Strong Structural Controllability and Zero Forcing

In this chapter, we study controllability and output controllability of systems defined over graphs. Specifically, we consider a family of state-space systems, where the state matrix of each system has a zero/non-zero structure that is determined by a given directed graph. Within this setup, we investigate under which conditions all systems in this family are controllable, a property referred to as strong structural controllability. Moreover, we are interested in conditions for strong structural output controllability. We will show that the graph-theoretic concept of zero forcing is instrumental in these problems. In particular, as our first contribution, we prove necessary and sufficient conditions for strong structural controllability in terms of so-called zero forcing sets. Second, we show that zero forcing sets can also be used to state both a necessary and a sufficient condition for strong structural output controllability. In addition to these main results, we include interesting results on the controllability of subfamilies of systems and on the problem of leader selection.</p