Emmanuel Blanchard

Parameter estimation for mechanical systems via an explicit representation of uncertainty

Emmanuel D. Blanchard et al. — Wed, 19 Oct 2011 15:34:11 PDT

Purpose – The purpose of this paper is to propose a new computational approach for parameter estimation in the Bayesian framework. A posteriori probability density functions 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 a priori 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. This is believed to be the first time the polynomial chaos theory has been applied to Bayesian estimation.

A polynomial chaos-based kalman filter approach for parameter estimation of mechanical systems

Emmanuel D. Blanchard et al. — Wed, 19 Oct 2011 15:34:09 PDT

Mechanical systems operate under parametric and external excitation uncertainties. The polynomial chaos approach has been shown to be more efficient than Monte Carlo for quantifying the effects of such uncertainties on the system response. Many uncertain parameters cannot be measured accurately, especially in real time applications. Information about them is obtained via parameter estimation techniques. Parameter estimation for large systems is a difficult problem, and the solution approaches are computationally expensive. This paper proposes a new computational approach for parameter estimation based on the extended Kalman filter (EKF) and the polynomial chaos theory for parameter estimation. The error covariances needed by EKF 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 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. The main advantages of this method are an accurate representation of uncertainties via polynomial chaos, a computationally efficient update formula based on EKF, and the ability to provide a posteriori probability densities of the estimated parameters. The method is able to deal with non-Gaussian parametric uncertainties. The paper identifies and theoretically explains a possible weakness of the EKF with approximate covariances: numerical errors due to the truncation in the polynomial chaos expansions can accumulate quickly when measurements are taken at a fast sampling rate. To prevent filter divergence, we propose to lower the sampling rate and to take a smoother approach where time-distributed observations are all processed at once. We propose a parameter estimation approach that uses polynomial chaos to propagate uncertainties and estimate error covariances in the EKF framework. Parameter estimates are obtained in the form of polynomial chaos expansion, which carries information about the a posteriori probability density function. The method is illustrated on a roll plane vehicle model.

Non-dimensionalized Closed-form Parametric Analysis of Semiactive Vehicle Suspensions using a Quarter-car Model

Mehdi Ahmadian et al. — Wed, 19 Oct 2011 15:34:07 PDT

This article provides a non-dimensionalised closed-form analysis of semi-active vehicle suspensions, using a quarter-car model. The derivation of the closed-form solutions for three indices that can be used for ride comfort, vehicle handling, and stability are presented based on non-dimensionalised suspension parameters. The behaviour of semi-active vehicle suspensions is evaluated using skyhook, groundhook, and hybrid control policies, and compared with passive suspensions. The relationship between vibration isolation, suspension deflection, and road holding is studied, using three performance indices based on the mean square of the sprung mass acceleration, rattle space, and tyre deflection, respectively. The results of the study indicate that the hybrid control policy yields significantly better comfort than a passive suspension, without reducing the road-holding quality or increasing the suspension displacement for typical passenger cars. The results also indicate that for typical passenger cars, the hybrid control policy results in a better compromise between comfort, road holding and suspension travel requirements than both the skyhook and groundhook control methods.

Ride performance analysis of half-car model for semi-active system using RMS as performance criteria

Sany Ihsan et al. — Wed, 19 Oct 2011 15:34:05 PDT

The work aims to study the root mean square (RMS) responses to acceleration input for four state variables: the m_S vertical acceleration, the m_S pitch angular acceleration and the front and rear deflections of the suspensions. A half-car two degree-of-freedom model of semi-active control scheme is analyzed and compared with the conventional passive suspension system. Frequency response of the transfer function for the heave, pitch of the sprung mass and suspension deflections are initially compared and then mean square analysis is utilized to see the effect of semi-active scheme. Results indicate that significant improvements were achieved in the sprung mass heave and pitch responses using semi-active control scheme. However results for the rear and front suspension deflection show that there are limiting values of damping coefficient beyond which, the semi-active scheme becomes disadvantageous than the passive system.

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

Emmanuel D. Blanchard et al. — Wed, 19 Oct 2011 15:34:02 PDT

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.