Abstract
The operational modal analysis methods based on output-only measurements are well-known and applied in linear systems. However, they are not so well developed for nonlinear systems. Thus, this paper proposes an approach for nonlinear system identification using output-only data. In the conventional Volterra series, the outputs of the system are computed by multiple convolutions between the excitation force and the Volterra kernels. However, in this paper at least two time series measured in different placements are used to compute the multiple convolutions and the excitation signals are not required. The novel kernels identified can be used to characterize nonlinear behavior through a model using only output data. A numerical example based on a Duffing oscillator with two degrees-of-freedom (2DOF) and experimental vibration data from a buckled beam with hardening nonlinearities are used to illustrate the proposed method. The prediction results using output-only data are similar to the conventional Volterra kernels based on input and output data.
Issue Section:
Research Papers
References
1.
Kerschen
, G.
, Worden
, K.
, Vakakis
, A. F.
, and Golinval
, J.
, 2006
, “Past, Present and Future of Nonlinear System Identification in Structural Dynamics
,” Mech. Syst. Signal Process.
, 20
(3
), pp. 505
–592
.2.
Falco
, M.
, Liu
, M.
, Nguyen
, S. H.
, and Chelidze
, D.
, 2014
, “Nonlinear System Identification and Modeling of a New Fatigue Testing Rig Based on Inertial Forces
,” ASME J. Vib. Acoust.
, 136
(4
), p. 041001
.3.
Pasquali
, M.
, Lacarbonara
, W.
, and Marzocca
, P.
, 2014
, “Detection of Nonlinearities in Plates Via Higher-Order-Spectra: Numerical and Experimental Studies
,” ASME J. Vib. Acoust.
, 136
(4
), p. 041015
.4.
Manktelow
, K. L.
, Leamy
, M. J.
, and Ruzzene
, M.
, 2014
, “Analysis and Experimental Estimation of Nonlinear Dispersion in a Periodic String
,” ASME J. Vib. Acoust.
, 136
(3
), p. 031016
.5.
Virgin
, L.
, 2000
, Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration
, 1st ed., Cambridge University Press
, Cambridge, UK
.6.
Worden
, K.
, and Tomlinson
, G. R.
, 2001
, Nonlinearity in Structural Dynamics
, Institute of Physics
, London
.7.
Adams
, D. E.
, 2002
, “Frequency Domain ARX Model and Multiharmonic FRF Estimators for Non-Linear Dynamic Systems
,” J. Sound Vib.
, 250
(5
), pp. 935
–950
.8.
Lenaerts
, V.
, Kerschen
, G.
, and Golinval
, J.
, 2003
, “ECL Benchmark: Application of the Proper Orthogonal Decomposition
,” Mech. Syst. Signal Process.
, 17
(1
), pp. 237
–242
.9.
da Silva
, S.
, Cogan
, S.
, Foltête
, E.
, and Buffe
, F.
, 2009
, “Metrics for Nonlinear Model Updating in Structural Dynamics
,” J. Braz. Soc. Mech. Sci. Eng.
, 31
(1
), pp. 27
–34
.10.
Isasa
, I.
, Hot
, A.
, Cogan
, S.
, and Sadoulet-Reboul
, E.
, 2011
, “Model Updating of Locally Non-Linear Systems Based on Multi-Harmonic Extended Constitutive Relation Error
,” Mech. Syst. Signal Process.
, 25
(7
), pp. 2413
–2425
.11.
Billings
, S.
, 1980
, “Identification of Nonlinear Systems—A Survey
,” IEEE Proc. D: Control Theory Appl.
, 127
(6
), pp. 272
–285
.12.
Sjöberg
, J.
, Zhang
, Q.
, Ljung
, L.
, Benveniste
, A.
, Deylon
, B.
, Glorennec
, P.
, Hjalmarsson
, H.
, and Juditsky
, A.
, 1995
, “Nonlinear Black-Box Modeling in System Identification: A Unified Overview
,” Automatica
, 31
(12
), pp. 1691
–1724
.13.
Ljung
, L.
, 2007
, System Identification: Theory for the User
, 2nd ed., Prentice Hall
, Upper Saddle River, NJ
.14.
Wills
, A.
, Schön
, T. B.
, Ljung
, L.
, and Ninness
, B.
, 2013
, “Identification of Hammerstein–Wiener Models
,” Automatica
, 49
(1
), pp. 70
–81
.15.
Billings
, S.
, and Fakhouri
, S.
, 1977
, “Identification of Nonlinear Systems Using the Wiener Model
,” Electron. Lett.
, 13
(17
), pp. 502
–504
.16.
Chen
, S.
, and Billings
, S. A.
, 1989
, “Representation of Non-Linear Systems: The NARMAX Model
,” Int. J. Control
, 49
(3
), pp. 1013
–1032
.17.
Mohamed Vall
, O. M.
, and M'hiri
, R.
, 2008
, “An Approach to Polynomial NARX/NARMAX Systems Identification in a Closed-Loop With Variable Structure Control
,” Int. J. Autom. Comput.
, 5
(3
), pp. 313
–318
.18.
Noël
, J.
, and Kerschen
, G.
, 2013
, “Frequency-Domain Subspace Identification for Nonlinear Mechanical Systems
,” Mech. Syst. Signal Process.
, 40
(2
), pp. 701
–717
.19.
Noël
, J.
, Marchesiello
, S.
, and Kerschen
, G.
, 2014
, “Subspace-Based Identification of a Nonlinear Spacecraft in the Time and Frequency Domains
,” Mech. Syst. Signal Process.
, 43
(12
), pp. 217
–236
.20.
Hot
, A.
, Kerschen
, G.
, Foltête
, E.
, and Cogan
, S.
, 2012
, “Detection and Quantification of Non-Linear Structural Behavior Using Principal Component Analysis
,” Mech. Syst. Signal Process.
, 26
, pp. 104
–116
.21.
Ibnkahla
, M.
, 2002
, “Statistical Analysis of Neural Network Modeling and Identification of Nonlinear Systems With Memory
,” IEEE Trans. Signal Process.
, 50
(6
), pp. 1508
–1517
.22.
Ibnkahla
, M.
, 2003
, “Nonlinear System Identification Using Neural Networks Trained With Natural Gradient Descent
,” J. Appl. Signal Process.
, 12
, pp. 1229
–1237
.23.
Jafarian
, A.
, Measoomy
, S.
, and Abbasbandy
, S.
, 2015
, “Artificial Neural Networks Based Modeling for Solving Volterra Integral Equations System
,” Appl. Soft Comput.
, 27
, pp. 391
–398
.24.
Worden
, K.
, Manson
, G.
, and Tomlinson
, G.
, 1997
, “A Harmonic Probing Algorithm for the Multi-Input Volterra Series
,” J. Sound Vib.
, 201
(1
), pp. 67
–84
.25.
da Silva
, S.
, Cogan
, S.
, and Foltête
, E.
, 2010
, “Nonlinear Identification in Structural Dynamics Based on Wiener Series and Kautz Filters
,” Mech. Syst. Signal Process.
, 24
(1
), pp. 52
–58
.26.
Schetzen
, M.
, 1980
, The Volterra and Wiener Theories of Nonlinear Systems
, Wiley
, New York.27.
Rugh
, W. J.
, 1981
, Nonlinear System Theory—The Volterra/Wiener Approach
, Johns Hopkins University Press
, Baltimore, MD.28.
Korenberg
, M.
, and Hunter
, I.
, 1996
, “The Identification of Nonlinear Biological Systems: Volterra Kernel Approaches
,” Ann. Biomed. Eng.
, 24
(4
), pp. A250
–A268
.29.
Zhang
, Q.
, Suki
, B.
, Westwick
, D.
, and Lutchen
, K.
, 1998
, “Factors Affecting Volterra Kernel Estimation: Emphasis on Lung Tissue Viscoelasticity
,” Ann. Biomed. Eng.
, 26
(1
), pp. 103
–116
.30.
Björsell
, N.
, Suchnek
, P.
, Händel
, P.
, and Rönnow
, D.
, 2008
, “Measuring Volterra Kernels of Analog-to-Digital Converters Using a Stepped Three-Tone Scan
,” IEEE Trans. Instrum. Meas.
, 57
(4
), pp. 666
–671
.31.
Narayanan
, S.
, and Poon
, H. C.
, 1973
, “An Analysis of Distortion in Bipolar Transistors Using Integral Charge Control Model and Volterra Series
,” IEEE Trans. Circuit Theory
, 20
(4
), pp. 341
–351
.32.
Gruber
, J.
, Ramirez
, D.
, Limon
, D.
, and Alamo
, T.
, 2013
, “Computationally Efficient Nonlinear Min-Max Model Predictive Control Based on Volterra Series Models: Application to a Pilot Plant
,” J. Process Control
, 23
(4
), pp. 543
–560
.33.
Gruber
, J.
, Bordons
, C.
, and Oliva
, A.
, 2012
, “Nonlinear MPC for the Airflow in a PEM Fuel Cell Using a Volterra Series Model
,” Control Eng. Pract.
, 20
(2
), pp. 205
–217
.34.
Lee
, D.
, 2011
, “Short-Term Prediction of Wind Farm Output Using the Recurrent Quadratic Volterra Model
,” IEEE
Power and Energy Society General Meeting
, San Diego, CA, July 24–29.35.
Kaizer
, A.
, 1987
, “Modeling of the Nonlinear Response of an Electrodynamic Loudspeaker by a Volterra Series Expansion
,” Audio Eng. Soc.
, 35
(6
), pp. 421
–432
.36.
Novák
, A.
, 2007
, “Identification of Nonlinear Systems: Volterra Series Simplification
,” Acta Polytech.
, 47
(4–5
), pp. 72
–75
.37.
Cunha
, T.
, Lima
, E.
, and Pedro
, J.
, 2010
, “Validation and Physical Interpretation of the Power-Amplifier Polar Volterra Model
,” IEEE Trans. Microwave Theory Tech.
, 58
(12
), pp. 4012
–4021
.38.
Ogunfunmi
, T.
, 2007
, “Adaptative Nonlinear System Identification: The Volterra and Wiener Model Approaches
,” Signal and Communication Technology
, Springer
, Berlin.39.
Silva
, W.
, 2005
, “Identification of Nonlinear Aeroelastic Systems Based on the Volterra Theory: Progress and Opportunities
,” Nonlinear Dyn.
, 39
(1), pp. 25
–62
.40.
Chatterjee
, A.
, 2009
, “Crack Detection in a Cantilever Beam Using Harmonic Probing and Free Vibration Decay
,” 27th International Modal Analysis Conference (IMAC-XXVII
), Orlando, FL, Feb. 9–12.41.
Chatterjee
, A.
, 2011
, “Nonlinear Dynamics and Damage Assessment of a Cantilever Beam With Breathing Edge Crack
,” ASME J. Vib. Acoust.
, 133
(5
), p. 051004
.42.
Tawfiq
, I.
, and Vinh
, T.
, 2003
, “Contribution to the Extension of Modal Analysis to Non-Linear Structure Using Volterra Functional Series
,” Mech. Syst. Signal Process.
, 17
(2
), pp. 379
–407
.43.
da Silva
, S.
, 2011
, “Non-Linear Model Updating of a Three-Dimensional Portal Frame Based on Wiener Series
,” Int. J. Non-Linear Mech.
, 46
(1
), pp. 312
–320
.44.
Shiki
, S. B.
, Lopes
, V.
, Jr., and da Silva
, S.
, 2012
, “Model Updating of the Non-Linear Vibrating Structures Through Volterra Series and Proper Orthogonal Decomposition
,” International Conference on Noise and Vibration Engineering
(ISMA 2012
), Leuven, Belgium, Sept. 17–19, pp. 2199–2212.45.
Shiki
, S. B.
, da Silva
, S.
, Santos
, F. L. M.
, and Peeters
, B.
, 2014
, “Characterization of the Nonlinear Behavior of a F-16 Aircraft Using Discrete-Time Volterra Series
,” International Conference on Noise and Vibration Engineering
(ISMA
), Leuven, Belgium, Sept. 15–17, pp. 3143–3152.46.
Cafferty
, S.
, and Tomlinson
, G.
, 1997
, “Characterization of Automotive Dampers Using Higher Order Frequency Response Function
,” J. Automob. Eng.
, 211
(3
), pp. 181
–203
.47.
Peng
, Z.
, Lang
, Z.
, Billings
, S.
, and Tomlinson
, G.
, 2008
, “Comparisons Between Harmonic Balance and Nonlinear Output Frequency Response Function in Nonlinear System Analysis
,” J. Sound Vib.
, 311
(1–2
), pp. 56
–73
.48.
Chatterjee
, A.
, 2010
, “Identification and Parameter Estimation of a Bilinear Oscillator Using Volterra Series With Harmonic Probing
,” Int. J. Non-Linear Mech.
, 45
(1
), pp. 12
–20
.49.
Chatterjee
, A.
, and Vyas
, N. S.
, 2003
, “Nonlinear Parameter Estimation in Rotor-Bearing System Using Volterra Series and Method of Harmonic Probing
,” ASME J. Vib. Acoust.
, 125
(3
), pp. 299
–306
.50.
Wiener
, N.
, 1958
, Nonlinear Problems in Random Theory
, Wiley
, New York
.51.
Bedrosian
, E.
, and Rice
, S.
, 1971
, “The Output Properties of Volterra Systems (Nonlinear Systems With Memory) Driven by Harmonic and Gaussian Inputs
,” Proc. IEEE
, 59
(12
), pp. 1688
–1707
.52.
Kautz
, W. H.
, 1954
, “Transient Synthesis in the Time Domain
,” IRE Trans. Circuit Theory
, 1
(3
), pp. 29
–39
.53.
Wahlberg
, B.
, 1994
, “System Identification Using Kautz Models
,” IEEE Trans. Autom. Control
, 39
(6
), pp. 1276
–1282
.54.
Björsell
, N.
, Isaksson
, M.
, Händel
, P.
, and Rönnow
, D.
, 2010
, “Kautz–Volterra Modelling of Analogue-to-Digital Converters
,” Comput. Stand. Interfaces
, 32
(3
), pp. 126
–129
.55.
Hansen
, C.
, Shiki
, S. B.
, Lopes Junior
, V.
, and da Silva
, S.
, 2014
, “Non-Parametric Identification of a Non-Linear Buckled Beam Using Discrete-Time Volterra Series
,” 9th International Conference on Structural Dynamics
(EURODYN
), Porto, Portugal, June 30–July 2, pp. 2013–2018.56.
Brincker
, R.
, Zhang
, L.
, and Andersen
, P.
, 2001
, “Modal Identification of Output-Only Systems Using Frequency Domain Decomposition
,” Smart Mater. Struct.
, 10
(3
), p. 441
.57.
Caldwell
, R. A.
, and Feeny
, B. F.
, 2014
, “Output-Only Modal Identification of a Nonuniform Beam by Using Decomposition Methods
,” ASME J. Vib. Acoust.
, 136
(4
), p. 041010
.58.
Poulimenos
, A. G.
, and Fassois
, S. D.
, 2009
, “Output-Only Stochastic Identification of a Time-Varying Structure Via Functional Series TARMA Models
,” Mech. Syst. Signal Process.
, 23
(4
), pp. 1180
–1204
.59.
Spiridonakos
, M.
, and Fassois
, S.
, 2009
, “Parametric Identification of a Time-Varying Structure Based on Vector Vibration Response Measurements
,” Mech. Syst. Signal Process.
, 23
(6
), pp. 2029
–2048
.60.
Raz
, G. M.
, and Van Veen
, B.
, 2000
, “Blind Equalization and Identification of Nonlinear and IIR Systems: A Least Squares Approach
,” IEEE Trans. Signal Process.
, 48
(1
), pp. 192
–200
.61.
Tan
, H.-Z.
, Huang
, Y.
, and Fu
, J.
, 2008
, “Blind Identification of Sparse Volterra Systems
,” Int. J. Adapt. Control Signal Process.
, 22
(7
), pp. 652
–662
.62.
Fernandes
, C. A.
, Favier
, G.
, and Mota
, J. C. M.
, 2009
, “Blind Identification of Multiuser Nonlinear Channels Using Tensor Decomposition and Precoding
,” Signal Process.
, 89
(12
), pp. 2644
–2656
.63.
Cherif
, I.
, Abid
, S.
, and Fnaiech
, F.
, 2012
, “Nonlinear Blind Identification With Three-Dimensional Tensor Analysis
,” Math. Probl. Eng.
, 2012
, p. 22
.64.
Van Vaerenbergh
, S.
, Vía
, J.
, and Santamaría
, I.
, 2008
, “Adaptive Kernel Canonical Correlation Analysis Algorithms for Nonparametric Identification of Wiener and Hammerstein Systems
,” EURASIP J. Adv. Signal Process.
, 2008
(123
), pp. 1
–13
.65.
Xu
, G.
, Liu
, H.
, Tong
, L.
, and Kailath
, T.
, 1995
, “A Least-Squares Approach to Blind Channel Identification
,” IEEE Trans. Signal Process.
, 43
(12
), pp. 2982
–2993
.66.
Abed-Merain
, K.
, 1997
, “Blind System Identification
,” Proc. IEEE
, 85
(8
), pp. 1310
–1322
.67.
Kalouptsidis
, N.
, and Koukoulas
, P.
, 2003
, “Blind Identification of Volterra-Hammerstein Systems
,” IEEE Workshop on Statistical Signal Processing
(SSP
), St. Louis, MO, Sept. 28–Oct. 1, pp. 202
–205
.68.
Tan
, H.-Z.
, and Aboulnasr
, T.
, 2006
, “Tom-Based Blind Identification of Nonlinear Volterra Systems
,” IEEE Trans. Instrum. Meas.
, 55
(1
), pp. 300
–310
.69.
Volterra
, V.
, 1959
, Theory of Functionals and of Integral and Integro-Differential Equations
, Dover Publications
, New York.70.
Shiki
, S. B.
, Lopes
, V.
, Jr., and da Silva
, S.
, 2014
, “Identification of Nonlinear Structures Using Discrete-Time Volterra Series
,” J. Braz. Soc. Mech. Sci. Eng.
, 36
(3
), pp. 523
–532
.71.
Shiki
, S. B.
, Hansen
, C.
, and da Silva
, S.
, 2014
, “Nonlinear Features Identified by Volterra Series for Damage Detection in a Buckled Beam
,” MATEC Web of Conferences—2nd International Conference on Structural Nonlinear Dynamics and Diagnosis
(CSNDD14
), Agadir, Morocco, Vol. 16, p. 02003
.72.
da Silva
, S.
, Dias Júnior
, M.
, and Lopes Junior
, V.
, 2009
, “Identification of Mechanical Systems Through Kautz Filter
,” J. Vib. Control
, 15
(6
), pp. 849
–865
.73.
da Silva
, S.
, 2011
, “Non-Parametric Identification of Mechanical Systems by Kautz Filter With Multiple Poles
,” Mech. Syst. Signal Process.
, 25
(4
), pp. 1103
–1111
.74.
Heuberger
, P. S.
, Hof
, P. M. V. D.
, and Wahlberg
, B.
, 2005
, Modelling and Identification With Rational Orthogonal Basis Functions
, 1st ed., Springer
, London
.75.
Tomlinson
, G.
, Manson
, G.
, and Lee
, G.
, 1996
, “A Simple Criterion for Establishing an Upper Limit to the Harmonic Excitation Level of the Duffing Oscillator Using the Volterra Series
,” J. Sound Vib.
, 190
(5
), pp. 751
–762
.76.
Thouverez
, F.
, 1998
, “A New Convergence Criteria of Volterra Series for Harmonic Inputs
,” 16th International Modal Analysis Conference
(IMAC XVI
), Santa Barbara, CA, Feb. 2–5, pp. 723–727.77.
Chatterjee
, A.
, and Vyas
, N.
, 2000
, “Convergence Analysis of Volterra Series Response of Nonlinear Systems Subjected to Harmonic Excitation
,” J. Sound Vib.
, 236
(2
), pp. 339
–358
.78.
Peng
, Z.
, and Lang
, Z.
, 2007
, “On the Convergence of the Volterra-Series Representation of the Duffing's Oscillators Subjected to Harmonic Excitations
,” J. Sound Vib.
, 305
(12
), pp. 322
–332
.79.
Sakellariou
, J.
, and Fassois
, S. D.
, 2002
, “Nonlinear ARX (NARX) Based Identification and Fault Detection in a 2DOF System With Cubic Stiffness
,” International Conference on Noise and Vibrations Engineering
, Leuven, Belgium, Sept. 16–18.80.
Billings
, S.
, 2013
, Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domain
, Wiley
, Chichester, UK.81.
Tang
, B.
, Brennan
, M.
, Lopes
, V.
, Jr., da Silva
, S.
, and Ramlan
, R.
, 2015
, “Using Nonlinear Jumps to Estimate Cubic Stiffness Nonlinearity: An Experimental Study
,” Proc. Inst. Mech. Eng., Part C
, epub.Copyright © 2016 by ASME
You do not currently have access to this content.
