Distributing the Kalman Filter for Large-Scale Systems

This paper derives a \emph{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 $n_l\ll n$) sub-systems that are obtained after spatially decomposing the large-scale system. The resulting sub-systems overlap, which along with an assimilation procedure on the local Kalman filters, preserve an $L$th order Gauss-Markovian structure of the centralized error processes. The information loss due to the $L$th order Gauss-Markovian approximation is controllable as it can be characterized by a divergence that decreases as $L\uparrow$. 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) centralized Riccati 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 $L$th order Gaussian-Markovian structure on the centralized error processes. Nowhere storage, communication, or computation of $n-$dimensional vectors and matrices is needed; only $n_l \ll n$ dimensional vectors and matrices are communicated or used in the computation at the sensors

Harmonic-Copuled Riccati Equations and its Applications in Distributed Filtering

The coupled Riccati equations are cosisted of multiple Riccati-like equations with solutions coupled with each other, which can be applied to depict the properties of more complex systems such as markovian systems or multi-agent systems. This paper manages to formulate and investigate a new kind of coupled Riccati equations, called harmonic-coupled Riccati equations (HCRE), from the matrix iterative law of the consensus on information-based distributed filtering (CIDF) algortihm proposed in [1], where the solutions of the equations are coupled with harmonic means. Firstly, mild conditions of the existence and uniqueness of the solution to HCRE are induced with collective observability and primitiviness of weighting matrix. Then, it is proved that the matrix iterative law of CIDF will converge to the unique solution of the corresponding HCRE, hence can be used to obtain the solution to HCRE. Moreover, through applying the novel theory of HCRE, it is pointed out that the real estimation error covariance of CIDF will also become steady-state and the convergent value is simplified as the solution to a discrete time Lyapunov equation (DLE). Altogether, these new results develop the theory of the coupled Riccati equations, and provide a novel perspective on the performance analysis of CIDF algorithm, which sufficiently reduces the conservativeness of the evaluation techniques in the literature. Finally, the theoretical results are verified with numerical experiments.Comment: 14 pages, 4 figure

Distributed infinite-horizon optimal control of continuous-time linear systems over network

This article deals with the distributed infinite-horizon linear-quadratic-Gaussian optimal control problem for continuous-time systems over networks. In particular, the feedback controller is composed of local control stations, which receive some measurement data from the plant process and regulates a portion of the input signal. We provide a solution when the nodes have information on the structural data of the whole network but takes local actions, and also when only local information on the network are available to the nodes. The proposed solution is arbitrarily close to the optimal centralized one (in terms of cost index) when a design parameter is set sufficiently large. Numerical simulation validate the theoretical results