A Polynomial-Chaos-Based Bayesian Approach for Estimating Uncertain Parameters of Mechanical Systems

Abstract

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.

Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

“…First, the holding constraint is active and then the system transitions to the flat 12 constraints may be defined as, (45) represent scaling factors of the conditions from a standard deviation perspective. The uncertain holding constraint (46) result in a reduced set of efficient operations on the eviation sense. This means a subset of the systems still not satisfy the constraints.…”

“…Sandu and coworkers introduced its application to multibody dynamical systems in [27,28,[36][37][38][39][40]. Significant work has been done applying it as a foundational element in parameter [23][24][25][26][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] and state estimation [60,61], as well as system identification [62]. Relatively recent work has applied gPC to both classical and optimal control system design [41,63,64].…”

Polynomial chaos-based parameter estimation methods applied to a vehicle system

“…Blanchard et al [1] showed that the maximum likelihood estimate is given by that realization of the parameters (that value of ξ) which maximizes the a posteriori probability P[θ|z], or, equivalently, minimizes −log(P[θ|z]), where z is the latest, yet-to-beused set of observations. This value of ξ corresponding to the maximum likelihood estimate is given by…”

“…where F K 1 , F K 2 , F C 1 , and F C 2 are defined in equations (32) and (33). In these equations, the variables are expressed versus their position at equilibrium (if the added mass M is not in the middle, static deflections occur).…”

“…Xiu and Karniadakis [11] extended the approach to general formulations based on Wiener-Askey polynomials family, and applied it to fluid mechanics [12][13][14]. Sandu et al [3,4,15,16] applied for the first time the polynomial chaos method to multibody dynamic systems, terramechanics [17,18], and parameter estimation in the time domain for fixed parameters [19,20]. In their groundbreaking work, Soize and Ghanem [21] described mathematical settings for characterizing problems for which random uncertainties have arbitrary probability densities.…”