On non-parametric identification of multi-channel non-linear systems by multiscale expansions

The paper deals with the problem of reconstruction of nonlinearities in a certain class of nonlinear dynamical systems of the multichannel form. Each channel of the system has a nonlinearity being embedded in complex dynamics. The system dynamics is of the block oriented form containing dynamic linear subsystems and other "nuisance" nonlinearities. The a priori information about the system nonlinearities is very limited excluding the standard parametric approach to the problem. The multiresolution idea, being the fundamental concept of the modern wavelet theory, is adopted and multiscale expansions associated with a large class of scaling functions are applied to construct nonparametric identification techniques of the nonlinearities. The pointwise convergence properties of the proposed identification algorithms are established. Conditions for the convergence are given and for nonlinearities satisfying the local Lipschitz condition, the rate of convergence is evaluated. These accuracy results reveal that our estimates are able to separate the estimation problem related to each channel. This is a surprising result since the input signals are dependent with completely unknown the joint probability density function.