Controllability Degree of Directed Line Networks: Nodal Energy and Asymptotic Bounds
Shiyu Zhao, Fabio Pasqualetti · 2018
This paper studies the controllability degree of dynamical networks with continuous-time dynamics. To quantify the controllability degree of a network, we introduce a new notion termed nodal energy, which is the control energy required to change the state of a single node while keeping the final states of the other nodes unchanged. Since it is extremely challenging to analyze the nodal energy of general networks, this paper focuses on a special class of directed line networks with single control nodes. This choice allows us to derive the explicit expression of the nodal energy of different network nodes, and hence, reveal novel controllability properties of complex networks. Our analysis shows that (i) differently from their discrete-time counterpart, directed line networks with continuous-time dynamics are always difficult to control, as the control energy grows linearly or even exponentially with the network cardinality, (ii) the numerical investigation of the controllability degree of line networks is unreliable, because the condition number of the controllability Gramian grows exponentially fast as the network cardinality increases, and (iii) nodal energies may be inversely related to the graphical distance from the control node because, depending on the network weights, distant nodes may require far less nodal energy than immediate neighbors.