Distributing the Kalman Filter for Large-Scale Systems (0708.0242v2)

Abstract: 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.

• The paper introduces a distributed Kalman filter method that exploits spatial decomposition to lower computational burdens.
• It employs a DICI algorithm for iterative local matrix inversion, ensuring the accuracy of centralized solutions.
• The approach integrates overlapping sensor data via consensus averaging to achieve robust state estimation in large sensor networks.

Overview of Distributed Kalman Filtering for Large-Scale Systems

The paper "Distributing the Kalman Filter for Large-Scale Systems" by Usman A. Khan and Jos M. F. Moura tackles the challenges of implementing Kalman filters in large-scale dynamical systems. It introduces a distributed algorithm aimed at managing extensive computations and communication burdens that arise in centralized procedures.

Core Contributions

The central theme of the paper is the derivation of a distributed Kalman filter designed to estimate states in large-scale systems monitored by a network of sensors. The authors propose deploying local Kalman filters on spatially decomposed sub-systems, which are notably lower-dimensional than the original system. These sub-systems overlap, and the authors preserve the Gauss-Markovian structure of the centralized error processes by employing an approximation method, the order of which is adaptable and leads to a lower bound on sub-system dimensions.

A novel assimilation procedure for local error covariances is introduced, utilizing a distributed iterate collapse inversion (DICI) algorithm. The DICI algorithm facilitates computation of the Riccati and Lyapunov equations iteratively, requiring only local communication and computation of reduced-order quantities. Observations from overlapping sub-systems are integrated using bipartite fusion graphs and consensus averaging, ensuring coherency with a centralized Kalman filter's characteristics.

Numerical Results

The paper offers strong numerical results demonstrating that this distributed algorithm achieves coherence with centralized solutions without requiring the storage, communication, or computation of high-dimensional matrices and vectors. This is crucial for practical applications in sensor networks over large geographical areas.

Algorithmic Insights

• Spatial Decomposition: By leveraging the sparse and localized nature of large-scale systems, the algorithm decomposes the global system into overlapping sub-systems. Each sensor in the network applies a local Kalman filter to its respective sub-system, greatly reducing both computational and communication overhead.
• Distributed Matrix Inversion: The DICI algorithm innovatively tackles the matrix inversion problem, reducing the complexity from operations on $n \times n$ matrices to those that involve lower-dimensional matrices corresponding to the distributed sub-systems.
• Observation Fusion: The authors implement observation fusion using bipartite fusion graphs, an effective method for handling observations on overlapping states, thereby ensuring the system's global dynamics are accurately represented.

Implications and Future Directions

This research presents significant implications for the implementation of distributed Kalman filters in large-scale dynamical systems. The proposed method ensures robustness and scalability while maintaining computational feasibility in resource-constrained environments such as sensor networks.

Theoretical advancements in distributed filtering, as outlined in this paper, pave the way for further exploration of efficient algorithms in large-scale estimation problems. Future developments might include extending these methods to handle non-linear dynamics or improving communication efficiency for real-time applications.

Conclusion

The distributed Kalman filter proposed by Khan and Moura provides a sophisticated solution for large-scale systems, emphasizing both the theoretical underpinnings and practical implementation considerations. This approach efficiently balances the considerations of distribution, computation, and communication, marking a noteworthy contribution to the field of signal processing and estimation theory.