On the Identification of Wiener Systems Having Saturation-like Functions with Positive Slopes

Rimantas Pupeikis

doi:10.15388/informatica.2005.089

Abstract

The aim of the given paper is the development of an approach for parametric identification of Wiener systems with piecewise linear nonlinearities, i.e., when the linear part with unknown parameters is followed by a saturation-like function with unknown slopes. It is shown here that by a simple data reordering and by a following data partition the problem of identification of a nonlinear Wiener system could be reduced to a linear parametric estimation problem. Afterwards, estimates of the unknown parameters of linear regression models are calculated by processing respective particles of input-output data. A technique based on ordinary least squares (LS) is proposed here for the estimation of parameters of linear and nonlinear parts of the Wiener system, including the unknown threshold of piecewise nonlinearity, too. The results of numerical simulation and identification obtained by processing observations of input-output signals of a discrete-time Wiener system with a piecewise nonlinearity by computer are given.

Highlights

The aim of the given paper is the development of an approach for parametric identification of Wiener systems with piecewise linear nonlinearities, i.e., when the linear part with unknown parameters is followed by a saturation-like function with unknown slopes
One of important types of hybrid systems met in practice is piecewise affine Wiener systems
It is known that the piecewise affine (PWA) system consists of some subsystems, among which switchings occur at occasional time moments

Summary

Introduction

A lot of physical systems are naturally described as Wiener systems, i.e., when the linear system is followed by a static nonlinearity (Billings and Fakhouri, 1977; Bloemen et al, 2001; Glad and Ljung, 2000; Greblicki, 1994; Hagenblad, 1999; Hunter and Korenberg, 1986; Kalafatis et al, 1997; Ljung, 1999; Pupeikis et al, 2003; Roll, 2003; Wigren, 1993). Assuming the nonlinearity to be piecewise linear, one could let the nonlinear part of the Wiener system be represented by different regression functions with some parameters that are unknown beforehand. In such a case, observations of an output of the Wiener system could be partitioned into distinct data sets according to different descriptions. There arises a problem, first, to find a way to partition the available data, second, to calculate the estimates of parameters of regression functions by processing particles of observations to be determined, and, third, to get the unknown threshold.

Statement of the Problem

The Data Reordering

Numerical Simulation

Conclusions

Findings

144 References