Group consensus of multiple integrator agents under general topology

The group consensus algorithm for multiple integrator agents with the presence of negative couplings among different clusters is considered in this note which aims mainly to address the general concern that whether the group consensus can be reached from any initial states if the underlying topology of each cluster (subgroup) has a directed spanning tree and the couplings within each cluster, compared to those among different clusters, are sufficiently strong. In addition to the group convergence analysis, both the lower bounds for such strengths and the the group convergence rate can be explicitly specified as well. Moreover, this note also analyzes the factors that decide the final consensus value of each cluster.