Gossip and Distributed Kalman Filtering: Weak Consensus under Weak Detectability

TL;DR

This paper introduces the Gossip Interactive Kalman Filter (GIKF) for distributed state estimation in sensor networks, demonstrating its stability and weak consensus under weak detectability conditions despite random communication.

Contribution

The paper develops GIKF, a novel distributed Kalman filtering algorithm that achieves weak consensus and bounded error under minimal detectability assumptions.

Findings

GIKF error process remains stochastically bounded.

Network achieves weak consensus with error covariance convergence.

Results hold despite random communication and network instability.

Abstract

The paper presents the gossip interactive Kalman filter (GIKF) for distributed Kalman filtering for networked systems and sensor networks, where inter-sensor communication and observations occur at the same time-scale. The communication among sensors is random; each sensor occasionally exchanges its filtering state information with a neighbor depending on the availability of the appropriate network link. We show that under a weak distributed detectability condition: 1. the GIKF error process remains stochastically bounded, irrespective of the instability properties of the random process dynamics; and 2. the network achieves \emph{weak consensus}, i.e., the conditional estimation error covariance at a (uniformly) randomly selected sensor converges in distribution to a unique invariant measure on the space of positive semi-definite matrices (independent of the initial state.) To prove these results, we interpret the filtered states (estimates and error covariances) at each node in the GIKF as stochastic particles with local interactions. We analyze the asymptotic properties of the error process by studying as a random dynamical system the associated switched (random) Riccati equation, the switching being dictated by a non-stationary Markov chain on the network graph.