Based on the Jordan form of system matrix, this paper discusses algebraic conditions for the controllability of the multiagent network system with directed graph from two aspects: leader-follower network attribute and coupling input disturbance. Leader-follower network attribute refers to the topology structure and information communication among agents. Coupling input disturbance includes the number of external coupling inputs and the selection of leader nodes. When the leader-follower network attribute is fixed, the selection method of coupling input disturbance is studied for the controllability, and when the coupling input disturbance is known, we derive necessity and sufficiency conditions to determine the controllability. The reliability of theoretical results is verified by numerical examples and model simulation. Besides, the generally perfect controllability is introduced, that is, the system is always controllable regardless of the number and the locations of leaders. In practical engineering applications, the perfectly controllable topology can improve the system fault tolerance and accelerate the commercialization process, which has a profound significance for promoting the modernization process.

In this paper, the optimal topology structure is studied for hybrid-weighted leader-follower multiagent systems (MASs). The results are developed by taking advantage of linear quadratic regulator (LQR) theory. We show that the multiagent star composite structure is the optimal topology which can enable the MAS to achieve the bipartite consensus. In particular, we prove that the optimal topology corresponding to the multiagent system with the first-order static leader and the second-order dynamic leader is, respectively, a hybrid-weighted star composite structure and an unevenly hybrid-weighted star composite structure. The results of the paper indicate that, in addition to the necessary information communication between leader and followers, the information exchange among followers increases the control cost of the system.