Nonlinear system identification by the Haar multiresolution analysis

The paper deals with the problem of reconstruction of nonlinearities in a certain class of nonlinear systems of composite structure from their input-output observations when prior information about the system is poor, thus excluding the standard parametric approach to the problem. The multiresolution idea, being the fundamental concept of modern wavelet theory, is adopted, and the Haar multiresolution analysis in particular is applied to construct nonparametric identification techniques of nonlinear characteristics. The pointwise convergence properties of the proposed identification algorithms are established. Conditions for the convergence are given; and for nonlinearities satisfying a local Lipschitz condition, the rate of convergence is evaluated, With applications in mind, the problem of data-driven selection of the optimum resolution degree in the identification procedure, essential for the multiresolution analysis, is considered as well. The theory is verified by computer simulations.