Optimal Kalman Consensus Filter for Weighted Directed Graphs

The distributed estimation problem has proven to be a highly relevant topic today, due to its applicability in a wide variety of scenarios that do not accommodate a centralized supervisor. Decentralized algorithms can offer enhanced robustness and resilience to system failures and cyber-attacks. A seminal work on the topic was the development of the Kalman Consensus Filter (KCF), and recently the issue of the suboptimality of the KCF was addressed. However, in the KCF scheme, the sensor network is modeled as an unweighted undirected graph. This fact has been shown to severely degrade performance when certain assumptions on the network topology are not met, such as in the case of limited observability. Subsequent contributions in the field have implemented consensus based filters on directed graphs, but they either employ heuristic choices for the consensus gains or entail the sharing of information matrices between neighbors. In this paper, we address these issues by proposing an optimal distributed state estimation algorithm for weighted directed graphs. The proposed scheme is shown to be more versatile and offers critical performance improvements in scenarios where the KCF performs poorly. Specifically, we highlight the efficacy of the proposed algorithm in the presence of naïve sensors, through illustrative numerical examples.