Controllability and controllable subspace of multi-agent systems with directed interconnection

In this paper, we investigate the controllability of leader-follower multi-agent systems, where the interconnection is directed and weighted. First, we give algebraic criteria to check controllability. Then we employ balanced partition to classify the interconnection graphs, and study the controllable subspaces with given nontrivial balanced partition. Moreover, we also reveal the equivalence of flow invariance and existence of balanced partition and give a necessary and sufficient graph condition of structural controllability. In addition, some illustrative examples are given.