A Machine Learning Approach to the Identification of Dynamical Nonlinear Systems

The aim of this paper is to present a general machine learning approach to the identification of nonlinear systems, using the observed input-output finite datasets. The approach is derived representing the input and output signals in the feature space by the principal component analysis (PCA), thus transforming the nonlinear time dependent identification problem to the regression of a nonlinear input-output function. To face this problem an effective machine learning technique based on particle-Bernstein polynomials has been used to model the input-output relationship that describes the system. The approach has been validated by identifying two real world nonlinear systems, in the fields of speech signals and nonlinear audio amplifiers.

I. Introduction

Nonlinear system identification (NSI) refers to the problem of building a mathematical relation between input u and output y of an unknown dynamical system [1]–[7]. A large number of different approaches have been proposed in the literature over the last decades to face this problem. Among these the Lee-Shetzen method [8], [9] that identifies the Volterra kernels of nonlinear systems stimulated by random inputs with assigned statistics, is one of the most popular. To overcome calculation of multidimensional Volterra kernels a cascaded nonlinear identification model, with a static nonlinear element followed by a time-varying element (Hammerstein model [10]), and with a time-varying linear block followed by a static nonlinear element (Wiener model [11]), has been proposed. In the discrete-time domain one of the most successful approach for nonlinear system identification is the NARMAX model [12] (and its derivatives NARX [13] and NARMA [14]), in which the system is modelled in terms of a nonlinear functional expansion of lagged inputs, outputs and prediction errors. NARMAX models have shown to be very effective in many real-world applications [15] –[21], as they are powerful, efficient and unified representations of a wide variety of nonlinear systems. However a major difficulty in system identification using NARMAX model is selecting a model that is parsimonious in the number of parameters and represents the dynamics of the system adequately. Even though various methods, such as polynomials, multilayer perceptrons, wavelet ANNs and radial basis functions, have been used to build NARMAX models, the choice of an adequately model remains a bottleneck.