Nonparametric identification of a Wiener system using a stochastic excitation of arbitrarily unknown spectrum

A Wiener system consists of two sequential sub-systems: (i) a linear, dynamic, time-invariant, asymptotically stable sub-system, followed by (ii) a nonlinear, static (i.e. memoryless), invertible sub-system. Both sub-systems will be identified non-parametrically in this paper, based on observations at only the overall Wiener systems input and output, without any observation of any internal signal inter-connecting the two sub-systems, and without any prior parametric assumption on either sub-system. This proposed estimation allows the input to be temporally correlated, with a mean/variance/spectrum that are a priori unknown (instead of being white and zero-mean, as in much of the relevant literature). Moreover, the nonlinear sub-systems input and output may be corrupted additively by Gaussian noises of non-zero means and unknown variances. For the above-described set-up, this paper is first in the open literature (to the best of the present authors knowledge) to estimate the linear dynamic sub-system non-parametrically. This presently proposed linear system estimator is analytically proved as asymptotically unbiased and consistent. Moreover, the proposed nonlinear sub-systems estimate is assured of invertibility (unlike earlier methods), asymptotic unbiasedness, and pointwise consistence. Furthermore, both sub-systems estimates finite-sample convergence is also derived analytically. Monte Carlo simulations verify the efficacy of the proposed estimators and the correctness of the derived convergence rates.