Modal strong structural controllability for networks with dynamical nodes

In this article, a new notion of modal strong structural controllability is introduced and examined for a family of LTI networks. These networks include structured LTI subsystems, whose system matrices have the same zero/nonzero/arbitrary pattern. An eigenvalue associated with a system matrix is controllable if it can be directly influenced by the control inputs. We consider an arbitrary set \Delta, and we refer to a network as modal strongly structurally controllable with respect to \Delta if, for all systems in a specific family of LTI networks, every \lambda\in\Delta is a controllable eigenvalue. For this family of LTI networks, not only is the zero/nonzero/arbitrary pattern of system matrices available, but also for a given \Delta, there might be extra information about the intersection of the spectrum associated with some subsystems and \Delta. Given a set \Delta, we first define a \Delta-network graph, and by introducing a coloring process of this graph, we establish a correspondence between the set of control subsystems and the so-called zero forcing sets. We also demonstrate how with \Delta={0} or \Delta=C\{0}, existing results on strong structural controllability can be derived through our approach. Compared to relevant literature, a more restricted family of LTI networks is considered in this work, and then, the derived condition is less conservative.