Limits on the network sensitivity function for homogeneous multi-agent systems on a graph

This paper explores some tradeoffs and limits of performance in feedback control of interconnected multi-agent systems, focused on the network sensitivity functions. We consider the interaction topology described by a directed graph and we prove that the sensitivity transfer functions between every pair of agents, arbitrarily connected, can be derived using a version of Mason´s direct rule. An analysis of the role of cycles shows that these structures influence and considerably limit the behavior of the system. We also derive a networked version of Bode´s integral formula, showing that it still holds for multi-agent systems.