Asynchronous Consensus Protocols Using Nonlinear Paracontractions Theory

Several consensus protocols have been proposed in the literature and their convergence properties studied via a variety of methods. In all these methods, the communication topologies play a key role in the convergence of consensus processes. In this note, based on asynchronous iteration methods for nonlinear paracontractions, we establish a new result which shows that consensus is reachable under directional and time-varying topologies by using asynchronous nonlinear protocols. Our result makes use of the confluent iteration graph which unifies various communication assumptions and contributes to a fundamental understanding of convergent consensus processes. This result extends existing ones in the literature and has many potential applications. As an illustration, we consider a special case of our model and discuss the robot rendezvous problem via a center-of-gravity algorithm.