Wiener–Haar Expansion for the Modeling and Prediction of the Dynamic Behavior of Self-Excited Nonlinear Uncertain Systems

Abstract

This paper deals with the modeling and the prediction of the dynamic behavior of uncertain nonlinear systems. An efficient method is proposed to treat these problems. It is based on the Wiener–Haar chaos concept resulting from the polynomial chaos theory and it generalizes the use of the multiresolution analysis well known in the signal processing theory. The method provides a powerful tool to describe stochastic processes as series of orthonormal piecewise functions whose weighting coefficients are identified using the Mallat pyramidal algorithm. This paper shows that the Wiener–Haar model allows an efficient description and prediction of the dynamic behavior of nonlinear systems with probabilistic uncertainty in parameters. Its contribution, compared to the representation using the generalized polynomial chaos model, is illustrated by evaluating the two models via their application to the problems of the modeling and the prediction of the dynamic behavior of a self-excited uncertain nonlinear system.