Title: "A Tale of Heavy Tails: Kalman Filtering with information loss"

Abstract: The area of estimation and control where information/control actions are received over lossy networks has been extremely active over the last 15 years.  One of the key results is that the second moment of the estimation error in Kalman filtering (linear state estimation algorithm) for unstable systems becomes unbounded if the measurement packets are randomly lost at a probability higher than a certain threshold. While many of the subsequent studies have largely focused on the investigation of second moments only, a more fundamental quantity - the estimation/prediction error itself has not been investigated deeply under such scenarios involving information loss. 

In this talk, we will 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.

First we will 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  will further provide sufficient conditions for the existence of such stationary distribution. Moreover, we will 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, where the exponent of the power-law can be explicitly computed. Finally, we will 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.

Please contact Molly Kruko @ mkruko@mit.edu for the Zoom link.