A Classification of Nodes for Structural Controllability

In this paper we consider (large and complex) interconnected networks. We assume that each state/node of the network can be selected to act as a steering node, meaning that such node then is influenced by its own individual control. This control may influence other nodes indirectly through steering node which it is controlling. The goal in this paper is to select steering nodes such that the overall system (in)directly becomes structurally controllable, where " structurally " means that the system is controllable for almost all of its numerical realizations. Obviously, it may happen that not every state needs to act as a steering node. In fact, in configurations where the overall system is required to be structurally controllable, some states may never need to be steering node, while some other states always have to be a steering node. In this paper, we aim to achieve structural controllability and we present a classification of the associated steering nodes as being essential (always required to be present), useful (present in certain configurations) and useless (never necessary in whatever configuration). The classification is based on two types of decomposition that naturally show up in the context of the two conditions (connection condition and rank condition) for structural con-trollability. The underlying methods are related to well-known and efficient network algorithms. The main result of the paper is the characterization of useless, useful and essential steering nodes in order to obtain a system that is structurally controlable. The results are illustrated by means of examples.