Controllability of multi-agent systems with directed and weighted signed networks

This paper studies the controllability of multi-agent systems with signed networks, which are represented by directed weighted signed graphs. The adjacency weights of network depict the property of interactions. Positive weight means cooperative interaction while negative weight antagonistic interaction. First, by using switching equivalent transformation, the controllability of three kinds of signed networks is studied: structurally balanced, anti-balanced, and strictly unbalanced. It is shown that the controllability of the structurally balanced network is equivalent to that of the associated underlying network. Then, a graph-theoretic necessary condition for controllability is given by virtue of almost equitable partition of directed weighted signed networks. Furthermore, some necessary and sufficient conditions are given for the controllability of generic linear multi-agent systems. In addition, several examples are presented to illustrate the theoretical results.