Kalman Filtering over Gilbert-Elliott Channels: Stability Conditions and the Critical Curve

TL;DR

This paper analyzes the stability of Kalman filtering over Gilbert-Elliott channels with Markovian packet drops, establishing conditions for stability, a critical failure-recovery rate curve, and linking different stability notions.

Contribution

It introduces relaxed stability conditions, a linear matrix inequality approach, and characterizes a critical failure-recovery rate curve for Kalman filter stability over Markovian channels.

Findings

Peak-covariance stability condition derived

Existence of a critical failure-recovery rate curve identified

Lower bound for critical failure rate established

Abstract

This paper investigates the stability of Kalman filtering over Gilbert-Elliott channels where random packet drop follows a time-homogeneous two-state Markov chain whose state transition is determined by a pair of failure and recovery rates. First of all, we establish a relaxed condition guaranteeing peak-covariance stability described by an inequality in terms of the spectral radius of the system matrix and transition probabilities of the Markov chain. We further show that that condition can be interpreted using a linear matrix inequality feasibility problem. Next, we prove that the peak-covariance stability implies mean-square stability, if the system matrix has no defective eigenvalues on the unit circle. This connection between the two stability notions holds for any random packet drop process. We prove that there exists a critical curve in the failure-recovery rate plane, below which the Kalman filter is mean-square stable and no longer mean-square stable above, via a coupling method in stochastic processes. Finally, a lower bound for this critical failure rate is obtained making use of the relationship we establish between the two stability criteria, based on an approximate relaxation of the system matrix.