An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension

Abstract

A new solution for the problem of selecting poles of the two-parameter Kautz functions in Volterra models is proposed. In general, a large number of parameters are required to represent the Volterra kernels, although this difficulty can be overcome by describing each kernel using a basis of orthonormal functions, such as the Kautz basis. This representation has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a non-linear static mapping represented by the Volterra series. The resulting Wiener/Volterra model can be truncated into fewer terms if the Kautz functions are properly designed. The underlying problem is how to select the arbitrary complex poles that fully parametrize these functions. This problem has been approached in previous research by minimizing an upper bound for the error resulting from the truncation of the kernel expansion. The present paper goes even further in that each multidimensional kernel is decomposed into a set of independent Kautz bases, each of which is parametrized by an individual pair of conjugate Kautz poles intended to represent the dominant dynamic of the kernel along a particular dimension. An analytical solution for one of the Kautz parameters, valid for Volterra models of any order, is derived. A simulated example is presented to illustrate these theoretical results. The same approach is then used to model a real non-linear magnetic levitation system with oscillatory behaviour.

Optimization of Volterra models with asymmetrical kernels based on generalized orthonormal functions

“…The use of multiple independent orthonormal bases for the expansion of Volterra kernels subsumes that some asymmetric kernels can be better represented by multiple bases than can their symmetric counterparts be represented by a single basis. This issue has been previously addressed in [6] and [10] in the contexts of Laguerre and Kautz bases, respectively. As far as the authors know, it has not been addressed in the context of GOBF so far.…”

“…For this reason, the optimal selection of (G)OBF poles has been extensively discussed in the literature (e.g. see [5], [6], [8], [10], [7] and references therein).…”

Asymmetric Volterra Models Based on Ladder-Structured Generalized Orthonormal Basis Functions

“…Since the poles of the (G)OBFs are free-design parameters, their optimal selection constitutes an important stage of the model identification procedure. For this reason, the optimal selection of orthonormal basis poles for (G)OBF Volterra models has been given particular attention in the literature (e.g., see [11], [17]- [24] and references therein).…”

“…Here in this paper, we extend the work in [11] in the following ways: (i) our formulation is derived in such a way that each multidimensional Volterra kernel is decomposed into a set of independent orthonormal bases (rather than a single, common basis), each of which is parameterized by an individual set of poles responsible for representing the dominant dynamic of the kernel along a particular dimension (asymmetric model). Using independent bases for each kernel dimension is expected to reduce the series truncation error when the dominant dynamics along the multiple dimensions of the kernel are significantly different from one another [22], [24], [25]. 1 The method in [11] uses instead a single basis for expanding each kernel along all its dimensions (symmetric model); (ii) our formulation is based on a ladder-structured GOBF architecture that is characterized by having only real-valued parameters to be estimated, regardless of whether the GOBF poles encoded by these parameters are real-or complex-valued.…”

Active Vibration Control Using a Kautz Filter

“…Kautz filters have found several applications, e.g., acoustic and audio [20], circuit theory [17], experimental modal analysis in mechanical systems [2,[12][13][14], vibration control [6], model reduction [4], robust control [18], predictive control [19], general system identification [5,16,22,25], non-linear system identification with Volterra models [7][8][9][10][11], etc. Although it may seem that the mathematical and theoretical aspects of Kautz filters are more interesting for academic purposed, some practical applications can be found in the literature.…”