Global Synchronization of Sampled-Data Invariant Systems on Exponential Lie Groups

We examine the synchronization of networks of identical continuous-time agents on a matrix Lie group, controlled by a discrete-time controller with constant sampling periods and directed, weighted communication graphs with a globally reachable node. We present a smooth, distributed, nonlinear discrete-time control law that achieves global synchronization, for any sampling period, on exponential matrix Lie groups, which include simply connected nilpotent Lie groups as a special case. Synchronization is generally asymptotic, but if the Lie group is nilpotent, then synchronization is achieved at an exponential rate. We first linearize the synchronization error dynamics at identity, and show that the proposed controller achieves local exponential synchronization on any Lie group. Building on the local analysis, we show that if the Lie group is exponential, then synchronization is global. We provide conditions for finite-time synchronization when the communication graph is unweighted and complete.