In this study, a new computational approach for parameter identification 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. The new parameter estimation method is illustrated on a four degreeoffreedom roll plane model of a vehicle in which the vertical stiffnesses of the tires are estimated from periodic observations of the displacements and velocities across the suspensions. The results obtained with this approach are close to the actual values of the parameters even when only measurements with low sampling rates are available. The accuracy of the estimations has been shown to be sensitive to the number of terms used in the polynomial expressions and to the number of collocation points, and thus it may become computationally expensive when a very high accuracy of the results is desired. However, the noise level in the measurements affects the accuracy of the estimations as well. Therefore, it is usually not necessary to use a large number of terms in the polynomial expressions and a very large number of collocation points since the addition of extra precision eventually affects the results less than the effect of the measurement noise. 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.

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.

Parameter estimation for large systems is a difficult problem, and the solution approaches are computationally expensive. The polynomial chaos approach has been shown to be more efficient than Monte Carlo for quantifying the effects of uncertainties on the system response. This article compares two new computational approaches for parameter estimation based on the polynomial chaos theory for parameter estimation: a Bayesian approach, and an approach using an extended Kalman filter (EKF) to obtain the polynomial chaos representation of the uncertain states and the uncertain parameters. The two methods are applied to a non-linear 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. When using appropriate excitations, the results obtained with both approaches are close to the actual values of the parameters, and both approaches can work with noisy measurements. The EKF approach has an advantage over the Bayesian approach: the estimation comes in the form of a posteriori probability densities of the estimated parameters. However, it can yield poor estimations when dealing with non-identifiable systems, and it is recommended to repeat the estimation with different sampling rates in order to verify the coherence of the results with the EKF approach. The Bayesian approach is more robust, can recognize non-identifiability, and use regularization techniques if necessary.

Advanced Vehicle Dynamics LabCenter for Vehicle Systems and Safety, Virginia Tech, Blacksburg, VA 24061-0238 ABSTRACT 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.

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