A family of degenerate robust Kalman filters

A well-established way to address model uncertainty in state estimation problems is through minimax formulations, where the optimal filter is obtained against the least favorable model within an ambiguity set. The latter is defined by placing an upper bound on a divergence between the actual and nominal state-space models. In this work, we extend the robust Kalman filtering framework to the case in which the ambiguity set is defined using the tau-divergence family, and the Riccati recursion evolves on the cone of positive semi-definite matrices. The resulting family of robust Kalman filters, parameterized by tau, includes as a special case a degenerate robust Kalman filter previously introduced in the literature. Moreover, it generalizes the existing class of robust Kalman filters based on the tau-divergence, whose Riccati recursion is restricted to the cone of positive definite matrices. We derive the corresponding least favorable model, study the convergence properties of the Riccati iteration under constant model parameters, and present numerical experiments showing that the proposed estimator provides a flexible framework where an appropriate tuning of the parameter tau can lead to improved transient performance.