Observation of Periodic Systems: Bridge Centralized Kalman Filtering and Consensus-Based Distributed Filtering

TL;DR

This paper investigates the observation problem of linear periodic systems using a sensor network, establishing a relationship between distributed consensus filtering and centralized Kalman filtering, and analyzing the trade-offs involved.

Contribution

It bridges consensus-based distributed filtering and centralized Kalman filtering for periodic systems, providing convergence conditions and performance gap analysis.

Findings

CMDF performance can be characterized by a symmetric periodic positive semidefinite solution.

The performance gap between CMDF and CKF depends on fusion steps and network weights.

Estimation error covariance of CMDF exponentially converges to that of CKF with increasing fusion steps.

Abstract

Compared with linear time invariant systems, linear periodic system can describe the periodic processes arising from nature and engineering more precisely. However, the time-varying system parameters increase the difficulty of the research on periodic system, such as stabilization and observation. This paper aims to consider the observation problem of periodic systems by bridging two fundamental filtering algorithms for periodic systems with a sensor network: consensus-on-measurement-based distributed filtering (CMDF) and centralized Kalman filtering (CKF). Firstly, one mild convergence condition based on uniformly collective observability is established for CMDF, under which the filtering performance of CMDF can be formulated as a symmetric periodic positive semidefinite (SPPS) solution to a discrete-time periodic Lyapunov equation. Then, the closed form of the performance gap between CMDF and CKF is presented in terms of the information fusion steps and the consensus weights of the network. Moreover, it is pointed out that the estimation error covariance of CMDF exponentially converges to the centralized one with the fusion steps tending to infinity. Altogether, these new results establish a concise and specific relationship between distributed and centralized filterings, and formulate the trade-off between the communication cost and distributed filtering performance on periodic systems. Finally, the theoretical results are verified with numerical experiments.