A Tight Lower Bound on the Controllability of Networks with Multiple Leaders

In this paper we study the controllability of networked systems with static network topologies using tools from algebraic graph theory. Each agent in the network acts in a decentralized fashion by updating its state in accordance with a nearest-neighbor averaging rule, known as the consensus dynamics. In order to control the system, external control inputs are injected into the so called leader nodes, and the influence is propagated throughout the network. Our main result is a tight topological lower bound on the rank of the controllability matrix for such systems with arbitrary network topologies and possibly multiple leaders

Controllability Gramian spectra of random networks

We propose a theoretical framework to study the eigenvalue spectra of the controllability Gramian of systems with random state matrices, such as networked systems with a random graph structure. Using random matrix theory, we provide expressions for the moments of the eigenvalue distribution of the controllability Gramian. These moments can then be used to derive useful properties of the eigenvalue distribution of the Gramian (in some cases, even closed-form expressions for the distribution). We illustrate this framework by considering system matrices derived from common random graph and matrix ensembles, such as the Wigner ensemble, the Gaussian Orthogonal Ensemble (GOE), and random regular graphs. Subsequently, we illustrate how the eigenvalue distribution of the Gramian can be used to draw conclusions about the energy required to control random system

On the controllability of networks with nonidentical linear nodes

"The controllability of dynamical networks depends on both network structure and node dynamics. For networks of linearly coupled linear dynamical systems the controllability of the network can be determined using the well-known Kalman rank criterion. In the case of identical nodes the problem can be decomposed in local and structural contributions. However, for strictly different nodes an alternative approach is needed. We decomposed the controllability matrix into a structural component, which only depends on the networks structure and a dynamical component which includes the dynamical description of the nodes in the network. Using this approach we show that controllability of dynamical networks with strictly different linear nodes is dominated by the dynamical component. Therefore even a structurally uncontrollable network of different nn dimensional nodes becomes controllable if the dynamics of its nodes are properly chosen. Conversely, a structurally controllable network becomes uncontrollable for a given choice of the node’s dynamics. Furthermore, as nodes are not identical, we can have nodes that are uncontrollable in isolation, while the entire network is controllable, in this sense the node’s controllability is overwritten by the network even if the structure is uncontrollable. We illustrate our results using single-controller networks and extend our findings to conventional networks with large number of nodes.

Strong Structural Controllability of Systems on Colored Graphs

This paper deals with structural controllability of leader-follower networks. The system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero, while the remaining entries take nonzero values. The network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix, the system is controllable in the classical sense. In this paper we introduce a more general notion of strong structural controllability which deals with the situation that given nonzero entries in the system's pattern matrix are constrained to take identical nonzero values. The constraint of identical nonzero entries can be caused by symmetry considerations or physical constraints on the network. The aim of this paper is to establish graph theoretic conditions for this more general property of strong structural controllability.Comment: 13 page