Wiener system identification with generalized orthonormal basis functions

TL;DR

This paper introduces a method for identifying Wiener systems using a finite set of generalized orthonormal basis functions, with proven convergence and fast rate, improving system approximation accuracy.

Contribution

It proposes a novel approach to Wiener system identification employing GOBFs with convergence analysis and practical estimation of pole locations.

Findings

Convergence of the estimated output to the true output in probability.

Fast convergence rates proportional to the number of excited frequencies.

Effective approximation of Wiener systems with finite GOBFs.

Abstract

Many nonlinear systems can be described by a Wiener-Schetzen model. In this model, the linear dynamics are formulated in terms of orthonormal basis functions (OBFs). The nonlinearity is modeled by a multivariate polynomial. In general, an infinite number of OBFs is needed for an exact representation of the system. This paper considers the approximation of a Wiener system with finite-order infinite impulse response dynamics and a polynomial nonlinearity. We propose to use a limited number of generalized OBFs (GOBFs). The pole locations, needed to construct the GOBFs, are estimated via the best linear approximation of the system. The coefficients of the multivariate polynomial are determined with a linear regression. This paper provides a convergence analysis for the proposed identification scheme. It is shown that the estimated output converges in probability to the exact output. Fast convergence rates, in the order $O_p({N_F}^{-n_{rep}/2})$, can be achieved, with $N_F$ the number of excited frequencies and $n_{rep}$ the number of repetitions of the GOBFs.