Distributing the Kalman filter for large-scale systems

This paper presents a 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(1)-dimensional subsystems, n(l) << n, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an Lth order Gauss-Markov approximation to the centralized filter. We quantify the information loss due to this Lth-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 subsystems, hence, providing a criterion for subsystem 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 required; only n(l) << 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.