On stability of the Kalman filter for discrete time output error systems

The stability of the Kalman filter is classically ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. This paper studies the case of discrete time output error (OE) systems, in which the process noise is totally absent. The classical stability analysis assuming the controllability regarding the process noise is thus not applicable. It is shown in this paper that the uniform complete observability is sufficient to ensure the stability of the Kalman filter applied to time varying OE systems, regardless of the stability of the OE systems. Though the continuous time case has been studied recently, the results on continuous time systems cannot be directly transposed to discrete time systems, because of a difficulty related to the observability of the discrete time filter error dynamics system.