(240b) Computation of Arrival Cost for Moving Horizon Estimation Via Unscented Kalman Filtering

Moving horizon estimation (MHE) alleviates the computational burden of solving a full information estimation problem by considering a finite horizon of the measurement data. However, it is non-trivial to determine the arrival cost. A commonly used approach for computing the arrival cost is to use a first order Taylor series approximation of the nonlinear model and then apply an extended Kalman filter (EKF). Unscented Kalman filters (UKF) avoid the linearization in the Kalman filter update formula by an unscented nonlinear transformation. By carefully choosing a set of sigma points, which capture the true mean and covariance of a given distribution and then passing the means and covariances of estimated states through a nonlinear transformation, UKF is capable of estimating the posterior means and covariances accurately to an order higher than two. Therefore UKF can improve upon EKF performance for nonlinear systems as has been demonstrated for some applications [1].

These advantages of UKF over EKF form the motivation of this work where a MHE with computation of the arrival cost via UKF (uMHE) is proposed. The unscented transformation and a set of selected sigma points are used to compute the covariances and then the arrival cost. The selection procedure for the sigma points is the same as the one used for UKF if the constraints are inactive, however, a modification is used that satisfies the state variable constraints when the constraints are active [2]. Linearization of the model is not required for the presented approach.

The case study shows that the presented uMHE performed better than the MHE via the commonly used EKF (eMHE) for all investigated horizon lengths and measurement noise levels. The advantages of uMHE over eMHE are clearly visible for small horizon lengths. Since the performance of both MHEs improves as the horizon length grows, the advantages of the uMHE over the eMHE decrease for large N. However, as the computational burden also grows with the length of the horizon, this restricts how large a value of N can be chosen. The case study illustrated that the proposed uMHE has a better performance than eMHE and can be a promising alternative for approximating the arrival cost for MHE.

References:

[1] S. J. Julier and J. K. Uhlmann. Unscented Filtering and Nonlinear Estimation. Proceedings. of the IEEE, Vol. 92: 401–422, 2004

[2] P. Vachhani, S. Narasimhan, and R Rengaswamy. Robust and reliable estimation via unscented recursive nonlinear dynamic data reconciliation. Journal of Process. Control.,16:1075-1086, 2006.