Harmonic-Coupled Riccati Equation and its Applications in Distributed Filtering

The coupled Riccati equations (CREs) are a set of multiple Riccati-like equations whose solutions are coupled with each other through matrix means. They are a fundamental mathematical tool to depict the inherent dynamics of many complex systems, including Markovian systems or multi-agent systems. This paper investigates a new kind of CREs called harmonic-coupled Riccati equations (HCREs), whose solutions are coupled using harmonic means. We first introduce the specific form of HCREs and then analyze the existence and uniqueness of its solutions under the conditions of collective observability and primitiveness of coupling matrices. Additionally, we manage to find an iterative law with low computation-complexity to obtain the solutions to HCREs. Based on this newly established theory, we greatly simplify the steady-state estimation error covariance of consensus-on-information-based distributed filtering (CIDF) into the solutions to a discrete-time Lyapunov equation (DLE). This leads to a significant conservativeness reduction of traditional performance evaluation techniques for CIDF. The obtained results are remarkable since they not only enrich the theory of CREs, but also provide a novel insight into the synthesis and analysis of CIDF algorithms. We finally validate our theoretical findings through several numerical experiments.