Hammerstein System Identification With the Nearest Neighbor Algorithm

The nonlinear characteristic in a Hammerstein system, i.e., a system in which a nonlinear memoryless subsystem and a linear dynamic are connected in a cascade, is recovered with the nonparametric nearest neighbor regression estimate. The a priori information is nonparametric, both the nonlinear characteristic and the impulse response 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 asymptotic bias and variance of the proposed estimate. As a result, the optimal rate of convergence is established that additionally is independent of the shape of the input density. Results of numerical simulations are also presented.