Convergence results in distributed Kalman filtering

The paper studies the convergence properties of the estimation error processes in distributed Kalman filtering for potentially unstable linear dynamical systems. In particular, it is shown that, in a weakly connected communication network, there exist (randomized) gossip based information dissemination schemes leading to a stochastically bounded estimation error at each sensor for any non-zero rate γ̄ of inter-sensor communication (the rate γ̄ is defined to be the average number of inter-sensor communications per signal evolution epoch). A gossip-based information exchange protocol, the M-GIKF, is presented, in which sensors exchange estimates and aggregate observations at a rate γ̄ > 0, leading to desired convergence properties. Under the assumption of global (centralized) detectability of the signal/observation model (necessary for a centralized estimator having access to all sensor observations at all times to yield bounded estimation error), it is shown that the distributed M-GIKF leads to a stochastically bounded estimation error at each sensor. The conditional estimation error covariance sequence at each sensor is shown to evolve as a random Riccati equation (RRE) with Markov modulated switching. The RRE is analyzed through a random dynamical system (RDS) formulation, and the asymptotic estimation error at each sensor is characterized in terms of an associated invariant measure µγ̄ of the RDS.