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

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

Gérard Favier Thomas Bouilloc 

Laboratoire I3S /CNRS /UNS 2000, route des Lucioles – Les Algorithmes – bât. Euclide B BP 121 – 06903 Sophia Antipolis Cedex

Corresponding Author Email: 
favier@i3s.unice.fr
Page: 
27-51
|
DOI: 
https://doi.org/10.3166/TS.27.27-51
Received: 
1 October 2009
|
Accepted: 
15 May 2010
|
Published: 
28 February 2010
| Citation

OPEN ACCESS

Abstract: 

Volterra models are widely used in various application areas. Their usefulness is mainly due to their ability to approximate with an arbitrary precision any fading memory nonlinear system, and their property of linearity with respect to parameters, the kernels coefficients. The main drawback of these models is their parametric complexity needing to estimate a huge number of parameters. Considering Volterra kernels as symmetric tensors, we use their PARAFAC decomposition to derive the Volterra-Parafac models inducing a substantial parametric complexity reduction. We show that these models can be viewed as a set of Wiener models in parallel. Then, we apply the extended Kalman filter for recursively identifying such Volterra-Parafac models. Some simulation results illustrate the effectiveness of the proposed identification method, in the case of cubic Volterra systems.

RÉSUMÉ

Les modèles de Volterra sont très utilisés dans de nombreux domaines d’application du fait qu’ils permettent de représenter, avec une précision arbitraire, tout système non linéaire de mémoire finie. Ils possèdent de plus la propriété d’être linéaires vis-à-vis de leurs paramètres, les coefficients des noyaux. Le principal inconvénient de ces modèles est leur complexité paramétrique qui nécessite d’estimer un très grand nombre de paramètres. Cet article présente une nouvelle méthode permettant de réduire cette complexité paramétrique en considérant les noyaux de Volterra d’ordre supérieur à un comme des tenseurs symétriques et en les décomposant à l’aide de la décomposition PARAFAC. Les modèles de Volterra-Parafac ainsi obtenus peuvent être vus comme une série de modèles de Wiener mis en parallèle. En exploitant cette nouvelle formulation des modèles de Volterra, nous proposons un algorithme d’identification récursif basé sur le filtre de Kalman étendu. Des résultats de simulation illustrent le comportement de la méthode d’identification proposée, dans le cas de systèmes de Volterra cubiques.

Keywords: 

Volterra models, nonlinear system modeling and identification, Volterra kernels, PARAFAC decomposition, extended Kalman filter, tensors

MOTS-CLÉS

modèles de Volterra, modélisation et identification de systèmes non linéaires, noyaux de Volterra, décomposition PARAFAC, filtre de Kalman étendu, tenseurs

Extended Abstract
1. Introduction
2. Quelques Rappels Sur Les Tenseurs
3. Les Modèles De Volterra-Parafac
4. Estimation Paramétrique Des Modèles De Volterra-Parafac Basée Sur Le Filtre De Kalman Étendu
5. Résultats De Simulation
6. Conclusion
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