Using the equitable partition technique to determine the non-controllability of dynamic networks

This paper considers the controllability of dynamic networks. Networks consist of a set of dynamic agents and links that describe the interconnection behavior between agents. These links can be described either by static gains, which we call static gain networks, or they can be described by dynamic transfer functions in which case they are dynamic gain networks. We consider a subset of one or more agents that act as control inputs to regulate the other remaining agents. A network can be described by a graph consisting of nodes and weighted edges. In previous work, tests for controllability for static graphs have been developed, and the results of controllability have been obtained. These results are based on the Laplacian matrices of the graph and some other graphical tools. This paper will consider a large class of networks where the links are dynamic systems. The main contribution of this paper is to test the controllability, based on extending the equitable partition technique to a dynamic gain networks. This technique was previously used for static gain networks. Finally, we illustrate our results with a practical application in robotics.