Distributed PI Control for Consensus of Multi-agent Systems with Time-delay Under Directed Topology

This paper considers the consensus problem of multi-agent systems with time delay under directed topology. A novel proportional-integral (PI) control protocol with a time delay is proposed for each agent to ensure the consensus. The problem of consensus is converted to a stability problem by system transformation. Sufficient conditions are obtained based on the proposed Lyapunov functional candidate using Lyapunov stability theory, graph theory and matrix theory. The proposed condition has less conservatism by adding a triple-integral term. Finally, numerical simulations are provided to illustrate the effectiveness of the results.