Multi-agent Systems with Compasses

TL;DR

This paper introduces a relaxed convexity condition for agreement protocols in multi-agent systems, utilizing shared reference directions akin to magnetic compasses, and establishes convergence criteria for both cooperative and antagonistic interactions.

Contribution

It relaxes the convexity assumption by using tangent cones based on local hyperrectangles, interpreted as shared reference directions or compasses, to guarantee convergence.

Findings

Exponential state agreement under joint quasi-strong connectivity.

Asymptotic convergence with componentwise agreement in absolute values.

New geometric condition broadens applicability of agreement protocols.

Abstract

This paper investigates agreement protocols over cooperative and cooperative--antagonistic multi-agent networks with coupled continuous-time nonlinear dynamics. To guarantee convergence for such systems, it is common in the literature to assume that the vector field of each agent is pointing inside the convex hull formed by the states of the agent and its neighbors, given that the relative states between each agent and its neighbors are available. This convexity condition is relaxed in this paper, as we show that it is enough that the vector field belongs to a strict tangent cone based on a local supporting hyperrectangle. The new condition has the natural physical interpretation of requiring shared reference directions in addition to the available local relative states. Such shared reference directions can be further interpreted as if each agent holds a magnetic compass indicating the orientations of a global frame. It is proven that the cooperative multi-agent system achieves exponential state agreement if and only if the time-varying interaction graph is uniformly jointly quasi-strongly connected. Cooperative--antagonistic multi-agent systems are also considered. For these systems, the relation has a negative sign for arcs corresponding to antagonistic interactions. State agreement may not be achieved, but instead it is shown that all the agents' states asymptotically converge, and their limits agree componentwise in absolute values if and in general only if the time-varying interaction graph is uniformly jointly strongly connected.