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

Abstract

Abstract For a Hammerstein system subject to a stochastic input that is spectrally coloured, this study is first in the open literature (to the present authors' best knowledge) to estimate its linear dynamic subsystem. This estimation is achieved without any prior knowledge nor any prior/simultaneous estimation of the preceding non‐linear static subsystem. This proposed estimator can handle any temporally self‐correlated input despite its potentially unknown spectrum, unknown variance and unknown mean—unlike the common assumption that the input is white and zero‐mean. This proposed estimator needs observations only of the Hammerstein system's overall input and consequential output, but not any observation of any intrasubsystem signal. Furthermore, this proposed estimator can handle a linear subsystem whose input and/or output are each corrupted additively by stationary (and possibly coloured) noises of unknown probability distributions, of unknown non‐zero means and of unknown autocovariances. The proposed estimate is analytically proved herein as asymptotically unbiased and as pointwise consistent; and the estimate's finite‐sample convergence rate is also derived analytically.

Keywords

• linear systems
• nonlinear dynamical systems
• estimation theory
• stochastic systems
• statistical distributions
• stochastic processes