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

Finite-dimensional discrete-time Volterra models, also called truncated Volterra series expansions or non-recursive polynomial models, are now widely used in various fields of applications. Their usefulness is mainly due to their ability to approximate to an arbitrary accuracy any fading memory nonlinear system, and also to their property of linearity with respect to the parameters, the kernels coefficients. Moreover, they represent a nonlinear extension of the very popular finite impulse response (FIR) linear model, with guaranteed stability in the bounded-input bounded-output (BIBO) sense. They are interpretable in terms of multidimensional convolutions which makes easy the derivation of their z-transform and Fourier transform representations, and therefore their frequency-domain interpretation. The main drawback of these models concerns their parametric complexity implying the need to estimate a huge number of parameters. This number increases rapidly with the system nonlinearity order and memory. There are two main approaches for reducing the number of free parameters in Volterra models. One approach consists in considering block-oriented nonlinear models constituted by a cascade of linear dynamic subsystems and memoryless (static) nonlinearities. Another approach consists in expanding the Volterra kernels onto orthonormal basis functions (OBFs) like Laguerre, Kautz or GOBF functions. Such kernel expansions can lead to parsimonious modeling when the basis functions are adequately chosen. However, the determination of such basis functions for a given system is a difficult task. In this paper which is an extended version of our paper presented at GRETSI 2009, we consider a new approach for reducing the parametric complexity of Volterra models that consists in using the PARAFAC decomposition of the kernels viewed as symmetric tensors. This approach is briefly summarized hereafter. Let us consider a P -order Volterra model for a causal, stable, finite memory, timeinvariant, single-input single-output system, described by the following input-output relationship: