Heterogeneity and scalability in group agreement protocols: Beyond small gain and passivity approaches

It is shown in this paper how by introducing interconnection symmetries as in bidirectional communication schemes, 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 and other related convexifications in the complex plane, that can lead to decentralized certificates by means of their dual interpretation. 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 that exploit the interconnection structure. Examples are given of networks comprised of agents with heterogeneous higher order dynamics and also with non-identical input and communication delays. Special cases of the results recover small gain related delay independent stability, but can also lead to less conservative delay dependent conditions that are fully decentralized and can be necessary and sufficient. Time domain interpretations are finally discussed for certain formulations of the problem.