Parameter estimation for mechanical systems via an explicit representation of uncertainty

Abstract

Purpose

The purpose of this paper is to propose a new computational approach for parameter estimation in the Bayesian framework. A posteriori probability density functions are obtained using the polynomial chaos theory for propagating uncertainties through system dynamics. The new method has the advantage of being able to deal with large parametric uncertainties, non‐Gaussian probability densities and nonlinear dynamics.

Design/methodology/approach

The maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. Direct stochastic collocation is used as a less computationally expensive alternative to the traditional Galerkin approach to propagate the uncertainties through the system in the polynomial chaos framework.

Findings

The new approach is explained and is applied to very simple mechanical systems in order to illustrate how the Bayesian cost function can be affected by the noise level in the measurements, by undersampling, non‐identifiablily of the system, non‐observability and by excitation signals that are not rich enough. When the system is non‐identifiable and an a priori knowledge of the parameter uncertainties is available, regularization techniques can still yield most likely values among the possible combinations of uncertain parameters resulting in the same time responses than the ones observed.

Originality/value

The polynomial chaos method has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters. This is believed to be the first time the polynomial chaos theory has been applied to Bayesian estimation.

Citation

Blanchard, E., Sandu, A. and Sandu, C. (2009), "Parameter estimation for mechanical systems via an explicit representation of uncertainty", Engineering Computations, Vol. 26 No. 5, pp. 541-569. https://doi.org/10.1108/02644400910970185