On connectivity, observability, and stability in distributed estimation

We introduce a new model of social learning and distributed estimation in which the state to be estimated is governed by a potentially unstable linear model driven by noise. The state is observed by a network of agents, each with its own linear noisy observation models. We assume the state to be globally observable, but no agent is able to estimate the state with its own observations alone. We propose a single consensus-step estimator that consists of an innovation step and a consensus step, both performed at the same time-step. We show that if the instability of the dynamics is strictly less than the Network Tracking Capacity (NTC), a function of network connectivity and the observation matrices, the single consensus-step estimator results in a bounded estimation error. We further quantify the trade-off between: (i) (in)stability of the parameter dynamics, (ii) connectivity of the underlying network, and (iii) the observation structure, in the context of single timescale algorithms. This contrasts with prior work on distributed estimation that either assumes scalar dynamics (which removes local observability issues) or assumes that enough iterates can be carried out for the consensus to converge between each innovation (observation) update.