Hammerstein system with a stochastic input of arbitrary/unknown autocorrelation – nonparametric estimator of the static nonlinear subsystem

Abstract

This study proposes the first estimator in the open literature (to the present authors' best knowledge) to nonparametrically estimate a Hammerstein system's nonlinear static subsystem when excited by an input that is temporally self-correlated with an unknown spectrum, an unknown variance and an unknown mean (instead of the input as commonly presumed to be white and zero-mean). This proposed nonparametric estimator is analytically proved here to be asymptotically unbiased and pointwise consistent. The proposed estimate's associated finite-sample convergence rate is also derived analytically. | INTRODUCTIONA Hammerstein system comprises two subsystems in series: (i) a nonlinear, static (memoryless) subsystem, trailed by (ii) a linear, dynamic, time-invariant, asymptotically stable subsystem. Please see Figure 1 for a schematic showing these two subsystems and their associated signals, which are described in great details in Section 1.This study proposes a new estimate for the nonlinear static subsystem's input-output nonlinearity m(⋅). This estimator relies on observations of only the input and the output of the overall Hammerstein system (i.e. {(U n , Y n ), ∀n}), but not on any observation of any intrasystem signal/noise (e.g. {W n , Z n , V n , X n , P n , ∀n}).This proposed estimator also does not require any prior/simultaneous identification of the nonlinear static subsystem.This is an open access article under the terms of the Creative Commons Attribution-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited and no modifications or adaptations are made.

Hammerstein system with a stochastic input of arbitrary/unknown autocorrelation: Identification of the dynamic linear subsystem

“…For the Hammerstein system's other constituent subsystem (i.e. the non-linear static subsystem), a non-parametric estimator has been developed in a companion study [77] for a similar signal/noise statistics model as indicated herein. These two subsystems' estimators are algorithmically independent from each other, in the sense that each may be computed without computing the other, or that the two may be computed simultaneously in parallel.…”