Consensus in networks of anticipatory agents under transmission delays

Abstract

This thesis examines the dynamics of a coupled system of linear delay differential equations, addressing the normalized consensus problem on undirected and connected graphs of anticipatory agents in the presence of a fixed information transmission delay. The anticipation rule, a first-order linear extrapolation, enables agents to predict the present states of their neighbours using past information, thereby introducing an additional delayed term into the formulation and resulting in a system of delay differential equations with two discrete delays. The main result of this study is the necessary and sufficient condition for the anticipatory consensus protocol under transmission delays to reach consensus. Simulations indicate that the convergence rate of the anticipatory protocol is superior to that of the protocol without anticipatory agents, both under transmission delays. As a natural extension, the findings are applied to the Kuramoto model of coupled phase oscillators to determine the local stability of synchronized states. It is demonstrated that the delay margin for achieving local stability is inversely proportional to the coupling strength between agents. Furthermore, it is shown that the synchronized frequency of the extended model remains the same as that of the original Kuramoto model, contrasting with other extended versions that involve single delays.