A Polynomial Chaos Based Bayesian Approach for Estimating Uncertain Parameters of Mechanical Systems – Part I: Theoretical Approach
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This is the first part of a two-part article. A new computational approach for parameter estimation is proposed based on the application of the polynomial chaos theory. 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. In the new approach presented in this paper, 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. This approach is applied to very simple mechanical systems in order to illustrate how the cost function can be affected by undersampling, non-identifiablily of the system, non-observability, and by excitation signals that are not rich enough. When the system is non-identifiable, 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. This is illustrated using a simple spring-mass system. Possible applications of this theory to the field of vehicle dynamics simulations include the estimation of mass, inertia properties, as well as other parameters of interest. In the second part of this article, this new parameter estimation method is illustrated on a nonlinear four-degree-of-freedom roll plane model of a vehicle in which an uncertain mass with an uncertain position is added on the roll bar.