Parameter estimation for mechanical systems via an explicit representation of uncertainty

Abstract

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 identification study of frequency response data for a trilayer conjugated polymer actuator displacement model

“…The three uncertain parameters are the diffusion coefficient (D), the resistance (R), and the double-layer thickness (δ). The parameter estimation method used in this study was a Bayesian approach similar to the one developed by Blanchard et al [13,14] which has been proven to identify zones of nonidentifiability [13]. Since the full model used in [11] is not suitable for real-time control purposes, it is approximated by a reduced form (second-order) of the model [11].…”

“…6-8 (grids of 200x200x200) were between 15 and 30 minutes. With the polynomial chaos theory, results for a similar resolution would probably easily be obtained within a few seconds or even less [13,14], which would also enable the use of higher sample frequencies if needed.…”

“…This can be visualized when looking at the shape of the Bayesian cost function. The cost function can yield very similar values when a first parameter varies (this parameter is unidentifiable) while it yields very different values when another (identifiable) parameter varies, as explained in [13]. In that case, the stochastic controller would only need to take the parameters that cannot be estimated into account.…”

“…The simulation involves the formation of a multidimensional grid, calculating the value of the cost function at every point considered, and finding the minimum of those calculated values. The main advantage of this parameter estimation is that the quality of the maximum likelihood estimate is related to the shape of the Bayesian cost function, with a sharp minimum indicating an accurate estimate [13]. This will be useful in order to design better controllers for artificial muscles.…”

Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

“…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].…”