Analytical Bounds for an Interval Kalman Filter

This article is concerned with analytical developments of results first introduced by Lu et al. (2019). These developments are devoted to the optimization of upper bounds of the interval covariance matrices appearing in the interval Kalman filter. The proposed study is mainly highlighted through two aspects. First, the optimization is further performed by considering a class of upper bounds and minimizing the traces of these bounds in two stages (in terms of a gain matrix and then with respect to a scalar parameter). Second, this article provides conditions under which the optimal trace value is controlled, and hence, the proposed algorithm in the work by Lu et al. (2019), namely the optimal upper bound interval Kalman filter (OUBIKF), is ensured to perform with stability (i.e., without width explosion of the resulting interval estimators). Also under these conditions, the OUBIKF algorithm, having a similar structure of the standard Kalman filter, is ensured to get a smaller trace upper bound of the covariance matrices in the correction step than the one in the prediction step. Numerical simulations based on an automotive model are performed to illustrate the developed results.