Nonlinear system modeling and identification using Volterra‐PARAFAC models

Abstract

International audienceDiscrete-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

Volterra kernels of bilinear systems have tensor train structure

“…As a direct discretization is typically hard or too costly for practical use, several alternative models are derived using the Volterra kernels computed from these equations. Our results highlight the benefits of exploiting the natural TT structure described above, as opposed to employing the approaches of [5], [6].…”

Identification de modèles de Volterra basée sur la décomposition PARAFAC de leurs noyaux et le filtre de Kalman étendu

“…L'inconvénient majeur de la solution proposée réside dans le volume de calcul nécessité par la résolution de l'équation de Riccati. Une solution alternative consiste à utiliser un algorithme du type LMS (Favier et al, 2010). Parmi les autres perspectives de ce travail, nous envisageons de développer des algorithmes d'estimation adaptatifs avec mise à jour partielle sélective des coefficients Parafac comme cela a été proposé pour les filtres linéaires RIF standards (Douglas, 1997), (Dogançay et al, 2001).…”

Efficient Approximation of Volterra Series for High-Dimensional Systems

“…The potential impact of scalable Volterra identification extends across a diverse set of scientific and engineering disciplines where high-dimensional nonlinear interactions are prevalent. [2,4,5,7] The Volterra series as a universal approximator provides an interpretable representation with the ability to also capture memory effects. [13,14] This interpretability can be pivotal for numerous domains where understanding the underlying mechanisms is equally important to predictive accuracy, such as in analysis of physiological systems [1,3] or in advanced control systems where stability guarantees can depend on the explicit structure of a model.…”