The S-hull approach to consensus
loannis Lestas, Glenn Vinnicombe · 2007
It is shown in the paper how by introducing interconnection symmetries by use of bidirectional communication, stability of consensus protocols on arbitrary topologies can be guaranteed in a decentralized and scalable way, despite the presence of higher order heterogeneous dynamics. The analysis is centred round the notion of an S-hull a relaxed convex hull in the complex plane. In the case of linear dynamics the certificates derived include, as special cases, passivity and dissipativity approaches, though a wider class of dynamics is allowed by employing convexification arguments. As a corollary, it is shown that in the presence of heterogeneous delays in the relative information (in which case stability is delay dependent) a simple scaling can guaranty stability on an arbitrary topology. The condition becomes necessary and sufficient when the delays are homogeneous. Time domain interpretations are finally discussed for certain formulations of the problem.