Average Consensus by Graph Filtering: New Approach, Explicit Convergence Rate and Optimal Design

TL;DR

This paper introduces a graph signal processing approach to analyze and design consensus protocols in multi-agent systems, providing explicit convergence rates and optimal protocol design methods for uncertain networks.

Contribution

It presents a novel graph spectrum filtering framework for consensus analysis, deriving new conditions, convergence rates, and optimal design strategies for uncertain multi-agent networks.

Findings

New necessary and sufficient conditions for consensus.

Explicit formulas for convergence rates.

Optimal protocol design methods.

Abstract

This paper revisits the problem of multi-agent consensus from a graph signal processing perspective. Describing a consensus protocol as a graph spectrum filter, we present an effective new approach to the analysis and design of consensus protocols in the graph spectrum domain for the uncertain networks, which are difficult to handle by the existing time-domain methods. This novel approach has led to the following new results in this paper: 1) New necessary and sufficient conditions for both finite-time and asymptotic average consensus of multi-agent systems. 2) Direct link between the consensus convergence rate and the periodic consensus protocols. 3) Conversion of the fast consensus problem to the problem of polynomial design of graph spectrum filter. 4) A Lagrange polynomial interpolation method and a worst-case optimal interpolation method for the design of periodic consensus protocols for the MASs on uncertain graphs. 5) Explicit formulas for the convergence rate of the designed protocols. Several numerical examples are given to demonstrate the validity, effectiveness and advantages of these results.