Consensus Robustness under PI Protocols with Undirected Graphs

This paper concerns robust consensus problems for continuous-time first-order multi-agent systems with respect to uncertain gain and phase of the agent’s frequency response varying within certain ranges. We consider dynamic output feedback control protocol in the form of proportional-integral (PI) control. The agents can in general be nonminimum phase and unstable systems, which are interconnected by an undirected graph network. We seek to determine the largest ranges of gain and phase variations, referred to as the gain consensus margin (GCM) and the phase consensus margin (PCM), respectively, so that consensus can be achieved robustly within these ranges. Our main results consist of explicit analytical expressions of the GCM and PCM, which demonstrate how the agent’s unstable pole and nonminimum phase zero, as well as the network connectivity may fundamentally confine the gain and phase variation ranges so that consensus can or cannot be maintained.