Scholarly Works, Computational Science Laboratory

Permanent URI for this collection

https://hdl.handle.net/10919/102075

Browse

Browsing Scholarly Works, Computational Science Laboratory by Author "Blanchard, Emmanuel"

Now showing 1 - 4 of 4
  • Thumbnail Image
    Parameter Estimation for Mechanical Systems Using an Extended Kalman Filter
    Blanchard, Emmanuel; Sandu, Adrian; Sandu, Corina (Department of Computer Science, Virginia Polytechnic Institute & State University, 2008-09-01)
    This paper proposes a new computational approach based on the Extended Kalman Filter (EKF) in order to apply the polynomial chaos theory to the problem of parameter estimation, using direct stochastic collocation. The Kalman filter formula is used at each time step in order to update the polynomial chaos of the uncertain states and the uncertain parameters. The main advantage of this method is that the estimation comes in the form of a probability density function rather than a deterministic value, combined with the fact that simulations using polynomial chaos methods are much faster than Monte Carlo simulations. The proposed method is applied to 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. A major drawback was identified: the EKF can diverge when using a high sampling frequency, which might prevent the use of enough data to obtain accurate results when a low sampling frequency is necessary. When applying the polynomial chaos theory to the EKF, numerical errors can accumulate even faster than in the general case due to the truncation in the polynomial chaos expansions, which is illustrated on a simple example. An alternative EKF approach which consists of applying the filter formula on all the observations at once usually yields better results, but can still sometimes fail to produce very accurate results. Therefore, using different sampling rates in order to verify the coherence of the results and comparing the results to a different approach is strongly recommended.
  • Thumbnail Image
    Polynomial Chaos Approaches to Parameter Estimation and Control Design for Mechanical Systems with Uncertain Parameters
    Blanchard, Emmanuel (Virginia Tech, 2010-03-26)
    Mechanical systems operate under parametric and external excitation uncertainties. The polynomial chaos approach has been shown to be more efficient than Monte Carlo approaches for quantifying the effects of such uncertainties on the system response. This work uses the polynomial chaos framework to develop new methodologies for the simulation, parameter estimation, and control of mechanical systems with uncertainty. This study has led to new computational approaches for parameter estimation in nonlinear mechanical systems. The first approach is a polynomial-chaos based Bayesian approach in which maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. The second approach is based on the Extended Kalman Filter (EKF). The error covariances needed for the EKF approach are computed from polynomial chaos expansions, and the EKF is used to update the polynomial chaos representation of the uncertain states and the uncertain parameters. The advantages and drawbacks of each method have been investigated. This study has demonstrated the effectiveness of the polynomial chaos approach for control systems analysis. For control system design the study has focused on the LQR problem when dealing with parametric uncertainties. The LQR problem was written as an optimality problem using Lagrange multipliers in an extended form associated with the polynomial chaos framework. The solution to the Hâ problem as well as the H2 problem can be seen as extensions of the LQR problem. This method might therefore have the potential of being a first step towards the development of computationally efficient numerical methods for Hâ design with parametric uncertainties. I would like to gratefully acknowledge the support provided for this work under NASA Grant NNL05AA18A.
  • Thumbnail Image
    A Polynomial Chaos Based Bayesian Approach for Estimating Uncertain Parameters of Mechanical Systems – Part I: Theoretical Approach
    Blanchard, Emmanuel; Sandu, Adrian; Sandu, Corina (Department of Computer Science, Virginia Polytechnic Institute & State University, 2007)
    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.
  • Thumbnail Image
    A Polynomial Chaos Based Bayesian Approach for Estimating Uncertain Parameters of Mechanical Systems – Part II: Applications to Vehicle Systems
    Blanchard, Emmanuel; Sandu, Adrian; Sandu, Corina (Department of Computer Science, Virginia Polytechnic Institute & State University, 2007)
    This is the second part of a two-part article. In the first part, a new computational approach for parameter estimation was proposed based on the application of the polynomial chaos theory. The maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. In this part, the 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. The value of the mass and its position are estimated from periodic observations of the displacements and velocities across the suspensions. Appropriate excitations are needed in order to obtain accurate results. For some excitations, different combinations of uncertain parameters lead to essentially the same time responses, and no estimation method can work without additional information. 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. When using appropriate excitations, the results obtained with this approach are close to the actual values of the parameters. 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.