The controllability of weighted and directed networks with antagonistic interactions

This paper studies the controllability problem for weighted digraphs with antagonistic interactions. Based on the graph partition theory, the L-equitable partition(L-EP) is proposed. Using L-equitable partition, a graph-theoretic necessary condition is proposed for the controllability of the multi-agent system and an algorithm is given for the computation of the partition. Besides, the upper bound on the controllable subspace is derived by utilizing the coarsest L-EP under leaders V
_l
when the system is uncontrollable.