Delay-induced discrete-time consensus

Distributed coordination algorithm (DCA) is a classic algorithm for information fusion via local interactions, which has been widely applied in distributed computing, sensor networks, and mobile robots. Consensus is a typical behavior of DCA, which implies the convergence to some identical value for all the states. It is well known that under certain conditions DCA generates consensus with exponential rate. Though it has been proved in existing literature that time-delays trigger consensus in continuous-time DCA, it is still unknown how to trigger consensus via time-delays in discrete-time DCA. This paper introduces time-delays to discrete-time DCA which originally cannot realize consensus, and proves that if the introduced delay is co-prime with the period of the communication topology, then consensus happens. Specifically, we found that there does not exist a critical threshold of time-delay for consensus in discrete-time DCA. The proposed method in this paper is totally a graphical method, which does not rely on the analysis of eigenvalues.