Consensus Protocol-Based Reachable Nodes in the Controllability of Directed Graphs

In this article, we discuss the controllability of reachable nodes in directed graphs over consensus protocol. A node is called a reachable node if there is a path from an input to this node, otherwise unreachable node. The rows in the controllability matrix associated with unreachable nodes are shown to be zero rows, and a prerequisite for controllability is that all nodes are reachable. A method for constructing controllable directed graphs is provided later in this article. If the in-degrees of all nodes are distinct, and there are neither sibling nodes nor unreachable nodes, the system is controllable. A subsystem composed of reachable nodes is called a reachable subsystem. All reachable nodes are proved to be controllable if and only if the reachable subsystem is controllable. For a reachable subsystem with a tree graph, a necessary and sufficient condition for controllability is given. Besides, if the reachable subsystem with a tree graph is uncontrollable, a graphical method is given to identify the dimension of controllable subspace.