Emmanuel Blanchard

A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems

Background. 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. Method of Approach. 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. Results. The main advantages of this method are an accurate representation of uncertainties via polynomial chaoses, a computationally efficient update formula based on EKF, and the ability to provide aposteriori 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 a set of time-distributed observations are all processed at once. Conclusions. We propose a parameter estimation approach that uses polynomial chaoses to propagate uncertainties and estimate error covariances in the EKF framework. Parameter estimates are obtained in the form of a polynomial chaos expansion which carries information about the aposteriori probability density function. The method is illustrated on a roll plane vehicle model.

Parameter estimation for mechanical systems via an explicit representation of uncertainty

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 sy...

A POLYNOMIAL­CHAOS­BASED BAYESIAN APPROACH FOR ESTIMATING UNCERTAIN PARAMETERS OF MECHANICAL SYSTEMS

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.

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

Abstract 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.

Non-dimensionalised closed-form parametric analysis of semi-active vehicle suspensions using a quarter-car model

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.

Comparison Between a Polynomial-Chaos-Based Bayesian Approach and a Polynomial-Chaos-Based EKF Approach for Parameter Estimation With Application to Vehicle Dynamics

Many parameters in mechanical systems cannot be measured physically with good accuracy, which results in parametric and external excitation uncertainties. This paper compares two new computational approaches for parameter estimation. 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 one uses an Extended Kalman Filter (EKF) to obtain the polynomial chaos representation of the uncertain states and the uncertain parameters. The two methods are illustrated on a nonlinear four degree of freedom roll plane vehicle model, where an uncertain mass with an uncertain location is added on the roll bar. Both approaches can work with noisy measurements and yield results close to the actual values of the parameters, except when different combinations of uncertain parameters lead to essentially the same time response than the measured response. In that case, the aposteriori probability densities of the estimated parameters obtained with the EKF approach cannot be trusted. The Bayesian approach identifies that problem since the Bayesian cost function has an entire region of minima, and can use regularization techniques to yield most likely values in that region based on apriori knowledge.Copyright

Polynomial-chaos-based numerical method for the LQR problem with uncertain parameters in the formulation

This paper proposes a polynomial chaos based numerical method providing an optimal controller for the linear-quadratic regulator (LQR) problem when the parameters in the formulation are uncertain, i.e., a controller minimizing the mean value of the LQR cost function obtained for a certain distribution of the uncertainties which is assumed to be known. The LQR problem is written as an optimality problem using Lagrange multipliers in an extended form associated with the polynomial chaos framework, and an iterative algorithm converges to the optimal answer. The algorithm is applied to a simple example for which the answer is already known. Polynomial chaos based methods have the advantage of being computationally much more efficient than Monte Carlo simulations. The Linear-Quadratic Regulator controller is not very well adapted to robust design, and the optimal controller does not guarantee a minimum performance or even stability for the worst case scenario. Stability robustness and performance robustness in the presence of uncertainties are therefore not guaranteed. However, this is a first step aimed at designing more judicious controllers if combined with other techniques in the future. The next logical step would be to extend this numerical method to H2 and then H-infinity problems.Copyright

Non-Dimensional Analysis of the Performance of Semiactive Vehicle Suspensions

An analytical study that evaluates the response characteristics of a two-degree-of freedom quarter-car model using passive and semi-active dampers is provided as an extension to the results published by Chalasani for active suspensions. The behavior of a semi-actively suspended vehicle is evaluated using the hybrid control policy, and compared to the behavior of a passively suspended vehicle. The relationship between vibration isolation, suspension deflection, and road-holding is studied for the quarter-car model. Three performance indices are used as a measure of vibration isolation (which can be seen as a comfort index), suspension travel requirements, and road-holding quality. These indices are based on the mean square responses to a white noise velocity input for three motion variables: the vertical acceleration of the sprung mass, the deflection of the suspension, and the deflection of the tire, respectively. The results of this study indicate that the hybrid control policy yields better comfort than a passive suspension, without reducing the road-holding quality or increasing the suspension displacement for typical passenger cars.© 2007 ASME

Ride Performance Analysis of Semiactive Suspension Systems Based on a Full-Car Model

This paper extends the results for active suspensions obtained by Chalasani in 1986, by evaluating the potential of semiactive suspensions for improving ride performance of passenger vehicles. Numerical simulations are performed on a seven-degree-of-freedom full vehicle model in order to confirm the general trends found for a quarter-car model, used by the authors in an earlier study. This full car model is used not only to study the heave, but also the pitch and roll motions of the vehicle for periodic and discrete road inputs. The behavior of a semi-actively suspended vehicle is evaluated using the hybrid control policy, and compared to the behavior of a passively-suspended vehicle. The results of this study obtained with the periodic inputs indicate that the motion of the quarter-car model is not only a good approximation of the heave motion of a full-vehicle model, but also of the pitch and roll motions since both are very similar to the heave motion. The results obtained with the discrete road input show that, for the example used in this study, the hybrid configuration clearly yields better results than the passive configuration when the objective is to minimize different deflections, angles, and accelerations at the same time.Copyright