Finite-Time Distributed State Estimation over Time-Varying Graphs: Exploiting the Age-of-Information

We study the problem of collaboratively estimating the state of a discrete-time LTI process by a network of sensor nodes interacting over a time-varying directed communication graph. Existing approaches to this problem either (i) make restrictive assumptions on the dynamical model, or (ii) make restrictive assumptions on the sequence of communication graphs, or (iii) require multiple consensus iterations between consecutive time-steps of the dynamics, or (iv) require higher-dimensional observers. In this paper, we develop a distributed observer that operates on a single time-scale, is of the same dimension as that of the state, and works under mild assumptions of joint observability of the sensing model, and joint strong-connectivity of the sequence of communication graphs. Our approach is based on the notion of a novel "freshness-index" that keeps track of the age-of-information being diffused across the network. In particular, such indices enable nodes to reject stale information regarding the state of the system, and in turn, help achieve stability of the estimation error dynamics. Based on the proposed approach, the estimate of each node can be made to converge to the true state exponentially fast, at any desired convergence rate. In fact, we argue that finite-time convergence can also be achieved through a suitable selection of the observer gains. Our proof of convergence is self-contained, and employs simple arguments from linear system theory and graph theory.