Analysis of the Compressed Distributed Kalman Filter Over Markovian Switching Topology
研究了随机切换通信拓扑下高维稀疏状态向量的分布式估计问题，提出压缩分布式卡尔曼滤波算法，利用压缩感知和协方差交叉融合规则，在弱于传统条件的压缩协同激励下给出估计误差上界。
This article investigates the distributed estimation problem of an unknown high-dimensional sparse state vector for a stochastic dynamic system. The communication topology randomly switches, and the switching law is governed by a time-homogeneous Markovian chain. By means of the compressed sensing (CS) theory and a diffusion strategy, we propose a compressed distributed Kalman filter (CDKF). That is, each sensor first compresses the original high-dimensional regression data. Then, the covariance intersection fusion rule is utilized to obtain a distributed Kalman filter (DKF) estimate in the compressed low-dimensional space. Afterward, the original high-dimensional sparse state vector can be well recovered by a reconstruction technique. In terms of stability analysis, one of the main difficulties lies in analyzing the product of nonindependent and nonstationary random matrices in the context of time-varying communication topologies. Relying on the stochastic stability theory, the Markov chain theory, and the CS theory, we establish the upper bound for the estimation error under the compressed cooperative excitation condition, which is much weaker than the traditional uncompressed collective observability conditions used in the existing literature. Finally, we provide a simulation example to illustrate the performance of the proposed algorithm.