Distributing the Kalman Filter for Large-Scale Systems

Abstract

This paper derives a distributed Kalman filter to estimate a sparsely connected, large-scale, n−dimensional, dynamical system monitored by a network of N sensors. Local Kalman filters are implemented on the (n l −dimensional, where n l n) sub-systems that are obtained after spatially decomposing the large-scale system. The resulting subsystems overlap, which along with an assimilation procedure on the local Kalman filters, preserve an Lth order Gauss-Markovian structure of the centralized error processes. The information loss due to the Lth order GaussMarkovian approximation is controllable as it can be characterized by a divergence that decreases as L ↑. The order of the approximation, L, leads to a lower bound on the dimension of the sub-systems, hence, providing a criterion for sub-system selection. The assimilation procedure is carried out on the local error covariances with a distributed iterate collapse inversion (DICI) algorithm that we introduce. The DICI algorithm computes the (approximated) centralizedRiccati and Lyapunov equations iteratively with only local communication and low-order computation. 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 that is coherent with the centralized Kalman filter with an Lth order Gaussian-Markovian structure on the centralized error processes. Nowhere storage, communication, or computation of n−dimensional vectors and matrices is needed; only n l n dimensional vectors and matrices are communicated or used in the computation at the sensors.

Network Navigation: Theory and Interpretation

“…However, realizing this benefit in practical implementation incurs associated costs such as additional communication and more complicated information fusion: (i) the communication among nodes required for cooperation can jeopardize the benefits of cooperation if such communication is not properly designed [1], [13], and (ii) the non-diagonal structure of the above EFIMs implies strong correlation in agents' position estimates and hence hinders the development of efficient distributed information fusion algorithms for medium-and largescale networks [1], [36], [37]. Hence, for realistic network design and operation, it is crucial to develop efficient communication strategies.…”

Control principles of complex systems

“…We also discuss the parameter identifiability problem, which is a special case of the observability problem. Finally, we introduce a graphical approach to identify the minimum set of sensor nodes that assure the observability of nonlinear systems Khan and Doostmohammadian, 2011;Khan and Moura, 2008;Letellier and Aguirre, 2005;Letellier et al, 2006;Letellier and Aguirre, 2010;Siddhartha and van Schuppen, 2001) and its application to metabolic networks .…”

Particle filtering with applications in networked systems: a survey

“…A fully distributed Kalman filter has been proposed in [106] for sparsely connected, large-scale systems. The global dynamic model is decomposed into low-dimensional subsystems for which local filters are designed.…”