Hybrid protocols for leader-follower consensus of multi-agent systems with distributed delays

The primary focus of this paper is to investigate the leader-follower consensus problem of multi-agent systems (MASs) with discrete and distributed delays in complex domains. We propose a new hybrid consensus protocol that incorporates a continuous-time protocol based on the communication topology of follower agents, along with an event-triggered pinning impulsive control (ETPIC) protocol. Using the Lyapunov functional method in complex domains, we establish delay-dependent sufficient conditions for leader-follower consensus of delayed complex-valued MASs. Our results demonstrate that the proposed hybrid protocol can ensure leader-follower consensus even when the size of discrete and distributed delays exceeds the length of intervals between two consecutive triggering instants. Furthermore, we prove that the Zeno phenomenon can be excluded under the proposed control protocol. In particular, as a special case, we derive the leader-follower consensus result for delay-free complex-valued MASs based on a reduced hybrid control protocol. Two numerical examples are presented to validate the effectiveness of the proposed control scheme.