My Favorite Application Using Eigenvalues: Convergence of Consensus in Multiagent Networks

This paper discusses the consensus problem for distributed multiagent networks–an important problem in cooperative control and computer science. In particular it examines the convergence properties of one solution to the average consensus problem for networks of dynamic agents. We first consider fixed-topology networks with instantaneous agent-to-agent communication, determine the conditions sufficient for convergence, and relate the spectral properties of the network to the convergence rate. We next consider networks with dynamic topology and instantaneous communication, not only determining the conditions sufficient for convergence, but also bounding the convergence with the spectral properties of the network. Finally, we examine fixed networks in which there may be a delay in communication and find that maximum allowable delay in communication is bounded by the spectral properties of the network. 1 History and Motivation Technological advances over the last fifteen years have inspired a broad interest in autonomous and semi-autonomous vehicles. As the abilities of the individual vehicle grew and computation and communication capabilities improved, the capacity of networks of vehicles to cooperatively achieve a goal beyond the ability of the individual attracted several researchers [1]. In fact, there already exists a broad spectrum of applications for multiagent systems, including flight formation of unmanned aerial vehicles (UAVs), congestion control in communication networks, flocking, and distributed sensor networks [2].