The Weighted Nearest Neighbor Estimate for Hammerstein System Identification

Abstract

This paper concerns the nonparametric identification problem for a class of nonlinear discrete-time dynamical systems that is characterized by its cascade structure. This is a Hammerstein system being a series connection of a nonlinear memoryless element followed by a linear dynamic system. The input–output training data generated from the system are dependent and they do not reveal the strong mixing property. The nonlinear part of the system is recovered with the weighted $k$ -nearest neighbor regression estimate. The a priori information is nonparametric, both the nonlinear characteristic and the impulse response of the linear part are completely unknown and can be of any form. Local and global properties of the estimate are examined. Whatever the probability density of the input signal, the estimate converges at every continuity point of the characteristic as well as in the global sense. We derive the formulas for asymptotic bias and the variance and evaluate the corresponding rate of convergence. The convergence rate is independent of the shape of the input density and is proved to be optimal. These results allow us to find a set of optimal nonnegative weights that further improve the accuracy of our estimation algorithm. Our findings are supported by simulation experiments.