Purpose -To propose a new computational approach for parameter estimation in the Bayesian framework. Aposteriori PDFs 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 apriori 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. To the best of our knowledge, it is the first time the polynomial chaos theory has been applied to Bayesian estimation. Paper type Research paper = probability distribution of the multidimensional random variable . K = standard deviation for the stiffness distribution M = standard deviation for the mass distribution meas = standard deviation of the measurement oscillation frequency obs = frequency of the forcing function (radians s -1 ) n = natural frequency (radians s -1 ) obs = measured oscillation frequency (radians s -1 ) = multi-dimensional random variable = space of possible value for the unknown variables = generalized Askey-Wiener polynomial chaoses Subscripts nom = nominal value, i.e., value obtained when 0 Superscripts ref = reference value used to generate observations 4 uncertainties through covariance matrices at the same time. In order to approximate PDFs propagated through the system, linearization using the EKF (Blanchard et al., 2007b) and Monte Carlo techniques using the EnKF (Saad et al., 2007) are common approaches. 10 mismatch J is usually the most important component of the cost function, but apriori J is useful when mismatch Jdoes not contain enough information in order to find a clear minimum value for our cost function. This is illustrated in the next section of this article.
Mass-spring system with uncertain initial velocityThis section applies the Bayesian approach to the simple mass-spring system shown in Figure 1.