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

Abstract

This paper revisits the problem of multiagent 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: 1) explicit connection between the time-varying consensus protocol and the graph filter; 2) new necessary and sufficient conditions for both finite-time and asymptotic average consensus of multiagent systems (MASs); 3) direct link between the consensus convergence rate and periodic consensus protocols, and conversion of fast consensus problem to the polynomial design of the graph filter; 4) two explicit design methods of the periodic consensus protocols with a predictable convergence rate for MASs on uncertain graphs; and 5) explicit formulas for the convergence rate of designed protocols. Several numerical examples are given to demonstrate the validity, effectiveness, and advantages of these results.