Harmonic-Copuled Riccati Equations and its Applications in Distributed Filtering

TL;DR

This paper introduces harmonic-coupled Riccati equations (HCRE) derived from distributed filtering algorithms, proving their existence, uniqueness, and convergence, and demonstrating their application in analyzing the steady-state behavior of estimation errors in multi-agent systems.

Contribution

It formulates a new class of coupled Riccati equations with harmonic mean coupling, establishes their theoretical properties, and links them to distributed filtering performance analysis.

Findings

HCRE solutions exist and are unique under certain conditions.

The iterative law of CIDF converges to the HCRE solution.

Steady-state estimation error covariance can be characterized by a Lyapunov equation.

Abstract

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.