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

Abstract

An improved approach to determine exact search directions for the optimization of Volterra models based on Generalized Orthonormal Bases of Functions (GOBF) is proposed. The proposed approach extends the work in [7], where a novel, exact technique for optimizing the GOBF parameters (poles) for Volterra models of any order was presented. The proposed extensions take place in two different ways: (i) the formulation here is derived in such a way that each multidimensional kernel of the model 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 intended for representing the dominant dynamic of the kernel along a particular dimension; and (ii) a novel, more computationally efficient method to analytically and recursively calculate the search directions (gradients) for the bases poles is derived. A simulated example is presented to illustrate the performance of the proposed approach. A comparison between the proposed method, which uses asymmetric kernels with multiple orthonormal bases, and the original method, which uses symmetric kernels with a single basis, is presented.

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

“…, n b , are the basis poles. These are the functions used in [11] and [25]. However, there are some shortcomings associated with the identification of models based on these functions when the dynamic characteristics of the system to be modeled (and, accordingly, the nature of the poles that should be used) are unknown, namely:…”

“…The results obtained from the approach proposed here have also been compared with the ones provided by the method in [25]. Such method also proposes the optimization of asymmetric Volterra models, but using Takenaka-Malmquist (rather than ladder-structured) orthonormal functions.…”

“…The results of the proposed methodology are once again compared with the ones provided by the methods in [11] and [25]. Those models were simulated by varying the number of real-and complex-valued poles arbitrarily included into their Takenaka-Malmquist orthonormal bases.…”

“…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.…”

“…The problem of optimizing the free-design parameters of such models is elaborated in Section IV. In Section V, simulation examples are presented in which the results of the proposed method are compared against: (i) those obtained by the method in [11]; and also (ii) against those obtained by the method in [25], in which asymmetric GOBF Volterra models have also been proposed, but using Takenaka-Malmquist (rather than ladder-structured) orthonormal functions. Finally, the conclusions are addressed in Section VI.…”