M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, 1980.

S. A. Billings, Identification of nonlinear systems???a survey, IEE Proceedings D-Control Theory and Applications, pp.272-285, 1980.
DOI : 10.1049/ip-d.1980.0047

W. J. Rugh, Nonlinear System Theory: The Volterra/Wiener Approach, 1981.

A. Isidori, Nonlinear Control Systems, 1995.

V. J. Mathews and G. L. Sicuranza, Polynomial signal processing, 2000.

V. Marmarelis, Nonlinear Dynamic Modeling of Physiological Systems, 2004.
DOI : 10.1002/9780471679370

T. Ogunfunmi, Adaptive Nonlinear System Identification: The Volterra and Wiener model approaches, 2007.
DOI : 10.1007/978-0-387-68630-1

R. W. Brockett, Volterra series and geometric control theory, Automatica, vol.12, issue.2, pp.167-176, 1976.
DOI : 10.1016/0005-1098(76)90080-7

T. Koh and E. J. Powers, Second-order volterra filtering and its application to nonlinear system identification, IEEE Transactions on acoustics, speech, and signal processing, vol.33, issue.6, pp.1445-1455, 1985.

S. Boyd and L. Chua, Fading memory and the problem of approximating nonlinear operators with Volterra series, IEEE Transactions on Circuits and Systems, vol.32, issue.11, pp.1150-1161, 1985.
DOI : 10.1109/TCS.1985.1085649

J. H. De, M. Goulart, and P. M. Burt, Efficient kernel computation for Volterra filter structure evaluation, IEEE Signal Processing Letters, vol.19, issue.3, pp.135-138, 2012.

P. M. Burt, J. H. De, and M. Goulart, Evaluating the potential of Volterra-PARAFAC IIR models, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.2013-5745
DOI : 10.1109/ICASSP.2013.6638765

A. Carini and G. L. Sicuranza, A new class of FLANN filters with application to nonlinear active noise control, European Signal Processing Conference (EUSIPCO), pp.1950-1954, 2012.

G. Favier, A. Y. Kibangou, and T. Bouilloc, Nonlinear system modeling and identification using Volterra-PARAFAC models, International Journal of Adaptive Control and Signal Processing, vol.44, issue.3, pp.30-53, 2012.
DOI : 10.1109/82.558455
URL : https://hal.archives-ouvertes.fr/hal-00642363

A. Y. Kibangou and G. Favier, Tensor Analysis-Based Model Structure Determination and Parameter Estimation for Block-Oriented Nonlinear Systems, IEEE Journal of Selected Topics in Signal Processing, vol.4, issue.3, pp.514-525, 2010.
DOI : 10.1109/JSTSP.2009.2039175

M. Zeller and W. Kellermann, Fast and Robust Adaptation of DFT-Domain Volterra Filters in Diagonal Coordinates Using Iterated Coefficient Updates, IEEE Transactions on Signal Processing, vol.58, issue.3, pp.1589-1604, 2010.
DOI : 10.1109/TSP.2009.2037073

L. A. Azpicueta-ruiz, M. Zeller, A. R. Figueiras-vidal, J. Arenas-garcía, and W. Kellermann, Adaptive Combination of Volterra Kernels and Its Application to Nonlinear Acoustic Echo Cancellation, IEEE Transactions on Audio, Speech, and Language Processing, vol.19, issue.1, pp.97-110, 2011.
DOI : 10.1109/TASL.2010.2045185

C. Crespo-cadenas, J. Reina-tosina, M. J. Madero-ayora, and J. Cruzado, A New Approach to Pruning Volterra Models for Power Amplifiers, IEEE Transactions on Signal Processing, vol.58, issue.4, pp.2113-2120, 2010.
DOI : 10.1109/TSP.2009.2039815

B. F. Beidas, Intermodulation Distortion in Multicarrier Satellite Systems: Analysis and Turbo Volterra Equalization, IEEE Transactions on Communications, vol.59, issue.6, pp.1580-1590, 2011.
DOI : 10.1109/TCOMM.2011.042111.100320

L. Tan and J. Jiang, Adaptive Volterra filters for active control of nonlinear noise processes, IEEE Transactions on Signal Processing, vol.49, issue.8, pp.1667-1676, 2001.
DOI : 10.1109/78.934136

K. Shi and A. Redfern, Blind volterra system linearization with applications to post compensation of ADC nonlinearities, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.3581-3584, 2012.
DOI : 10.1109/ICASSP.2012.6288690

D. Franken, K. Meerkotter, and J. Waßmuth, Observer-based feedback linearization of dynamic loudspeakers with Ac amplifiers, IEEE Transactions on Speech and Audio Processing, vol.13, issue.2, pp.233-242, 2005.
DOI : 10.1109/TSA.2004.841043

A. Carini, G. Sicuranza, and V. J. Mathews, Equalization and linearization of nonlinear systems, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181), pp.1617-1620, 1998.
DOI : 10.1109/ICASSP.1998.681763

A. Germani, C. Manes, and P. Palumbo, Filtering of Stochastic Nonlinear Differential Systems via a Carleman Approximation Approach, IEEE Transactions on Automatic Control, vol.52, issue.11, pp.2166-2172, 2007.
DOI : 10.1109/TAC.2007.908347

J. Arroyo, E. Barocio, R. Betancourt, and A. R. Messina, A bilinear analysis technique for detection and quantification of nonlinear modal interaction in power systems, 2006 IEEE Power Engineering Society General Meeting, 2006.
DOI : 10.1109/PES.2006.1709380

T. Mukherjee, G. K. Fedder, D. Ramaswamy, and J. White, Emerging simulation approaches for micromachined devices, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol.19, issue.12, pp.1572-1589, 2000.
DOI : 10.1109/43.898833

C. Fager, J. C. Pedro, N. B. De-carvalho, and H. Zirath, Prediction of IMD in LDMOS transistor amplifiers using a new large-signal model, IEEE Transactions on Microwave Theory and Techniques, vol.50, issue.12, pp.2834-2842, 2002.
DOI : 10.1109/TMTT.2002.805187

S. A. Maas, Nonlinear Microwave and RF circuits, 2003.

A. Gothenberg and H. Tenhunen, Performance analysis of sampling switches in voltage and frequency domains using Volterra series, 2004 IEEE International Symposium on Circuits and Systems (IEEE Cat. No.04CH37512), 2004.
DOI : 10.1109/ISCAS.2004.1328307

S. B. Kuntze, A. J. Zilkie, L. Pavel, and J. Aitchison, Nonlinear State???Space Model of Semiconductor Optical Amplifiers With Gain Compression for System Design and Analysis, Journal of Lightwave Technology, vol.26, issue.14, pp.2274-2281, 2008.
DOI : 10.1109/JLT.2008.922212

H. Michalska, Generic nonlinear stabilization of systems with matching algebraic structure, Annual Reviews in Control, vol.35, issue.2, pp.215-234, 2011.
DOI : 10.1016/j.arcontrol.2011.10.010

M. Ekman, Suboptimal control for the bilinear quadratic regulator problem: Application to the activated sludge process, IEEE Transactions on Control Systems Technology, vol.13, issue.1, pp.162-168, 2005.

A. Asma and A. Kamel, Nonlinear System Identification based on Volterra and Laguerre Models, Proceedings of the 14th International Conference on Methods and Models in Automation and Robotics, pp.489-494, 2009.
DOI : 10.3182/20090819-3-PL-3002.00085

P. Chevalier, A. Oukaci, and J. Delmas, Third order widely nonlinear Volterra MVDR beamforming, Proceedings of the 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.2648-2651, 2011.
DOI : 10.1109/icassp.2011.5947029
URL : https://hal.archives-ouvertes.fr/hal-01126072

R. Boyer, R. Badeau, and G. Favier, Fast orthogonal decomposition of Volterra cubic kernels using oblique unfolding, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.4080-4083, 2011.
DOI : 10.1109/ICASSP.2011.5947249
URL : https://hal.archives-ouvertes.fr/hal-00576019

E. L. Batista and R. Seara, A fully LMS/NLMS adaptive scheme applied to sparse-interpolated Volterra filters with removed boundary effect, Signal Processing, vol.92, issue.10, pp.2381-2393, 2012.
DOI : 10.1016/j.sigpro.2012.02.011

R. R. Mohler, Bilinear Control Process, 1973.

C. Bruni, G. Dipillo, and G. Koch, Bilinear systems: An appealing class of "nearly linear" systems in theory and applications, IEEE Transactions on Automatic Control, vol.19, issue.4, pp.334-348, 1974.
DOI : 10.1109/TAC.1974.1100617

R. R. Mohler, An overview of bilinear system theory and applications, IEEE Transanctions on Systems, Man, and Cybernetics, vol.10, pp.683-688, 1980.

D. L. Elliott, Bilinear Control Systems, 2009.
DOI : 10.1023/b101451

R. W. Brockett, The early days of geometric nonlinear control, Automatica, vol.50, issue.9, pp.2203-2224, 2014.
DOI : 10.1016/j.automatica.2014.06.010

V. J. Mathews and J. Lee, Adaptive algorithms for bilinear filtering, SPIE Proceedings, pp.317-327, 1994.

F. Fnaiech and L. Ljung, Recursive identification of bilinear systems, International Journal of Control, vol.106, issue.2, pp.453-470, 1987.
DOI : 10.1080/00207178308933123

W. Favoreel, B. De-moor, and P. Van-overschee, Subspace identification of bilinear systems subject to white inputs, IEEE Transactions on Automatic Control, vol.44, issue.6, pp.1157-1165, 1999.
DOI : 10.1109/9.769370

S. Gibson, A. Wills, and B. Ninness, Maximum-likelihood parameter estimation of bilinear systems, IEEE Transactions on Automatic Control, vol.50, issue.10, pp.1581-1595, 2005.
DOI : 10.1109/TAC.2005.856664

E. D. Sontag, Y. Wang, and A. Megretski, Input Classes for Identifiability of Bilinear Systems, IEEE Transactions on Automatic Control, vol.54, issue.2, pp.195-207, 2009.
DOI : 10.1109/TAC.2008.2006927

A. J. Krener, Linearization and bilinearization of control systems, Allerton Conference on Circuit and System Theory, 1974.

J. Deutscher, Asymptotically exact input-output linearization using Carleman linearization, European Control Conference, 2003.

S. Irving and C. Joaquin, On stabilization of non linear systems by using Carleman linearization and periodic systems theory Computing Science and Automatic Control (CCE) Circuit analysis and optimization with automatically derived Volterra kernels, International Conference on Electrical Engineering IEEE International Symposium on Circuits and Systems (ISCAS), pp.491-494, 2000.

A. Bauer, Efficient algorithms for the computation and application of Volterra kernels in the behavior analysis of nonlinear circuits and systems, Proceedings of the 2005 European Conference on Circuit Theory and Design, 2005., 2005.
DOI : 10.1109/ECCTD.2005.1522990

S. Billings and K. Tsang, Spectral analysis for non-linear systems, Part II: Interpretation of non-linear frequency response functions, Mechanical Systems and Signal Processing, vol.3, issue.4, pp.341-359, 1989.
DOI : 10.1016/0888-3270(89)90042-3

A. V. Balakrishnan, ON THE CONTROLLABILITY OF A NONLINEAR SYSTEM, Proc. Nat. Acad. Sci. USA, pp.465-568, 1966.
DOI : 10.1073/pnas.55.3.465

X. Zhao and V. Marmarelis, On the relation between continuous and discrete nonlinear parametric models, Automatica, vol.33, issue.1, pp.81-84, 1997.
DOI : 10.1016/S0005-1098(96)00132-X

J. Vlach and K. Singhal, Computer methods of circuit analysis and design, 1994.

K. Meerktter and R. Scholz, Digital simulation of nonlinear circuits by wave digital filter principles, IEEE International Symposium on Circuits and Systems, pp.720-723, 1989.
DOI : 10.1109/ISCAS.1989.100452

T. B. Schn, System identification of nonlinear state-space models, Automatica, vol.47, issue.1, pp.39-49, 2011.
DOI : 10.1016/j.automatica.2010.10.013

A. J. Krener, Bilinear and Nonlinear Realizations of Input-Output Maps, SIAM Journal on Control, vol.13, issue.4, pp.827-834, 1975.
DOI : 10.1137/0313049

J. H. De and M. Goulart, DerivaçDerivaç?Derivação eficiente e utilizaçutilizaç?utilização de filtros de Volterra de referência na avaliaçavaliaç?avaliação de formalismos não-lineares, 2012.

J. H. De, M. Goulart, and P. M. Burt, Dec.) Carleman bilinearization and Volterra kernels Toolbox, 2016.
URL : https://hal.archives-ouvertes.fr/halshs-00720813

D. Franken, K. Meerkotter, and J. Waßmuth, Passive parametric modeling of dynamic loudspeakers, IEEE Transactions on Speech and Audio Processing, vol.9, issue.8, pp.885-891, 2001.
DOI : 10.1109/89.966091

B. W. Kernighan and D. M. Ritchie, The C programming language, 1988.

A. Soni, Control-relevant system identification using nonlinear Volterra and Volterra-Laguerre models, 2006.

P. M. Burt, J. H. De, and M. Goulart, The Sampling of Triangular Kernels

H. A. Barker and S. Ambati, Nonlinear sampled-data system analysis by multidimensional z transforms, Proc. IEE, pp.1407-1413, 1972.
DOI : 10.1049/piee.1972.0277

N. Vervliet, O. Debals, L. Sorber, M. Van-barel, and L. De-lathauwer, Mar.) Tensorlab 3.0 Krylov subspace methods for model order reduction of bilinear control systems, Systems & Control Letters, vol.59, issue.8, pp.443-450, 2010.

H. Tang, Y. H. Liao, J. Y. Cao, and H. Xie, Fault diagnosis approach based on Volterra models, Mechanical Systems and Signal Processing, vol.24, issue.4, pp.1099-1113, 2010.
DOI : 10.1016/j.ymssp.2009.09.001

E. E. Papalexakis, C. Faloutsos, and N. D. Sidiropoulos, Tensors for Data Mining and Data Fusion, ACM Transactions on Intelligent Systems and Technology, vol.8, issue.2, p.16, 2016.
DOI : 10.1145/2632048.2632102

M. S. Phillip and . Burt, 85) received the bachelor and doctor degrees in Electrical Engineering from Escola Politécnica da Universidade de São Paulo, Brazil, where he is an Associated Professor. From 1984 to 1992 he worked in the telecommunications industry as a development engineer he spent a sabbatical year at the INT His research interests include nonlinear signal processing, adaptive filtering and digital signal processing for telecommunications and audio, 2003.