Dynamic Consensus Networks: Dynamic Graph Definitions and Controllability Analysis Using the Behavioral Approach

Abstract

The focus of this paper is to study a generalization of consensus problems whereby the weights of network edges are no longer modeled as static gains, but instead are represented as dynamic systems coupling the nodes, which might also be more general than an integrator, leading to the notion of dynamic consensus networks. We transform each concept of static graph theory into dynamic terms, out of which a generalized dynamic graph theory naturally emerges. We present a framework for dynamic graphs and dynamic consensus networks. This framework introduces the idea of dynamic degree, adjacency, incident, and Laplacian matrices in a way that naturally extends these concepts from the static case. We consider controllability for dynamic consensus networks. The ideas developed for dynamic graph theory, in conjunction with the behavioral approach, lead to the development of a controllability analysis methodology for dynamic consensus networks. The controllability conditions obtained using the behavioral approach cannot be applied for the general dynamic networks, such as identical LTI nodes with dynamic edges or even in the more general case with heterogeneous nodes. This is because of scalability in such dynamic networks. Thus, we develop controllability conditions based on node and interconnection (edge) parameters that guarantee controllability of the overall dynamic network.