On the stability and optimality of distributed Kalman filters with finite-time data fusion

In this paper, we consider distributed estimation for discrete-time, linear systems, with finite-time data fusion of agent measurements between each time-step of the dynamics. Prior work in this context is related to average-consensus, where either the data fusion is implemented for an infinite time (in general) to reach average-consensus, or under restricted observability requirements (one-step and/or local), whereas, our results hold under the broadest observability conditions (n-step global observability, where n is the dimension of the dynamics). We show that after the finite-time data fusion on agent measurements, the observation map at each agent is a linear combination of the local observation maps. We then show that this new observation map is observable (if the data is fused for a sufficient number of iterations that we lower bound) resulting in a stable distributed estimator that can be implemented using semi-definite programming at each agent. We further characterize the performance of such distributed estimators by comparing the positive-definiteness of their corresponding information matrices. The centralized and distributed performance gap, although cannot be written in closed form, can be computed using the infinite horizon Kalman gain of each filter. Finally, we consider special cases under which the performance of these distributed estimators is equal to the performance of the centralized Kalman filter.