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Nonlinear system modeling and identification using Volterra‐PARAFAC models
Author(s) -
Favier Gérard,
Kibangou Alain Y.,
Bouilloc Thomas
Publication year - 2012
Publication title -
international journal of adaptive control and signal processing
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.73
H-Index - 66
eISSN - 1099-1115
pISSN - 0890-6327
DOI - 10.1002/acs.1272
Subject(s) - volterra series , nonlinear system identification , kernel (algebra) , nonlinear system , parametric statistics , kalman filter , orthonormal basis , system identification , mathematics , computer science , model order reduction , algorithm , mathematical optimization , data modeling , statistics , physics , quantum mechanics , database , combinatorics , projection (relational algebra)
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.