Average consensus for nonlinearly coupled agents: quadratic criteria

Many algorithms for multi-agent coordination are based on the fundamental idea of averaging. Probably the most widely known example is given by a class of consensus (agreement, rendezvous) procedures for first-order agents. Representing the agents state with points in the Euclidean space, the algorithm makes each point move to the relative interior of the convex hull spanned by the neighbors, hence the overall convex hull spanned by agents' states shrinks over time. For undirected or balanced network topology, such a protocol establishes average consensus, that is, the agents' states converge to the average of their initial values. Averaging algorithms are mimicked by many consensus protocols for high-order agents. Although the dynamics of the resulting network is no longer averaging (the convex hull spanned by the agents' states is no longer nested and may be unbounded), under certain conditions asymptotic averaging and high-order analogue of the average consensus properties still can be guaranteed. Most of results in the area assume special structure of the agents (e.g. their passivity) or the network (e.g. linear stationary couplings and graphs). In the present paper, we consider nonlinear algorithms for average consensus among identical agents of arbitrary order. The topology is assumed to be undirected but may switch, the couplings are symmetric yet uncertain, satisfying only some sector inequalities or conic quadratic constraints. Effective analytic criteria for consensus, extending the circle stability criterion, are obtained and their relations to well established approaches from the absolute stability theory are discussed.