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

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

“…Volterra-CP model. The first and simplest separable Volterra model, proposed in [Favier et al, 2012], represents the kernels by symmetric tensors of rank R n in the CP format, that is…”

A reduced-rank approach for implementing higher-order Volterra filters

“…In its standard form, the input-output relationship of the pth-order kernel is given by [1,25]. The first-order kernel of the Volterra filter is a linear kernel whose input-output relationship…”

“…All aforementioned approaches are, in some sense, based on the use of predefined forms of basis vectors for identifying and discarding the less significant coefficients of a Volterra filter. In contrast, the approaches from [20][21][22][23][24][25] involve the identification of particular basis vectors that can then be exploited aiming to reduce the complexity of a Volterra filter. These approaches are typically based on the definition of coefficient matrices, which are decomposed aiming to obtain the basis vectors.…”

Performance estimation for tensor CP decomposition with structured factors

“…In addition, unlike SVD, it is essentially unique under mild conditions. Therefore, it is naturally well suited for the analysis of data sets constituted by observations of a function of multiple discrete indices, as encountered in signal processing [9,10,11], data mining [12] and biomedical engineering [13] ; see [8,14] for other examples.…”