All-in-one: Certifiable Optimal Distributed Kalman Filter under Unknown Correlations

TL;DR

This paper introduces CO-DKF, a scalable and certifiably optimal distributed Kalman filter that guarantees optimality and stability using a novel SDP relaxation, outperforming existing methods especially in noisy, sparse networks.

Contribution

The paper presents the first certifiably optimal DKF using SDP relaxation, enabling real-time certification of optimality with only local neighbor information.

Findings

CO-DKF achieves optimality in MSE when the SDP relaxation is tight.

The method demonstrates global asymptotic stability.

Outperforms state-of-the-art DKF algorithms in noisy, sparse scenarios.

Abstract

The optimal fusion of estimates in a Distributed Kalman Filter (DKF) requires tracking of the complete network error covariance, problematic in terms of memory and communication. A scalable alternative is to fuse estimates under unknown correlations, doing the update by solving an optimisation problem. Unfortunately, this problem is NP-hard, forcing relaxations that lose optimality guarantees. Motivated by this, we present the first Certifiable Optimal DKF (CO-DKF). Using only information from one-hop neighbours, CO-DKF solves the optimal fusion of estimates under unknown correlations by a particular tight Semidefinite Programming (SDP) relaxation which allows to certify, locally and in real time, if the relaxed solution is the actual optimum. In that case, we prove optimality in the Mean Square Error (MSE) sense. Additionally, we demonstrate the global asymptotic stability of the estimator. CO-DKF outperforms other state-of-the-art DKF algorithms, specially in sparse, highly noisy setups.