Transfer function representation of cyclic consensus systems

This paper presents transfer functions of a class of cyclic consensus system in closed forms. Four different kinds of network topologies; directed, undirected, one, two or full reference agents, are considered. Each agent of consensus systems is assumed to satisfy a first-order scalar dynamics which is driven by a common consensus protocol and an independent exogenous input. It is shown that every SISO transfer function between the exogenous input of one agent and the state of another generally different agent, is always minimum phase. In addition, poles and zeros, system degrees and relative degrees of such SISO transfer functions are specified. These results are interpretated in relation to the controllability and closed loop performance of a networked system having one leader agent. Furthermore, our transfer function representations are applied to an investigation of stability margins for a closed loop cyclic consensus system.