U. A. Khan and J. M. F. Moura, "Distributing the Kalman filter
for large-scale systems," *IEEE Transactions on Signal
Processing*, vol. 56, Part 1, no. 10, pp. 4919"4935, Oct.
2008.

## Abstract

This paper presents a \emph{distributed} Kalman filter to estimate the state of a sparsely connected, large-scale, $n$-dimensional, dynamical system monitored by a network of~$N$ sensors. Local Kalman filters are implemented on $n_l$-dimensional sub-systems, $n_l\ll n$, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an $L$th order Gauss-Markov approximation to the centralized filter. We quantify the information loss due to this $L$th order approximation by the divergence, which decreases as $L$ increases. The order of the approximation, $L$, leads to a bound on the dimension of the sub-systems, hence, providing a criterion for sub-system selection. The (approximated) centralized Riccati and Lyapunov equations are computed iteratively with only local communication and low-order computation by a distributed iterate collapse inversion (DICI) algorithm. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter. Nowhere in the network, storage, communication, or computation of~$n$-dimensional vectors and matrices is needed; only~$n_l \ll n$ dimensional vectors and matrices are communicated or used in the local computations at the sensors. In other words, knowledge of the state is itself distributed.