Feedback Generation in Rendezvous Problems Using Synthetic Dynamics

Rendezvous problems belong to the class of consensus problems where it is desired that vehicles arrive at a desired position simultaneously. 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. Stability is analyzed with respect to communication topology. The current methods do not characterize the behavior of the collective system, which is necessary to generate feedback between the vehicles. In this paper we focus on generating feedback between multiple agents such that rendezvous is achieved. We treat the rendezvous problem in the set invariance setting and define rendezvous on the positions of the 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. 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.