Strong Structural Controllability of Directed Graphs via Zero Forcing Sets

This paper presents several conditions for strong structural controllability of a directed structured network, which is determined by the non-zero/zero patterns of the weights of directed edges. First, we introduce the existing notion of zero forcing sets with the necessary and sufficient condition for strong structural controllability. Based on the zero forcing sets, we introduce the necessary condition that a strongly structurally controllable graph with the minimum number of m-inputs can be decomposed into m-disjoint controllable paths and bridge edges between them. With a disjoint controllable path as a basic graph, we explore the properties of a strongly structurally controllable graph. Then, we present several merging rules that can maintain the strong structural controllability based on the notion of zero forcing sets.