Cone Invariance and Rendezvous of Multiple Agents

In this paper we present a dynamical systems framework for analyzing multi-agent rendezvous problems and characterize the dynamical behavior of the collective system. Recently, the problem of rendezvous has been addressed considerably in the graph theoretic framework, which is strongly based on the communication aspects of the problem. The proposed approach is based on set invariance theory and focusses on how to generate feedback between the vehicles, a key part of the rendezvous problem. The rendezvous problem is defined on the positions of the agents and the dynamics is modeled as linear first order systems. The proposed framework however is not fundamentally limited to linear first order dynamics and can be extended to analyze rendezvous of higher order agents. In the proposed framework, the problem of rendezvous is cast as a stabilization problem, with a set of constraints on the trajectories of the agents defined on the phase plane. We pose the n-agent rendezvous problem as an ellipsoidal cone invariance problem in the n dimensional phase space. Theoretical results based on set invariance theory and monotone dynamical systems are developed. The necessary and sufficient conditions for rendezvous of linear systems are presented in form of linear matrix inequalities. These conditions are also interpreted in the Lyapunov framework using multiple Lyapunov functions. Numerical examples that demonstrate application are also presented. Index Terms Multi-agent rendezvous, cooperative dynamical systems, monotone systems, cone invariance, non-negative matrices.