A Distributed Implementation of Steady-State Kalman Filter

This article studies the distributed state estimation in sensor network, where
$m$
sensors are deployed to infer the
$n$
-dimensional state of a linear time-invariant Gaussian system. By a lossless decomposition of the optimal steady-state Kalman filter, we show that the problem of distributed estimation can be reformulated as that of the synchronization of homogeneous linear systems. Based on such decomposition, a distributed estimator is proposed, where each sensor node runs a local filter using only its own measurement, alongside with a consensus algorithm to fuse the local estimate of every node. We prove that the average of local estimates from all sensors coincides with the optimal Kalman estimate, and under certain condition on the graph Laplacian matrix and the system matrix, the covariance of local estimation error is bounded and the asymptotic error covariance is derived. As a result, the distributed estimator is stable for each single node. We further show that the proposed algorithm has a low message complexity of
$\min (m,n)$
. Numerical examples are provided in the end to illustrate the efficiency of the proposed algorithm.