Consensus on Intervals of Communication Delay

This paper brings out a structured methodology for identifying intervals of communication time-delay where consensus in directed networks of multiple agents with high-order integrator dynamics is achieved. It is built upon the stability analysis of a transformed consensus problem which preserves all the nonzero eigenvalues of the Laplacian matrix of the associated communication topology graph. It is shown that networks of agents with first-order integrator dynamics can be brought to consensus independently of communication delay, on the other hand, for agents with second-order integrator dynamics, the consensus is achieved independently of communication delay only if certain conditions are satisfied. Conversely, if such conditions are not satisfied, it is shown how to compute the intervals of communication delay where multiple agents with second-order or higher-order can be brought to consensus. The paper is ended by showing an interesting example of a network of agents with second-order integrator dynamics which is consensable on the first time-delay interval, but as the time-delay increases, it loses consensability on the second time-delay interval, then it becomes consensable again on the third time-delay interval, and finally it does not achieve consensus any more on the fourth time-delay interval. This example shows the importance of analyzing consensus with time-delay in different intervals.