Optimality of ODEFTC for Linear Time-Varying Systems

Establish the optimality of the Optimal Distributed Estimation based on Fixed-Time Consensus (ODEFTC) algorithm for distributed state estimation of continuous-time linear time-varying systems subject to stochastic disturbances by proving that, at each node, the true error covariance of the ODEFTC estimates asymptotically matches the error covariance achieved by the centralized Kalman-Bucy filter.

Background

Distributed state estimation over sensor networks for continuous-time linear systems has been extensively studied, with the centralized Kalman-Bucy filter providing the optimal estimation solution when all measurements are available at a single processor. Consensus-based distributed estimators aim to replicate this performance across the network.

The ODEFTC algorithm generalizes prior LTI results to LTV systems using a fixed-time consensus mechanism to fuse information across nodes. Prior to this work, simulation evidence suggested that ODEFTC could achieve performance comparable to the centralized optimal filter at each node, but a formal proof existed only for the LTI case. Consequently, whether ODEFTC is optimal for LTV systems—i.e., whether it asymptotically recovers the centralized Kalman-Bucy performance in terms of error covariance at each node—was an unresolved question that this note sets out to resolve.