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

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.