Average-consensus Problem of Multi-agent Systems with Non-uniform and Asymmetric Time-delays

Abstract

This paper deals with consensus problems of networked multi-agent systems with non-uniform and asymmetric time-delays. It has been known that multi-agent systems without time-delays achieve average-consensus, which means that all the states of agents converge to the average of their initial states. However, our target systems do not necessarily achieve average-consensus because of the non-uniform and asymmetric time-delays on the networks. The goal of this paper is to derive the necessary and sufficient condition of the network structures under which these systems achieve average-consensus. First, we derive a conservative quantity and calculate a consensus value, which all the agents converge to. We show that the consensus value depends not only on the structures of the time-delay networks and the initial states of the agents, but also on the initial functions, which define the initial conditions for the time-delay systems. Then, the necessary and sufficient condition for average-consensus is derived, which is represented by the eigenvectors of the delay-transition and the delay-weight Laplacians. These Laplacians are newly introduced in this paper to express the structures of the time-delay networks. Finally, the validity of the analysis results are demonstrated by simulations with six-agent systems.