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Research Papers

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

[+] Author and Article Information
Lyes Nechak

MIPS Laboratory, 12, rue des frères Lumière, Mulhouse 68093, Francelyes.nechak@uha.fr

Sébastien Berger

MIPS Laboratory, 12, rue des frères Lumière, Mulhouse 68093, Francesebastien.berger@uha.fr

Evelyne Aubry

MIPS Laboratory, 12, rue des frères Lumière, Mulhouse 68093, Franceevelyne.aubry@uha.fr

J. Dyn. Sys., Meas., Control 134(5), 051011 (Jul 27, 2012) (11 pages) doi:10.1115/1.4006371 History: Received March 06, 2011; Revised February 21, 2012; Published July 26, 2012; Online July 27, 2012

## 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.

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## Figures

Figure 1

The recursive Mallat algorithm

Figure 2

Hultèn system

Figure 3

Realization of X1 corresponding to μ = 0.3 with the Legendre polynomial chaos model

Figure 4

Realization of X1 corresponding to μ = 0.315 with the Legendre polynomial chaos model

Figure 5

Evolution of X1 in the stochastic dimension modeled with the Legendre random variable

Figure 6

Realization of X1 corresponding to μ = 0.3 with the Wiener–Haar chaos model

Figure 7

Realization of X1 corresponding to μ = 0.315 with the Wiener–Haar chaos model

Figure 8

Realization of X1 corresponding to μ = 0.3 with the Wiener–Haar chaos model computed by Mallat algorithm and MC method

Figure 9

Realization of X1 corresponding to μ = 0.315 with the Wiener–Haar chaos model computed by Mallat algorithm and MC method

Figure 10

Evolution of X1 in the stochastic dimension estimated by the Wiener–Haar chaos model

Figure 11

The limit cycle (X1 , dX1 /dt) corresponding to μ = 0.3

Figure 12

The limit cycle (X1 , dX1 /dt) corresponding to μ = 0.315

Figure 13

L2 -norm of the Wiener–Haar expansion errors for successive resolution levels at different times

Figure 14

L2 -norm of the Wiener–Haar expansion error for successive resolution levels with respect to the MC solution at different times

Figure 15

The density functions of amplitudes X1 and dX1 /dt

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