A Study on Rate of Convergence Increase Due to Time Delay for a Class of Linear Systems

This paper characterizes fully how time delay affects the rate of convergence of a class of linear time-delayed systems. Contrary to the prevailing intuition that links time delay with system sluggishness, we show that for specific ranges of time delay, faster response can be achieved in the presence of delay. Specifically, we determine exactly for what values of delay the rate of convergence of our system of interest increases with delay. We also prove that the ultimate bound on the maximum achievable rate of convergence via time delay is e (Euler's number) times the delay free rate. We demonstrate our results by studying the convergence rate of the Laplacian static average consensus algorithm in the presence of time delay.