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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
Virgin,
L.
, 2000, Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration, 1st ed.,
Cambridge University Press,
Cambridge, UK.

Worden,
K.
, and
Tomlinson,
G. R.
, 2001, Nonlinearity in Structural Dynamics,
Institute of Physics,
London.

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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
Billings,
S.
, 1980, “
Identification of Nonlinear Systems—A Survey,” IEEE Proc. D: Control Theory Appl.,
127(6), pp. 272–285.

[CrossRef]
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.

[CrossRef]
Ljung,
L.
, 2007, System Identification: Theory for the User, 2nd ed.,
Prentice Hall,
Upper Saddle River, NJ.

Wills,
A.
,
Schön,
T. B.
,
Ljung,
L.
, and
Ninness,
B.
, 2013, “
Identification of Hammerstein–Wiener Models,” Automatica,
49(1), pp. 70–81.

[CrossRef]
Billings,
S.
, and
Fakhouri,
S.
, 1977, “
Identification of Nonlinear Systems Using the Wiener Model,” Electron. Lett.,
13(17), pp. 502–504.

[CrossRef]
Chen,
S.
, and
Billings,
S. A.
, 1989, “
Representation of Non-Linear Systems: The NARMAX Model,” Int. J. Control,
49(3), pp. 1013–1032.

[CrossRef]
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.

[CrossRef]
Noël,
J.
, and
Kerschen,
G.
, 2013, “
Frequency-Domain Subspace Identification for Nonlinear Mechanical Systems,” Mech. Syst. Signal Process.,
40(2), pp. 701–717.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
Ibnkahla,
M.
, 2003, “
Nonlinear System Identification Using Neural Networks Trained With Natural Gradient Descent,” J. Appl. Signal Process.,
12, pp. 1229–1237.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
Schetzen,
M.
, 1980,
*The Volterra and Wiener Theories of Nonlinear Systems*,
Wiley, New York.

Rugh,
W. J.
, 1981, Nonlinear System Theory—The Volterra/Wiener Approach,
Johns Hopkins University Press, Baltimore, MD.

Korenberg,
M.
, and
Hunter,
I.
, 1996, “
The Identification of Nonlinear Biological Systems: Volterra Kernel Approaches,” Ann. Biomed. Eng.,
24(4), pp. A250–A268.

[CrossRef]
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.

[CrossRef] [PubMed]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

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.

Novák,
A.
, 2007, “
Identification of Nonlinear Systems: Volterra Series Simplification,” Acta Polytech.,
47(4–5), pp. 72–75.

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.

Ogunfunmi,
T.
, 2007, “
Adaptative Nonlinear System Identification: The Volterra and Wiener Model Approaches,” Signal and Communication Technology,
Springer, Berlin.

Silva,
W.
, 2005, “
Identification of Nonlinear Aeroelastic Systems Based on the Volterra Theory: Progress and Opportunities,” Nonlinear Dyn.,
39(1), pp. 25–62.

[CrossRef]
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.

Chatterjee,
A.
, 2011, “
Nonlinear Dynamics and Damage Assessment of a Cantilever Beam With Breathing Edge Crack,” ASME J. Vib. Acoust.,
133(5), p. 051004.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

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.

Cafferty,
S.
, and
Tomlinson,
G.
, 1997, “
Characterization of Automotive Dampers Using Higher Order Frequency Response Function,” J. Automob. Eng.,
211(3), pp. 181–203.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
Wiener,
N.
, 1958, Nonlinear Problems in Random Theory,
Wiley,
New York.

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.

[CrossRef]
Kautz,
W. H.
, 1954, “
Transient Synthesis in the Time Domain,” IRE Trans. Circuit Theory,
1(3), pp. 29–39.

[CrossRef]
Wahlberg,
B.
, 1994, “
System Identification Using Kautz Models,” IEEE Trans. Autom. Control,
39(6), pp. 1276–1282.

[CrossRef]
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.

[CrossRef]
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.

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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
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.

[CrossRef]
Cherif,
I.
,
Abid,
S.
, and
Fnaiech,
F.
, 2012, “
Nonlinear Blind Identification With Three-Dimensional Tensor Analysis,” Math. Probl. Eng.,
2012, p. 22.

[CrossRef]
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.

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.

[CrossRef]
Abed-Merain,
K.
, 1997, “
Blind System Identification,” Proc. IEEE,
85(8), pp. 1310–1322.

[CrossRef]
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.

Tan,
H.-Z.
, and
Aboulnasr,
T.
, 2006, “
Tom-Based Blind Identification of Nonlinear Volterra Systems,” IEEE Trans. Instrum. Meas.,
55(1), pp. 300–310.

[CrossRef]
Volterra,
V.
, 1959, Theory of Functionals and of Integral and Integro-Differential Equations,
Dover Publications, New York.

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.

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.

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.

[CrossRef]
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.

[CrossRef]
Heuberger,
P. S.
,
Hof,
P. M. V. D.
, and
Wahlberg,
B.
, 2005, Modelling and Identification With Rational Orthogonal Basis Functions, 1st ed.,
Springer,
London.

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.

[CrossRef]
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.

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.

[CrossRef]
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.

[CrossRef]
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.

Billings,
S.
, 2013, Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domain,
Wiley, Chichester, UK.

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.