

turbulence level

dam

ping

rat

io (

%)


0 1 2 3 4 5 6 7−8

−6

−4

−2

0

2

4

6

8

turbulence level

dam

ping

rat

io (

%)


1 2 3 4 5 62

2.1

2.2

2.3

2.4

2.5

2.6

2.7Damping ratio: mode 5

turbulence level

dam

ping

rat

io (

%)

1 2 3 4 5 6−1

−0.5

0

0.5

1

1.5

2

2.5

3Damping ratio: mode 6

turbulence level

dam

ping

rat

io (

%)

Figure 4.10: Comparison of the damping ratio estimates (∗ : CLSF IO, : NLLS IO, : MLE FRF, – : exact parameter)

classic ML estimator starting from FRFs was used. This ML FRF method suffersfrom a bias on the damping ratios of the low frequent modes due to the leakageeffects in the FRF estimates. Furthermore the variance on the ML FRF estimatesis large even in the case of low turbulence levels. (The FRF driven ML did nottake into account the initial and final conditions of each block). From these resultsone can conclude that the FRF based methods are not recommended to use in thecase of short time records. Since the IO data driven methods easily model thetransient effects they do not suffer from bias errors in the case of low turbulence


levels. The CLSF IO method is also compared to a ML IO data driven estimator,under the assumption of white output noise. Although the ML IO estimator isconsistent under the assumption of only output noise, the results show a bias forincreasing noise levels. This can be explained by local minima. The CLSF IO

2 4 6 8 10

−40

−30

−20

−10

0

10

Freq. (Hz)

Am

pl. (

dB)

(a)

3 4 5 6 7 8 9 10

−50

−40

−30

−20

−10

0

Freq. (Hz)A

mpl

. (dB

)

(b)

Figure 4.11: Synthesized energy spectra (o: measured output spectra, full line: stochas-tic + deterministic energy, dashed line: deterministic contribution (artificial force), dot-ted line: stochastic contribution (turbulent forces))

resulted in unbiased estimates with the smallest uncertainty for all modes evenin the presence of high levels of turbulent forces. Interesting to notice is thatthe CLSF IO method estimates the damping ratios of the 3rd, 4th and 5th modevery accurately, although these modes are weakly excited by the artificial force.However, the combined approach takes advantage from a better data exploitation,since its was noted that these modes are well excited by the turbulent forces (thesemodes are mainly vibrating at the tail, nose and cabin as can be seen in Figure 4.9and thus less good excited by the artificial force). Figure 4.11 gives a comparisonbetween the deterministic and stochastic contribution in the response spectra fromwhich it is clear that modes 3,4 and 6 at respectively 4.6Hz, 5.1Hz and 8.4Hz havean important stochastic contribution from the turbulent excitation.

Other examples of the CLSF algorithm applied on measurements of a vibra-ting plate excited by a measurable force and unmeasurable acoustic excitation arepresented in [22], while [109] illustrates the application for a coupled structural-acoustical problem.

4.8.2 Measurement on an aluminium plate

An aluminium plate excited by a mini shaker was measured in 39 points with ascanning laser vibrometer. To obtain very accurate reference measurements thestructure was excited by 10 periods of pseudo random noise (each period contained


200 400 600 800 10000

5

10

15

20

25

Freq. (Hz)

mod

el o

rder

200 400 600 800 1000−60

−40

−20

0

20

40

Freq. (Hz)

Am

pl. (

dB)

exact FRFsynthezised FRF

Figure 4.12: H1 estimate (Hanning), stabilization diagram and synthesized FRF forthe classical LSCF method

200 400 600 800 10000

5

10

15

20

25

Freq. (Hz)

mod

el o

rder

200 400 600 800 1000−60

−50

−40

−30

−20

−10

0

10

20

30

40

50

Freq. (Hz)

Am

pl. (

dB)

exact FRFsynthezised FRF

Figure 4.13: H1 estimate (rectangular), stabilization diagram and synthesized FRF forthe Extended LSCF method

213 samples) and measured with a sample frequency of 213Hz. This results inleakage free FRFs with a frequency resolution of 1Hz and a total measurementtime of 10s for each measured point. These FRFs are considered as the ’exact’FRFs. In a second set of measurements the plate was excited by 211 samplesrandom noise and a sample frequency of 213Hz resulting in a measurement timeof 0.25s for each measured point. These measured sequences were divided into 2blocks to estimate the FRFs by the H1 technique leading to frequency resolutionof only 8Hz. It is clear that these FRFs are ’difficult data’ to handle because,

• leakage effects are present, since no periodic signals were used,

• only a very low frequency resolution is obtained

• the measurements are noisy, since only 2 blocks were averaged

The goal is to process this ’difficult’ data by both the classical LSCF andextended LSCF and to compare the synthesized models from this data to the’exact’ FRFs. Figures 4.12 and 4.13 show respectively the stabilization chart and


400 500 600 700

−30

−25

−20

−15

−10

−5

0

5

10

15

20

Freq. (Hz)

Am

pl. (

dB)

exact FRFExtended log MLEExtended LSCF

500 550 600 650 700

−30

−20

−10

0

10

20

30

Freq. (Hz)

Am

pl. (

dB)

exact FRFExtended log MLEExtended LSCF

Figure 4.14: Comparison between the Extended LSCF and Extended ML algorithm

a synthesized FRF for the classic and extended LSCF algorithm. If the leakageis taken into account in the model by F t

bi(ωk), the obtained results are bettershown by the synthesized FRF. Comparison of both stabilization diagrams clearlyindicates that the extended LSCF method stabilizes for each physical pole, whichwas not the case for the Classical LSCF (*: stable pole, +: unstable pole).

The extended LSCF outperforms the classical LSCF for these short time mea-surements, but the obtained models can still be improved by the extended MLalgorithms starting from the initial estimates by the extended LSCF. Figure 4.14clearly shows the more accurate models obtained by the logarithmic Advanced MLtechnique under the assumption of white noise on the outputs (i.e. logarithmicestimator with σHoi

(ωk) = |Hoi(ωk)|).

4.9 Conclusions

In this chapter the extensions of the LSCF and ML algorithms were developedto take into account the initial and final conditions of each block used by the H1

estimator. Special attention was paid to the implementation to results in a fast andmemory efficient algorithm by using the structure in the matrix equations. Therelationship between the implementation of FRF and IO data driven algorithm wasdiscussed. Finally, a combined deterministic-stochastic algorithm was proposedand applied on simulations in the framework of in flight test on aircrafts. Otherexamples illustrated the use of the extended implementation of the FRF drivenalgorithms.


4.10 Appendix

Assume that the discrete-time signals y(r) and f(r) are exactly known in the timeinterval [0, (N −1)Ts] and are unknown outside the interval with N the number ofsamples and ∆t the sample period. The discrete input and output signals satisfythe following difference equation:

n∑

m=0

an−my((r − n)∆t) =

n∑

m=0

bn−mf((r − m)∆t) (4.59)

Taken the discrete fourier transform (DFT) defined by

Y (zk) = DFT (y(r)) =1√N

N−1∑

r=0

y(r∆t)z−rk (4.60)

with zk = ejk/N . Consider the DFT of y(r − 1)

DFT (x(r − 1)) =1√N

N−1∑

r=0

y((r − 1)∆t)z−r

=z−1

√N

N−1∑

r=0

y(r∆t)z−r + y(−∆t) − zNy(N∆t)

= z−1Y (z) +y(−∆t) − y(N∆t)√

N(4.61)

since zNk = 1. The terms y(−1)−y(N+1)√

Nare often neglected in frequency-domain

identification methods. It is clear that these terms appear due to the fact thatinitial and final conditions are not equal. This is the case when non-periodic orarbitrary excitation signals are used or when periodic measurements are corruptedwith transients. In the case purely periodic signals (y(−∆t) = y(N∆t)) or timelimited signals (y(−∆t) = y(N∆t) = 0) are considered these terms disappear.Analog to Equation 4.61 we can show that for the more general case that

DFT (y(r − n)) = z−nY (z) +1√N

n∑

r=1

(y(−r∆t) − y((N − 1 + r)∆t))zr−n (4.62)

Taken the DFT of both sides of equation 4.59 and using the expression of Equation4.62 results in

A(z)Y (z) = B(z)F (z) + T (z) (4.63)

4.10. Appendix 105

with

A(z) =

n∑

m=0

amz−m (4.64)

B(z) =

n∑

m=0

bmz−m (4.65)

T (z) =

n−1∑

m=1

m∑

r=1

bm(f(−r∆t) − f((N − 1 + r)∆t))zr−m (4.66)

−n−1∑

m=1

m∑

r=1

am(y(−r∆t) − y((N − 1 + r)∆t))zr−m (4.67)

=n−1∑

m=0

tmz−m (4.68)

The extra polynomial T (z) takes into account the influence of the final and initialconditions. A more profound study can by found in [99].


Chapter 5

Identification of right andleft matrix fractionpolynomial models

This chapter presents several implementations of modal parameter estimation al-gorithms based on a generalization of the fast common-denominator based algo-rithms to identify right and left matrix fraction description models. It is shownthat, these algorithms results in more accurate modal parameters in the multipleinput case than their common-denominator counterparts. Both a poly-referenceleast squares and poly-reference maximum likelihood estimator are proposed basedon a right matrix fraction model. Next, a fast maximum likelihood estimator isdeveloped, which uses a scalar frequency weighting. Finally a left matrix fractionbased least squares estimator is proposed for both IO data and FRF driven iden-tification. It is shown that the IO data driven implementation can be interpretedas a combined stochastic-deterministic estimator. The applicability of the matrixfraction descriptions is shown by simulations and several real-life experiments anda comparison is made with the common-denominator based algorithms.

107

108Chapter 5. Identification of right and left matrix fraction polynomial models

5.1 Introduction

In the previous chapter the identification of common-denominator models was dis-cussed starting from both FRF and IO data. In this chapter the identification ofboth right and left matrix fraction polynomial models is introduced and discussedresulting in several well-performing modal analysis estimators. It is shown thata similar methodology to solve the identification problem in a time and memoryefficient way as developed in the previous chapter can be applied. A LS estimatorand mixed LS-TLS estimator is discussed, which fits a right matrix fraction de-scription model trough the FRF data. This, so-called poly-reference LSCF is todaycommercially known as the LMS PolyMAX estimator [91], [69]. Next, the ML es-timator, which optimizes the poly-reference LSCF estimates by an Gauss-Newtonalgorithm, is discussed. Unfortunately the initial ML implementation results in aslower implementation and therefore a fast version is developed by using a scalarfrequency weighting. Finally, a LS estimator is proposed based on a left matrixfraction description model and starting from IO data its combined interpretationis discussed in the OMAX framework.

The interest in RMFD models for modal analysis can be explained by thegrowing use of multiple input test setups for higher damped structures like fullytrimmed cars and airplanes. In fact, the well-known Least Squares Complex Ex-ponential (LSCE) modal parameter estimator can be seen as RMFD of whichonly the denominator coefficients are estimated in the time-domain. The devel-opment of frequency domain algorithms in the Acoustics and Vibration ResearchGroup of the Vrije Universiteit Brussel for modal analysis started with apply-ing common-denominator models for MIMO test setups. It was found that theidentified common-denominator models closely fitted the measured frequency re-sponse functions (FRFs). Nevertheless, when converting this model to a modalmodel by reducing the residues to a rank-one matrix using the singular valuedecomposition (SVD), the quality of the fit decreased [116]. This drawback ofcommon-denominator based models tends to be more important for highly dampedstructures, since in this case the natural frequencies are less explicit. Another the-oretically associated drawback of a common-denominator model is that closelyspaced poles will erroneously show up as a single pole. However, in practice thisproblem is solved by increasing the model order, which is typically done in EMA.It can be stated that the common-denominator model does not forces rank oneresidue matrices on the data and uses this extra freedom, which is not present inthe modal model, to obtain a close fit of the data. These two reasons motivate apoly-reference version of the LSCF method using a so-called RMFD. This poly-reference LSCF was initially presented in [48] and applied on industrial examplesin [86], while [101] discusses a RMFD LS estimator in the continuous-time. In thisapproach both the system poles and the participation factors are available fromthe denominator coefficients. The main advantage of the poly-reference RMFDimplementations is that an SVD decomposing the residues in mode shapes and

5.2. Frequency-domain identification of RMFD models 109

participation factors can be avoided. As a result, no loss of quality is encounteredfrom the conversion to the modal model and closely coupled poles can be sepa-rated more easily. A disadvantage of a RMFD is that the model order can onlybe increased in steps of Ni (the number of input forces) and that the method ismore sensitive to data inconsistencies due to e.g. mass loading effect over differentpatches [15].

Finally the LMFD, which forces also rank one residues on the measurements,is presented as an IO data and FRF driven estimator. In [63] a profound study isgiven for the LMFD model identification in the Laplace domain (i.e. a continuous-time model in the frequency domain) starting from FRF data.

5.2 Frequency-domain identification of RMFDmodels

5.2.1 Poly-reference Weighted Least Squares ComplexFrequency-domain estimator

The relationship between output o and all inputs can be modelled in the frequencydomain by means of a right matrix fraction description given by

Ho(ωk) = Bo(zk)A−1(zk) (5.1)

with Bo(z) ∈ C1×Ni the numerator row-vector polynomial and A(z) ∈ C

Ni×Ni

the denominator matrix polynomial defined by

Bo(zk) =

n∑

j=0

Bojzjk A(zk) =

n∑

j=0

Ajzjk (5.2)

The matrix coefficients Aj and Boj are the parameters to be estimated. Thesepolynomial coefficients are grouped together in one matrix

θ =

β1

...βNo

α

βo =

Bo0

...Bon

αo =

A0

...An

(5.3)

The linear LS equation error is obtained by right multiplication of Eq. 5.1 byA(zk) and taking the difference between the left and right part resulting in

Eo(ωk) = W−1o (ωk) (Bo(zk) − Ho(ωk)A(zk)) (5.4)


with W−1o (ωk) an additional scalar frequency-weighting for each output. The

weighted linear LS problem is found by minimizing the cost function given by

lp−LSCFFRF (θ) =

No∑

o=1

N∑

k=1

tr(

EHo (ωk)Eo(ωk)

)

(5.5)

=

No∑

o=1

N∑

k=1

Eo(ωk)EHo (ωk) (5.6)

which corresponds with solving

Jθ = 0 (5.7)

with J the Jacobian matrix given by

J =

Γ1 0 . . . 0 Υ1

0 Γ2 . . . 0 Υ2

......

. . ....

...0 0 . . . ΓNo

ΥNo

(5.8)

with

Γo =

W−1o (ω1)

[

1 z1 . . . zn1

]

W−1o (ω2)

[

1 z2 . . . zn2

]

...W−1

o (ωN )[

1 zN . . . znN

]

∈ CN×(n+1) (5.9)

Υo =

−W−1o (ω1)

[

1 z1 . . . zn1

]

⊗ Ho(ω1)

−W−1o (ω2)

[

1 z2 . . . zn2

]

⊗ Ho(ω2)...

−W−1o (ωN )

[

1 zN . . . znN

]

⊗ Ho(ωN )

(5.10)

∈ CN×Ni(n+1)

Note that the cost function Eq. is equivalent to

lp−LSCF (θ) = tr(

θHJHJθ)

(5.11)

Similar as in the common-denominator LSCF implementation, the number of fre-quencies N can be eliminated from the equations dimensions by starting from thenormal equations given by

JHJθ = 0 (5.12)

R1 · · · 0 S1

.... . .

...0 RNo

SNo

SH1 · · · SH

No

∑No

o=1 To

β1

...βNo

α

= 0 (5.13)


with Ro = ΓHo Γo ∈ C

(n+1)×(n+1), So = ΓHo Υo ∈ C

(n+1)×Ni(n+1) and To =

ΥHo Υo ∈ C

Ni(n+1)×Ni(n+1). Elimination of the βo coefficients

βo = −R−1o Soα (5.14)

results in the so-called reduced normal equations

No∑

o=1

(

To − SHo R−1

o So

)

α = Mα = 0 (5.15)

with M a square Ni(n + 1) matrix. Similar as for the reduced normal equationsfor the common-denominator model a constraint must be imposed to remove theparameter redundancy. This can be done, for instance, by imposing that one ofthe denominator coefficients is equal to the unity matrix INi

e.g. for An = INi,

for which the least squares solution is given by

α =

[ −M−1[1:nNi,1:nNi]

M[1:nNi,nNi+1:(n+1)Ni]

INi

]

(5.16)

and backwards substituting the α coefficients in Eq. 5.14 results in the β coeffi-cients. By the use of a singular value decomposition the reduced normal equationscan be solved in TLS sense resulting in a mixed LS-TLS solution for the original(full) normal equations Eq. 5.13. The submatrices Ro, So and To have a predefinedblock Toeplitz structure given by

[Ro]rs =

N∑

k=1

|Wk(ωk)|−2zs−rk (5.17)

[So]rj =N∑

k=1

|Wk(ωk)|−2Ho(ωk)zs−rk (5.18)

[To]ij =

N∑

k=1

|Wk(ωk)|−2HHo (ωk)Ho(ωk)zs−r

k (5.19)

with i = [(r−1)Ni +1 : rNi] j = [(s−1)Ni +1 : sNi] for both r, s = 1, 2, . . . n+1.Taken into account the Toeplitz structure and using the FFT algorithm to calculatethe matrix entries reduces both the memory requirements and computation time.The gain in calculation time, if 15log2(N) < 32n [131], by the use of the FFTalgorithm is typical less than for the common-denominator LSCF estimator, sincefor the same number poles the model order n is Ni times smaller.

5.2.2 Poly-reference Maximum Likelihood Estimator

By taking the uncertainty information on the estimated FRF data into accountduring the parametric identification, more accurate parameter estimates can be


obtained compared to the LS approach. The ML estimator minimizes the ML costfunction by a Gauss-Newton algorithm starting from initial parameters estimatesfrom the p-LSCF algorithm.

In this paragraph, it is shown that the use of the ML weighting requires thevec operator to transform the matrix of unknown parameters θ into a vector ofunknown parameters. This will increase the memory requirements and calculationtime, since the dimensions of the submatrices of the normal equations increase bya factor Ni. This type of implementation is defined as a vec implementation usinga matrix weighting.

Under the assumption that the noise on the FRFs is complex normally dis-tributed and the noise on the FRFs corresponding to different outputs is uncorre-lated, the (negative) log-likelihood function is given by [21], [37]

lp−mlFRF (θ) =

N∑

k=1

No∑

o=1

E∗o (ωk)C−1

o (ωk)ETo (ωk) (5.20)

with the equation error Eo(ωk) defined by

Eo(ωk) = Ho(ωk) − Bo(zk)A−1(zk) (5.21)

and the covariance matrix of Ho(ωk) defined by

Co(ωk) = E(

∆HHo (ωk)∆Ho(ωk)

)

(5.22)

with ∆Ho(ωk) ∈ C1×Ni the noise on the FRFs corresponding to output o. This

assumption of uncorrelated noise over the different outputs is exactly true in thecase of scanning vibrometer measurements or a roving hammer test. In the casethis noise assumption is violated, the ML estimator is still consistent and only someefficiency is lost. These covariance matrices for each output o can be obtainedfrom the non-parametric FRF identification. Similar as for the ML common-denominator, the weighted equation error is non-linear in the system parametersand a Gauss-Newton optimization algorithm is implemented to minimize the costfunction. The entries of the polynomial coefficients are grouped in a vector givenby

θ =

β1

...βNo

α

βo =

vec(Bo0)...

vec(Bon)

α =

vec(A0)...

vec(An)

(5.23)

and the Jacobian matrix J is defined by

J =

Γ1 0 . . . 0 Υ1

0 Γ2 . . . 0 Υ2

.... . .

...0 0 . . . ΓNo

ΥNo

(5.24)


with

Υo = −

−Co(ω1)−1/2[(Bo(z1)A(z1)

−1) ⊗ A(z1)−T ]

[

IN2izn1 . . . IN2

iz1 IN2

i

]

...

−Co(ωN )−1/2[(Bo(zN )A(zN )−1) ⊗ A(zN )−T ][

IN2izn

N . . . IN2izN IN2

i

]

∈ CNN2

i ×(n+1)N2i (5.25)

Γo =

−Co(ω1)−1/2A(z1)

−1[

INizn1 . . . INiz1 INi

]

...

−Co(ωN )−1/2A(zN )−1[

INiznN . . . INizN INi

]

∈ CNNi×(n+1)Ni (5.26)

The normal equations are given by

R1 · · · 0 S1

.... . .

...0 RNo

SNo

SH1 · · · SH

No

∑No

o=1 To

vec(∆β1)...

vec(∆βNo)

vec(∆α)

=

V1

...VNo

∑No

o=1 Wo

(5.27)

with Ro = ΓHo Γo ∈ C

Ni(n+1)×Ni(n+1), So = ΓHo Υo ∈ C

Ni(n+1)×Ni(n+1)N2i , To =

ΥHo Υo ∈ C

N2i (n+1)×N2

i (n+1), Vo = ΓHo Eo ∈ C

Ni(n+1)×1 and Wo = ΥHo Eo ∈

CN2

i (n+1)×1 and Eo defined by

Eo =

vec(Eo(ω1))...

vec(Eo(ωN ))

(5.28)

Elimination of the βo coefficients

vec(∆βo) = −R−1o (Sovec(∆α) + Vo) (5.29)

results in the so-called reduced normal equations

No∑

o=1

(

To − SHo R−1

o So

)

vec(∆α) = −No∑

o=1

(

Wo − SHo R−1

o Vo

)

(5.30)

Compared to the LS implementation, the dimensions of the reduced normal equa-tions are increased by a factor Ni resulting in a N3

i slower implementation foreach iteration. Similar as for the common-denominator ML estimator, the start-ing values are generated by the poly-reference LSCF estimator and optimized bythe Gauss-Newton implementation of the poly-reference ML estimator. The use ofthe predefined block Toeplitz structure of the submatrices Ro, So and To reducesboth the memory requirements and calculation time. Next, the uncertainties on


the estimated parameters can be estimated from the Cramr-Rao lower bound (theML implementation reaches this lower bound). In the appendix 5.7 it shown howthe uncertainty levels on the estimated poles can be obtained from the covariancematrix on the estimated denominator coefficients.

Similar as for the common-denominator ML estimator a more robust logarith-mic poly-reference ML estimator that is more suited to handle large dynamicalranges is given by the following cost function

lp−ml,logFRF (θ) =

N∑

k=1

No∑

o=1

E∗o (ωk)Clog,o(ωk)−1ET

o (ωk) (5.31)

with the equation error Eo(ωk) defined by

Eo(ωk) = log (Ho(ωk)) −(

Bo(zk)A−1(zk))

(5.32)

and with the covariance matrix defined by

Clog,o =

Ho1 0 . . . 00 Ho2 . . . 0...

.... . .

...0 0 0 HoNi

−H

Co

Ho1 0 . . . 00 Ho2 . . . 0...

.... . .

...0 0 0 HoNi

−1

(5.33)

in analogy with the scalar case (y = var(x) ⇒ var(y) = var(x)|x|2 and with the

submatrices of the Jacobian matrices given by

Υo = −

−Clog,o(ω1)−1/21./(

A−T (z1)NT (z1))

. ∗ [(Bo(z1)A(z1)−1) ⊗ A(z1)−T ]ZN2

i(1)

.

.

.

−Clog,o(ωN )−1/21./(

A−T (z1)NT (z1))

. ∗ [(Bo(zN )A(zN )−1) ⊗ A(zN )−T ]ZN2

i(N)

∈ CNN2

i ×(n+1)N2i (5.34)

Γo =

−Co,log(ω1)−1/21./(

A−T (z1)BT (z1))

. ∗ A(z1)−1ZNi(1)

.

.

.

−Co,log(ωN )−1/21./(

A−T (z1)BT (z1))

. ∗ A(zN )−1ZNi(N)

∈ CNNi×(n+1)Ni (5.35)

with ZN2i(k) =

[

IN2izn

k . . . IN2izk IN2

i

]

and ZNi(k) = [ INiz

nk . . . INizk INi ]

and the operators ./ and .∗ defined as respectively the element-wise division andproduct. Based on this weighting and the vec operator both a poly-reference IQMLand BTLS can be proposed to solve the identification iteratively with a weightingdepending on the parameters estimated in the previous iteration.

5.2.3 Fast Poly-reference Maximum Likelihood Estimator

The estimator proposed in this section uses a scalar weighting resulting in a matriximplementation i.e. the parameter θ are grouped in a matrix with Ni columns


similar as for the poly-reference LSCF and the use of the vec operator is avoided.This results in a N3

i faster implementation, but also in a small loss of efficiencycompared to the scalar implementation of the poly-reference ML with a matrixweighting. Consider the same cost function and equation error as for the poly-reference LSCF

lp−fMLFRF (θ) =

No∑

o=1

N∑

k=1

tr(

Eo(ωk)HEo(ωk))

(5.36)

and

Eo(ωk) = Wo(ωk)−1 (Bo(zk) − Ho(ωk)A(zk)) (5.37)

with the parameter dependent scalar weighting Wo(ωk) now defined by

W 2o (ωk) = E

(

∆Eo(ωk)∆EHo (ωk)

)

= E(

∆Ho(ωk)A(zk)AH(zk)∆HHo (ωk)

)

= tr(

AH(zk)Co(ωk)A(zk))

(5.38)

Since the equation error is non-linear in the parameters a Gauss-Newton approachis used to minimize the cost function. A necessary condition to be consistent isthat the expected value of the cost function is minimal in the exact parameters[98], [94]. As a result, for the fast poly-reference ML this is shown by

E (lp−fMLF RF (θe)) =

N∑

k=1

No∑

o=1

tr(

E((

BHoe(zk) − HH

o AHe (zk)(ωk)

)

(Boe(zk) − Ho(ωk)Ae(zk))))

W 2o (ωk)

(5.39)

where the subindex e indicates the polynomials with exact coefficients. Takinginto account that Ho(ωk) = Hoe(ωk) + ∆Ho(ωk) and Hoe(ωk) = Boe(zk)A−1

e (zk)results in

E (lp−fML(θe)) =N∑

k=1

No∑

o=1

tr(

EHoe(ωk)Eoe(ωk)

)

Wo(ωk)2(5.40)

+tr(

AH(zk)E(


)

A(zk))

Wo(ωk)2(5.41)

=N∑

k=1

No∑

o=1

tr(

EHoe(ωk)Eoe(ωk)

)

Wo(ωk)2+ 1 (5.42)

= 1 (5.43)

and Eoe the equation error in the exact parameters (which is zero) and thus thecost function is minimal in the exact parameters θe. The Jacobian matrix has thesame structure as for the poly-reference ML that uses a matrix weighting shown byEq.5.24. Since the weighting Wo(ωk) is independent of the numerator polynomials


Bo(z) the submatrix Γo is similar as for the poly-reference LSCF case

Γo = −

W−1o (f1)

[

1 z1 . . . zn1

]

W−1o (ω2)

[

1 z2 . . . zn2

]

...W−1

o (fN )[

1 zN . . . znN

]

∈ CN×(n+1) (5.44)

The submatrices Υo need some more attention since the weighting depends on thedenominator polynomial A(z) and thus

Υo =

[

1 z1 . . . zn1

]

⊗ ∂Eo(f1)∂A

[

1 z2 . . . zn2

]

⊗ ∂Eo(f2)∂A

...[

1 zN . . . znN

]

⊗ ∂Eo(fN )∂A

∈ CN×Ni(n+1) (5.45)

with

∂Eo(ωk)

∂A=

∂W (ωk)−1

∂A(Ho(ωk)A(zk) − Bo(zk)) + W

−1 ∂(Ho(ωk)A(zk) − Bo(zk))

∂A(5.46)

and

∂Wo(ωk)−1

∂A= −1

2W−3

o

∂Wo(ωk)

∂A(5.47)

= −1

2W−3

o

tr(

∂AH(zk)Co(ωk)A(zk))

∂A(5.48)

= −1

2W−3

o

(

Co(ωk)A(zk) + CHo A(zk)

)

(5.49)

∂ (Ho(ωk)A(zk) − Bo(zk))

∂A= Ho(ωk) (5.50)

Specified for coefficient An the derivative is given by

∂Eo(ωk)

∂An=

∂Eo(ωk)

∂Aznk (5.51)

The normal equations have a similar structure as for the matrix-weighted poly-reference ML estimator given by Eq. 5.27 and as a result, they also can be reducedto a compact formulation given by Eq. 5.29 and Eq. 5.30 with Ro = ΓH

o Γo ∈C

(n+1)×(n+1), So = ΓHo Υo ∈ C

(n+1)×(n+1)Ni , To = ΥHo Υo ∈ C

Ni(n+1)×Ni(n+1),

Vo = ΓHo Eo ∈ C

(n+1)×1 and Wo = ΥHo Eo ∈ C

Ni(n+1)×1 withET

o = [ETo (f1) ET

o (f2) . . . ETo (fN )]. Compared to the poly-reference ML, the

submatrices become Ni times smaller in the dimension of the rows and columnsresulting in N3

i faster algorithm, which can be significant in time for multiple inputtest with e.g. Ni > 3. Similar as for the vec implementation, a logarithmic versionof the fast ML algorithm can be developed to increase the robustness.


5.2.4 Fast Poly-reference IQML

A fast IQML estimator based on the scalar weighting functionW 2

o (ωk) = tr(

AH(zk)Co(ωk)A(zk))

works iteratively by minimizing in iteration jthe cost function given by

lp−fIQML =

No∑

o=1

N∑

k=1

(Ho(ωk)Aj(zk) − Boj(zk)) (Ho(ωk)Aj(zk) − Boj(zk))H

tr(

AHj−1(zk)Co(ωk)Aj−1(zk)

) (5.52)

with Aj(zk) and Boj(zk) the polynomials estimated in iteration j. Each iterationj can be considered as a weighted poly-reference LSCF with a frequency weight-ing |Wo(ωk)|2 = tr

(

AHj−1(zk)Co(ωk)Aj−1(zk)

)

. In [94], it is shown that the IQMLimproves the estimates of the least squares estimator, since its cost function con-verges to the fast ML cost function. Nevertheless, it can not be shown that theestimator is consistent and efficient.

5.2.5 Poly-reference WGTLS and fast BTLS Estimator

Consider the LS formulation in paragraph 5.2.1, which can be shortly denoted asJθ = 0. The (Weighted) Generalized Total Least Squares (WGTLS) solution forthis identification problem is then given by [54]

arg minJ,θ‖(J − J)S−1J ‖2

F subject to Jθ = 0 and θHθ = I (5.53)

The weighting matrix SJ is the square root of the covariance matrix of J i.e.CJ = SH

J SJ = E(

∆JH∆J)

, with ∆J = J − J the noise contribution on theJacobian matrix caused by uncertainty on the FRF data. The GTLS solution θ ofthis estimation problem is a consistent estimate. By elimination of J it is proventhat Eq. 5.53 is equivalent to minimizing the following cost function [95]

lp−WGTLSFRF =

θHJHJθ

θHSHJ SJθ

subject to θHθ = I (5.54)

of which the solution in practice is found by the Generalized Singular Value De-composition (GSVD) of the matrices J and SJ . Since JHJ is equivalent to thenormal matrix given by Eq. 5.13 and CJ = SH

J SJ , the minimization problem canbe rewritten as

lp−WGTLSFRF =

θHQθ

θHCJθsubject to θHθ = I (5.55)

with Q = JHJ . In practice the solution is found by means of a GeneralizedEigenvalue Decomposition (GEVD) that directly follows from the cost function

QV = λCJV (5.56)


where the solution θ is given by the Ni eigenvectors corresponding to the Ni

smallest eigenvalues.

The poly-reference WGTLS implementation is in strong analogy with theWGTLS implementation based on a common-denominator model [133], [138]. Thecovariance matrix on the Jacobian matrix J has the same block structure as thenormal matrix M

CJ = E(

∆JH∆J)

=

CR1· · · 0 CS1

.... . .

...0 CRNo

CSNo

CHS1

· · · CHSNo

∑No

o=1 CTo

(5.57)

under the assumption that the noise over the different output on the FRFs is un-correlated. This is exactly true for both Hammer and Scanning Laser Vibrometermeasurements since for those test setups the different outputs are measured in-dependent in time. For simultaneous accelerometer measurements in the case ofa shaker setup, important correlations can be introduced over the different out-puts [129], [126]. However, in order to obtain a fast and practically applicableimplementation these correlations must be neglected. Nevertheless, the resultsobtained by neglecting the correlations over the outputs, are in general more accu-rate then the classic poly-reference LS solution discussed in paragraph 5.2.1. Sincethe submatrices Γo of the Jacobian matrix defined by Eq. 5.9 are free from noise(under the assumption that the frequency weighting Wo(ωk) is exact) the matricesCRo

= E(

ΓHo Γo

)

and CSo= E

(

ΓHo Υo

)

equal the zero matrix. The subentries ofthe matrices To with i = [(r − 1)Ni + 1 : rNi], j = [(s − 1)Ni + 1 : sNi] for bothr, s = 1, 2, . . . N are given by

[To]ij =[

ΥHo Υo

]

ij=

N∑

k=1

|Wk(ωk)|−2HHo (ωk)Ho(ωk)zs−r

k (5.58)

and the subentries of the CToare calculated by

[CTo]ij =

[

∆ΥHo ∆Υo

]

rs=

N∑

k=1

|Wk(ωk)|−2Co(ωk)zs−rk (5.59)

with Co(ωk) = E(


)

the covariance matrix of the FRFs corre-sponding to output o. It can visually be checked that the matrices CTo

have ablock Toeplitz structure similar as the submatrices of the normal matrix Q. Thegeneralized total least squares problem is then explicitly written as

R1 · · · 0 S1

.

.

.. . .

.

.

.0 RNo SNo

SH1 · · · SH

No

∑Noo=1 To

β1

.

.

.βNoα

= λ

0 · · · 0 0

.

.

.. . .

.

.

.0 0 00 · · · 0 CT

β1

.

.

.βNoα

(5.60)


with CT =∑No

o=1 CTo. As for the LS implementation, the normal equations

can be reformulated to a reduced set of equations by elimination the numeratorcoefficients given by

βo = −R−1o Soα (5.61)

and substitution in last Ni(n+1) rows of the GTLS problem results in a compactGTLS formulation

No∑

o=1

(

To − SHo R−1

o So

)

α = λCT α (5.62)

from which the left-hand side is identical to the LS formulation. Once the coeffi-cients α are estimated, backwards substitution yields the coefficients βo. The useof this predefined block structure of the normal equations results in an N2

o fasterimplementation than solving Eq. 5.60 directly. A stabilization chart can be con-structed by solving the GEVD for an increasing model order of the denominatorpolynomial by increasing the size of M and CT with Ni(n + 1) column and rows

(M =∑No

o=1

(

To − SHo R−1

o So

)

).

An improvement in efficiency is obtained by solving the WGTLS problem iter-atively with the same frequency weighting Wo(ωk) as the fast poly-reference IQMLsolver based on the parameters α estimated in the previous iteration. This ap-proach is called the fast poly-reference BTLS estimator and improves the accuracyin each iteration, while each iteration results in consistent estimates.

5.2.6 RMF description for IO data

The RMFD model is less suitable for IO data driven identification since the modelcan only be linearized for Ni = 1 (which is of course identical to the common-denominator model)

Yo(ωk) = Bo(zk)A−1(zk)F (ωk) (5.63)

For the multiple input case, only non-linear equation errors in the parameterscan be formulated, resulting in an optimization problem. This complicates theidentification of starting values to start the Gauss-Newton algorithm. Thereforeno further attention is paid to IO data driven RMFD model identification in thisthesis.

5.2.7 From matrix coefficients to modal parameters

The poles and participation factors are found from the denominator polynomialcoefficients Aj . The companion matrix Ac, build from the coefficients, is given by


[52]

Ac =

A′

n−1 . . . A′

1 A′

0

I 0 0 0...

. . ....

...0 . . . I 0

(5.64)

with A′

j = −A−1n Aj . By solving an eigenvalue decomposition of the companion

matrix the poles and participation factors of the system are determined

(Ac − λrI) Vr = 0 (5.65)

where Vr is related to the rth modal participation vector according to

Vr =

λn−1r L[:,r]

...λrL[:,r]

L[:,r]

(5.66)

The poles λr, given by the eigenvalues of Ac, must be converted to the continuoustime domain and re-scaled according to the scaling procedure to cover the fullunit circle. The mode shapes Φ are obtained in a second step by considering thefrequency-domain formulation of the modal model. Once the physical poles aredistinguished from the mathematical ones by interpretation of the stabilizationchart, the mode shapes Φ, according to the a priori known poles and participationvectors, can be estimated directly by the well-known Least Squares Frequency-Domain (LSFD) estimator [118], [68]. This estimator estimates the mode shapesand upper and lower residues in a linear least squares sense from

H1(ω)...

HNo(ω)

=

Nm∑

r=1

(

φrLTr

jω − λr+

φrLHr

jω − λ∗r

)

− LR

ω2+ UR (5.67)

where the lower and upper residues respectively model the influence from the lowerand higher band poles. Under the assumption that the estimates of the poles λr

and the modal participation vectors Lr are consistent estimates, the linear least-squares estimation of the mode shapes is also consistent. An additional weightingin the LSFD improves the efficiency and results in an ML estimation of the modeshapes. Therefore in combination with the inconsistent poly-reference LSCF, fromwhich the poles and participation vectors are extracted, the LSFD is advised toestimate the mode shapes in a second step, instead of extracting the mode shapesdirectly from the poly-reference LSCF numerator coefficients estimates Boi. In thedifferent examples of paragraph 5.5 is shown that the LSFD improves the qualityof the modal model in combination with first poly-reference LSCF step.

5.3. Left Matrix Fraction Description 121

5.3 Left Matrix Fraction Description

The left matrix fraction description considers all response measurements simul-taneously. Based on the primary data, a distinction can be made between theidentification of na LMFD starting from IO data or FRF data. In [63], [68] theFrequency domain Direct Parameter Identification (FDPI) algorithm is presented,which uses a Laplace domain model to estimate only a denominator polynomialof order 2 and no numerator polynomial. In the next paragraph, a discrete-timeLMFD model identification algorithm is discussed identifying both a denominatorand numerator in the frequency-domain.

5.3.1 Linear Least Squares estimator for IO data

The LMFD model between the outputs Yo and inputs Fi is given by

Y (ωk) = A−1(zk)B(zk)F (ωk) (5.68)

with Y (ωk) ∈ CNo×1 a column vector containing all outputs and F (ωk) ∈ C

Ni×1 acolumn vector containing all inputs. The low-order polynomials A(zk) ∈ C

No×No

and B(zk) ∈ CNo×Ni are defined by

A(zk) =

n∑

j=0

Ajzjk Bo(zk) =

n∑

j=0

Bojzjk (5.69)

The matrix coefficients Aj and Boj are the parameters to be estimated. An addi-tional matrix polynomial T (zk) can be taken into account to model the transientphenomena

Y (ωk) = A(zk)−1B(zk)F (ωk) + T (zk) (5.70)

Since the transient contribution can be modelled by an extra input signal FNi+1(zk) =1, the following expressions do not consider the polynomial T (zk) explicitly. Alinear-in-the-parameters equation error is found by multiplying of Eq.5.68 by A(zk)resulting in

E(ωk) = A(zk)Y (ωk) − B(zk)F (zk) (5.71)

and a corresponding cost function

lrmfd =

N∑

k=1

tr(

E(ωk)EH(ωk))

(5.72)

The parameters are grouped together in θT = [βT αT ] with βT = [B0 . . . Bn] andαT = [A0 . . . An]. The minimization of the cost function in a least squares sense


can be formulated as

Jθ = 0 (5.73)

[Γ Υ]

[

βα

]

= 0 (5.74)

with

Γ =

−[

1 z1 . . . zn1

]

⊗ FT (ω1)

−[

1 z2 . . . zn2

]

⊗ FT (ω2)...

−[

1 zN . . . znN

]

⊗ FT (ωN )

∈ CN×Ni(n+1) (5.75)

Υ =

[

1 z1 . . . zn1

]

⊗ Y T (ω1)[

1 z2 . . . zn2

]

⊗ Y T (ω2)...

[

1 zN . . . znN

]

⊗ Y T (ωN )

∈ CN×No(n+1) (5.76)

The normal equations JHJθ = 0 can be solved by imposing a constraint to thecoefficients e.g. An = I. It should be noted that the number of estimated polesequals Nm = nNo. In the case of a large number of outputs (i.e. No > Nm),this means that a huge number of mathematical poles are estimated. A possiblesolution to prevent this is the use of a data condensation based on a SVD asdiscussed in paragraph 5.3.4. A smart choice of the parameter constraint resultsin stable physical poles and unstable mathematical poles as discussed in chapter9.

In the theoretical case that the number of outputs equals the number of modes,the Newton equation of motion is given by

(

Is2k + M−1C1sk + M−1K

)

Y (ωk) = M−1F (ωk) (5.77)

and can be modelled in the frequency-domain with a discrete-time model as

A(zk)Y (ωk) = B(zk)F (ωk) + T (zk) + E(ωk) (5.78)

with A(zk) a second order matrix polynomial modelling the dynamic stiffness andT (zk) the vector modelling the transients. This can be fitted in LMFD frameworkby choosing a parameter constraint B(zk) = I. The LMFD LS estimator is consis-tent with constraint An = I under the restriction that E(ωk) is circular complexnormal distributed Gaussian noise over the different spectral lines.

A physical interpretation is that E(ωk) represents unmeasurable forces, whichsatisfy the noise assumption. This combined stochastic-deterministic point of view,places the LMFD estimator in the OMAX framework as a combined IO datadriven estimator. Notice that in absence of measurable inputs, i.e. in the purely

5.3. Left Matrix Fraction Description 123

output-only case, the LMFD estimator can still be used as a data-driven stochasticestimator. Unfortunately, the use of the LMFD estimator as a stochastic andcombined deterministic-stochastic estimator is only applicable if the number ofmodes in the frequency band of interest is smaller than the number of outputs.

5.3.2 Linear Least Squares estimator for FRF data

The LMFD model for FRFs is given by

H(ωk) = A(zk)−1B(zk) (5.79)

with H(ω) ∈ CNo×Ni the FRF matrix. The the equation error is given by

E(ωk) = A(zk)H(ωk) − B(zk) (5.80)

is minimized by solving Eq. 5.74 with

Γ =

−[

1 z1 . . . zn1

]

⊗ INi

−[

1 z2 . . . zn2

]

⊗ INi

...−[

1 zN . . . znN

]

⊗ INi

∈ CNiN×Ni(n+1) (5.81)

Υ =

[

1 z1 . . . zn1

]

⊗ HT (ω1)[

1 z2 . . . zn2

]

⊗ HT (ω2)...

[

1 zN . . . znN

]

⊗ HT (ωN )

∈ CNiN×No(n+1) (5.82)

In theory an elimination procedure over the number of inputs can be used in asimilar way as the elimination over the number of outputs for the RMFD modelestimators. However, in practice the number of outputs exceeds the number ofinputs and the gain in calculation time by the elimination procedure in the caseof a LMFD is negligible.

5.3.3 From matrix coefficients to modal parameters

The poles and mode shapes are determined from the eigenvalues and eigenvectorsof the companion matrix defined by Eq. 5.64 in an analogous way as the determi-nation of the poles and participation factors in case of a RMFD. The participationfactors can then be estimated in a second step LS based on Eq. 5.67.


5.3.4 Data condensation

In many industrial applications an experimental modal analysis is characterizedby several hundreds of output measurements, while the frequency band of interesttypically contains 10-30 modes. Direct identification using the LMFD LS estimatorwould result in a long process time and many mathematical modes. Therefore, adata condensation is used to reduce artificially the number of outputs. Considerthe output data matrix Yd = [Y (ω1) . . . Y (ωN )] ∈ C

No×N for the data drivenidentification and Hd = [H(ω1) . . . H(ωN )] ∈ C

No×NiN for the FRF driven,the transformation matrix T is than given by the Nr first columns of U fromrespectively the SVD of Yd = USV H in the data driven case and of Hd = USV H

in the FRF driven case. The reduced data matrices Yr and Hr are given by

Yr = TYd (5.83)

Hr = TYd (5.84)

from which can be seen that the transformation matrix T reduces the number ofoutputs from No in the physical space to Nr in the condensed data space. After theidentification process the mode shapes must be re-transformed from the condenseddata space to the physical data space. Since the number of poles estimated bythe LMFD LS estimator equals nNr, a stabilization chart can be constructed bysolving the equations for an increasing Nr. Finally, it should be noticed that inanalogy to the poly-reference MLE, fast-MLE, GTLS, IQML, fast-BTLS differentestimators with similar schemes can be developed to identify LMFD models.

5.4 Output-Only

This chapter focussed mainly on the identification of modal parameters startingfrom FRF data. Only the LMFD LS estimators was discussed as a data drivenestimator. Nevertheless, all the FRF-estimators can start from power spectra or’positive’ power spectra in the case of an operational modal analysis.

5.5 Illustrating examples

The following examples will make a comparison of the quality of the estimatedpolynomials and modal models estimate by the common-denominator and poly-reference versions of both the LSCF and ML estimator.

5.5. Illustrating examples 125

Figure 5.1: Modal test of a body-in-white

5.5.1 Body-in-white

An impact hammer test was performed on a body-in-white (see figure 5.1) with3 fixed reference accelerometers, while the structure was excited in 28 locationsin 3 directions. By applying the reciprocity property the total measured FRFmatrix consists of 84 outputs and 3 inputs. A frequency band from 35Hz to62Hz is processed by the LSCF, ML, p-LSCF and p-ML estimators. For thecommon-denominator models the poles are estimated in the first step. Next, afterselection of the physical poles, the residues are estimated by the LSFD algorithmand decomposed in participation vectors and mode shapes. The poly-referencemodels estimate both the poles and participation factors in the first step and onlythe mode shapes are estimated in the second step by the LSFD estimator. Table5.1 compares the mean error and correlation between the measurements and theestimated polynomial models and between the measurements and the estimatedmodal models. The correlation C and error E are defined by

C =1

NoNi

No∑

o=1

Ni∑

i=1

∣

∣

∣

∑Nk=1 Hoi(ωk, θ)H∗

oi(ωk)∣

∣

∣

2

(

∑Nk=1 Hoi(ωk, θ)H∗

oi(ωk, θ))

(∑N

k=1 Hoi(ωk)H∗oi(ωk)

)

(5.85)

E =1

NoNi

No∑

o=1

Ni∑

i=1

∑Nk=1

∣

∣

∣Hoi(ωk, θ) − Hoi(ωk)

∣

∣

∣

2

∑Nk=1

∣

∣

∣Hoi(ωk, θ)∣

∣

∣

2(5.86)

with Hoi(ωk, θ) the synthesized FRF and Hoi(ω − k) the measured FRF. Sinceno noise information was available the logarithmic version of the ML estimatorswas used under the assumption of relative noise. It is clear that on the levelof the polynomial model the common-denominator ML estimator results in thebest quality. This can be explained by its inherent optimization and the extrafreedom in the common-denominator model compared to a RMFD with regard to


the rank of the residue matrices (The common-denominator model uses Ni × No

parameters to model each residue matrix, while the RMFD uses only No + Ni

parameters.). Nevertheless, it should be noticed that the overall differences for thedifferent algorithms are rather small for this data set. Nevertheless, transforming

Table 5.1: Comparison of model quality obtained by different algorithms for measure-ments on a body-in-white

algorithm error correlation error correlationpolynomial model modal model

p-ML 1.4% 98.7% 2.1% 98.04%ML 0.7% 99.3% 2.9% 97.4%

p-LSCF 2.7% 97.4% 2.5% 97.6%LSCF 5.8% 95.7% 3.24% 96.9%

the polynomial model to the modal model results in a larger loss of accuracy forthe common-denominator ML estimator than for the poly-reference ML estimator.More surprising is the small gain in accuracy for the LS estimators by switchingto the modal model. This can be explained, since the first LSCF step resultsin biased and inconsistent estimates of the polynomials. These error are slightlycompensated in the second LSFD step, which is consistent under the assumptionthat the already known modal parameters (obtained from in the first step) areconsistent. Table 5.2 shows the natural frequencies and damping ratios of themodes present in the frequency band.

5.5.2 Fully trimmed car

Using a MIMO test a fully trimmed Porsche was excited in 4 different locations byshakers (more details about the test setup can be found in [119]). The accelerationswere measured in 154 locations distributed over the car. Since no covariances wereavailable in the data set, the logarithmic implementation of both the p-ML and theML was used under the assumption of relative noise. A model order correspondingto 24 modes was used to identify the modal parameters in the frequency band from3Hz to 30Hz. Table 5.3 compares the mean errors and the mean correlation forthe different algorithms with the correlation between the synthesized FRF and themeasured FRF.

It is clear that the p-ML gives the best results in terms of the fit of the modal model,which can also be concluded from figure 5.2. For the least squares algorithms thep-LSCF method outperforms the LSCF algorithm. It should be mentioned thatboth the p-ML and the ML algorithm resulted in perfect fits of their polyno-mial models. Nevertheless, the common-denominator based algorithms LSCF and

5.5. Illustrating examples 127

Table 5.2: Natural frequencies and damping ratios estimated by p-ML

frequency (Hz) damping ratio (%)

40.17 1.3641.46 0.6043.31 0.6543.69 0.4544.66 0.4345.91 0.5949.96 0.6550.55 0.7253.73 0.1854.77 0.4258.84 1.2760.12 0.1160.63 0.2561.27 0.29

Table 5.3: Comparison of model quality obtained by different algorithms for measure-ments on a fully trimmed car

algorithm error correlation error correlation

polynomial model modal modelp-MLFD 0.99% 99.0 % 1.41% 98.7%

MLFD 0.42% 99.6 % 21.48% 85.47%p-LSCF 16.87% 86.9 % 9.92% 91.20%

LSCF 11.2 % 90.1 % 28.36% 81.94%

ML suffer an important loss in quality by converting the common-denominatormodel to the modal model by reducing the residues to a rank-one matrix using anSVD. Other examples [116] and [89] confirm this fact. This fact of loosing qual-ity by transferring common-denominator models into modal models tends to bemore problematic for highly-damped cases (damping ratios > 2%), which can beconcluded from a comparison of the ’body-in-white’ with the ’fully trimmed car’example. Table 5.4 gives the estimated natural frequencies and the correspondingdamping ratios. This proves that the p-ML method is suited to deal with highmodal densities and highly damped structures.

In fact, the p-ML optimizes the estimated model in a maximum likelihoodsense by optimizing the resonance frequencies, damping ratios, mode shapes and


Table 5.4: Natural frequencies and damping ratios estimated by p-ML

frequency (Hz) damping ratio (%)

4.05 5.804.29 7.494.72 6.556.04 4.298.59 6.8014.69 5.7515.74 8.3017.07 5.5418.34 5.0920.79 4.0121.82 3.0722.43 4.2925.15 3.0325.93 3.0326.93 6.00

participation factors. The common-denominator based ML algorithm optimizesthe model by optimizing the resonance frequencies, damping ratio and residues,while there is no guarantee that the residue matrices are of rank one. Insteadof optimizing the modal parameters, the ML algorithm uses this extra freedomin the model for fine tuning its parameters by using the freedom of not rank 1residue matrices and by consequence quality is lost by transforming the residuematrices to rank one matrices. Furthermore, it is clear for this example that apart of the errors of the p-LSCF estimator are compensated in the second stepwhen estimating the mode shapes by the LSFD. The quality of the modal modelis improved compared to the polynomial model by a reduction of the mean errorfrom 16.9% to 9.9%. Figure 5.3 illustrates the improvement of the p-LSCF by thesecond step LSFD algorithm by comparing the synthesized polynomial model withthe synthesized modal model. Nevertheless, the p-ML estimator outperforms thep-LSCF algorithm for both the polynomial and the modal model.

5.5.3 Villa Paso Bridge

To show the applicability of the presented LSCF, p-LSCF, ML and p-ML forOMA the operational measurements (only ambient excitation) on the Villa Pasobridge are processed and the parametric results are compared. Starting fromthe ’positive’ power spectra for two reference sensors the polynomial models areestimated and converted to the modal model. Since operational spectral density


functions are typically characterized by high noise levels both the LSCF and p-LSCF estimators result in larger errors, that are partially reduced in the secondstep by the LSFD estimator. Nevertheless in this case it is recommended to usethe ML estimator under the assumption of relative noise. Figure 5.4 illustratessome of the synthesized ’positive’ power spectra for the different algorithms.

Table 5.5: Comparison of model quality obtained by different algorithms for measure-ments on the Villa Paso bridge

algorithm error correlation error correlation

polynomial model modal modelp-MLFD 12.2% 88.2 % 11.4% 88.7%

MLFD 12.2% 88.3 % 14.5% 85.0%p-LSCF 243.5% 18.9 % 39.8% 60.6%

LSCF 149.7 % 43.1 % 30.3% 69.6%

5.6 Conclusions

In this chapter the use of a right and left matrix fraction description is proposedfor modal parameter estimation. The implementation of the poly-reference LSCFis based on similar approaches as the common-denominator LSCF to speed upthe algorithm and to reduce the memory requirements. Next, a consistent poly-reference maximum likelihood estimator is proposed to handle noisy data. Tospeed up this algorithm, a fast version is presented based on a scalar frequencyweighting. By several experiments, it was illustrated that modal models extractedfrom the poly-reference implementations outperform the ones extracted from thecommon-denominator model for MIMO measurements on highly damped struc-tures. Furthermore, the proposed poly-reference algorithms can also be applied toprocess power spectra in case of an OMA test. Finally, it was noticed the (poly-reference) LSCF estimates can be improved by estimating in a second step themode shapes (and participation vectors) by the LSFD.

5.7 Appendix: Confidence intervals on the esti-mated poles from the p-ML estimator

The poly-reference ML estimator (the vec implementation with matrix weight-ing) reaches the Cramer-Rao lower bound for the uncertainties on the estimated


parameters. The covariance matrix on the estimated parameters is given by

cov(θML) =(

JHJ)−1

(5.87)

with J the Jacobian matrix given by Eq. 5.24. Elimination of the covariancematrix on the coefficients α of the denominator only results in

cov(α) =

(

No∑

o=1

To −No∑

o=1

SoR−1o SH

o

)−1

(5.88)

(in fact this is the covariance matrix on the vec operators of the denominatorcoefficients) To calculate the uncertainties on the poles from the uncertainties onthe estimated matrix coefficients, consider the EVD of the companion matrix

Ac

λn−1r V[:,r]

...λrV[:,r]

V[:,r]

− λr

λn−1r V[:,r]

...λrV[:,r]

V[:,r]

= 0 (5.89)

and

Ac =

A′

n−1 . . . A′

1 A′

0

I 0 0 0...

. . ....

...0 . . . I 0

(5.90)

with A′

j = −A−1n Aj . It holds for the denominator polynomial A(zk) that

A(λr)Lr = 0 (5.91)

Consider now a perturbation ∆Aj on the coefficient Aj and Aj = Aj + ∆Aj .

This new set of coefficients A0, . . . , Aj , . . . , An defines a polynomial A(zk) whichsatisfies

A(zk) = A(zk) + ∆Ajzjk (5.92)

Furthermore, the new set of coefficients defines a perturbed companion matrix.The EVD on this perturbed companion matrix results in the perturbed eigenvaluesλr = λr + ∆λr and eigenvectors Vr = Vr + ∆Vr for which it holds that

A(λr + ∆λr)(Vr + ∆Vr) = 0 (5.93)

Substitution of Eq. 5.92 results in

(

A(λr + ∆λr) + ∆Aj(λr + ∆λr)j)

(Vr + ∆Vr) = 0 (5.94)

5.7. Appendix: Confidence intervals on the estimated poles from the p-ML estimator131

Applying a Taylor expansion on the polynomial A(zk) around λr, taking intoaccount that A(λr)Vr = 0 and neglecting all second order perturbations results in

(

A′(λr)∆λr + ∆Ajλjr

)

Vr = 0 (5.95)

with A′(z) =∑n−1

j=1 nAnzn−1 the first derivative with respect to z. Elimination ofthe perturbation on the eigenvalue λr resulting from the perturbation on coefficientAj is than given by

∆λr = − λjr

V Hr Vr

V Hr A′−1(λr)∆AjVr (5.96)

= − λjr

V Hr Vr

(

V Tr ⊗ V H

r A′−1(λr))

vec (∆Aj) (5.97)

The sensitivity between the perturbation ∆Aj and its influence on the pole λr is

then defined by Srj = − λjr

V Hr Vr

(

V Tr ⊗ V H

r A′−1(λr))

. The variances or uncertainties

on the poles λr are then given by

σ2λr

=[

Sr0 . . . Srn

]

cov(α)

SHr0...

SHrn

(5.98)

This sensitivity analysis to calculate the relation between the perturbation on amatrix coefficient and the poles is generalization of the scalar case presented in[44].


10 20 30

−100

−90

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

p−MLFD

10 20 30

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

MLFD

10 20 30

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

p−LSCF

10 20 30

−90

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

LSCF

0 10 20 30−100

−90

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

p−MLFD

0 10 20 30−100

−90

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

MLFD

0 10 20 30−100

−90

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

p−LSCF

0 10 20 30−100

−90

−80

−70

−60

−50

Freq (Hz)

Am

pl (

dB)

LSCF

Figure 5.2: Fully trimmed car. Comparison between the modal model obtained by thep-MLFD, MLFD, p-LSCF and LSCF algorithm for two FRFs. (cross: measurement, fullline: estimated modal model)

5.7. Appendix: Confidence intervals on the estimated poles from the p-ML estimator133

5 10 15 20 25

−80

−70

−60

−50

p−LSCF polynomial model

AM

PL.

(dB

)

FREQ. (Hz)5 10 15 20 25

−80

−70

−60

−50

p−LSCF modal model

AM

PL.

(dB

)

FREQ. (Hz)

5 10 15 20 25

−80

−70

−60

−50

p−ML polynomial model

AM

PL.

(dB

)

FREQ. (Hz)5 10 15 20 25

−80

−70

−60

−50

p−ML modal model

AM

PL.

(dB

)

FREQ. (Hz)

5 10 15 20 25

−90

−80

−70

−60

−50p−LSCF polynomial model

AM

PL.

(dB

)

FREQ. (Hz)5 10 15 20 25

−90

−80

−70

−60

−50p−LSCF modal model

AM

PL.

(dB

)

FREQ. (Hz)

5 10 15 20 25

−90

−80

−70

−60

−50p−ML polynomial model

AM

PL.

(dB

)

FREQ. (Hz)5 10 15 20 25

−90

−80

−70

−60

−50p−ML modal model

AM

PL.

(dB

)

FREQ. (Hz)

Figure 5.3: Fully trimmed car. Comparison between the polynomial and modal modelsobtained by the p-ML and p-LSCF algorithm for two FRFs. (cross: measurement, fullline: estimated modal model)


4 6 8 1020

40

60

80

Freq. (Hz)

Am

pl. (

dB)

p−MLFD

2 4 6 8 10

20

40

60

80

Freq. (Hz)

Am

pl. (

dB)

MLFD

4 6 8 1020

40

60

80

Freq. (Hz)

Am

pl. (

dB)

p−LSCF

2 4 6 8

30

40

50

60

70

80

Freq. (Hz)

Am

pl. (

dB)

LSCF

4 6 8

60

65

70

75

80

85

Freq. (Hz)

Am

pl. (

dB)

p−MLFD

2 4 6 8

60

65

70

75

80

85

Freq. (Hz)

Am

pl. (

dB)

MLFD

2 4 6 8 10

60

65

70

75

80

85

Freq. (Hz)

Am

pl. (

dB)

p−LSCF

2 4 6 8 10

60

70

80

Freq. (Hz)

Am

pl. (

dB)

LSCF

Figure 5.4: Villa Paso bridge. Comparison between the modal model obtained by thep-MLFD, MLFD, p-LSCF and LSCF algorithm for two FRFs. (cross: measurement, fullline: estimated modal model)

Chapter 6

DeterministicFrequency-domain SubspaceIdentification

Frequency-domain subspace algorithms estimate state-space models by means of ge-ometrical projections and can be considered as the frequency-domain counterpartsof time-domain subspace algorithms. This chapter gives a short introduction infrequency-domain subspace identification for structural engineering. Special atten-tion is paid to an extended state-space model to consider the effect of the initial andfinal conditions to avoid errors introduced by transients and leakage. This exten-sion makes it possible to estimate state-space models in the frequency domain fromnonperiodic signals without any approximation and under the same assumptionsas in the time domain. Next, the extended state-space model is further used in amixed non-parametric/parametric frequency response function (FRF) estimationprocedure to eliminate bias errors introduced by leakage. In this way, the esti-mated multiple input/ multiple output (MIMO) state space models can be validatedby accurate FRF estimates.

135

136 Chapter 6. Deterministic Frequency-domain Subspace Identification

6.1 Introduction

The previous two chapters were devoted to system identification for modal ana-lysis by means of polynomial models. These identification techniques such as e.g.LSCF, ML, BTLS, ... can all be related to an equation error, which is minimizedby a quadratic cost function and are well documented in [67], [98]. In parallel withfrequency-domain cost function related techniques, a class of so-called realizationalgorithms was developed to estimate state-space models by factorization of theHankel matrix build from the impulse response function [53], [144]. The ERAestimator is an implementation of a realization algorithm applied for mechanicalengineering [55], [56]. Meanwhile, identification techniques were developed in astochastic realization framework for the identification from output-only measure-ments [3]. These realization algorithms formed the basis for the development ofsubspace algorithms, which estimate the system matrices by means of projectionsof so-called input and output block Hankel matrices. Starting from the pure de-terministic framework [79], stochastic subspace algorithms were developed [120]and combined in so-called combined deterministic-stochastic framework [122]. Anoverview of subspace algorithms can be found in [6], [71] , while [123] gives a pro-found discussion in a general framework. Subspace algorithms have been appliedsuccessfully for several applications like control engineering, electrical engineering,financial engineering and mechanical engineering [28]. In the domain of modalanalysis, subspace identification has mainly been applied to modal parameter es-timation from output-only data [5], [78], [87], [85], [49]. Compared to the tra-ditional identification algorithms, subspace algorithms are non-iterative and as aresult they always yield a solution without the risk of convergence problems. Un-til recently, research efforts were and still are focussed on time-domain subspacemethods, while linear systems often are characterized in the frequency domain.Therefore, it is quite natural to consider subspace identification algorithms in thefrequency domain to identify models directly from input/output spectra, FRFs or(’positive’) power spectra. In [66] a basic projection frequency-domain subspacealgorithm is proposed in a deterministic framework, which does not require uni-form frequency grid and which allows to use a frequency weighting. By consideringthe covariances on the primary data as a frequency weighting, it is shown that thisprojection algorithm is strongly consistent [77], [92]. Another approach consistsin transforming the frequency-domain data to the time-domain by the IDFT andcombining this with classical time-domain techniques [77]. However, in this casethe estimated modal parameters must then be corrected to compensate for time-domain aliasing. This can be done by a correction of the participation factorson the level of the modal factors as shown in paragraph 3.9 or by a correctionof the system matrices [77]. In this chapter, the basic principles for frequency-domain subspace identification are briefly presented. Next, an extended modelis proposed in order to take into account the initial and final conditions. Basedon this extended formulation, a mixed non-parametric/parametric FRF estimatoris proposed for validation purposes [18]. In addition, the FRF driven frequency-

6.2. Basic Frequency-Domain Projection Algorithm 137

domain subspace algorithm is extended to consider the initial and final conditionsfor each block used by the H1 FRF estimator [17].

6.2 Basic Frequency-Domain Projection Algorithm

Consider the state space model given by

zkX(k) = AX(k) + BU(ωk) (6.1)

Y (ωk) = CX(k) + DU(ωk) (6.2)

with X(k) ∈ Cn×1, Y (k) ∈ C

No×1 and U(k) ∈ CNi×1 respectively the states,

responses and inputs for spectral line k. Recursive use of Eq. 6.1 and 6.2 gives

zpkY (ωk) = zp−1

k (CzkX(k) + zkDU(ωk)) (6.3)

= zp−1k (CAX(k) + CBU(ωk) + zkDU(ωk)) (6.4)

...

= CApX(k) + (CAp−1BU(ωk) + CAp−2BU(ωk)zk

+ . . . + CBU(ωk)zp−1k + DU(ωk)zp

k)U(ωk) (6.5)

Stocking Eq. 6.5 for p = 0, 1, . . . , r − 1 yields to

Y (ωk)zkY (ωk)

...zr−1k Y (ωk)

= OrX(k) + Γ

U(ωk)zkU(ωk)

...zr−1k U(ωk)

(6.6)

with Or the extended observability matrix defined by

Or =

CCA...

CAr−1

(6.7)

and the lower triangular block Toeplitz matrix Γ given by

Γ =

D 0 . . . 0CB D . . . 0...

......

...CAr−2B CAr−3B . . . D

(6.8)

The matrix formulation of Eqs. 6.8 for k = 1, 2, . . . , N leads to

Y = OrX + ΓU (6.9)


with X = [X(1) . . . X(N)] (with N the number of frequency lines in the chosenfrequency band), Y a complex Nor × N block Vandermonde matrix and U acomplex Nir × N block Vandermonde matrix each defined by

Y =

Y (ω1) Y (ω2) . . . Y (ωN )z1Y (ω1) z2Y (ω2) . . . zNY (ωN )

......

......

zr−11 Y (ω1) zr−1

2 Y (ω2) . . . zr−1N Y (ωN )

U =

U(ω1) U(ω2) . . . U(ωN )z1U(ω1) z2U(ω2) . . . zNU(ωN )

......

......

zr−11 U(ω1) zr−1

2 U(ω2) . . . zr−1N U(ωN )

In the case that one is interested in estimating real system matrices, Eq. 6.9 isconverted in a set of real equations

Yre = OrXre + ΓUre (6.10)

where ()re represents a matrix with the real and imaginary parts aside each other,for example

Yre = [Re(Y) Im(Y)] (6.11)

Eq. 6.10 with r larger than the model order divided by the number of outputsn/No (to have an extended observability matrix Or which is of rank n to extractthe system matrices A and C) is the basic equation in frequency-domain sub-space identification and illustrates that Y lies in the subspace spanned by the rowspaces defined by X and U. Frequency-domain subspace identification algorithmsbasically consists of a four step procedure. By projection of Y in a space U⊥

orthogonal to the row space U the inputs are eliminated and the extended observ-ability matrix can be determined. In the first step, a practical implementation ofthe orthogonal projection is obtained by the use of the QR-factorization [142]

[

Ure

Yre

]

=

[

RT11 0

RT12 RT

22

] [

QT1

QT2

]

(6.12)

RT22 = UΣV T (6.13)

with the orthogonal projection defined by

Yre/Ure,⊥ = OrXre/Ure,⊥ = RT

22QT2 (6.14)

The second step uses a SVD to estimate the extended observability matrix Or.For a model order n, Or is given by

Or = U[:,1:n] (6.15)

6.3. Starting from FRFs or power spectra 139

In a third step an estimate of A and C from Or is found in a LS sense

A = O+[1:No(r−1),:]O[No+1:Nor,:] and C = O[1:No,:] (6.16)

In the fourth and last step, given the estimates A and C, the system matrixes Band D are estimated in a least squares sense from

Y (ωk) = C(Izk − A)−1BU(ωk) + D (6.17)

This basic projection algorithm can also be used to estimate a continuous-timestate-space model by replacing the basic function zk by ωk [121]. For reasons ofnumerical conditioning the use of orthogonal polynomials in ωk are recommended[121].

6.3 Starting from FRFs or power spectra

Instead of starting from IO data the subspace algorithm can also start with FRFor (’positive’) power spectra data. Depending on the form of the primary data,the residues are forced to be of rank one or not. By replacing Y (ωk) by H(ωk) ∈C

No×Ni and U(ωk) by INia rank one residue model is estimated. In the case

that the FRFs are stacked under each other, no rank one model is forced on themeasurements and thus the mode shapes and participation vectors are obtainedfrom an SVD decomposition reducing to rank to 1. The stacked FRFs are givenby Hst(ωk) = [HT

1 (ωk) . . . HTNi(ωk)]T ∈ C

NoNi×1 with Hi(ωk) ∈ CNo×1 the FRFs

corresponding to input i. Similar the power spectra, corresponding to the differentreference sensors, can be stacked in one column. This approach can be useful fordata from low damped structures, with small data inconsistencies in the differentpatches e.g. caused by temperature effects, mass-loading effect.

6.4 A weighted frequency-domain projection al-gorithm

In order to make the basic projection algorithm, presented in the previous para-graph, consistent an additional frequency weighting should be taken into accountby considering the covariances on the primary data [77]. In [92] the asymptoticproperties of this weighted projection algorithms are studied when the true noisecovariance matrix is replaced by the sample noise covariance matrix obtained froma small number of repeated experiments. The theory was developed assuming thatthe input is exactly known and noise is only present on the outputs. The state-space model with output noise is given by

zkX(k) = AX(k) + BU(ωk) (6.18)

Y (ωk) = CX(k) + DU(ωk) + N(k) (6.19)


Consider C(ωk) as the covariance matrix of the output noise defined by

C(ωk) = E(

N(k)NH(k))

∈ CNo×No (6.20)

with N(k) ∈ CNo×1 the noise on the response measurements. The covariance

matrix of the Vandermonde matrix Y is then given by [77]

C = Re(

Z diag (C(ω1) . . . C(ωM )) ZH)

(6.21)

with

Z =

INoINo

. . . INo

z1INoz2INo

. . . zNINo

......

......

zr−11 INo

zr−12 INo

. . . zr−1N INo

(6.22)

Given the QR decomposition by Eq. 6.12 a consistent estimate for the extendedobservability matrix is given by [77]

C−1/2RT22 = UΣV T (6.23)

Or = C1/2U[:,1:n] (6.24)

The system matrices A,B,C and D are than determined in a similar way as forthe unweighted projection algorithm. However, in practice the use of deterministicsubspace algorithms with a frequency weighting to guarantee consistency is notstraight forward for a MIMO test setup with e.g. No = 100 and thus for each spec-tral line k a No × No covariance matrix. This would result in both large memoryrequirements and calculation times. Furthermore, to obtain a invertible covariancematrix at least No averages should be taken into account in the estimation of thecovariance matrix, which requires a long measurement time. Notice also the dif-ference with maximum likelihood identification where the frequency weighting hasno influence on the consistency property and only improves the efficiency. Hence,for the ML implementations the correlations between different outputs could beneglected to result in a fast implementation with only a small loss in efficiency.

In [74] another consistent frequency-domain algorithm based on an instrumen-tal variable approach is presented not requiring any a priori noise information.Although, from several experiments, simulations and testing of different imple-mentations from independent authors, this IV-base frequency-domain subspacealgorithm did not result in satisfactorily estimates. In fact, the classical projec-tion algorithm outperformed this IV approach is most cases.

6.5. Extended state-space model for initial and final conditions 141

6.5 Extended state-space model for initial and fi-nal conditions

Frequency-domain system identification generally assumes that the input and out-put signals are periodic or time limited within the observation window. This isnecessary to guarantee leakage-free spectra calculated through the discrete Fouriertransform. In the case that an arbitrary input signal is used, often a window suchas a Hanning window, is used to reduce errors introduced by leakage phenomena.In [99] and chapter 4 it is shown for single input/single output (SISO) systemsthat for a rational fraction polynomial model formulation the ’transient’ polyno-mial takes into account the initial and final conditions of the experiments andeliminates the bias error due to leakage. This key idea can be generalized forMIMO state space models [75], [76]. It will be shown that the frequency-domainstate-space identification methods e.g. frequency domain subspace identificationcan easily be extended to take into account the initial and final conditions of thestates.

6.5.1 State-Space Model for Arbitrary Signals

Consider a linear time-invariant state space model. Assume that the input andoutput samples are exactly known at discrete time instants tn = n∆t (samplingperiod ∆t) inside the time interval [0, (N − 1)∆t] and unknown outside this timeinterval. The discrete input un ∈ R

Ni×1 and output yn ∈ RNo×1 samples satisfy

following difference equation:

xn+1 = Axn + Bun (6.25)

yn = Cxn + Dun (6.26)

with xn ∈ Rn×1 (n the number of states (’n’ as a subindex stands for time sample

n), Ni number of inputs, No number of outputs). The discrete Fourier transfor-mation X(k) = DFT (xn) of state vector xn is defined as

X(k) =1√N

N−1∑

n=0

xnz−nk (6.27)

with zk = ej2πk/N . Equivalent expressions can be obtained for the DFT of thesignals un and yn as U(k) = DFT (un) and Y (k) = DFT (yn) with DFT () defined


by Eq. 6.27. The discrete Fourier transform of xn+1 is given by

DFT (xn+1) =1√N

N−1∑

n=0

xn+1z−nk (6.28)

=zk√N

N−1∑

n=0

xnz−nk +

zk√N

(xN − x0) (6.29)

= zkX(k) +zk√N

(xN − x0) (6.30)

with X(k) defined by Eq. 6.27. Taking the discrete Fourier transform of Eqs. 6.25and 6.2 leads to

zkX(k) = AX(k) + BU(k) + Tzk√N

(6.31)

Y (k) = CX(k) + DU(k) (6.32)

with T = x0 − xN . These frequency domain state space equations describe thesystem under the same assumption as their time-domain equivalents, althoughin many identification methods the contribution of T is neglected. A sufficientcondition for T to be zero is that the initial and final conditions are equal, whichis the case for periodic and time-limited signals.

6.5.2 Remarks on the extended state-space model

• Since a transient is nothing else than the output of a system caused by initialconditions of the states different from zero, the model proposed by equations(6.31) and (6.32) has exactly the same form as the model valid for periodicexcitations corrupted with transients.

• The exponential decay (stable system) of the transients in the outputs de-pends on the damping of the system. Therefore, lightly damped systems aremore sensitive for errors due to leakage and transients than highly dampedsystems. So, in case of lightly damped systems, it is certainly advised to usethe extended state-space formulation to avoid large errors on the dampingratios of the system.

• Notice that the extended formulation of the frequency domain state spacemodel introduces only one extra input in the first equation of the model, i.e.Eq. 6.31. Existing identification methods, e.g. frequency-domain subspacealgorithms, can easily be adapted to take into account this extra input.

6.5. Extended state-space model for initial and final conditions 143

6.5.3 A mixed non-parametric/parametric FRF estimatorfor validation

Consider a data set of measurements with arbitrary input signals. Different identi-fications methods such as prediction error methods, time-domain subspace meth-ods, frequency domain-subspace methods exist to estimate the state-space matricesA,B,C,D. However, an important step in the system identification process is thevalidation of the estimated model. A common approach consists in comparing theestimated model with the experimentally obtained FRFs.

Assuming that only noise is present on the response measurement and the inputsare noise free, the maximum likelihood estimate for the FRFs is given by theH1 estimator. Since we consider arbitrary input signals errors are introduced byleakage, even with the use of a window (e.g. Hanning window). Therefore, avalidation based on the comparison between synthesized transfer functions andmeasured FRFs can be misleading and lead to erroneous conclusions about themodel quality.

The goal is now to validate the estimated model A, B, C and D using an FRFestimate that is free from bias errors due to leakage. To estimate the FRFs fromthe measured time histories, they are divided in Nb blocks with Nb ≥ Ni. The useof Eqs. 6.31 and 6.32 leads to the following expression for the spectra of block b

Yb(ωk) =(

C(zkI − A)−1B + D

)

Ub(ωk) + C(zkI − A)−1Tb

zk√N

(6.33)

Right multiplying by UHb (ωk) and summation over all Nb blocks results in

NbSY U (ωk) =(

C(zkI − A)−1B + D

)

NbSUU (ωk) (6.34)

+ C(zkI − A)−1[ T1 . . . TNb ]

zk√N

U1(k)

...zk√N

UNb(k)

(6.35)

Since the matrices A, B, C, D are already identified in the first step the parametersTb for b = 1 . . . Nb can be estimated in a LS sense. In a next step the FRFs canbe estimated by taking into account the initial conditions in order to remove thebias introduced by leakage and transients

H1,e(ωk) =

SY U (ωk) − C(zkI − A)−1

[ T1 . . . TNb ]

zk√NNb

U1(k)

.

.

.zk√NNb

UNb(k)

SUU (ωk)−1

(6.36)

with H1,e the extended H1 FRF estimator. Eq. 6.36 represents a mixed non-parametric/parametric estimation of the FRFs, since the parametric compensation


for the initial/final conditions. As a model validation the synthesized FRFs fromthe estimated system parameters

H(zk) = C(zkI − A)B + D (6.37)

can be compared with the FRFs obtained by the mixed non-parametric/parametricestimation H1,e(ωk).

6.5.4 State-Space Model for FRF data

Similar as for the common denominator model an extended state-space model canbe formulated to start from FRF data. Starting from FRFs has the advantagethat the size of the initial data set is reduced and the influence of the noise on themeasurements is reduced by the averaging process. Nevertheless, in practice onehas to deal with a trade-off between the variance and bias on the FRF estimate.Given a data set with a limited amount of data, a choice must be made for thenumber of blocks Nb. A large number of blocks will result in a large reduction ofthe noise levels, but introduces a bias error caused by the leakage, since then onlya small number of data samples are present within a block. From Eq. 6.36 theextended parametric model for the H1 estimator is given by

H1(ωk) = C(zkI − A)−1[ B T1 T2 . . . TNb ]

INi

F t1(k)...

F tNB

(k)

+ D (6.38)

with F tb (k) = 1√

NNbzkUb(k)SUU (ωk)−1 and INi

a unity matrix of dimension Ni.

Equation (6.38) can be rewritten as a state space model

zkX(k) = AX(k) + B′U ′(k) (6.39)

H1(ωk) = CX(k) + D (6.40)

with B′ = [B T1 . . . TNb], U ′T = [INi F t,T

1 . . . F t,TNb

] and the state vector X(k) ∈R

n×(Nb+1). It is important to notice that this extended state space model fitsthe FRFs calculated by the H1 method exactly if a rectangular window is used.Furthermore, each extra block b used by the H1 estimator introduces n extraparameters Tb in the model, while the equivalent common-denominator approachin [14] and [24] introduces Non extra parameters in the model for each block likediscussed in chapter 4.


100 200 300 400 500

−6

−4

−2

0

2

4

6

x 10−4

n (samples)

resp

onse

y(n

)

Figure 6.1: Response of a 6-order discrete time system excited by random noise.

6.6 Simulation and Measurement examples

6.6.1 Simulations

Extended frequency-domain state-space model

The goal of the simulation is to illustrate that model ( Eqs.6.31 and 6.32) isexactly true without any approximation. Therefore, no disturbing noise is addedto the signals. A sixth-order MIMO system (see Appendix) is excited by twouncorrelated uniformly distributed noise sequences (see Fig. 6.1). Data samplesn = 0, 1, . . . 511 (N = 512) are used for the identification. The DFT spectra ofthe 3 responses and 2 inputs are used as primary data for the identification of adiscrete-time state-space model by the basic projection. The state-space model isidentified by the basic projection subspace algorithm for 2 cases: once by using aHanning window and no extra term T and once with a rectangular window andthe additional term T to model the initial and final conditions. The differencesbetween both identified models and the true model are shown in figure 6.2. It canbe seen that the accuracy of the estimates for the second case including T in thestate space model, are at the level of the arithmetic precision of the calculations(16 digits). Large errors remain if T is not included in the model.

Mixed non-parametric/parametric FRF estimation for validation

In real-life applications one can not validate the identified model by comparing theestimated model to the true model. Consider the model A, B, C and D identifiedfrom the measurements (identification was done with the term T included in the


0 0.1 0.2 0.3 0.4 0.5−400

−350

−300

−250

−200

−150

−100

−50

0

frequency (Hz)

erro

r (d

B)

Figure 6.2: Magnitude of the complex difference between the true and the estimatedmodel using the classic projection frequency-domain subspace (dashed line) and usingthe extended model (dotted line). The full line represents the exact FRF.

model). By comparing the synthesized transfer function with the FRFs estimatedby the H1 method (4 blocks, Hanning window) one concludes that still large errorsare present (see figure 6.3 (a)). This illustrates that the classic validation procedureis not robust for leakage, since we know the estimated model is of the accuracyat the level of the arithmetic precision of the calculations (15 digits) as shown byfigure 6.2. Eliminating the bias in the H1 estimator by estimating the Tb fromequation 6.35 and inserting in equation 6.36 leads to an unbiased H1,e estimate ofthe FRFs. Validation of the model with these FRFs leads to correct conclusionslike as shown in figure 6.3 (b).

Extended model for FRFs

In a second set of simulations the response measurements yn are corrupted bycolored noise yr,n. For each of the 100 Monte Carlo runs the ’4096’ time samplesare processed by the identifying the classic state-space model (classic model)andthe extended state-space model (extended model) from FRFs, obtained from adifferent numbers of blocks, as primary data. The mean square error (MSE) ofthe damping ratios from the 100 Monte Carlo runs is calculated as the

MSE =1

100

100∑

i=1

(di − de)2 (6.41)

with di the estimated damping ratio and de the exact damping ratio. The meansquare error is equal to the sum of the variance and square of the bias. Figure6.4 clearly shows the trade-off between variance and bias for the CSSM approach.The classical H1 approach reduces the noise levels on the FRFs by averaging, butthe bias due to leakage increases by reducing the block length. Based on this


0 0.1 0.2 0.3 0.4 0.5−110

−100

−90

−80

−70

−60

−50

−40

−30

frequency (Hz)

ampl

itude

(dB

)

(a)

0 0.1 0.2 0.3 0.4 0.5−400

−350

−300

−250

−200

−150

−100

−50

0

frequency (Hz)

ampl

itude

(dB

)

(b)

Figure 6.3: Validation by comparison between the synthesized transfer function (fullline) and the FRF (∗). The difference between both is indicated by the dashed line; (a)Traditional H1 approach, (b) Mixed non-parametric/parametric approach H1,e

3 5 7 9 11 13 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7mode 1

Nb

MS

E

3 5 7 9 11 13 150

0.02

0.04

0.06

0.08

0.1

0.12mode 2

Nb

MS

E

3 5 7 9 11 13 150

0.01

0.02

0.03

0.04

0.05mode 3

Nb

MS

E

Figure 6.4: MSE of the damping ratio estimates of the classic model approach () andextended model approach (∗) for different number of blocks.

trade-off the user has to the define the number of blocks. Notice that the optimalnumber of blocks is not a priori known and depends on the total measurementtime and the modal density in the measurements. The extended modal approachclearly improves the quality of the estimates, since the noise is reduced withoutintroducing bias errors caused by leakage. In this way the MSE reduces andbecomes much less dependent on the number of blocks.


6.6.2 Measurements on a subframe of a car

A modal analysis was performed on a subframe of car, which has to support theengine (see figure 3.1 (b)). Two shakers excite the structure with random forces. Asample rate of 2048Hz is used and 8K (1K=1024) samples are processed to extractthe model. The extended frequency domain state space model was estimated bythe frequency-domain projection subspace algorithm. To validate the estimatedmodel, the synthesized transfer functions are compared to the measured transferfunction. Large differences can be observed between the synthesized transfer func-tions and the FRFs due to the spectral leakage errors in the calculation of the H1

estimator (see figure 6.5 (a) and (b)). When the FRFs are compensated for leak-age and transients by equation 6.38, a comparison with the synthesized transfersfunctions shows a good agreement up to a SNR of 40dB. Even when records of 64Kdata samples are processed, a validation based on the classical H1 approach stillleads to misleading results, while the proposed validation technique shows again agood agreement up to a SNR of 40dB (see Fig 6.5 (c) and (d)).

Starting from 256K time samples of both acceleration and force measurementsreference FRFs are estimated, by dividing the time records in 8 blocks of 64Kresulting in a frequency domain resolution of 0.0156Hz. These FRFs can be as-sumed to be leakage free and are considered as the reference modal parameters.Next, only the first 8K data samples are used and divided in 4 equal blocks of each2K samples resulting in a frequency resolution of 1Hz. First, the FRFs are cal-culated by the H1 method with a Hanning window. From these FRFs the classicstate-space model (A,B,C and D matrices) is estimated with the basic projectionsubspace algorithm. Secondly the same 4 blocks of data are used to obtain theFRFs with a rectangular window. These FRFs are processed to estimate the ex-tended state-space model for FRFs (A,B,C,D and T1, . . . , TNb

. Table 6.1 showsthe good agreement between the natural frequencies and damping ratios obtainedfrom the reference FRFs and the estimates derived from the short data sequenceswith the extended state-space model (ESSM) approach. The use of a Hanningwindow and a classic state space identification clearly results in poor estimates,especially for the damping estimates. Figure 6.6 shows the stabilization diagramfor both the classic state-space model (CSSM) approach and the ESSM approach.Although, the ESSM approach does not use a Hanning window and as a result theFRF looks much noisier, all the poles clearly appear in the stabilization diagram.The CSSM approach suffers from estimating double poles, where only one physi-cal pole is present e.g. around 129Hz, 205Hz, 286Hz and 380Hz. This is typicallycaused by errors in the data (due to leakage) and makes the stabilization diagramconfusing for the end-user. Finally, figure 6.7 compares the synthesized FRFs ofboth approaches with the reference FRF from the leakage free measurement. Oneconcludes that although the extended model approach started from only 8K datasamples, averaged in 4 blocks of 2K samples, still a very good agreement is foundbetween the model and the reference FRFs, while a classic approach clearly failsdue to errors introduced by leakage.


340 360 380 400

−50

−40

−30

−20

−10

0

10

20

frequency (Hz)

ampl

itude

(dB

)

(a)

320 340 360 380 400

−60

−50

−40

−30

−20

−10

0

10

20

frequency (Hz)

ampl

itude

(dB

)

(b)

340 360 380 400

−60

−50

−40

−30

−20

−10

0

10

20

frequency (Hz)

ampl

itude

(dB

)

(c)

340 360 380 400

−60

−50

−40

−30

−20

−10

0

10

20

frequency (Hz)

ampl

itude

(dB

)

(d)

Figure 6.5: Validation by comparison between the synthesized transfer function (fullline) and the FRF (∗). The difference between both is indicated by the dashed line;(a) traditional H1 approach (8K), (b) mixed non-parametric/parametric approach H1,e

(8K), (c) traditional H1 approach (64K), (d) mixed non-parametric/parametric approachH1,e (64K)

6.7 Conclusions

This chapter introduces frequency-domain subspace algorithm to identify discrete-time state-space models from both IO data and FRFs. Next, an extended frequency-domain state-space model is introduced to allow the identification starting fromarbitrary signals without introducing systematic errors caused by leakage and tran-sients. Therefore the initial and final conditions of the states have to be taken intoaccount. It is shown that this can be realized by using an additional input signal.Moreover, it is shown how the state space model can be validated by using an ex-tended FRF estimator based on a mixed non-parametric/parametric approach that


150 200 250 300 350 400

−50

−40

−30

−20

−10

0

10

20

30

40

50

Freq. (Hz)

mod

el o

rder

(a)

150 200 250 300 350 400

−30

−20

−10

0

10

20

30

40

Freq. (Hz)

mod

el o

rder

(b)

Figure 6.6: Stabilization diagrams (a) classic state-space model (b) extended state-space model +: unstable pole, ∗: stable pole

150 200 250 300 350 400−70

−60

−50

−40

−30

−20

−10

0

10

frequency (Hz)

ampl

itude

(dB

)

Figure 6.7: Comparison between the reference FRF (full line) , synthesized FRF fromthe classic state-space model () and from the extended state-space model (∗)

removes bias errors caused by transients and leakage. This validation procedureis illustrated by both a simulation and a measurement example. In addition, anextended formulation is proposed to estimate state-space models together with theinitial/final conditions of each data block used by the H1 estimator. Simulationshave shown, that this approach makes the quality of the estimated parameters lessdependent on the choice of the number of blocks to estimate the FRFs.

6.8. Appendix 151

Table 6.1: Natural frequencies and damping ratios of the subframe

fref fESSM fCSSM dref dESSM dCSSM

129.5 129.5 130.3 0.122 0.116 0.202147.8 147.9 148.1 0.081 0.081 -0.002175.9 176.3 176.0 1.138 1.118 0.208205.6 205.6 204.8 0.106 0.112 0.110241.0 241.0 241.4 0.115 0.110 0.160252.9 252.9 253.0 0.095 0.091 0.102286.3 286.3 286.7 0.221 0.224 0.291291.2 291.5 1.006 0.801329.3 329.3 329.6 0.164 0.164 0.214331.3 331.3 331.3 0.130 0.134 0.277356.1 356.1 355.8 0.099 0.099 0.212359.7 359.7 359.5 0.101 0.101 0.306379.7 379.7 379.8 0.142 0.143 0.061402.6 402.6 402.3 0.098 0.097 0.148

6.8 Appendix

The system matrices used for the simulations are given by

A =

0.6541 −0.8108 −0.0009 −0.0042 0.0003 0.00180.6977 0.6512 0.0004 0.0031 −0.0001 −0.0013−0.0007 0.0050 −0.0211 −1.0106 0.0131 −0.04360.0001 −0.0001 0.9599 −0.0270 −0.0079 0.0165−0.0003 0.0019 −0.0118 −0.0442 −0.4961 0.90790.0002 −0.0016 0.0027 0.0176 −0.7944 −0.4636

B =

−0.0018 −0.00250.0010 0.0014−0.0014 0.00000.0004 −0.00000.0007 −0.0010−0.0001 0.0002

C =

−0.1735 −0.0301 −0.2569 0.0922 0.1920 0.0696−0.2453 −0.0435 −0.0054 −0.0114 −0.2753 −0.0786−0.1736 −0.0318 0.2650 −0.0744 0.1976 0.0408

D =

0.1327 −0.0085−0.0085 0.1322−0.0006 −0.0085

1.0e − 003


Chapter 7

Stochastic frequency-domainsubspace identification

Until now frequency-domain subspace algorithms are limited to identify determinis-tic models from input/output or FRF measurements. In this chapter, a frequency-domain subspace algorithm is presented to identify stochastic state-space modelsin a consistent way from the spectra of the given output data. The relation toand the analogy with time domain stochastic subspace models is established. Theapplicability of the method is shown by both simulations and several measurementexamples.

153

154 Chapter 7. Stochastic frequency-domain subspace identification

7.1 Introduction

Identification methods which identify state-space models by geometrical operationsof the input and output sequences are commonly known as subspace methodsand have received much attention in the literature. In [124] a frequency-domainsubspace algorithm for the identification starting from power spectra is proposedbased on the deterministic basic projection algorithm.

In time-domain subspace identification, several algorithms were developed toidentify stochastic models from output-only measurements [120]. In [123] it isshown how well know methods like the Principal Component algorithm (PC),the Unweighted Principal Component algorithm (UPC) and the Canonical vari-ate algorithm (CVA) fit in this framework of stochastic time-domain subspaceidentification.

However, until today frequency-domain subspace identification is limited toestimate deterministic models from FRFs, IO data or power spectra. In thischapter a consistent stochastic frequency domain subspace identification methodis presented that directly starts from the output-only spectra, without forming thepower density matrices. The relation and analogy to the time domain stochasticsubspace identification [123] is established.

The main advantage of this stochastic frequency domain subspace algorithmis that the identification problem can be solved in a single step, while frequency-domain prediction error methods (PEM) result in a non-linear estimation ap-proach, which require iterative optimization methods (e.g. Gauss-Newton) [97],[98]. In that framework, the main problems arise from convergence difficultiesand/or the existence of local minima. Subspace algorithms typically use numeri-cally robust and time efficient QR and SVD decompositions to optimize compu-tation time and memory usage.

7.2 A first approach

Consider a proper (order of the numerator ≤ order of the denominator of theequivalent polynomial transfer function), stable nth order multiple input-multipleoutput stochastic system. The frequency domain state-space equations of thisdiscrete-time system are given by


Yk = CXk + Vk (7.2)

with Yk ∈ CNo×1 the vector of the output spectra at spectral line k, Xk ∈ C

n×1

the state vector at frequency lines k and zk = ei2πk/N (k = 1, 2, . . . , N) covering

7.2. A first approach 155

the unit circle. The vectors Wk ∈ Cn×1 and Vk ∈ C

No×1 contain respectively thespectral frequency lines of the process and the measurement noise.

Assumption 1: Wk and Vk are zero mean circular complex independent andidentically distributed noise sources with a covariance matrix

E[(

Wk

Vk

)

(

WHk V H

k

)

]

=

(

Q SSH R

)

(7.3)

where E is the expected value.

Recursive use of the Eqs. 7.1 and 7.2 results in

zpkYk = zp−1

k (CzkXk + Vkzk) (7.4)

= zp−1k (CAXk + CWk + Vkzk) (7.5)

...

= CApXk + CAp−1Wk + CAp−2Wkzk + . . . + CWkzp−1k + Vkzp

k(7.6)

Stocking equation 7.6 for p = 0, 1, . . . , r − 1 gives

Yk

zkYk

...zr−1k Yk

= OrXk + Γ

Wk

zkWk

...zr−1k Wk

+

Vk

zkVk

...zr−1k Vk

(7.7)

with Or the extended observability matrix

Or =

CCA...

CAr−1

(7.8)

and Γ a lower triangular block Toeplitz matrix.

Γ =

0 0 . . . 0 0C 0 . . . 0 0...

......

......

CAr−2 CAr−3 . . . C 0

(7.9)

Matrix formulation of Eq. 7.7 for k = 1, 2, . . . , N yields

Y = OrX + ΓW + V (7.10)

with Y ∈ CNor×N , W ∈ C

nr×N and V ∈ CNor×N block Vandermonde matrices


defined by

Y =

Y1 Y2 . . . YN

z1Y1 z2Y2 . . . zNYN

......

......

zr−11 Y1 zr−1

2 Y2 . . . zr−1N YN

W =

W1 W2 . . . WN

z1W1 z2W2 . . . zNWN

......

......

zr−11 W1 zr−1

2 W2 . . . zr−1N WN

V =

V1 V2 . . . VN

z1V1 z2V2 . . . zNVN

......

......

zr−11 V1 zr−1

2 V2 . . . zr−1N VN

and X ∈ Cn×N matrix defined by

X =[

X1 X2 . . . XN

]

Eq. 7.10 with r larger than the model order divided by the number of outputs n/No

is the basic equation in frequency-domain subspace identification. In deterministicfrequency-domain subspace algorithms the basic equation Y = OrX + ΓU is inclose analogy to Eq. 7.10. The term ΓU can be removed by the use of (Π)⊥U ,where (Π)⊥U is the geometric projection onto the orthogonal complement of therow space of the input Vandermonde matrix U. However, in the stochastic casethis projection can not be used since the matrices W and V are then unknown.

Consider the matrix

L =

zr1Y H

1 zr−11 Y H

1 . . . z1YH1

zr2Y H

2 zr−12 Y H

2 . . . z2YH2

......

. . ....

zrNY H

N zr−1N Y H

N . . . zNY HN

(7.11)

with r > n/No, than the following theorem holds:

Theorem 1: Under Assumption 1

YL

N→ OrXL

Nw.p. 1 for N → ∞ (7.12)

(w.p.1 stands for with probability one)

7.2. A first approach 157

Proof : To prove the strong consistency it needs to be shown that

a.s. limN→∞

(

1

NYL − 1

NOrXL

)

= 0 (7.13)

Taking into account equation (7.10) it is sufficient to show that

a.s. limN→∞

1

NWL = 0

a.s. limN→∞

1

NVL = 0

where a.s.lim stands for the as sure limit [98]) Multiplication of the ith row of Wwith the jth column of L leads to

(WL)ij =N∑

k=1

zi−1k Wkzr−j+1

k Y Hk (7.14)

and elimination of Xk in equations (7.1) and (7.2) gives

Y Hk = WH

k

(

Iz−1k − AH

)−1CH + V H

k

= WHk zk

(

I − AHzk

)−1CH + V H

k

= WHk zk

(

I + AHzk + AH2

z2k + . . .

)

CH + V Hk (7.15)

= WHk

+∞∑

l=1

Alzlk + V H

k

with Al =(

AH)l−1

CH . The Taylor expansion (I−AHz)−1 = I +AHz+AH2

z2 +

AH3

z3 + . . . holds since |z| = 1 if |eig(AH)| < 1. This is equivalent to |eig(A)| < 1and hence the system must be stable. Substitution of (7.16) in (7.14) results in

(WL)ij =

+∞∑

l=1

(

N∑

k=1

zr+i+l−jk WkWH

k Al

)

+

N∑

k=1

zr+i−jk WkV H

k (7.16)

Under Assumption 1, (WL)ij/N converges w.p.1 to its expected value for N → ∞(strong law of large numbers for independent and identically distributed randomvariables, [70]):

a.s. limN→∞

1

N(WL)ij = lim

N→∞

1

NE(

(WL)ij

)

(7.17)

= limN→∞

(

1

N

+∞∑

l=1

QAl

(

N∑

k=1

zr+i+l−jk

)

+S

N

N∑

k=1

zr+i−jk

)

(7.18)

= 0 (7.19)

This last equation is due to∑N

k=1 zr+i+l−jk = 0 and

∑Nk=1 zr+i−j

k = 0 sincerespectively r + i + l − j ∈ N0 and r + i − j ∈ N0 and we assume that zk


k = 1, . . . N covers the full unit circle (i = 1, . . . , r and j = 1, . . . , r) and thus∑N

k=1 zr+i+l−jk = 0. Except for the values of l such that r + i + l − j = qN with

q ∈ N0 the term∑N

k=1 zr+i+l−jk = N is different from zero. So it still needs to

be proven that the terms with r + i + l − j = qN in 7.18 are zero for N → ∞.Consider thus r + i + l − j = qN and thus the corresponding term

limN→∞

1

N

N∑

k=1

QAlzr+i+l−jk = lim

N→∞QAHqN−r−i+j−1

(7.20)

since Al = AHl−1

. Substituting the eigenvalue decomposition AHl

= TΛT−1, withΛ a diagonal matrix containing the discrete poles, results in

limN→∞

QTΛqN−r−i+j−1T−1 = 0 (7.21)

under the assumption that all poles are stable. Similarly one can shown that

a.s. limN→∞

1

N(VL)ij =

1

N

+∞∑

l=1

SHAl

(

N∑

k=1

zr+i+l−jk

)

+ R

N∑

k=1

zr+i−jk

= 0 (7.22)

which concludes the proof.

This observation forms a the basis for stochastic frequency domain subspaceidentification. It shows that the terms ΓW and V in the basic equation (7.10) canbe removed by right multiplication with L and that a strongly consistent estimateof Or is obtained.

In a next step a singular value decomposition of YL gives an estimate of theextended observability matrix Or. This is a commonly-used step in subspacealgorithms

YL = UΣV T (7.23)

An estimate of Or for model order n is then given by

Or = U[:,1:n] (7.24)

In a last step an estimate of A and C are obtained from Or in a LS sense

A = O†[1:No(r−1),:]O[No+1:Nor,:] and C = O[1:No,:] (7.25)

7.3 A second approach

In a second approach a normalized multiplication is proposed to obtain a relation-ship with the Kalman filter state estimate. The second and main theorem of thischapter is now discussed and proven.

7.3. A second approach 159

Theorem 2 Under Assumption 1

YL(LHL)−1LHf → OrXf w.p.1 for N → ∞ (7.26)

with Xf a vector of the spectra of the equivalent time domain Kalman filter stateestimate at spectral line f and Lf a part of L containing the information at spectralline f .

Proof : From the stochastic frequency domain model given by Eq.7.1 one canform the following relation

1

N

N∑

k=1

zkXkz∗kX

Hk =

1

N

N∑

k=1

XkXHk (7.27)

=A

N

N∑

k=1

XkXHk A

H+

A

N

N∑

k=1

XkWHk +

1

N

N∑

k=1

WkXHk A

H+

1

N

N∑

k=1

WkWHk

Since the Taylor expansion Xk = (Izk − A)−1Wk ≃ z−1k (I + Az−1

k + A2z−2k +

. . .)Wk holds since |zk| = 1 for |eig(A)| < 1 (stable system) the expected value

limN→∞ E ( 1N

∑Nk=1 XkWH

k ) = 0 (follow the lines of the proof of Theorem 1).

Similarly it can be shown that limN→∞ E ( 1N

∑Nk=1 WkXH

k ) = 0 and as a resultthe expected value of equation (7.28) is equal to the following Lyapunov equation

Σ = AΣAH + Q (7.28)

with Σ = E ( 1N

∑Nk=1 XkXH

k ). Defining Λi = E ( 1N

∑Nk=1 zi

kYkY Hk ),

G = E ( 1N

∑Nk=1 zkXkY H

k ) it can be shown from the state-space model that

Λ0 = CΣCH + R (7.29)

G = AΣCH + S (7.30)

Λi = CAi−1G (7.31)

Note that the elements Λi form the Markov parameters of the deterministic statespace system given by A,G,C,Λ0. This observation is very closely related to thestochastic time-domain theory, since Λi = E (

∑Nk=1 zi

kYkY Hk ) can be interpreted

as the Inverse Fourier Transformation (IFT) of the power spectra of the responses.It is a well known result that the output correlation sequences can be consideredas Markov parameters.

To prove the strong consistency it needs to be shown that

a.s. limN→∞

(

1

NYL(

1

NLHL)−1LH

f − OrXf

)

= 0 (7.32)

with Lf = [zrfY H

f zr−1f Y H

f . . . zfY Hf ] holds for each spectral line f and zf =

ei2πf/N with f = pN and p a fixed percentage between 0 and 100%. We start


with the first two products and apply the strong law of large numbers

a.s. limN→∞

1

N(YL)ij = lim

N→∞E(

1

N(YL)ij

)

= limN→∞

E(

1

N

N∑

k=1

zr+i−jk YkY H

k

)

= limN→∞

Λr+i−j (7.33)

Combining equations (7.31) and (7.33) results in

a.s. limN→∞

1

NYL = lim

N→∞

CCA...

CAr−1

[

Ar−1G Ar−2G . . . G]

(7.34)

= limN→∞

Or∆r (7.35)

So far, it is proven that Y 1N L( 1

N LHL)−1LHf → Or∆r(

1N LHL)−1LH

f w.p.1 for

N → ∞, so it still has to be shown that ∆r(LHL)−1LH

f → Xf w.p.1 for N → ∞.To prove this, a result from stochastic time domain subspace and Kalman filtertheory is used. In [123] it is proven that

xi+q = ∆tM−1t

yq

...yi+q−1

(7.36)

with ∆t and M defined as

∆t =[

Ai−1G′ Ai−2G′ . . . G′ ] (7.37)

Mt =

Λ′0 Λ′

−1 . . . Λ′1−r

Λ′1 Λ′

0 . . . Λ′2−r

......

. . ....

Λ′r−1 Λ′

r−2 . . . Λ′0

(7.38)

with G′ = E (xn+1yHn ) and Λ′

i = E (yn+iyHn ). The sequence xi+q is the forward

Kalman filter state sequence and yi+q is generated by a stochastic time domainprocess

xi+q+1 = Axi+q + wi+q (7.39)

yi+q = Cxi+q + vi+q (7.40)

Since Xk and Yk satisfy the frequency domain equivalent of equations (7.39) and

(7.40), their Inverse Fourier Transform (IFT) xi+q = 1√N

∑Nk=1 zi+q

k Xk and yi+q =

7.3. A second approach 161

1√N

∑Nk=1 zi+q

k Yk fulfill (7.39), (7.40) and thus equation (7.36) holds. Equation

(7.36) indicates that the Kalman filter generating the estimate of xi+q only uses ioutput measurements yq, . . . , yi+q−1. Using the definition of the IFT leads to

N∑

k=1

zi+qk Xk = ∆tM

−1t

∑Nk=1 zq

kYk

...∑N

k=1 zq+i−1k Yk

(7.41)

with Yk and Xk respectively the frequency spectra of output yi and the Kalmanstate estimate xi. Multiplying equation (7.41) by z−q−i

f , substituting i + q by mand making the summation for m = 1, 2, . . . , N leads to

N∑

m=1

(

z−mf

N∑

k=1

zmk Xk

)

= ∆tM−1t

N∑

m=1

z−mf

∑Nk=1 zm−i

k Yk

...∑N

k=1 zm−1k Yk

(7.42)

Since zk = ei2πk/N and zf = ei2πf/N

1

N

N∑

m=1

z−mf zm

k = δkf (7.43)

1

N

N∑

m=1

z−mf zm−i

k = z−ik δkf (7.44)

with δkf = 1 for k = f and δkf = 0 for k 6= f . Substituting equations (7.43) and(7.44) in (7.42) results in

Xf = ∆tM−1t LH

f (7.45)

This is an important observation for stochastic frequency domain subspace algo-rithms, since it shows that the spectra of the forward Kalman filter state estimatecan be expressed as a function of the output spectra. The only thing left toprove Eq. 7.26, is to show that ∆t and M−1

t are respectively equal to ∆ anda.s. limN→∞( 1

N LHL)−1. Since

∆ =[

Ai−1G Ai−2G . . . G]

(7.46)

a.s. limN→∞

1

NLHL −

Λ0 Λ−1 . . . Λ1−r

Λ1 Λ0 . . . Λ2−r

......

. . ....

Λr−1 Λr−2 . . . Λ0

= 0 (7.47)

the proof reduces to showing that G = G′ and Λi = Λ′i. (the proof of equation

(7.47) is similar that of equation (7.33)). Using the definition of the IFT one


obtains

G′ = E (xn+1yHn ) (7.48)

=1

N

N∑

n=1

E (xn+1yHn ) (7.49)

=1

NE

N∑

n=1

1√N

N∑

k1=1

Xk1zn+1k1

1√N

N∑

k2=1

Y Hk2

z−nk2

(7.50)

=1

NE 1

N

N∑

k1=1

(

Xk1Yk2

(

N∑

n=1

zn+1k1

z−nk2

))

(7.51)

= E (1

N

N∑

k=1

zkXkY Hk ) (7.52)

= G (7.53)

since Eq. 7.44 holds and similar it can be shown that Λ′i = Λi. This concludes the

proof of the theorem 2.

The major difference between the first approach based on equation (7.12) andthe second approach based on (7.26), is that the second approach provides us anestimate of the frequency spectra of the states (for f = 1, . . . N). Given thestates, one can then easily obtain the covariance matrices Q,R and S as shownin paragraph7.6. An estimate of the extended observability matrix and the stateestimate is obtained again from an SVD

YL(LHL)−1LH = USV H (7.54)

In paragraph 7.6 it is shown how the observability matrix and the state spectrumis obtained from U , S and V .

7.4. Connection to time domain stochastic subspace identification 163

7.4 Connection to time domain stochastic sub-space identification

In time domain stochastic subspace algorithms one starts from a Hankel matrix,which is divided in a past and a future submatrix as follows

yo y1 . . . yN−1

......

. . ....

yr−2 yr−1 . . . yN+r−3

yr−1 yr . . . yN+r−2

yr yr+1 . . . yN+r−1

yr+1 yr+2 . . . yN+r

......

. . ....

y2r−1 y2r . . . yN+2r−2

def=

[

yp

yf

]

(7.55)

The main theorem of [123] for stochastic subspace identification states that

yf/yp = Orx (7.56)

for N → ∞ with x the forward Kalman filter state sequence. In frequency domainstochastic subspace identification, the Hankel matrix is replaced by a Vandermondematrix

Y =

z−r1 Y1 z−r

2 Y2 . . . z−rN YN

z−r+11 Y1 z−r+1

2 Y2 . . . z−r+1N YN

......

......

z−11 Y1 z−1

2 Y2 . . . z−1N YN

Y1 Y2 . . . YN

z1Y1 z2Y2 . . . zNYN

......

......

zr−11 Y1 zr−1

2 Y2 . . . zr−1N YN

def=

[

Y−Y+

]

(7.57)

and the equivalent main theorem is given by

Y+/Y− = OrX (7.58)

for N → ∞ and with X the DFT spectrum of the forward Kalman filter state se-quence. In the case N → ∞, F(yn+j) = zj

kYk with Yk = F(yn) = 1√N

∑Nn=1 ynz−n

k

holds and thus the Vandermonde matrix can be considered as the DFT of the rowsof the Hankel matrix.


7.5 Geometrical Interpretation

It is well known that subspace algorithms are related to geometrical operations.Basic equations (7.26) can be considered as the projection of the row space of Y+

onto the row space of Y− with Y− = LH .

Y+/Y− = Y+YH− (Y−YH

− )−1Y− (7.59)

= OrX (7.60)

In [123] a fast projection algorithm is proposed based on a QR decomposition.Consider the following decomposition

[

BA

]

=

[

RB 0RAB RA

] [

QA

QB

]

(7.61)

then the projection of A onto B is given by

A/B = RABQA (7.62)

7.6 Final Algorithms

Basically, the numerical implementation of the frequency-domain stochastic sub-space identification methods are similar to their time-domain counterparts.

1. Build the Vandermonde matrices Yf and Yp given the data Yk for a valuer > n/No.

2. Calculate the projection Yf/Yp by means of the QR-factorization[

Yp

Yf

]

=

[

RHB 0

RHAB RH

A

] [

QA

QB

]

(7.63)

Yf/Yp = RHABQA (7.64)

3. Calculate the SVD of projection

Yf/Yp = USV H (7.65)

4. Determine the extended observability matrix Or and the state estimate X.For a chosen order n (in theory the order n is equal to the rank of S)

Or = W−11 U1S

1/21 (7.66)

X = S1/21 V H

1 W−12 (7.67)

with U1 = U[:,1:n], S1 = S[1:n,1:n] and V1 = V[1:n,:]. In paragraph 7.7 differentchoices for the weighting matrices are briefly discussed.

7.6. Final Algorithms 165

5. The estimates for the A and C matrices can be obtained in two differentways (by not using the states or not)

(a) The A and C matrices can be obtained from the extended observabilitymatrix.

A = O+[1:No(r−1),:]O[No+1:Nor,:] and C = O[1:No,:] (7.68)

(b) Another possibility exists of solving the following equation for A and Cin a LS sense, since Xk and Yk are known

[

zkXk

Yk

]

=

[

AC

]

Xk +

[

ρWk

ρVk

]

(7.69)

6. In the last step the covariance matrices Q, S and R are estimated. Twopossible approaches can be used.

(a) The first and most straight forward approach, results in a guaranteedpositive definite covariance matrix. If the A and C matrices are ob-tained from the observability matrix (step 5a) the spectra Wk and Vk

can be calculated from equations 7.1 and 7.2 since X is already known.When the A and C matrices are obtained from the LS problem (step5b) the residues ρWk

and ρVkcan be considered as Wk and Vk. The

covariance matrix is then estimated by

[

Q SSH R

]

=1

N

N∑

k=1

[(

Wk

Vk

)

(

WHk V H

k

)

]

(7.70)

(b) The second approach exists of estimating the matrices G, Σ and Λ0.

G =1

N

N∑

k

zkXkY Hk (7.71)

Σ =1

N

N∑

k

XkXHk (7.72)

Λ0 =1

N

N∑

k

YkY Hk (7.73)

By using equations 7.28-7.30 the covariance matrices R, S and Q canthen be calculated.

The total identification procedure is schematically given by figure 7.1 for the caseof (5a) and (6a).


Estimate X by projection (QR and SVD)

+ -Y / Y rO X=

k k kk

k k

z X WAX

Y VC

= + Least-squares

• Frequency band selection, scaling

• Construct the Vandermonde matrices

A,C natural frequencies, dampingratios and mode shapes (EVD)

Figure 7.1: Schematic overview of the stochastic frequency-domain subspace algorithm

7.7 Remarks

• Equation 7.54 can be replaced by

W1YL(LHL)−1LHW2 = USV H (7.74)

with W1 ∈ CNor×Nor and W2 ∈ C

N×N such that W1 is of full rank andrank(LH) = rank(LHW2). For time domain stochastic subspace it is shownin [123] that special choices of W1 and W2 correspond to well known algo-rithms like the Principal Component algorithm (PC), the Unweighted Prin-cipal Component algorithm (UPC) and the Canonical Variate Algorithm(CVA). Analogously, the frequency counterpart of this weighing matricescorrespond to the frequency counterparts of the PC, UPC and CVA algo-rithms.

• Working in the frequency-domain has some specific advantages compared totime domain methods. One advantage is the easy pre-filtering of the datae.g. disturbing harmonics can simply be removed [97]. Although

∑Nk zr

k =0 is not fulfilled, since the disturbing harmonics are eliminated, the biasintroduced is negligible if N is large w.r.t. the number of spectral DFTlines that have been removed [97]. A second advantage is the easy frequency

7.7. Remarks 167

band selection, which is an important advantage for applications like modalanalysis. The chosen frequency band is re-scaled to cover the full unit circleuniformly to ensure consistency and the numerical stability is improved sincethe basis functions zr are orthogonal.

• In case that Nor > N it follows from Eq. 7.59 that the projection Y+/Y− =Y+, which makes the QR operation unnecessarily and the SVD decompo-sition can start directly from Y+. Physically it means that Y+ totally liesin the row space spanned by Y−. In practice this occurs for measurementswith a high spatial density e.g. scanning laser vibrometer measurements anda limited frequency band e.g. r = 10, No = 100 and N = 500.

• In the case that the excitation signal is a multisine with a constant am-plitude spectrum and a random phase and only one input is used (e.g. aloudspeaker), the assumption for the excitation signals Wk and Vk are stillvalid. Notice that in this case no probability limit for N → ∞ is needed toprove the main theorem 7.26. Indeed, in this case the following expressionsare valid, without the use of their expected value

N∑

k=0

zlkVkV H

k = 0 (7.75)

N∑

k=0

zlkWkWH

k = 0 (7.76)

N∑

k=0

zlkWkV H

k = 0 (7.77)

for l ∈ N0 , since VkV Hk = R, WkWH

k = Q and WkV Hk = S for the excitation

of a state space model with a flat multisine signal. As a result, the proposedtechnique is exact for this case and no uncertainty is present on the esti-mated parameters for a finite N . This can easily be understood since thepower spectra density matrix is exactly given by Syy(ωk) = YkY H

k for thisdeterministic excitation signal and as a result the model given by Eq. 2.62exactly fits YkY H

k . In practice this means that the modal parameters i.e.natural frequencies, damping ratios and mode shapes can be exactly deter-mined from an output-only experiments from a finite amount of data by theuse of deterministic multisine excitation. A specific application is found inlaboratory testing of small and light structures which can not be excited byclassic hammer or shaker excitation. In this case an unmeasurable acousticexcitation is often applied in the test setup.

• The proof of the theorems in this chapter are given for a complex state-spacemodel. In the case that one is interested to fit a state-space model with realvalued system matrices on the measurements, the matrices Y+ and Y− must


be converted respectively to Yre+ and Yre

− in accordance with Eq. 6.11. Themain theorem is than given by

Yre+ /Yre

− → OrXref w.p.1 for N → ∞ (7.78)

with zk covering the half unit circle. In the case of real system matrices thedouble order is required compared to a complex model for estimating thesame amount of poles, i.e. n = 2Nm.

• The algorithm can be slightly adapted to reduce the calculation time byreplacing the QR decomposition by a QR decomposition without the explicitcalculation of the Q.

• In some cases one is interested to use only a few outputs as references in-stead of all outputs. This reduces the dimensions of the problems and byconsequence the calculation time. Furthermore, some sensors are of higherquality in terms of the structural information than others (i.e. they arenot placed in nodes of certain modes). If these ’high quality’ sensors arechosen as the reference sensors no information is lost and the identificationmay even result in better estimates [87]. In fact the row space given byY− can be considered as a reference space on which the measurement Y+

are projected. So the use of only the reference outputs to construct thereference subspace Yr

− transforms the proposed algorithm in a reference-based stochastic frequency-domain subspace algorithm in analogy with thetime-domain counterpart [87]. This reference based subspace speeds up thealgorithm, without any loss of information (if the reference sensors are wellchosen).

• In the case that short time sequences are transformed to the frequency do-main leakage errors are introduced resulting in biased estimates. Therefore,an additional term T must be taken into account in the state-space model,resulting in a combined deterministic-stochastic model, which is discussed inthe next chapter.

7.8 Simulation and Measurement example

7.8.1 Simulation

The simulations are done in the frequency domain (N=1024) for a discrete stochas-tic system described by the matrices given in appendix 7.10. The system containstwo poles and two outputs. Figure 7.2 shows a comparison of the estimated modelrepresented by the estimated power density function

E (Y Y H) = C(Iz−A)−1Q(Iz∗−AH)−1CH+R+C(Iz−A)−1S+SH(Iz∗−AH)−1CH

(7.79)

7.8. Simulation and Measurement example 169

0 0.2 0.4 0.6 0.8 1−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

frequency line (Hz)

ampl

itude

(dB

)

0 0.2 0.4 0.6 0.8 1−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

frequency line (Hz)

ampl

itude

(dB

)Figure 7.2: Comparison between the estimated model (full line) and the output spectra(dots)

and the spectra of the outputs. Figure 7.3 compares the exact absolute value ofboth poles with the estimates of 1000 Monte Carlo runs. The difference between

0 200 400 600 800 10000.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

Monte Carlo run

abs(

pole

)

Figure 7.3: Comparison between the exact magnitude of the poles and the estimatedmagnitude of the poles of the 1000 Monte Carlo runs

the estimated power spectra and the true power spectra satisfies the 95% limitgiven by 2σ, with σ the standard deviation of the 1000 runs (see Figure 7.4)


0 0.2 0.4 0.6 0.8 1−160

−140

−120

−100

−80

−60

−40

−20

Frequency (Hz)

Am

plitu

de (

dB)

Figure 7.4: Comparison between the true system (full line), the error between the true(dotted line) and estimated system and the 95% confidence limit (dashed line) from the1000 Monte Carlo runs.

200 250 300 350

5

10

15

20

Freq. (Hz)

mod

el o

rder

Figure 7.5: Subframe. Stabilization chart for the stochastic frequency-domain subspacealgorithm. Notice that all identified poles are stable (positive damping), ∗: stable pole.

7.8.2 Measurement example: Subframe of an car

The proposed technique was applied to fit a stochastic model through the mea-surements on the subframe of a car starting from the output-only spectra. Theaccelerations at 23 different positions are measured at a sampling rate of 2048 Hzduring 10 s, while the structure was excited by two shakers with random forces. Be-fore the actual modelling task starts, the frequency band of interest between 190Hzand 390Hz is re-scaled to cover the half unit circle uniformly (real system matricesA and C are estimated). Notice that the identification starts from output-onlyspectra and the solution obtained from the stochastic state-space identification


Table 7.1: Natural frequencies and damping ratios of the subframe.

fIO(Hz) dIO(%) fO(Hz) dOf (%)206.1 0.20 206.3 0.22241.9 0.12 241.7 0.16254.4 0.10 254.6 0.15287.1 0.32 287.3 0.28329.4 0.33 328.9 0.33333.0 0.14 332.9 0.15357.8 0.13 357.9 0.12359.9 0.17 360.1 0.14380.0 0.31 380.3 0.26

is consistent for a stable system. Since output-only spectra can be modelled byboth a stable and an unstable system (both resulting in the same fit of the powerspectra), the solution obtained by the proposed subspace algorithm always resultsin stable poles for N → ∞. This is clearly illustrated by the stabilization diagramin figure 7.5, which only shows stable poles. Figure 7.6 compares the synthesizedpower spectra with the output spectra. Table 7.1 shows a good agreement betweenthe natural frequencies fIO and damping dIO ratios obtained from deterministicmodelling from input/output measurements (i.e. the 23 responses and the 2 elec-trical generator signals) and the natural frequencies fO and damping ratios dO

obtained from the output-only analysis.

7.8.3 Flight flutter testing

The proposed algorithm can be used to process in-flight vibration measurementswithout the use of the excitation signals. This is e.g. the case when the airplaneis excited by the natural turbulence only or even when excitation is applied butunmeasurable. Nevertheless, sources from NASA report that the natural excita-tion is often band limited [10] and many time is lost during flight to search forsufficient turbulent force levels. Therefore, the use of artificial excitation is stilloften used and nowadays a fly-by-wire system allows to inject perturbation signalsin the control loops of the flap mechanisms of the wings to excite the structure. Inthis case the rotational vibration of the flaps is often measured as an input signal,while the dynamical vibration behavior is already partially included in this rota-tional vibration measurement. (One can discuss whether this rotational vibrationhas to be considered as an input or output signal.) Therefore, it can be betterto start from the vibration measurements only, without such an input. In thisexample, no use is made of the input signals and the algorithm starts only fromthe response measurements. The ambient excitation forces are in this case both


200 250 300 350

0

10

20

30

40

50

60

Freq. (Hz)

Am

pl. (

dB)

240 250 260 270 280 290

5

10

15

20

25

30

35

40

45

50

55

Freq. (Hz)

Am

pl. (

dB)

200 250 300 350

0

10

20

30

40

50

60

Freq. (Hz)

Am

pl. (

dB)

200 210 220 230 240 250 26010

20

30

40

50

60

Freq. (Hz)

Am

pl. (

dB)

200 250 300 350

−10

0

10

20

30

40

50

Freq. (Hz)

Am

pl. (

dB)

325 330 335 340 345 350 355 3600

10

20

30

40

50

Freq. (Hz)

Am

pl. (

dB)

Figure 7.6: Subframe. Comparison between the synthesized power spectral densityfrom the estimated model (full line) and the output spectra (×).

the present turbulent forces and the artificial excitation. Figure 7.7 illustrates thestabilization chart from the analysis of 12 acceleration measurements during flighton a military aircraft. Similar as for the subframe all identified poles are stable. Tomake a distinction between the physical and mathematical poles, one has to rely


4 5 6 7 8 9 10

2

4

6

8

10

12

14

Freq. (Hz)

mod

el o

rder

Figure 7.7: In-flight test. Stabilization chart for the stochastic frequency-domain sub-space algorithm. Notice that all identified poles are stable (positive damping), ∗: stablepole.

on both a priori known insights in the dynamical behavior (from e.g. simulationsand ground vibration tests) and criteria based on both mathematics and physics.In [141], [132] the use of pole-zero criteria combined with the uncertainty levelson the estimated poles is presented to select the physical poles, while [135] pro-poses an automated interpretation of the stabilization chart by the use of clusteralgorithms and physical parameters such as the mode complexity and the modalassurance criterium [52]. Some different methods based on a heuristic approachfor an automatic interpretation of the stabilization chart are proposed in [108],while in [35] some new techniques are presented for mode selection. Figure 7.8illustrates the synthesized power spectra of 4 different acceleration measurements.

7.8.4 Villa Paso bridge

The vibrations on the Villa Paso bridge were measured by 10 accelerometers in6 different patches during ambient excitation by the traffic. Transformation ofthe measured acceleration sequences to the frequency domain by the DFT anda frequency band selection from 1Hz to 6.5Hz provides the primary data for thealgorithm. Figure 7.9 illustrates the stabilization chart for a model order of 20modes. The use of simple criteria such as the relative damping and frequency de-viations between the poles estimated for subsequent orders guides the user troughthe selection of the physical modes. Figure 7.10 illustrates the synthesized powerspectral densities for 2 different outputs.


4 5 6 7 8 9 10

20

25

30

35

40

45

50

55

60

65

70

Freq. (Hz)

Am

pl. (

dB)

4 5 6 7 8 9 10

15

20

25

30

35

40

45

50

55

60

65

Freq. (Hz)

Am

pl. (

dB)

4 5 6 7 8 9 10

15

20

25

30

35

40

45

50

55

60

65

Freq. (Hz)

Am

pl. (

dB)

4 5 6 7 8 9 10

10

20

30

40

50

60

70

Freq. (Hz)

Am

pl. (

dB)

Figure 7.8: In-flight test. Comparison between the power spectral density synthesizedfrom the estimated model (full line) and the output spectra (×)

2 3 4 5 6

2

4

6

8

10

12

14

16

18

20

Freq. (Hz)

mod

el o

rder

Figure 7.9: Stabilization chart for the stochastic frequency-domain subspace algorithm.Notice that all identified poles are stable (positive damping), ∗: stable pole.

7.9. Conclusion 175

1 2 3 4 5 6

20

30

40

50

60

70

Freq. (Hz)

Am

pl. (

dB)

1 2 3 4 5 6

10

20

30

40

50

60

70

80

Freq. (Hz)

Am

pl. (

dB)

Figure 7.10: Comparison between the power spectra synthesized from the estimatedmodel (full line) and the output spectra (×)

7.9 Conclusion

In this chapter a frequency-domain subspace algorithm was presented to identifystochastic state-space models from output measurements only. Different fromother frequency domain output-only algorithms is that the proposed algorithmstarts directly from the spectra, while other frequency-domain algorithms usedfor OMA start from (’positive’) power spectra. Therefore, the main advantageof the proposed algorithm is that it can operate from a limited amount of data.Furthermore the relationship and analogy to the stochastic time-domain subspaceidentification is made. Finally the algorithm is applied on both simulations andreal measurements such as in-flight flutter measurements and operational bridgemeasurements. Compared to its time-domain counterpart the main advantage isthe physical interpretation of the frequency domain, the simple pre-filtering andband selection.


7.10 Appendix

The model used for the simulation is given by

A =

[

−0.8455 + 0.4839i −0.0217 + 0.0057i0.0056 + 0.0253i −0.1531 − 0.9131i

]

C =

[

−0.2918 0.3204−0.2848 − 0.0068i −0.3364 + 0.0073i

]

Q =

[

0.5207 −0.0750 − 0.2295i−0.0750 + 0.2295i 0.1120

]

10−5

R =

[

0.0200 0.0682 − 0.0374i0.0682 + 0.0374i 0.3024

]

10−6

S =

[

0.2966 − 0.1275i 0.7726 − 0.9887i0.0135 + 0.1491i 0.3246 + 0.4829i

]

10−6

Chapter 8

Combined frequency-domainsubspace identification

Until recently frequency-domain subspace algorithms were limited to identify deter-ministic models from input/ouput measurements. In the previous chapter, a con-sistent frequency-domain subspace identification method was proposed to identifystochastic models from output-only spectra. In this chapter, a combined deterministic-stochastic frequency-domain subspace algorithm is presented to estimate modelsfrom input/output spectra, frequency response functions or power spectra. The re-lation with time-domain subspace identification is elaborated. It is shown by bothsimulations and real-life test examples that the presented method outperforms tra-ditional frequency-domain subspace methods. The proposed algorithm outperformsthe classic projection frequency-domain subspace algorithms.

177

178 Chapter 8. Combined frequency-domain subspace identification

8.1 Introduction

In general three different types of subspace identification algorithms exist in thetime-domain [123]:

• Identification of deterministic models given by

xn+1 = Axn + Bun

yn = Cxn + Dun

Deterministic algorithms only estimate the dynamics between the measuredoutputs yn and the measured inputs un. Their frequency-domain equivalents,discussed in chapter 6, can be made consistent in the case of colored outputnoise by introducing the covariances of output noise Ck = E

(

NkNHk

)

as afrequency weighting in the projection algorithm

zkXk = AXk + BUk

Yk = CXk + DUk + Nk (8.1)

with Nk zero mean complex circular independent noise. Remind that, al-though this algorithm is consistent, dynamics excited by ambient forces arenot identified, since no information is extracted from Nk. Therefore, thisalgorithm has no combined deterministic-stochastic meaning in the OMAXframework.

• Identification of stochastic models [120] given by

xn+1 = Axn + wn

yn = Cxn + vn

Stochastic models estimate the dynamics of a system without the use ofartificially applied forces un, but instead considers all unmeasured forcesto be white zero mean (random) noise wn and vn. These type of modelshave their applications in the OMA framework and their frequency-domainequivalents is presented in chapter 7.

• Identification of combined deterministic-stochastic models given by [122]

xn+1 = Axn + Bun + wn

yn = Cxn + Dun + vn

white wn and vn white zero mean noise sources. Under the assumption thatthe measured input forces un are uncorrelated with the unmeasurable in-puts wn and vn combined stochastic-deterministic subspace algorithms areconsistent. The combined algorithms identify the dynamics excited by the

8.2. Theoretical Aspects 179

measured inputs, the dynamics excited by unmeasurable forces as well asthe coupled dynamics i.e. modes excited by both the measurable and un-measurable forces. This combined interpretation of this subset of subspacealgorithms fits in the framework of IO-data driven OMAX identification. Inthis chapter a frequency-domain counterpart for these combined stochastic-deterministic algorithms is developed and its applicability is shown by seve-ral examples and simulations. This combined algorithm can also be used toobtain consistent estimates from noisy FRFs and (positive) power spectra.

8.2 Theoretical Aspects

8.2.1 System description

Consider a proper, stable nth order multiple-input/multiple-output combined deter-ministic-stochastic system. The term combined means in the context of this the-sis that the deterministic, the stochastic and the coupled deterministic-stochasticdynamics are considered simultaneously. The frequency-domain state-space equa-tions of a combined discrete-time system is given by

zkXk = AXk + BUk + Wk (8.2)

Yk = CXk + DUk + Vk (8.3)

with Yk ∈ CNo×1 and Uk ∈ C

Ni×1 respectively the vectors of the output and inputspectra at spectral line k, Xk ∈ C

n×1 the state vector at frequency spectral linesk and zk = ei2πk/N (k = 1, 2, . . . , N) covering the full unit circle. The vectorsWk ∈ C

n×1 and Vk ∈ CNo×1 contain respectively the process and the output

measurement noise at spectral line k.

Assumption 1: Wk and Vk are zero mean circular complex independent andidentically distributed (over k) noise sources with covariance matrix

E[(

Wk

Vk

)

(

WHk V H

k

)

]

=

(

Q SSH R

)

(8.4)

where E is the expected value.

The goal of this chapter is to estimate the unknown matrices A,B,C,D,Q,R, S(up to a similarity transformation) from the given spectra Yk and Uk and assumingthe data is generated by an unknown combined deterministic-stochastic system.


8.2.2 Main theorem

Consider the input and output Vandermonde matrices U ∈ C2rNi×N and Y ∈

C2rNo×N respectively given by

U =

z−r1 U1 z−r

2 U2 . . . z−rN UN

z−r+11 U1 z−r+1

2 U2 . . . z−r+1N UN

......

......

z−11 U1 z−1

2 U2 . . . z−1N UN

U1 U2 . . . UN

z1U1 z2U2 . . . zNUN

......

......

zr−11 U1 zr−1

2 U2 . . . zr−1N UN

def=

[

U−U+

]

(8.5)

and similarly for

Y =def=

[

Y−Y+

]

(8.6)

with r > n/No. Both the input and output Vandermonde matrices are divided ina ’positive’ U+, Y+ and a ’negative’ U−, Y− Vandermonde matrices as definedabove. The main theorem of this chapter is then given by

Theorem 1 Under Assumption 1

Y+/U+

[

U−Y−

]

→ OrX for N → ∞ (8.7)

with Or the extended observability matrix given by

Or =

CCA...

CAr−1

(8.8)

and X = [ X1 X2 . . . XN ] the spectra of the forward Kalman filter stateestimates. The oblique projection A/BC is defined by [123]

A/BC = A[

CH BH]

[

CCH CBH

BCH BBH

]†

[:,1:r]

C (8.9)

and the geometrical interpretation of this oblique projection is given in [123].

Proof : Consider the Inverse Discrete Fourier Transform (IDFT) of both the


output and input spectra given by

y(n) = IDFT (Yk) =1√N

N∑

k=1

Ykznk (8.10)

u(n) = IDFT (Uk) =1√N

N∑

k=1

Ukznk (8.11)

From the definition of the IDFT it follows that if y(n) = DFT (Y ) then y(n+m) =DFT (zmY ). Taking the IDFT from the basic model equations (8.2) and (8.3)results in the following time-domain state space model

x(n + 1) = Ax(n) + Bu(n) + w(n) (8.12)

y(n) = Cx(n) + Du(n) + v(n) (8.13)

and w(n) = 1√N

∑Nk=1 Wkzn

k , v(n) = 1√N

∑Nk=1 Vkzn

k zero mean, white noise vector

sequences. The input and output block Hankel matrices are then given by

u−r u−r+1 . . . uN−r

......

. . ....

u−2 u−1 . . . uN−2

u−1 u0 . . . uN−2

u0 u1 . . . uN−1

u1 u2 . . . uN

......

. . ....

ur−1 ur . . . uN+r−2

def=

[

up

uf

]

(8.14)

and similarly for

[

yp

yf

]

.

The main theorem for combined deterministic-stochastic subspace identificationin the time-domain, presented in [123], states that

yf/uf

[

up

yp

]

→ Orx for N → ∞ (8.15)

with x = [x0 . . . xN ]T the forward Kalman filter state sequence estimate. Considerthe Discrete Fourier Transformation matrix

F =1√N

z01 z0

2 . . . z0N

z−11 z−1

2 z−1N

.... . .

...

z−(N−1)1 z

−(N−1)2 . . . z

−(N−1)N

(8.16)


From the definition of the DFT and IDFT it follows that

Y− = ypF

Y+ = yfF

U− = upF

U+ = ufF (8.17)

and F is orthogonal, i.e. FFH = I. Applying the property A/BC = D ⇔[AF ]/[BF ][CF ] = DF if FFH = I, which can be proven by the definition of theoblique projection, for the time domain theorem results in

Y+/U+

[

U−Y−

]

= [yfF]/[ufF]

[

[upF][ypF]

]

(8.18)

= yf/uf

[

up

yp

]

F (8.19)

= OrxF (8.20)

= OrX (8.21)

which ends the proof.

The main theorem Eq. 8.15 in time-domain subspace is proven to be consistent,when at least one of the following conditions is satisfied [122]:

• The system is purely deterministic, i.e. wn = 0 and vn = 0

• The input signal is white noise.

• r → ∞, since for r → ∞ the non-steady state Kalman filter converges toa steady state Kalman filter. This is intuitively clear, since by the timethe Kalman filter is in steady state, the effect of the initial conditions hasdied out. In practice r ≥ n/No + 8 already results in good estimates formechanical systems [122].

8.2.3 From states to system matrices

Once the states Xk are estimated by the oblique projection according the maintheorem Eq. 8.7 the system matrices A, B, C and D can be estimated in a LSsense

[

zkXk

Yk

]

=

[

A BC D

] [

Xk

Uk

]

+

[

ρWk

ρVk

]

(8.22)

Since it is assumed that no input noise is present and the state spectra Xk areconsistent estimates, the least squares estimate of the system matrices is consistent.

8.3. Practical Implementation 183

The covariance matrix of the stochastic sources Wk and Vk is then obtained fromthe residuals ρWk

and ρWkas

[

Q SSH R

]

=1

N

N∑

k=1

[(

ρWk

ρVk

)

(

ρHWk

ρHVk

)

]

(8.23)

8.2.4 Taking into account effects of transients and leakage

Until now it was assumed that the given spectra Yk and Uk obey the frequency-domain state-space model. Since the spectra of the outputs and inputs are ob-tained from the DFT of the measured time signals Yk = 1√

N

∑N−1n=0 y(n)z−n

k and

Uk = 1√N

∑N−1n=0 u(n)z−n

k the proposed model is only true if x(N) = x(0) or

N = ∞. In all other cases the extra term T must be taken into account inthe state space model (according to paragraph 6.5.1). The extended combineddeterministic-stochastic frequency-domain state-space model is then given by

zkXk = AXk + BUk + Tzk + Wk (8.24)

Yk = CXk + DUk + Vk (8.25)

where T = x(N)−x(0)√N

models the influence of the initial and final conditions. These

extra parameters T (T ∈ Cn×1) model the non-steady state response of the system.

Taking into account these additional parameters makes the frequency model robustfor leakage and transients effects in case a rectangular window (uniform window) isused for the calculation of the spectra. This observation generalizes the stochasticfrequency-domain subspace identification method proposed in chapter 7, since theinfluence of leakage and transients can be modelled by a combined deterministic-stochastic model with input Uk = 1√

Nzk

zkXk = AXk + BUk + Wk (8.26)

Yk = CXk + DUk + Vk (8.27)

and B represents the transient term T . As a conclusion the presented algorithmin this chapter allows to identify combined deterministic-stochastic models fromIO data and purely stochastic models from output-only spectra, without sufferingfrom transient and leakage errors.

8.3 Practical Implementation

In this section the implementation of the different algorithm steps is discussed:


1. Form the Vandermonde matrices Y+, Y−, U+ and U− from the measuredspectra Yk and Uk. The inputs can be extended by zk to model the tran-sients in the measurements. In case one wants to estimate real systemmatrices A, B, C and D, the Vandermonde matrices V are replaced by[ Re(V) Im(V) ] (V = Y+,Y−,U+ and U−).

2. Calculate the RQ decomposition as

Y+

U+

U−Y−

=

R11 0 0 0R21 R22 0 0R31 R32 R33 0R41 R42 R43 R44

Q1

Q2

Q3

Q4

(8.28)

then[

U−Y−

]

= RaQ with Ra =

[

R31 R32 R33 0R41 R42 R43 R44

]

(8.29)

U+ = RbQ with Rb =[

R21 R22 0 0]

(8.30)

Y+ = RcQ with Rc =[

R11 0 0 0]

(8.31)

(8.32)

with QT = [ QT1 QT

2 QT3 QT

4 ]. The oblique projection Y+/U+

[

U−Y−

]

is then given by

Y+/U+

[

U−Y−

]

= Rc

[

I − RHb

[

RbRHb

]†Rb

] [

Ra

[

I − RHb

[

RbRHb

]†Rb

]]†RaQ (8.33)

A profound discussion of this expression for the oblique projection is givenin [123].

3. In the third step the SVD of the projection is calculated as

Y+/U+

[

U−Y−

]

= USV H (8.34)

4. Determine the extended observability matrix Or and the state estimate X.For a chosen order n (in theory the order n is equal to the rank of S)

Or = U1S1/21 (8.35)

X = S1/21 V H

1 (8.36)

with U1 = U[:,1:n], S1 = S[1:n,1:n] and V1 = V[1:n,:]. In the case that real

system matrices are estimated X ′ = S1/21 V H

1 and X = X ′[:,1:N ]+iX ′

[:,N+1:2N ].

5. Knowing Yk, Uk and Xk the system matrices are obtained by solving the LSproblem 8.22 and the covariance matrix of the process and noise is obtainedby 8.23

The total identification procedure is schematically given by figure 8.1.

8.4. Remarks 185

Estimate X by the oblique projection (QR and SVD)

-+ -

-+

UY / Y

U YrO X

=

k k k k

k k k

z X X WA B

Y Y VC D

= + Least-squares

• Frequency band selection, scaling

• Construct the Vandermonde matrices

A,C,B,C natural frequencies, damping ratios, mode shapes andmodal participation factors (EVD)

Figure 8.1: Schematic overview of the stochastic frequency-domain subspace algorithm

8.4 Remarks

• Equation (8.34) can be replaced by

W1 Y+/U+

[

U−Y−

]

W2 = USV H (8.37)

with W1 ∈ CNor×Nor and W2 ∈ C

N×N such that W1 is of full rank andrank([ U− Y− ]T ) = rank([ U− Y− ]T W2). For time-domain com-bined deterministic stochastic subspace it is shown in [123] that specialchoices of W1 and W2 correspond to well-known algorithms like the N4SID(Numerical algorithms for Subspace State Space System Identification),MOESP (Multivariable Output-Error State Space) and the Canonical vari-ate algorithm (CVA). Analogously the frequency counterpart of this weighingmatrices correspond to the frequency counterparts of the N4SID, MOESPand CVA algorithms.

• The presented algorithm can also be used as a combined deterministic-stochastic frequency-domain subspace method processing Frequency ResponseFunctions (FRFs) or positive power spectra. By simple substitution of Yk =


Hk or Yk = G+yy,k and Uk = Ik with Hk ∈ C

No×Ni and G+yy,k ∈ C

No×Nref

respectively the FRF matrix and power spectra matrix at spectral line k andIk a unity matrix with dimensions Ni. Since the inputs Uk are consideredwhite in this case, the projection results in a consistent estimate of the statespectra and the observability matrix. Compared to the consistent determin-istic projection algorithm of paragraph 6.4 the advantage is that no a-prioriknown covariance matrix of the noise has be taken into account.

• The same motivations to perform the identification in the frequency do-main as demonstrated for the stochastic subspace identification are valid forthe combined identification: i.e. easy pre-filtering of the data and simplefrequency band selection. The chosen frequency band is re-scaled to coveruniformly the full unit circle (half unit circle in case of real system matrices)to ensure consistency and the numerical stability is improved, since the basisfunctions zr are orthogonal. In practice, it is advised to cover only 90% tomake a trade-off between consistency and model errors (since the frequencyband of the underlying continuous-time domain model is approximated by adiscrete-time domain model).

8.5 Simulations and Real-life measurement exam-ples

8.5.1 Monte Carlo Simulation

Lets consider a discrete 6th order system excited by 2 Gaussian random sequences.The A, B, C and D matrices are given in appendix 8.7. The first input is a stan-dard gaussian distributed sequence with standard deviation 1, while the secondinput has a standard deviation of 0.5. All simulations are performed in the time-domain to include the non-steady state responses. During the identification processthe second input U2,k is considered as unknown and by consequence the identifi-cation process can be considered as a combined deterministic-stochastic problem.The response in the frequency-domain can be considered as a sum of three contri-butions i.e. the deterministic part, the transient part and the stochastic part

Yk =[

C(Izk − A)−1B[:,1] + D[:,1]

]

U1,k + C(Izk − A)−1Tzk

+ C(Izk − A)−1Wk + Vk (8.38)

with in this simulation Wk = B[:,2]U2,k and Vk = D[:,2]U2,k. To examine theconsistency properties (asymptotic unbiasedness), Monte Carlo simulations areperformed for 4 different cases of the total number of time samples Nt (N = Nt/2)in order to check if the expected value of the estimated model is equal to the truemodel for the number of frequency lines N going to infinity. As an error measurethe mean square relative error (MSRE) over the different frequencies between the

8.5. Simulations and Real-life measurement examples 187

0 0.1 0.2 0.3 0.4 0.5−180

−160

−140

−120

−100

−80

−60

−40

−20

frequency (Hz)

Am

pltit

ude

(a)

0 0.1 0.2 0.3 0.4 0.5−160

−140

−120

−100

−80

−60

−40

−20

frequency (Hz)

Am

pltit

ude

(b)

Figure 8.2: Mean and variance of the estimated system for N=250 ( exact system, full

line¯G, dashed line σ2

G: a) Combined deterministic-stochastic frequency domain subspace

algorithm (A1) b) Classical frequency-domain projection subspace algorithm (A3). Onecan clearly notice the bias of algorithm A3.

Table 8.1: MSRE for Monte Carlo simulations comparing the presented combined algo-rithm including transient effects (A1), combined algorithm not including transient effect(A2) and the classical frequency domain subspace projection algorithm (A3). From thesimulation results it is clear that the MSRE of A1 and A2 decreases if the number offrequency lines N increases, while A3 remains biased

N=150 N=250 N=400 N=600A1 0.0097 0.0054 0.0042 0.0027A2 0.0373 0.0074 0.0066 0.0041A3 0.2810 0.2132 0.2760 0.3404

mean estimated model (for the 100 runs and 100 estimated models) and the truemodel is calculated

|E|2 =1

N

N∑

k=1

|¯Gk − Gk

Gk|2 (8.39)

with Gk the ’true’ system between the output and the first input and¯Gk the mean

of the 100 estimated models. This MSRE is calculated for N = 150, 250, 400 and600 given in table 8.1. The presented subspace model is compared with the classicalprojection frequency domain projection algorithm. The presented algorithm wasapplied including the estimation of the transient T by considering an conditionalinput zk and without estimating the transient effects. The identification startsfrom the spectra of the response Yk and the first input U1,k. All spectra arecalculated using a rectangular window. From table 8.1 it is clear that the proposed


combined algorithm (A1,A2) outperforms the classical projection algorithm (A3).Furthermore, the error is further reduced by taking into account the transientseffect T to compensate for leakage and the non-steady state behavior (A1 versusA2). This is more important for small Nt, since for large Nt both leakage and the

transient effect have less influence. In figure 8.2, the mean estimated system¯G and

the variance σ2G

over the 100 simulations are shown for the different identificationcases for N = 250. Assume now that, both inputs are unknown and thus onlythe spectra of the outputs are known. In this case the identification procedurebecomes a stochastic identification problem. The influence of the final and initialconditions can still be modelled by considering the parameters T and the inputzk. The output can be considered as the sum of transient effects and a stochasticcontribution as shown in Eq. 8.40.

Yk = C(Izk − A)−1Tzk + C(Izk − A)−1Wk + Vk (8.40)

The presented algorithm can still estimate the system matrices C and A. This isimpossible by the classical frequency-domain projection subspace algorithm, sinceit models only a deterministic model between the input and output spectra. Fromthe 100 simulations the mean value and the standard deviation of the estimates ofthe magnitude R and angle θ of the second pole p2 = Reiθ with R = 0.9858 and θ =1.5938 is shown in Table 8.2. (similar for the other poles). The simulations show

Table 8.2: Monte Carlo simulations for a stochastic identification for different numbersof spectral lines N

N=300 N=600 N=900R 0.9826 0.9840 0.9851σR 0.0061 0.0044 0.0034θ 1.5950 1.5944 1.5939σθ 0.0070 0.0043 0.0039

clearly that the differences between the mean values of the 100 times estimatedmagnitude and phase of the pole p2 and the exact value decreases for an increasingnumber of spectral lines (consistent). The variances on the estimated R and θdecrease for an increasing number of spectral lines (consistency). Figure 8.3 showsthe output spectra and the estimated stochastic model from this spectra for 1simulation.

8.5.2 Flight flutter simulation

To show the applicability of the combined deterministic-stochastic algorithm, thesame in-flight simulation data based on a GVT test of an aircraft (as used in para-graph 4.8.1) are processed to compare the combined common-denominator (CLSF)estimator with both the deterministic and combined deterministic-stochastic


0 0.1 0.2 0.3 0.4 0.5−120

−100

−80

−60

−40

−20

0

20

40

frequency (Hz)

ampl

itude

(dB

)

Figure 8.3: Output spectra (dots), estimated stochastic model (full line)

frequency-domain subspace estimators. For each of the 6 increasing turbulence lev-els 10 different runs are simulated and processed. Figure 8.4 illustrates the meanvalues and standard deviations of the estimated damping ratios by the different al-gorithms for the 6 turbulence levels. The classic deterministic projection algorithmis less accurate than both combined algorithms. For the first two modes, i.e. wingmodes as shown in figure 4.9, the larger uncertainty and bias can be explainedby the influence of leakage. Both combined algorithms included an additionalpolynomial T (z) or term T to model the initial and final conditions, while thiswas not the case for the deterministic subspace algorithm. Furthermore it is clearthat the damping ratios of the tail modes 3 and 4 (which are only well excited bythe unknown turbulent forces), can only be accurately identified by the combineddeterministic-stochastic algorithms. Both the combined common-denominator al-gorithm and the combined subspace algorithm result in a comparable accuracy ofthe estimated damping ratios. Nevertheless, it should be noted that compared tothe combined common-denominator algorithm, the combined subspace algorithm:

• forces rank 1 residues on the measurements.

• no optimization procedure is required.

• the construction of stabilization diagram is possible in a fast way.

• both the amplitude and phase of the modes excited by the unknown forcescan be determined.

This makes the combined frequency-domain subspace algorithm in the general casesuperior to the common-denominator based combined algorithm.


1 2 3 4 5 6

1.2

1.3

1.4

1.5

1.6

1.7

1.8


turbulence level

dam

ping

rat

io

1 2 3 4 5 6

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95


turbulence level

dam

ping

rat

io

1 2 3 4 5 6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8


turbulence level

dam

ping

rat

io

1 2 3 4 5 6

1

1.5

2

2.5


turbulence level

dam

ping

rat

io

1 2 3 4 5 6

2.1

2.2

2.3

2.4

2.5

2.6


turbulence level

dam

ping

rat

io

1 2 3 4 5 6

1

1.5

2

2.5

3

3.5

4


turbulence level

dam

ping

rat

io

Figure 8.4: Comparison of the damping ratio estimates (∗ : CLSF IO, : determin-istic frequency-domain subspace , : combined frequency-domain subspace, – : exactparameter)

8.5.3 Flight flutter measurements

To show the applicability of the combined deterministic-stochastic frequency do-main the same real-life in-flight vibration measurements as introduced in para-graph 7.8.3 are now analyzed. In this case the measured signal of the angle per-


4 5 6 7 8 9 10

2

4

6

8

10

12

14

Freq. (Hz)

mod

el o

rder

(a)

4 5 6 7 8 9 10

2

4

6

8

10

12

14

Freq. (Hz)

mod

el o

rder

(b)

4 5 6 7 8 9 10

2

4

6

8

10

12

14

sf f

fff o

fofso

ffossf

ssfs sff

sfsf o s f o

ofssf f f f f

fffffffs f

f fsssf s s f os

fosffos f f f o

sffssos of of f f

f ofssfoos o s s f f

f fffofof o ff f f o

Freq. (Hz)

mod

el o

rder

(c)

4 5 6 7 8 9 10

2

4

6

8

10

12

14

sf f

sff

sofsd

ffofsf

ssosfs o

sffsff o

ossss s ss o

dsffs f ss s

sssss s ss s

s s ss s ssso sf

s s ss sssss do

ssss fos sf ssf

s s sss s f ssss d f

Freq. (Hz)

mod

el o

rder

(d)

Figure 8.5: Comparison between stabilization charts obtained by the deterministicand combined subspace algorithm: (a) deterministic algorithm (b) combined algorithm(c) deterministic algorithm after applying relative criteria (d) combined algorithm afterapplying relative criteria.

turbation of the flaps is used as the input force signal. The stochastic contributionin the responses is caused by the turbulent forces acting on the airplane duringflight. Starting from the input and output spectra in a frequency band from 3Hz to11Hz, a state-space model with real system matrices is estimated by both the clas-sical projection and combined deterministic-stochastic frequency-domain subspacealgorithms. Figure 8.5 compares the stabilization charts for both the (determin-istic) projection algorithm and the combined deterministic stochastic approaches.Based on the relative differences between the eigenfrequencies and damping ratiosestimated for subsequent model orders n, the poles are labelled in the stabilizationdiagram. The symbols s, f , d and o respectively mean

• s: both relative damping ratio difference < 15% and relative eigenfrequency


Table 8.3: Natural frequencies and damping ratios identified by the deterministic andcombined deterministic-stochastic frequency-domain algorithm.

fdet (Hz) ddet(%) fcom (Hz) dcom(%)4.08 1.95 4.09 3.064.60 4.07 4.69 3.395.27 3.46 5.16 5.59/ / 5.35 2.39

6.06 0.17 6.01 5.198.07 2.95 8.07 3.278.80 2.20 8.71 3.259.07 2.91 9.17 3.60/ / 9.87 3.81

difference < 3%

• f : relative damping ratio difference ≥ 15% and relative eigenfrequency dif-ference < 3%

• d: relative damping ratio difference < 15% and relative eigenfrequency dif-ference ≥ 3%

• o: relative damping ratio difference ≥ 15% and relative eigenfrequency dif-ference ≥ 3%

After applying these criteria it is clear that the stabilization diagram of the com-bined algorithm results in a easier interpretation of the chart than the determin-istic algorithm. Table 8.3 clearly shows that some of the natural frequencies anddamping ratios are not identified by the deterministic approach. Furthermore,large deviations in the estimated damping ratios can be observed between the de-terministic and combined identified parameters. From the aircraft manufacturer itwas confirmed that all the 9 poles identified with the combined are the only 9 polesin the frequency band and the damping ratios were in very close agreement withthe manufacturers specification. From figure 8.6, which compares the synthesizedFRFs for both the deterministic and combined algorithm with the FRFs obtainedfrom the H1 estimator, it is clear that the deterministic algorithm results in muchlarger model errors than the combined algorithm.

8.5.4 ABS-function driven identification

The combined deterministic-stochastic algorithm can also start from FRFs, posi-tive power spectra or both simultaneously as primary data to estimate the system


4 6 8 10−20

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pl. (

dB)

Figure 8.6: Comparison between the FRFs (×) and the synthesized FRFs obtainedby the classic projection algorithm (left) and the combined deterministic-stochastic fre-quency domain subspace algorithm (right). The classical projection approach results inlarge bias errors. (times: measured FRFs, full line: synthesized FRFs)

matrices and modal parameters. The combined algorithm was used in chapter 3process both the simulation data and measurements obtained on the Villa Pasobridge. The next two examples show that the algorithm is also capable to startfrom FRF data from modal testing in the laboratory which illustrates its capabil-


ity to handle a large number of outputs, several inputs, frequency band selectionand large model orders.

Subframe of an engine

10 20 30 40

10−2

10−1

100

101

102

MS

RE

FRF

Figure 8.7: Comparison between the mean square relative errors for the 46 FRFs.( classical projection frequency-domain subspace method, ∗ combined deterministic-stochastic frequency domain subspace algorithm)

The accelerations were measured at 23 response locations and two randominputs were applied by electrodynamic shakers. Before the actual identificationtask started, the frequency band of interest between 210 Hz and 410 Hz is re-scaledto cover 90% of the full unit circle (complex model). A model order of 30 was used.Both the models identified by the classical projection algorithm and the combinedalgorithm, are validated with a high quality validation data set by means of theMSRE over all frequencies per FRF. Figure 8.7 shows the MSRE for the 46 FRFsfor both identification approaches. The presented combined approach resultedin an average MSRE over all FRFs of 0.2, while the classical projection basedsubspace method resulted in a mean MSRE of 5.4. Figure 8.8 compares someidentified FRFs for both identification approaches with a high quality validationdata set.

Laser vibrometer measurements on a car door

To illustrate that this combined frequency-domain subspace algorithm is capableto process FRF measurements with high modal and high spatial density measure-ments were performed by a scanning laser vibrometer on the door of a car. TheFRF matrix has 79 responses, 1 input and a frequency resolution of 0.31Hz. Afrequency band from 46Hz up to 120Hz is modelled by a real state-space model


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Am

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Figure 8.8: Comparison between the measured FRFs and the synthesized FRFs by theclassical projection algorithm (left) and the combined deterministic-stochastic frequencydomain subspace algorithm (right). The classical projection approach results in largebias errors. (×: measured FRFs, full line: synthesized FRFs)

containing 50 modes, with the frequencies covering 90% of the half unit circle.Figure 8.9 shows the high quality of the synthesized FRFs for 4 different FRFmeasurements. Chapter 9 discusses in more detail how the physical poles can bedistinguished from the mathematical poles by a small adaption of the algorithm


to obtain clear stabilization charts based on the sign of the damping ratio.

8.6 Conclusion

This chapter presented a combined deterministic-stochastic frequency domain sub-space algorithm resulting in consistent estimates of the system matrices and thenoise covariance matrix. Furthermore, it is shown that the proposed algorithmis the frequency-domain counterpart of the well-known combined deterministic-stochastic time-domain subspace algorithms. The proposed algorithm can be usedto process output-only measurements with a transient term, input-output measure-ments, frequency response functions and (’positive’) power spectra. In the caseof input-output data, the algorithm can be considered in the OMAX framework.The applicability is illustrated by means of both simulations and several real-lifemeasurements. The method identifies simultaneously the deterministic dynam-ics, the stochastic dynamics and the coupled stochastic-deterministic dynamics

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Figure 8.9: Comparison between the measured FRFs and the synthesized FRFs by thecombined deterministic-stochastic frequency domain subspace algorithm. (×: measuredFRFs, full line: synthesized FRFs)

8.7. Appendix 197

an therefore outperforms classical deterministic subspace projection algorithms.Furthermore, the algorithm can take advantage of the frequency-domain by asimple frequency band selection. The proposed combined stochastic-deterministicfrequency-domain subspace algorithm is very promising to process all types offrequency-domain data.

8.7 Appendix

The 6th order model used for the simulation is given by

A =

−0.0103 0.9928 0.0377 −0.1746 −0.0367 0.0365−0.9409 −0.0070 0.2141 0.2065 0.1868 −0.04670.0076 −0.1240 0.6504 −0.8057 −0.1547 0.05380.1601 0.0689 0.6680 0.6289 −0.1739 −0.0240−0.0136 −0.0056 −0.0103 0.0168 −0.2689 0.79850.0003 −0.0024 −0.0315 −0.0193 −0.9110 −0.6742

C =[

−0.3971 −0.0433 −0.3410 −0.0192 −0.3572 −0.0234]

B =

−1.2561 −1.03950.1651 0.0782−0.7939 −0.66350.7051 0.7799−0.1138 0.5930−0.6482 −1.4204

10−3

D =[

−0.3937 −0.5708]

10−3

(8.41)


Chapter 9

The secrets behind clearstabilization charts

This chapter analyzes the influence of the parameter constraint and basis functionused in the implementations for Modal Parameter Estimation (MPE) methods onthe quality of the stabilization diagram. It is shown that by a proper choice of theparameter constraint/basis function a distinction can be made between physical andmathematical poles based on the sign of the real part of the pole. As a result, thequality of stabilization diagrams can be improved. This approach can be applied forseveral well-known modal parameter estimation methods and the user can benefitfrom this. Several MPE applications in aerospace, automotive and civil engineeringare studied in this chapter, showing the implication of the constraint/basis functionand the specific advantages and disadvantages of obtaining an easy-to-interpretstabilization diagram.

199

200 Chapter 9. The secrets behind clear stabilization charts

9.1 Introduction

In the previous chapters proposed several advanced modal parameter estimation al-gorithms are proposed. In modal analysis, one typically uses an identification orderthat is guaranteed to be larger than necessary to be sure all dynamics of the struc-ture are described. Unfortunately, as the model order of the model is increased,so will the number of identified poles. This results in the introduction of so-calledmathematical poles, which have no direct physical interpretation. Therefore, animportant step in modal parameter estimation (MPE), where user interaction isrequired, is the interpretation of the stabilization diagram to make the distinctionbetween physical and mathematical modes. This chapter goes into the mathemat-ical background of several MPE algorithms such as e.g. Least Squares ComplexExponential (LSCE) method [52], [72], Least Squares Complex Frequency domain(LSCF)[46] and frequency-domain subspace methods in order to discover the crit-ical and underestimated importance of the parameter constraint/basis function inthe algorithms. It will be shown and explained how the choice of a constraint/basisfunction in solving an identification problem has its influence on the separationbetween physical and mathematical modes. In a next step, it is explained how thismathematical key idea can be exploited in several MPE algorithms and its effecton the stabilization diagram is illustrated. Both the advantages and disadvantagesof the choice of the constraints/basis function are discussed.

In several papers it was noticed that methods like the LSCF [116], [113], [114]and the more recent poly-reference LSCF (also called PolyMAX algorithm) [46],[89] produce very clear stabilization diagrams. This chapter explains why thesemethods have such nice properties for EMA applications. Furthermore, this keyidea behind the construction of clear diagrams is generalized and applied for otherwell known MPE algorithms such as e.g. the LSCE algorithm and frequency-domain subspace algorithms.

This chapter starts with a short introduction to the LS identification problem,next the influence of a mathematical constraint is briefly explained by means ofa simple single input-single output (SISO) Auto-Regressive model (AR) [67]. Itis shown that the stochastic model for output-only data can be described exactlyby stable, unstable or a mix of stable and unstable poles. Depending on the con-straint applied to the coefficients, the stochastic data is modelled by a stable orunstable model. The analogy with state-space models and stochastic frequency-domain subspace identification algorithms is given. In a next step, the conclusionsfor this AR model and stochastic subspace identification are extrapolated to Auto-Regressive eXogeneous (ARX) and combined deterministic-stochastic state-spacemodels. It is shown how MPE methods like LSCE, LSCF, p-LSCF and the sub-space algorithms fit in this framework. Finally, the demonstrated influence of theconstraint on the quality of the identified models and stabilization charts is dis-cussed and demonstrated by several real life applications. The theory developedin this chapter is closely related to the results given in [62], where a distinction


between physical and mathematical poles for the estimation of damped sinusoidsis given based on the sign of the damping for an ARX time-domain model. Inthis chapter a frequency-domain counterpart is developed and the influence of theconstraint or basis function is investigated. Finally the results are applied for se-veral modal parameter estimation techniques and illustrated by several practicalexamples.

9.2 Theoretical Aspects

9.2.1 LS Identification

The goal of this section is to show the influence of the constraint for a LeastSquares (LS) problem in identification algorithms. Consider an over-determinedset of equations

Jθ = E (9.1)

with θ a set of unknown parameters, J ∈ Cn×m (n > m) the information ma-

trix and assume E ∈ Cn×1 to be vector with complex circular independent and

identically distributed noise on the equations. To solve this set of equations in aLS sense, a constraint must be applied on the coefficients in order to remove theparameter redundancy. For example consider, the j-th parameter fixed to one i.e.θj = 1. In that case the least squares solution is given by

J θ = −Jj + E

θ = −J†Jj + J†E (9.2)

with J = J[:,1...j−1,j+1...m], θ = θ[1...j−1,j+1...m], Jj the j-th column of J and J†

the pseudo inverse of the matrix J given by J† =(

JH J)−1

JH (·H is the complex

conjugate transpose (Hermitian) of a matrix). It can simply be shown that theestimate for the parameters is consistent if J†E → 0 for n → ∞. In the nextsubsections a more profound discussion is given about the influence of the choiceof the constraint on the stability of the poles.

9.2.2 Influence of the constraint on an AR model

Consider a SISO autoregressive model given by

a0y(n) + a1y(n − 1) + . . . ary(n − r) = e(n) (9.3)

with r the order and aj complex coefficients (real coefficients are considered asa special case of complex coefficients). Taking the DFT of Eq. 9.3 leads to the


frequency-domain counter part

(a0 + a1z−1k + . . . + arz

−rk )Yk = Ek (9.4)

with Yk = 1√N

∑N−1n=0 y(n)z−n

k , Ek = 1√N

∑N−1n=0 e(n)z−n

k and zk = ei2πk/N . Notice

that both leakage and transients are neglected. Taking into account these effectswould introduce an extra polynomial as discussed in chapter 3. The poles of thisAR-model are given by the solutions of the characteristic equation given by

aozr + a1z

r−1 + . . . ar = 0 (9.5)

Frequency-domain solution

Since the only assumption on Ek in Eq. 9.4 is that Ek is circular complex in-dependent and identically distributed noise, the phases of Eq. 9.4 contain noinformation. Therefore, Eq. 9.4 is totally equivalent with

|a0 + a1z−1k + . . . + arz

−rk ||Yk| = |Ek| (9.6)

with |x| the absolute value of x and as a result, every set of estimated parameters[a0 a1 . . . ar] that satisfies Eq. 9.6 is a solution of equation 9.4. The polynomialA(z) can be written as

A(z) = z−r(a0zr + a1z

r−1 + . . . ar) (9.7)

= a0z−r(z − λ1)(z − λ2) . . . (z − λr) (9.8)

with λj = eσj+iωj and σj , ωj respectively the damping and natural frequency.The magnitude of the polynomial A(z) is given by

|A(z)| = |a0||z − λ1||z − λ2| . . . |z − λr| (9.9)

In appendix 9.6, it is proven that

|zk − λj ||zk − λ

j |=

|zk − eσj+iωj ||zk − e−σj+iωj | = Rj (9.10)

with λj = eσj+iωj , λj defined by e−σj+iωj and Rj a constant independent of zk.

(Applying the operator · to a pole changes the sign of the damping and thus thestability, while the natural frequency remains unchanged.) As a consequence, every

term (zk − λj) of A(z) can be replaced by (zk − λj ) without any consequence on

the validity of Eq. 9.6. The conclusion is that Eq. 9.4 has several solutions, sincethe sign of the damping σj does not change the amplitude of the polynomial. Inthe context of this chapter, we are particularly interested in two special solutions:

• The solution with only stable poles (σj < 0) given by

As(z) = asz−r(z − λs1)(z − λs

2) . . . (z − λsr) (9.11)


• The solution with only unstable poles (σj > 0) given by

Au(z) = auz−r(z − λu1 )(z − λu

2 ) . . . (z − λur ) (9.12)

with λuj = (λs

j).

Consider the identification process of the coefficients aj of A(z) starting from thespectra of the response Yk. This identification can be formulated as an over-determined set of equations in the same way as in Eq. 9.1.

Y1 Y1z−11 . . . Y1z

−r1

Y2 Y2z−12 . . . Y2z

−r2

......

. . ....

YN YNz−1N . . . YNz−r

N

a0

a1

...ar

=

E1

E2

...EN

(9.13)

Eq. 9.13 can be solved in a least squares sense according to Eq. 9.2.

Depending on the constraint applied, the LS solution of Eq. 9.13 resultsin stable or unstable poles.

• The constraint a0 = 1 results in a consistent estimate of the poly-nomial As(z) with only stable poles.

• The constraint ar = 1 results in a consistent estimate of the poly-nomial Au(z) with only unstable poles.

The proof of this statement is given in appendix 9.7. A stable, unstable or mixedstable-unstable AR process in the frequency-domain can be identified as a stableor unstable system that fulfils Eq. 9.4 depending on the choice of the constraint.This observation plays a major role in the next sections.

Example Consider a system given by the coefficients a0 = 1, a1 = 0.5589−0.9710iand a2 = −0.6941 − 0.8036i. From the characteristic equation Eq. (9.5) thepoles in the discrete-time domain can be calculated, i.e. λ1 = 0.5243 + 0.8166iand λ2 = −1.0832 + 0.1544i. The continuous time domain poles are given byln(λ1) = σ1 + iω1 = −0.03 + i and σ2 + iω2 = 0.09 + 3i (∆t = 1). The first pole isstable |λ1| < 1 or σ1 < 0 and the second pole is unstable. Consider the spectrumYk = A(zk)−1Ek for k = 1, . . . , N (zk covers the full unit circle) and Ek zeromean circular complex independent and identically distributed noise e.g. real andcomplex part of Ek white independent Gaussian noise. From the spectrum Yk thepoles λ1 and λ2 are estimated (N=1024). Figure 9.1(a) illustrates one of the outputspectra Y . A Monte Carlo simulation was done for 100 runs for both constraintsa0 = 1 and a2 = 1. Figure 9.1(b) shows the estimated damping values σ for bothconstraints. It is clear that both constraints estimate exactly the same magnitudeof the damping values, but different signs. The constraint a0 = 1 estimates stable


0.2 0.4 0.6 0.8 1

−40

−30

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0

10

20

30

frequency (Hz)

ampl

itude

(dB

)

(a)

20 40 60 80 100

−0.1

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0

0.05

0.1

Monte Carlo runs

dam

ping

val

ue s

igm

a

(b)

Figure 9.1: (a) Spectra of the response Y(k); (b) Monte Carlo runs: full line exactdamping values, estimated damping values σ for a0 = 1 (∗), estimated damping valuesσ for a2 = 1 ()

poles (σ < 0) for every run, the constraint a2 = 1 estimates unstable poles for allruns.

Changing the constraint from a0 = 1 to ar = 1 is equivalent with replacingz−1 by z which can be seen from

(1 + a1z−1k + . . . + arz

−rk )Yk = Ek (9.14)

Replacing the basis function z−1k by zk results in

(1 + a1zk + . . . + arzrk)Yk = Ek (9.15)

and multiplication by z−r does not change the assumption made for Ek (since thephase of Ek is random) thus

(z−r + a1z−r+1k + . . . + ar)Yk = E

′

k (9.16)

with E′

k = z−rEk. This corresponds to a constraint for the highest order coefficientin z−1 fixed to one or with ar = 1 in Eq. 9.4. Thus replacing z−1 by zk in theidentification process of the AR model results in a change of the sign of the dampingvalues σj . Finally, it was observed that solving the identification problem in a TLSsense by imposing the norm-2 of the coefficients fixed to 1 results in damping valuesσj = 0 and thus biased estimates.

Time-domain Solution

Under the assumption that the system is in steady-state, the time-domain identi-fication has the same properties as the frequency-domain identification. It is well


Table 9.1: Stability of the poles in function of the constraint and basis function

z−1 za0 = 1 unstable stablear = 1 stable unstable

known that the constraint a0 = 1 results in a consistent estimation of the polesof a stable system [67]. Consider the identification problem of the coefficientsa0, a1, . . . ar in Eq. 9.3 starting from the time responses y(n). The set LS ofequations is now given by

y(r) y(r − 1) . . . y(0)y(r + 1) y(r − 2) . . . y(1)

......

. . ....

y(N) y(N − 1) . . . y(N − r)

a0

a1

...ar

=

e(r)e(2)

...e(N)

(9.17)

The constraint a0 = 1 corresponds with

y(n) + a1y(n − 1) + . . . ary(n − r) = e(n) (9.18)

The solution of the least squares problem will be consistent if e(n) is only corre-lated with y(n) and uncorrelated with y(n − j) with j ∈ N0. These assumptionscorresponds to looking forward in time. On the other hand the constraint ar = 1corresponds with

a0y(n) + a1y(n − 1) + . . . y(n − r) = e(n) (9.19)

Solving the least squares problem with this constraint implies that e(n) can onlybe correlated with y(n − r) and uncorrelated with y(n − i) with i = 0, . . . , r − 1in order to be consistent and this corresponds to looking backward in time. Sincethe difference between both constraints corresponds to reversing the time axis,it is clear that the natural frequencies for both constraints are the same and thedamping values differ from sign. Table 9.1 gives an overview of the poles in functionof the constraint and basis function.

9.2.3 Stochastic State-Space models

Consider the stochastic state-space model from chapter 7 given by


Yk = CXk + Vk (9.21)


with Wk and Vk are zero mean circular complex independent and identically dis-tributed noise sources. The output spectra can also be written as

Yk = C (Izk − A)−1

Wk + Vk (9.22)

in which a matrix denominator of order 1 can be recognized as (Izk − A), with thehighest order coefficient of the basis function z fixed to the unity matrix. Similar asfor the AR model, the stochastic spectra can be described by a state-space modelwith stable, unstable or mixed stable-unstable system poles. In chapter 7, it isshown that the developed stochastic subspace algorithm estimates a state-spacemodel with stable poles, since the Taylor expansion (I −Az)−1 = I +Az +A2z2 +A3z3 + . . . holds for |z| = 1 if |eig(A)| < 1, which is equivalent to stable systempoles. This is in close analogy with the AR model. The stabilization charts offigures 7.7 and 7.9 for the test examples explained in paragraphs 7.8.3 and 7.8.4illustrate that stable poles are identified.

Replacing now z by z−1 results in a Taylor expansion (I −Az−1)−1 = I +Az−1 +A2z−2 + A3z−3 + . . . which is only valid for unstable poles. Thus, replacing zk

by z−1k in the block Vandermonde matrices Y+ and Y− results in a strongly

consistent estimate of an unstable state-space model. This observation for thestochastic state-space identification can be summarized as:

Depending on the basis function, the stochastic state-space model iden-tified by the stochastic frequency-domain subspace algorithm results instable or unstable poles.

• The basis function zk results in a consistent estimate of a state-space model with only stable poles.

• The basis function z−1k results in a consistent estimate of a state-

space model with only unstable poles.

9.2.4 Extrapolation to ARX models

In this section an autoregressive model with an exogenous input (ARX) is con-sidered. Next, it will be shown how some MPE estimation methods fall into thiscategory of ARX models. An ARX process is described by the following equation

a0y(n) + a1y(n − 1) + . . . ary(n − r) = b0f(n) + b1f(n − 1) + . . . brf(n − r) + e(n)(9.23)

and the frequency-domain equivalent is given by

(a0 + a1z−1k + . . . + arz

−rk )Yk = (b0 + b1z

−1k + . . . + brz

−rk )Fk + Ek (9.24)

Dividing both parts of this equation by the polynomial A(z) results in

Yk =B(zk)

A(zk)Fk +

1

A(zk)Ek (9.25)


The response Yk can be considered as the combination of a deterministic contribu-tion and a stochastic contribution. Consider the identification of an ARX modelstarting from the data Yk and Fk by solving the following overdetermined set ofequations

Y1 . . . Y1z−r1 −F1 . . . −F1z

−r1

Y2 . . . Y2z−r2 −F2 . . . −F2z

−r2

.... . .

......

. . ....

YN . . . YNz−rN −FN . . . −FNz−r

N

a0

...ar

b0

...br

=

E1

E2

...EN

(9.26)

Since the model order is not a priori known, a model of order s > r (s estimatedorder, r order of the underlying model) is estimated and we can factorize A(z) asA(z) = Ar(z)At(z), with r and t indicating the order (s = r + t). Furthermore,we assume that the stochastic contribution 1

Ar(zk)Ek is of a lower amplitude level

than the deterministic contribution Br(zk)Ar(zk)Fk (e.g. 30dB lower). It is clear that the

estimated model has t = s−r extraneous mathematical poles, besides the r systempoles. The r system poles are determined both in damping ratio (magnitude andsign) and natural frequency, since not only the magnitude of Br(z)/Ar(z) but alsothe phase plays a role. In [62] it is shown that the t extraneous poles are determinedby the stochastic AR part. So whether the t extraneous poles are stable or unstablejust depends on the choice of the constraint in solving the least squares problemlike shown in paragraph 9.2.2. This facilitates making the distinction between ther deterministic poles from the t stochastic (also called mathematical) poles.

Example Consider a system given by the coefficients a0 = 1, a1 = 0.3805 −0.9456i and a2 = −0.5797 − 0.6712i. From the characteristic equation Eq. 9.5the poles in the discrete time-domain can be calculated λ1 = 0.5243 + 0.8166iand λ2 = −0.9048 + 0.1290i. The continuous time-domain poles are given byln(λ1) = σ1 + iω1 = −0.03 + i and ln(λ2) = σ2 + iω2 = −0.09 + 3i.

Consider the response spectrum Yk = B(zk)A(zk) (1 + 0.05Vk)Fk for k = 1, . . . , N (zk

covers the full unit circle) and Fk the force as the excitation signal. Five per-cent relative noise is added to the simulated model by Vk, which is zero meancircular complex normally distributed noise (real and complex part of Nk whiteindependent normal distributed Gaussian noise). From the spectrum Yk and Fk

the polynomials A(z) and B(z) are estimated with model order n = 9. A total of1024 spectral lines N was used for the simulation.

A Monte Carlo simulation is done for 100 runs for both constraints a0 = 1 anda9 = 1. Figure 9.2 shows the estimated damping values σ for both constraints inthe discrete Z domain and the Laplace domain. For all 100 runs the estimateswith constraint a10 = 1 result in two stable poles, the system poles p1 and p2 andeight unstable extraneous (mathematical) poles. While in the constraint a0 = 1


−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

(a)

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−2

−1

0

1

2

3

4

damping ratio sigma

natu

ral f

requ

ency

om

ega

(b)

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

(c)

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

−2

−1

0

1

2

3

damping ratio sigma

natu

ral f

requ

ency

om

ega

(d)

Figure 9.2: The poles estimated in a Monte Carlo simulation of an ARX process: (a),(b):results for constraint a0 = 1 in discrete Z-domain and continuous Laplace domain; (c),(d):results for constraint a9 = 1 in discrete Z-domain and continuous Laplace domain. (∗estimated poles, × exact pole). It is clear that the constraint a9 = 1 allows an easyclassification between the system poles and extraneous poles based on the damping.

results for every run in ten stable poles. From this example, it is clear that thedistinction between the deterministic system poles and the stochastic, extraneouspoles is very easy based on the sign of the damping ratio, if one chooses the rightconstraint. Equivalent results are obtained by the estimation of the coefficients ofan ARX model in the time-domain.

9.2.5 Combined deterministic-stochastic frequency domainsubspace identification

Similar as for the transition of an AR model to an ARX model, a deterministicinput can be added in the stochastic state-space model resulting in a combineddeterministic-stochastic state-space model. Under the assumption that the deter-ministic and stochastic dynamics are uncoupled, the stability of the poles of thestochastic dynamics depends on the choice of the basis function i.e. z or z−1. Thus

9.3. Application for Modal Parameter Estimation methods 209

the use of z−1 as a basis function instead of z results in unstable mathematicalpoles modelling the noise (stochastic contribution) and stable system poles mod-elling the deterministic contribution i.e. the structure. Based on the sign of thedamping ratio, a clear stabilization chart can be constructed to easily distinguishthe mathematical from the physical poles.

0 0.1 0.2 0.3 0.4 0.50

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Freq. (Hz)

mod

el o

rder

(a)

0 0.1 0.2 0.3 0.4 0.50

5

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15

20

25

30

Freq. (Hz)m

odel

ord

er

(b)

Figure 9.3: Combined frequency-domain subspace: (a) basis function zk; (b) basisfunction z−1

k (∗: stable, ×: unstable)

Example Consider the FRFs, defined by the state-space matrices given in the ap-pendix of chapter 8, as primary data for the combined frequency-domain subspacealgorithm. The simulated FRFs consist of 512 spectral lines and their correspond-ing zk values cover the upper half unit circle and the z−1

k the lower half unit circle.Since for real-life measurement the models order is not a priori known a modelwith 30 modes is estimated. Figure 9.3 compares the stabilization charts for thecombined subspace algorithm with both basis function zk and z−1

k . This clearlyillustrates that in the second case a distinction between mathematical and physicalpoles can be made easily based on the sign of the damping.

9.3 Application for Modal Parameter Estimationmethods

Today MPE methods for modal analysis applications must be able to process largedata sets (e.g. several 100 outputs and several inputs) with a high modal densityin a short time. In spite of the fact that LS based identification algorithms asthe LSCF, p-LSCF (PolyMAX), LSCE are not consistent, they are often used andpreferred for there speed. Since all these methods are basically LS algorithms, thechoice of the constraint has the same influence as for the ARX model.


9.3.1 Least Squares Frequency-domain (LSCF) algorithm

The LSCF method curvefits a common-denominator model on the FRFs or I/Odata in a least squares sense as discussed in chapter 3. In [116] it was alreadynoticed that this method resulted in very clear stabilization diagrams. In thischapter, the reason behind this remarkable result is investigated in more depth.The identification problem is solved by minimizing the following cost function

lLSCFFRF =

N∑

k=1

No∑

o=1

Ni∑

i=1

|Eoi,k|2 (9.27)

with the equation error given by

Eoi,k = A(zk)Hoi(ωk) − Boi(zk) (9.28)

Rewriting this equation as

Hoi =Boi(zk)

A(zk)− 1

A(zk)Eoi,k (9.29)

shows the structure of an ARX model with input Fk = 1. Since this equation erroris linear-in-the-parameters ,the unknown polynomial coefficients are estimated ina (linear) LS sense. The equations to solve are then given by

Jθ ≃ 0 (9.30)

with J the Jacobian matrix. Elimination and substitution results in the compactnormal equations given by Eq. 4.28, i.e.

No∑

o=1

(

−SoHRo

−1So + To

)

α = Mα = 0 (9.31)

Solving these compact normal equations with the constraint an = 1, i.e. thecoefficient corresponding with zn, results in unstable stochastic poles according totable 9.1 (so a clear stabilization diagram), while the parameter constraint a0 = 1results in stable stochastic poles. The easy to interpret stabilization charts areconstructed in a fast manner by solving the compact normal equations as

α[1:m−1] = D−1[1:m−1],[1:m−1]D[1:m−1],m (9.32)

for increasing model orders m from 1 to n.

9.3.2 Poly-reference Least Squares Frequency-domain (p-LSCF, PolyMAX) algorithm

The p-LSCF (PolyMAX) algorithm and its application was presented as a newfuture standard for modal testing [48], [89], because of its speed and the clear


stabilization charts produced by the algorithm. The p-LSCF, discussed and pre-sented in more detail in chapter 5, fits a right matrix fraction model on the FRFsin a least squares sense. Similar as for the LSCF algorithm a constraint must beapplied to the matrix coefficients of the denominator polynomial to remove theparameter redundancy. Eqs. 5.15 and 5.16 solve the compact normal equations.With the constraint An, i.e. the matrix coefficient of zn

k , fixed to the unity matrixINi

, the mathematical poles which model the noise on the FRFs become unstableand again clear stabilization charts are obtained.

9.3.3 Least Squares Complex Exponential (LSCE) method

A well-known and industrial standard is the poly-reference Least Squares ComplexExponential algorithm (LSCE) [11],[52]. The poles and participation factors areobtained from the solutions of the following matrix finite difference equation oforder r

r∑

j=0

Ajz−jk = 0 (9.33)

with Aj ∈ RNi×Ni . These matrix coefficients are obtained by solving the equations

given by

〈h(n)〉oA0 + 〈h(n − 1)〉oA1 + . . . 〈h(n − r)〉oAr = 0 (9.34)

for different n and with 〈h(n)〉o = [ho1(n) . . . hoNi(n)] and hoi(n) the impulse

response function between output o and input i at time instant n. The parametersare obtained by solving the following set of NoN finite difference equations in aleast squares sense.

〈h(r)〉1 . . . 〈h(1)〉1 〈h(0)〉1...

. . ....

〈h(r)〉No. . . 〈h(1)〉No

〈h(0)〉No

.... . .

...〈h(N)〉1 . . . 〈h(N − r + 1)〉1 〈h(N − r)〉1

.... . .

...〈h(N)〉No

. . . 〈h(N − r + 1)〉No〈h(N − r)〉No

A0

...Ar−1

Ar

= 0 (9.35)

In literature this least squares problem is classically solved with the constraintA0 = INi

leading to both stable physical and mathematical poles according to thetheory explained in this chapter. However by changing the constraint into Ar =INi

the LSCE algorithm is transformed into a MPE leading to clear stabilizationdiagrams, since the mathematical poles are identified as unstable poles. In fact,the LSCE algorithm is the time-domain counterpart of the poly-reference LSCF.Both methods have the same properties and result in very similar stabilizationdiagrams if the same parameter constraint is applied.


9.3.4 Frequency-domain subspace algorithms

It was already shown in paragraph 9.2.5 that the frequency-domain state-spacemodels also use a constraint, which can simply be changed by replacing zk by z−1

k .Using z−1

k as a basis function results in the stochastic contribution of the data tobe modelled by unstable poles and thus in this way clear stabilization charts canbe obtained.

9.3.5 Coupled stochastic-deterministic dynamics

The clear stabilization charts are obtained by the fact that an intelligent choiceof the constraint/basis function results in the fact that the stochastic contribu-tion in the measurements is modelled by unstable poles. Since the system polesare assumed to be stable, this facilitates the distinction between the physical andmathematical poles. As long as the dynamics between the deterministic contri-bution and the stochastic contribution are separated, this does not influence theestimation of the poles of the deterministic contribution. Unfortunately, in prac-tical modal analysis experiments the dynamics of the deterministic and stochasticcontribution are often coupled, i.e. the noise model contains some poles that arealso present in the deterministic contribution. Consider an OMAX experiment,since the structure is excited by both measurable forces and unmeasurable forces,the deterministic and stochastic contribution in the responses have coupled dynam-ics. Even in the case that FRFs are estimated from the IO data of this OMAXexperiment, the dynamics of the noise on the FRFs will be coupled with the dy-namics of the deterministic contribution. In all these cases, where the stochasticand deterministic dynamics are coupled, a contradiction will appear in the algo-rithm, if one is interested to build a clear stabilization chart based on the dampingratios. The chosen constraint/basis function tries to force the poles in the stochas-tic contribution to be unstable, while the deterministic contribution tries to forcethese coupled poles to be stable. Depending on the relative contribution in dataof the stochastic versus the deterministic part of the coupled physical pole, theestimated pole will be biased i.e. pulled to the unstable region and the dampingwill be underestimated. For measurements with high SNR (30-40dB) this effectis negligible. However, for low SNR as e.g. for in-flight flutter measurements orpower spectral density functions, this effect can seriously affect the quality of theestimated model.

Therefore, if the combined stochastic-deterministic subspace algorithm on IOdata is applied in the OMAX framework, the basis function must always be zk

in order to avoid bias errors on the damping estimates. In this case other tech-niques must be utilized to distinguish the physical from the mathematical poles[135], [108], [35]. Also the LSCF, p-LSCF and LSCE algorithm tend to estimatemore accurate models in the case of noisy data, when the constraint is chosento estimate stable stochastic poles. Therefore, it is suggested to choose the one


constraint/basis function to obtain clear stabilization diagrams and the other con-straint/basis function to obtain more accurate damping estimates. Afterwardsboth sets of poles must be correlated (in many cases the frequency of the physicalpoles are almost identically for both constraints/basis functions). Another possi-bility for noisy measurements is to construct the stabilization chart by means ofthe LSCF or p-LSCF and next further optimize these estimates by the frequency-domain ML estimators. Notice that the estimators based on an optimizationalgorithm as the ML estimators are independent of the constraint and as a resulttheir damping estimates will be unbiased.

Data driven output-only algorithms always need to estimate stable poles, sincethey model a stochastic process. The advantage of a clear stabilization diagramfor OMA can only be used in the case that power spectra are used as primary data.However, power spectra are also characterized by high noise levels (compared toFRFs) and a 4-quadrant symmetry (see Eq. 2.62) and thus they contain bothstable and unstable physical poles. Nevertheless, in the identification it oftenhappens that only one of both is identified depending on the constraint. If theconstraint is chosen to obtain clear stabilization diagrams, the algorithm tend togive more weight to the unstable poles and only the modes that are very clearlypresent in the data will be identified as both a stable and unstable. As a result, onlythe modes that are clearly present will appear as physical poles in the diagram.Therefore it is advised to transform the power spectra in ’positive’ power spectra,since in that case the physical poles will only appear as stable poles.

The only algorithm which is capable of constructing a clear stabilization dia-gram starting from noisy data, without introducing a bias on the damping ratios,is the weighted subspace algorithm. Since this algorithm uses the covariance ma-trix of the noise on the data as a weighting to be consistent all the dynamics inthe stochastic contribution are cancelled by the weighting. This means that nocoupled stochastic-deterministic dynamics need to be modelled and thus the choiceof the basis function does not affect the estimates of the physical poles.

9.3.6 Remarks

• In this chapter it is shown that by choosing an optimal constraint/basis func-tion, the mathematical poles (used for modelling the noise) become unstable,while the physical poles are stable. Another possibility, is to transform thedata in such way that the physical poles become unstable and the mathemat-ical poles remain stable. This can be done by taking the complex conjugateof the FRFs or by reversing the time axis for time-domain data. In this case,the constraint for stable mathematical poles leads to an easy interpretationof the stabilization diagram, since the only unstable poles will be the phys-ical ones. After the identification the sign of the damping of these unstablepoles can be easily reversed to obtain the true physical poles.


• The influence of the constraint/basis function on the damping ratio estimateof the poles holds only for discrete models in the time- or frequency-domain.Therefore, discrete-time models are preferred compared to continuous-timemodels in terms of the quality and interpretation of the stabilization dia-grams. Furthermore the basis in zk or z−1

k must cover uniformly the halfunit circle in the case that real system matrices are estimated and the fullunit circle in the case that complex matrices are estimated. The bi-lineartransformation which exactly maps a discrete-time model on a continuous-time underlying model does not result in a uniform distribution and thusdoes not result in clear stabilization diagrams.

9.4 Application of experimental structural test-ing

To study the influence of the constraint/basis function on the stabilization diagramseveral data sets were analyzed, ranging from ground vibration tests of commercialand military aircrafts, modal tests of a body-in-white, automotive subparts, a fullytrimmed car, operational data from bridges, ... . In this chapter the influence ofthe constraint is now illustrated by four data sets: a car door (low damped data), afully trimmed car (highly damped data), in-flight measurements (noisy data) andthe Villa Paso bridge measurement (operational data). In [135] an automated poleselection procedure is presented based on the clear stabilization charts obtainedby a proper choice of the parameter constraint.

9.4.1 Measurements on a car door

A modal test on the door of a car is performed by measuring the vibration responsein 79 outputs by a scanning laser Doppler vibrometer, while the door was excited byan electrodynamical shaker(cfr. the laser vibrometer measurements processed inparagraph 8.5.4). Since this test setup is a single input case, the LSCF and p-LSCFresult the same estimates. The LSCF, LSCE and combined subspace algorithmsare compared for both constraints/basis functions. A frequency band from 40Hz to120Hz, with a high modal density, is modelled. The model order for the differentidentification algorithms is chosen in order to estimate 50 modes. Figures 9.4 and9.5 show the stabilization diagrams and a synthesized FRF for the LSCF an LSCEalgorithms. It is clear that for the constraint a49 = 1 it is very easy to distinguishthe physical from the mathematical poles based on the damping value. Subspacemethods have the tendency to place their mathematical poles close to the physicalpoles. This fact complicates the interpretation of the diagram. Therefore, changingthe constraint in the state-space modal results in easy to interpreted stabilizationcharts. Figure 9.6 illustrates the influence of changing the constraint in frequency-

9.4. Application of experimental structural testing 215

50 60 70 80 90 100 110

5

10

15

20

25

30

35

40

45

Freq. (Hz)

mod

el o

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(a)

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5

10

15

20

25

30

35

40

45

Freq. (Hz)

mod

el o

rder

(b)

50 60 70 80 90 100 110 120

−140

−130

−120

−110

−100

−90

Freq. (Hz)

Am

pl. (

dB)

(c)

50 60 70 80 90 100 110 120−150

−140

−130

−120

−110

−100

−90

Freq. (Hz)

Am

pl. (

dB)

(d)

Figure 9.4: LSCF processing car door data: (a) stabilization diagram for constrainta0 = 1 (∗: stable, ×: unstable); (b) stabilization diagram for constraint a49 = 1 (∗stable, × unstable); (c) synthesized FRF for constraint a0 = 1 (×: measurement, fullline: model); (d) synthesized FRF for constraint a49 = 1 (×: measurement, full line:model).

domain subspace identification by replacing z by z−1 in the state-space equations.Both figures 9.4 and 9.6 show that for this high quality data set (SNR ∼ 30dB) thesynthesized FRFs (synthesized polynomial/state-space model for maximal order)from the modal parameters are in close agreement with the measurements for bothconstraints/basis functions.

Nevertheless, in common practice the LSCE method, subspace identification algo-rithms and many MPE algorithms are still applied with a constraint/basis functionresulting in stable mathematical poles. In that case, a distinction is made by arelative comparison between each pole and the corresponding poles for the lowerorder estimate. This approach often results in more confusing diagrams and there-fore an experienced user is often required to decide whether some poles are physicalor mathematical.


50 60 70 80 90 100 110

5

10

15

20

25

30

35

40

45

50

Freq. (Hz)

mod

el o

rder

(a)

50 60 70 80 90 100 1100

5

10

15

20

25

30

35

40

45

Freq. (Hz)

mod

el o

rder

(b)

Figure 9.5: LSCE processing car door data: (a) stabilization diagram for constrainta0 = 1 (∗: stable, ×: unstable); (b) stabilization diagram for constraint a49 = 1 (∗stable, × unstable)

9.4.2 Measurements on a fully trimmed car

During a MIMO modal test a fully trimmed Porsche was excited in 4 different lo-cations by shakers. The accelerations were measured in 154 DOFs spread all overthe car. The FRFs are estimated by the H1 method [72]. A total of 616 FRFs isprocessed by the LSCF, p-LSCF and LSCE method. Figures 9.7, 9.8 and 9.9 com-pare respectively the stabilization diagrams obtained by different constraints forrespectively the p-LSCF, LSCE and LSCF algorithm. It is clear that the choice ofthe constraint has a major influence on the interpretation of the stabilization dia-gram. Furthermore, it can be observed that both the LSCE and the p-LSCF resultin similar stabilization diagrams. The effect of forcing rank 1 residue matrices onthe measurements by the poly-reference LSCF clearly results in the identificationof more stable poles compared to the common-denominator based LSCF.

9.4.3 In-flight aircraft measurements

Flutter data is typically characterized by high noise levels caused by turbulence andthe limited amount of data. Typical SNR on the FRFs is between 5−20 dB. Duringflight , the airplane is artificially excited by the flaps by injecting an excitationsignal in the fly-by-wire system. In this case, the accelerations were measuredat 13 locations on a military aircraft. Figure 9.10 compares the LSCF for bothconstraints. The constraint a25 = 1 results in a clear stabilization chart, but 3of the 9 physical poles (given by table 8.6) do not appear as stable poles. Fromthe synthesized FRF it is also clear that some damping ratios are underestimated.On the contrary, the constraint a0 = 1 results in a better fit of the model onthe data, but the distinction between the mathematical and physical poles in the

9.4. Application of experimental structural testing 217

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45

Freq. (Hz)

mod

el o

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(a)

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45

Freq. (Hz)

mod

el o

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(b)

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40

45

50

Freq. (Hz)

mod

el o

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(c)

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45

Freq. (Hz)

mod

el o

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(d)

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−110

−100

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Freq. (Hz)

Am

pl. (

dB)

(e)

50 60 70 80 90 100 110

−140

−130

−120

−110

−100

−90

Freq. (Hz)

Am

pl. (

dB)

(f)

Figure 9.6: Combined subspace algorithm processing car door data: (a) stabilizationdiagram for basis function zk (∗: stable, ×: unstable); (b) stabilization diagram for basisfunction z−1

k (∗ stable, × unstable); (c) zoom on figure (a); (d) zoom on figure (b); (e)synthesized FRF for basis function zk (×: measurement, full line: model); (f) synthesizedFRF for basis function z−1

k (×: measurement, full line: model).


5 10 15 20 25

10

20

30

40

50

60

Freq (Hz)

mod

el o

rder

(a)

5 10 15 20 25 300

10

20

30

40

50

60

frequency (Hz)

mod

el o

rder

(b)

5 10 15 20 25

−55

−50

−45

−40

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−25

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−15

Freq. (Hz)

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dB)

(c)

5 10 15 20 25 30

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−55

−50

−45

−40

−35

−30

−25

−20

−15

Freq. (Hz)

Am

pl. (

dB)

(d)

Figure 9.7: p-LSCF processing data of a fully trimmed car: (a) stabilization diagram forconstraint a0 = 1 (∗: stable, ×: unstable); (b) stabilization diagram for constraint a12 = 1(∗ stable, × unstable); (c) synthesized FRF for constraint a0 = 1 (×: measurement, fullline: model); (d) synthesized FRF for constraint a12 = 1 (×: measurement, full line:model)

stabilization diagrams becomes difficult. In this case, it is advised to use the MLand p-ML algorithms [46], [21], which optimize the LSCF and p-LSCF estimates ina Maximum Likelihood sense. The combined subspace algorithm results in a clearstabilization diagram for the basis function z−1

k resulting in 7 of the 9 physicalpoles, but the damping ratios are largely underestimated as can be seen from thesynthesized FRF in figure 9.11.

Similar conclusions can be made for the combined subspace algorithm appliedfor both choices of the basis function. The clear stabilization chart misses twostable poles and the synthesized FRF clearly illustrates the bias on the dampingestimates.


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mod

el o

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(b)

Figure 9.8: LSCE processing data of a fully trimmed car: (a) stabilization diagramfor constraint a0 = 1 (∗: stable, ×: unstable); (b) stabilization diagram for constrainta12 = 1 (∗ stable, × unstable).

9.4.4 Villa Paso bridge

The last example illustrates the influence of the constraint/basis function for boththe combined subspace algorithm in figure 9.12 and the LSCF algorithm in figure9.13 starting from ’positive’ power spectra (output-only processing) of the VillaPaso bridge. Similar as for the other examples, an easy interpretation of thestabilization chart is facilitated by the right choice of the constraint/basis function.

9.5 Conclusions

In this chapter it is shown that the parameter constraint/basis function in theidentification of modal parameters of the structure play an important role withrespect to the interpretation of the stabilization diagram. A proper choice of thisconstraint/basis function forces the mathematical poles to be unstable, allowing asimple distinction between physical and mathematical poles based on the sign ofthe damping value. This methodology is applicable for different modal parameterestimation methods based on discrete-time models in both frequency- and time-domain identification. However, for data with large noise levels, the proposedtechnique to obtain clear stabilization charts must be applied with caution, sincethe damping values will be underestimated. Therefore, the clear stabilizationchart can still be used as a pole selection tool, but the modal parameter estimatesmust be further optimized, e.g. in a maximum likelihood sense. The influenceof the constraint/basis function has been illustrated by several experimental caseexamples and this for several MPE algorithms


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stabilization chart − LSCF

frequency (Hz)

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(c)

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−30

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pl. (

dB)

(d)

Figure 9.9: LSCF processing data of a fully trimmed car: (a) stabilization diagram forconstraint a0 = 1 (∗: stable, ×: unstable); (b) stabilization diagram for constraint a50 = 1(∗ stable, × unstable); (c) synthesized FRF for constraint a0 = 1 (×: measurement, fullline: model); (d) synthesized FRF for constraint a50 = 1 (×: measurement, full line:model).

9.6 Appendix 1

Consider a complex value λ = eσ+iω and its counterpart λ defined by λ =e−σ+iω . The ratio of the distance between of a complex value zk = eiθk lying onthe unit circle and λ and the distance between zk and λ is then independent ofθk.

Proof :

|zk − λ||zk − λ| =

|1 − Rei(ω−θk)||1 − 1

Rei(ω−θk)| (9.36)

9.6. Appendix 1 221

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0

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el o

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(c)

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−10

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0

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pl. (

dB)

(d)

Figure 9.10: LSCF processing in-flight aircraft data: a) stabilization diagram for con-straint a0 = 1 (∗: stable, ×: unstable); b) stabilization diagram for constraint a24 = 1 (∗stable, × unstable); c) synthesized FRF for constraint a0 = 1 (∗: measurement, full line:model); d) synthesized FRF for constraint a49 = 1 (∗: measurement, full line: model)

with R = eσ. Taking the complex conjugate of the denominator does not changethe amplitude, thus

|zk − λ||zk − λ| =

|1 − Rei(ω−θk)||1 − 1

Rei(ω−θk) |(9.37)

=|1 − Q||1 − 1

Q | (9.38)

= |Q| (9.39)

= R (9.40)

with Q = Rei(ω−θk). This ends the proof.


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pl. (

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(d)

Figure 9.11: Combined subspace algorithm processing in-flight aircraft data: (a) stabi-lization diagram for basis function zk (∗: stable, ×: unstable); (b) stabilization diagramfor basis function z−1

k (∗ stable, × unstable); (c) synthesized FRF for basis functionzk (×: measurement, full line: model); (d) synthesized FRF for basis function z−1

k (×:measurement, full line: model).

9.7 Appendix 2

The set of equations for estimating the coefficients aj from an AR process is givenby

Y1 Y1z−11 . . . Y1z

−r1

Y2 Y2z−12 . . . Y2z

−r2

......

. . ....

YN YNz−1N . . . YNz−r

N

a0

a1

...ar

=

E1

E2

...EN

(9.41)

This set of equations can be written as

Jθ = E (9.42)

9.7. Appendix 2 223

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(a)

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(d)

Figure 9.12: LSCF algorithm processing operational bridge data: (a) stabilizationdiagram for the constraint a0 = 1 (∗: stable, ×: unstable); (b) stabilization diagram forthe constraint a24 = 1 (∗ stable, × unstable); (c) synthesized FRF for the constrainta0 = 1 (×: measurement, full line: model); (d) synthesized FRF for the constrainta24 = 1(×: measurement, full line: model)

It will be proven that the constraint a0 = 1 leads to a strongly consistent estimateof a stable system, while the constraint ar = 1 leads to a strongly consistentestimate of an unstable system. Under the noise assumption that Ek is zero meancircular complex independent and identically distributed noise with covarianceE (Ek) = R, it has to be proven that

J†Jj + J†E → θ w.p. 1 for N → ∞ (9.43)

(w.p.1 = with probability one and N the number of spectral lines) holds for astable system under the constraint a0 = 1 and for an unstable system for theconstraint ar = 1. To prove this it is sufficient to show that

J†E → 0 w.p. 1 for N → ∞ (9.44)


1 2 3 4 5 6 7 80

5

10

15

20

25

Freq. (Hz)

mod

el o

rder

(a)

1 2 3 4 5 6 7 80

5

10

15

20

25

Freq. (Hz)

mod

el o

rder

(b)

1 2 3 4 5 6 7 865

70

75

80

85

90

95

100

105

Freq. (Hz)

Am

pl. (

dB)

(c)

1 2 3 4 5 6 7 865

70

75

80

85

90

95

100

105

Freq. (Hz)

Am

pl. (

dB)

(d)

Figure 9.13: Combined subspace algorithm processing operational bridge data: (a)stabilization diagram for basis function zk (∗: stable, ×: unstable); (b) stabilizationdiagram for basis function z−1

k (∗ stable, × unstable); (c) synthesized FRF for basisfunction zk (×: measurement, full line: model); (d) synthesized FRF for basis functionz−1

k (×: measurement, full line: model)

This means that

a.s. limN→∞

JHE = 0 (9.45)

In the case of the constraint a0 = 1, the matrix J is equal to

J =

Y1z−11 . . . Y1z

−r1

Y2z−12 . . . Y2z

−r2

.... . .

...YNz−1

N . . . YNz−rN

(9.46)

9.7. Appendix 2 225

Thus

a.s. limN→∞

(

JHE)

j=

N∑

k=1

zjkY ∗

k Ek (9.47)

=

N∑

k=1

zjk(A(zk)−1)∗E∗

kEk (9.48)

=N∑

k=1

zjk

∞∑

n=0

bnznk EH

k Ek (9.49)

where the Taylor expansion (A(z)∗)−1=∑∞

n=0 bnzn holds on the unity circle

(|zk| = 1) for A(z) having stable poles. Under the noise assumption, JHj converges

w.p.1 to its expected value (strong law of large numbers for independent andidentically distributed random variables [70])

a.s. limN→∞

(

JHE)

j= lim

N→∞E(

JHj

)

(9.50)

= limN→∞

( ∞∑

n=0

bnR

(

N∑

k=1

zj+nk

))

(9.51)

The terms in this last Eq. 9.51 are zero for n + j ∈ N0 and n + j 6= qN withq ∈ N0 0, since

∑Nk=1 zk = 0 (zk (k = 1, 2, . . . N) covers the full unit circle). In the

case that n + j = qN the terms are also zero since

limN→∞

∞∑

n=0

bnR

(

N∑

k=1

zj+nk

)

= limN→∞

∞∑

n=0

bqN−jRN (9.52)

and the coefficients |bqN−j | ≤ K|λ|qN−jmax . Thus it holds that

|∞∑

n=0

bqN−jRN | ≤ RKN |λ|N−jmax

∞∑

q=0

|λ|mNmax (9.53)

≤ RKN |λ|N−jmax

1

1 − |λ|Nmax

(9.54)

≤ O(N |λ|N−jmax ) (9.55)

which converges to zero for A(z) containing stable poles and limN→∞. This con-cludes the proof that the constraint a0 = 1 results in a strongly consistent estimateof a stable solution of 9.4. In a similar way one can show that the constraint ar = 1results in a strongly consistent solution with only unstable poles.


Chapter 10

Conclusions

10.1 Summary and main contributions

In this thesis, the applicability of frequency-domain identification methods in thefield of modal analysis has been improved and generalized. Special attention hasbeen paid to deal with noisy data from short measurement sequences and to ob-tain a maximal data exploitation. The techniques developed in this thesis areapplicable for an Experimental Modal Analysis (EMA) and an Operational ModalAnalysis (OMA) and for a combination of both i.e. an Operational Modal Analysiswith eXogenous inputs (OMAX). Several illustrating examples from automotive,aerospace and civil engineering are discussed to demonstrate the performance ofthe developed techniques. It is now worthwhile to summarize the most importantresults achieved in the different chapters.

Chapter 2 discusses the different parametric models and their relation to themodal model. These mathematical models form the basis of the identificationalgorithms developed in this thesis. The OMAX concept is introduced to obtaina maximal data exploitation by considering the vibration response as the sumof both a deterministic contribution from the measurable forces and a stochasticcontribution from the unmeasurable ambient forces. Based on the length of ofthe observation window and the desired data reduction, the identification canstart from the raw Output/Input spectra or from Averaged Based Spectral (ABS)functions, i.e. FRFs in the case of an EMA, power spectra in the OMA case andboth simultaneously for an OMAX approach.

The non-parametric identification of ABS functions has been revised in chap-ter 3. Special attention is paid to the reduction of the noise levels, without intro-ducing bias errors due to leakage. The classic approach to estimate ABS functions

227

228 Chapter 10. Conclusions

is to average the spectra of several adjacent time blocks of the time signals incombination with the use of a Hanning window. By using only a few time blocks,an initial estimate of the FRFs is obtained, which suffers less from leakage errors.Next, an additional noise reduction is obtained by the application of a rectangu-lar window. This rectangular time window applied on the initial estimates of theImpulse Response Functions (IRF) or correlations further reduces the noise levels,while the leakage errors can exactly be compensated in a parametric way by acorrection factor on the finally estimated modal participation factors. Using bothsimulations and measurements, it is demonstrated that this approach outperformsthe classic one. In particular, the accuracy of the estimated damping ratios isgreatly improved. Finally, the use of ’positive’ power spectra and their benefitsfor the OMA case are discussed.

Parametric identification is discussed starting from chapter 4. In chapter 4both the LSCF and ML common-denominator estimators are extended in order totake into account the initial/final conditions of every time block averaged by theH1 FRF estimator to prevent errors caused by leakage. The close analogy betweenthe implementation of FRF driven and IO driven algorithms is demonstrated and acombined deterministic-stochastic algorithm is proposed, which, starting from IOdata, can be considered in the OMAX framework. The ML estimators proposedin this thesis are always consistent under their noise assumptions, however, takinginto account the correct variances of the noise on the data as a frequency-domainweighting improves only the efficiency. In the case that, no a priori noise infor-mation is available, different weighting functions are proposed. These new contri-butions in Modal Parameter Estimation (MPE) based on common-denominatormodels are illustrated by simulations and measurement examples. An introductionto and the current challenges of in-flight flutter analysis on aircrafts are given.

In the multiple input case, common-denominator models do not force theresidues to be rank one, resulting in a loss of quality when decomposing the es-timated residues into mode shapes and modal participation factors. Therefore,chapter 5 has been devoted to the generalization of the identification of common-denominator models to right and left matrix fraction description models. Doingthis, special attention was paid to obtain an optimal memory/computation effi-ciency. Similar as for the common-denominator models, starting from the normalequations results in significant reduction of the problem and the exploitation ofthe structure of the normal equations results in a fast implementations. Sincethe least-squares implementation of the right matrix fraction description modelidentification, also referred to as poly-reference LSCF or PolyMAX algorithm, isnot consistent, a ML approach is proposed. Under the assumption of uncorrelatednoise over the different outputs (this assumption only influences the efficiency ofthe algorithm and not its consistency) the ML implementation, which uses a thecovariance matrix of the noise on the primary data for each output as a weightingfunction, improves the accuracy of the estimates. Furthermore, it is shown how theuncertainty levels on the final poles can be estimated. Since this ML implementa-

10.1. Summary and main contributions 229

tion is rather slow for large amounts of data, a faster implementation is proposedbased on a scalar weighting. Several experimental examples, illustrated the gainof accuracy by using a right matrix fraction description compared to the common-denominator models. Especially, for applications characterized by high damping,a large improvement is obtained. In the case, the measurements on highly dampedstructures are contaminated with noise, the use of the poly-reference ML estimatoris advised. Finally, the use and interpretation of a left matrix fraction descriptionin the OMAX framework is discussed.

Chapter 6 introduces frequency-domain subspace algorithms to estimate state-space models from IO data, FRFs or powers spectra. State-space models havethe advantage that the number of outputs No, inputs Ni and order n can bechosen independently and that they can be converted to a modal model withoutintroducing an extra error (residues of rank one). Subspace algorithms estimatethe parameters, without the need for minimizing a cost function in an optimizationprocedure. By means of geometrical projections, calculated by the use of a QRdecomposition the system matrices are estimated. Novel in this chapter, is theextension of the state-space model to take into account the intial/final conditionsto deal with transient effects and prevent errors caused by leakage. Based on thisprinciple a mixed non-parametric/parametric FRF estimator is proposed to removeleakage errors. This improved measured FRFs can be used to validate an estimatedstate-space model by comparison to the synthesized FRFs. Similar to the common-denominator models, the state-space models can also be extended to start fromFRFs without suffering from transient effects and leakage. These extensions ofthe IO and FRF model, together with the validation procedure, are evaluatedby simulations and an experimental example. The proposed frequency-domainsubspace algorithms in literature only estimate the deterministic contribution inthe data and they are not consistent, if no a priori noise information is taken intoaccount. By using the full covariance matrix on the primary data as a frequencyweighting, the identification of the deterministic model is consistent. However,this approach becomes impractical for a larger number of outputs and still has nocombined interpretation in the OMAX framework. In chapter 8 this drawback issolved by the development of a consistent combined frequency-domain subspacealgorithm, without the need for a priori noise information.

In chapter 7 a stochastic frequency-domain subspace algorithm is presentedand proven to be consistent. This estimator starts from output-only spectra toestimate the natural frequencies, damping ratios and mode shapes. This approachis able to process short data sequences for the case of in-flight flutter test dataand combines the advantages of frequency-domain algorithms (e.g. a simple bandselection and easy pre-filtering) with the advantages of subspace algorithms (noneed for an optimization procedure).

A combined deterministic-stochastic frequency-domain subspace algorithm isdeveloped in chapter 8 and proven to be consistent based on its relation to time-domain subspace algorithms. This algorithm can deal with noisy IO data, FRF


data or power spectra, without the need for a Gauss-Newton optimization. Thecombined interpretation for IO data considers the vibration responses as the sum ofa deterministic and stochastic contribution respectively related to the measurableinputs and unmeasurable inputs, which fits in the OMAX framework. Finally,this algorithm also allows to start from output-only spectra in combination withan additional term to model the initial/final conditions to remove leakage errors.From simulations and several experimental examples it is shown that this algorithmis applicable for different test cases (EMA, OMA, OMAX) and for all possibleprimary data (IO data, Output-only spectra, FRFs and power spectra).

Finally, it is shown that clear stabilization charts are obtained by a properchoice of the constraint on the parameters for the least-squares problem or bya proper choice of the basis function of the state-space models for the subspacealgorithms. This allows to make a distinction between mathematical and physicalpoles based on the sign of the damping. It is proven that a purely stochastic systemcan be modelled by stable poles, unstable poles or by a combination of both.Depending on the constraint/basis function, the stochastic algorithm identifies ina consistent way only stable or only unstable poles. This principle, results inan easy-to-interpret stabilization diagram, facilitating the manual pole selectionand forms the basis for several automated interpretations of the diagram. In thischapter, it is shown how this principle can be applied for MPE methods such as theLSCF, poly-reference LSCF, LSCE and subspace algorithms. However, it is shownthat in the case of coupled deterministic-stochastic dynamics the damping valuesare underestimated and therefore this principle must be applied with caution.Several illustrative examples demonstrate the differences between the choices ofthe constraints and basis functions on the stabilization diagram and synthesizedspectral functions.

10.2 Future research

The identification approaches in the framework of the OMAX concept presentedin this thesis, consider the influence of the unmeasurable forces to be a stochas-tic contribution in the vibration response. Nevertheless, in some applications theunmeasurable forces are deterministic forces e.g. harmonics of rotating machin-ery and impact forces. Therefore, the modal parameter estimation techniques canbe extended to take into account these contributions. A first approach, wheretest measurements are corrupted with harmonics is presented in [127]. In a nextstep, these unmeasurable forces can be identified in both magnitude and locationby applying a force identification process [84], simultaneously with the combinedidentification technique [128]. New identification techniques can be developed toconsider, all three contributions i.e. unmeasurable deterministic forces, unmea-surable stochastic forces and measurable forces in a single model. Based on thiscombined identified model, the loads can then be estimated and located in a next

10.2. Future research 231

step.

In this thesis, only uncertainty intervals on the estimated parameters are con-sidered for the ML estimators, which consider all stochastic contributions in themeasurements as measurement noise (and thus they do not extract system infor-mation from the stochastic contribution on the primary data). A topic for furtherresearch, is therefore the determination of uncertainties on the estimated modelparameters by the use of a combined deterministic-stochastic subspace algorithm.The uncertainty levels on the estimated poles can be used as a tool to make adistinction between physical and mathematical poles. A simplified approximatedapproach to estimate uncertainty levels is to consider the states obtained by theoblique projection as exact and next the noise levels on the estimated system ma-trices can be obtained from the least-squares problem given by Eq. 8.22. Finally,the noise levels on the system poles are obtained from a sensitivity analysis of theeigenvalue decomposition of the A matrix. The uncertainty on the modal param-eters can then be used in modal applications such as for a weighting in modalupdating procedures [111].

Other identification algorithms based on e.g. a left and right matrix fractiondescriptions can be extended to take into account unmeasurable forces. In thebroad field of system identification, the recent evolutions for modelling non-linearand time-varying systems can be further investigated for their application in me-chanical engineering in the context of an OMAX framework. In [34] a recursivestochastic subspace algorithm is presented based on simulations for its applicationto in-flight tests. Further research for its applicability for real-life testing must bedone to show the practical robustness.

Finally, the proposed identification algorithms can be applied for other appli-cations in mechanical engineering such as e.g. acoustical modal analysis, transferpath analysis, processing of large scale optical measurements and for other engi-neering fields, where frequency-domain system identification in general is usefulwith in particular electrical and control engineering. Furthermore the usefulnessof the OMAX approach in other engineering fields can be investigated.


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