Consensus Robustness Margins of First-Order Agents Over Undirected Graphs

This article 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 by PI protocols within these ranges. Our main results consist of explicit analytical expressions of the GCM and PCM achievable by PI protocols, which demonstrate how the agent's unstable pole and nonminimum phase zero, as well as the network connectivity may confine the gain and phase variation ranges so that consensus can or cannot be maintained.