Consensus of Multi-Agent Systems with Unbounded Time-Varying Delays

In multi-agent systems with increasing communication distances, the communication delay is time-varying and unbounded. In this paper, we describe the multi-agent system with increasing communication distances as the discrete-time system with non-distributed unbounded time-varying delays and study the consensus problem of the system via the distributed control. This paper uses a time-delay system to model the discrete-time system, and the maximum delay in the time-delay system tends to infinity as time goes on. Furthermore, caused by this property, most of convergence analysis methods for bounded time-delay systems are ineffective. Hence, for any finite integer k>0, the finite-dimensional augmented model of the time-delay system is built in the interval [0,k] to study the system state. Under the weaker topological assumption that the topology containing a spanning tree, the system is proved to achieve a consensus if the growth rate of the maximum delay satisfies some mild constraints, which also are constraints on the growth rate of the maximum communication distance between agents. Furthermore, we characterize that the rate of the system achieving a consensus and the growth rate of the maximum delay are negatively correlated. In other words, the rate of the system achieving a consensus and the growth rate of the maximum communication distance between agents are negatively correlated.