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Nonlinear system modeling and identification using Volterra-PARAFAC models

Authors

  • Gérard Favier,

    Corresponding author
    • Laboratoire I3S, University of Nice Sophia Antipolis, CNRS, Les Algorithmes - Bât. Euclide B, 2000 Route des lucioles, B.P. 121-06903 Sophia Antipolis, Cedex, France
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  • Alain Y. Kibangou,

    1. Gipsa-Lab, Systems Control Department, University Joseph Fourier, CNRS, 961 rue de la Houille Blanche, B.P. 46-38402 Grenoble, Cedex, France
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  • Thomas Bouilloc

    1. Laboratoire I3S, University of Nice Sophia Antipolis, CNRS, Les Algorithmes - Bât. Euclide B, 2000 Route des lucioles, B.P. 121-06903 Sophia Antipolis, Cedex, France
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Gérard Favier, Laboratoire I3S, University of Nice Sophia Antipolis, CNRS, Les Algorithmes - Bât. Euclide B, 2000 Route des lucioles, B.P. 121-06903 Sophia Antipolis, Cedex, France.

E-mail: favier@i3s.unice.fr

SUMMARY

Discrete-time Volterra models are widely used in various application areas. Their usefulness is mainly because of their ability to approximate to an arbitrary precision any fading memory nonlinear system and to their property of linearity with respect to parameters, the kernels coefficients. The main drawback of these models is their parametric complexity implying the need to estimate a huge number of parameters. Considering Volterra kernels of order higher than two as symmetric tensors, we use a parallel factor (PARAFAC) decomposition of the kernels to derive Volterra-PARAFAC models that induce a substantial parametric complexity reduction. We show that these models are equivalent to a set of Wiener models in parallel. We also show that Volterra kernel expansions onto orthonormal basis functions (OBF) can be viewed as Tucker models that we shall call Volterra-OBF-Tucker models. Finally, we propose three adaptive algorithms for identifying Volterra-PARAFAC models when input–output signals are complex-valued: the extended complex Kalman filter, the complex least mean square (CLMS) algorithm and the normalized CLMS algorithm. Some simulation results illustrate the effectiveness of the proposed identification methods. Copyright © 2011 John Wiley & Sons, Ltd.

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