Non-parametric identification of upper bound covariance matrices for min-sup Robust Kalman Filter: Application to the AR case

The min-sup type Robust Kalman Filter (RKF) introduced in Azhmyakov [2002] guarantees a robust estimate in uncertain linear dynamic systems under relatively weak assumptions related to the state and observation noises. In particular, it is supposed that the system and observation noises have some unknown probability distribution functions from some classes of centered distributions with bounded covariances with the known upper bound matrices. In our paper, we address the identification problem for upper-bound matrices in RKF, in the case of scalar observations. We use a novel Penalized Uncoverage (PU) function and an advanced optimization technique for this purpose. The novel PU-RKF methodology we develop in this paper is applied to robust state estimation in the stationary autoregressive model. We finally compare computationally our new PU-RKF algorithm with a classical approach involving a combination of the maximum-likelihood estimation and Kalman Filter (ML-KF) for Gaussian and some non-Gaussian noises.