Heavy-tails in Kalman filtering with packet losses

Abstract

In this paper, we study the existence of a steady-state distribution and its tail behaviour for the estimation error arising from Kalman filtering for unstable linear dynamical systems. Although a large body of literature has studied the problem of Kalman filtering with packet losses in terms of analysis of the second moment, no study has addressed the actual distribution of the estimation error. First we show that if the system is strictly unstable and packet loss probability is strictly less than unity, then the steady-state distribution (if it exists) must be heavy tail, i.e. its absolute moments beyond a certain order do not exist. Then, by drawing results from Renewal Theory, we further provide sufficient conditions for the existence of such stationary distribution. Moreover, we show that under additional technical assumptions and in the scalar scenario, the steady-state distribution of the Kalman prediction error has an asymptotic power-law tail, i.e. P[|e|>s]∼s−α, as s → ∞, where α can be explicitly computed. We further explore how to optimally select the sampling period assuming an exponential decay of packet loss probability with respect to the sampling period. In order to minimize the expected value of the second moment or the confidence bounds, we illustrate that in general a larger sampling period will need to be chosen in the latter case as a result of the heavy tail behaviour.