A Unified Wavelet Based Modelling Framework for Nonlinear System Identification: the WANARX Model Structure

A new unified modelling framework based on the superposition of additive submodels, functional components and wavelet decompositions is proposed for nonlinear system identification. A nonlinear model, which is often represented using a multivariate nonlinear function, is initially decomposed into a number of functional components via the well known analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (Nonlinear Autoregressive with eXogenous inputs) model for representing dynamic input-output systems. By expanding each functional component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and multiresolution wavelet decompositions, the multivariate nonlinear model can then be converted into a linear-in-the-parameters problem, which can be solved using least-squares type methods. An efficient model structure determination approach based upon a forward orthogonal least squares ( OLS) algorithm, which involves a stepwise orthogonalization of the regressors and a forward selection of the relevant model terms based on the error reduction ration (ERR), is employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to as a Wavelet-based ANOVA decomposition of the NARX model or simply WANARX model and can be applied to represent high-order and high dimensional nonlinear systems