# Agreement and consensus problems in groups of autonomous agents with linear dynamics

## Abstract

We study two recent consensus problems in multi-agent coordination with linear dynamics. In Saber and Murray an agreement problem was studied which has linear continuous-time state equations and a sufficient condition was given for the given protocol to solve the agreement problem; namely that the underlying graph is strongly connected. We give sufficient and necessary conditions which include graphs that are not strongly connected. In addition, Saber and Murray show that the protocol solves the average consensus problem if and only if the graph is strongly connected and balanced. We show how multi-rate integrators can solve the average consensus problem even if the graph is not balanced. We give lower bounds on the rate of convergence of these systems which are related to the coupling topology. Saber and Murray also considered the case where the coupling topology changes with time but remain a balanced graph at all times. We relate this case of switching topology to synchronization of nonlinear dynamical systems with timevarying coupling and give conditions for solving the consensus problem even when the graphs are not balanced. Jadbabaie et al. study a model of leaderless and follow-the-leader coordination of autonomous agents using a discrete-time model with time-varying linear dynamics and show coordination if the underlying undirected graph is connected across intervals. Mureau extended this to directed graphs which are strongly connected across intervals. We prove that coordination is possible even if the graph is not strongly connected. © 2005 IEEE.