Nonparametric approach to Wiener system identification

Abstract

A Wiener system, i.e., a system consisting of a linear dynamic subsystem followed by a memoryless nonlinear one is identified. The system is driven by a stationary white Gaussian stochastic process and is disturbed by Gaussian noise. The characteristic of the nonlinear part can be of any form. The dynamic subsystem is asymptotically stable. The a priori information about both the impulse response of the dynamic part of the system and the nonlinear characteristics is nonparametric. Both subsystems are identified from observations taken at the input and output of the whole system. The kernel regression estimate is applied to estimate the invertible part of the nonlinearity. An estimate to recover the impulse response of the dynamic part is also given. Pointwise consistency of the first and consistency of the other estimate is shown. The results hold for any nonlinear characteristic, and any asymptotically dynamic subsystem. Convergence rates are also given.