  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.
