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This paper is a comprehensive study of a long observed phenomenon of increasein the stability margin and so the rate of convergence of a class of linearsystems due to time delay. We use Lambert W function to determine (a) in whatsystems the delay can lead to increase in the rate of convergence, (b) theexact range of time delay for which the rate of convergence is greater thanthat of the delay free system, and (c) an estimate on the value of the delaythat leads to the maximum rate of convergence. For the special case when thesystem matrix eigenvalues are all negative real numbers, we expand our resultsto show that the rate of convergence in the presence of delay depends only onthe eigenvalues with minimum and maximum real parts. Moreover, we determine theexact value of the maximum rate of convergence and the corresponding maximizingtime delay. We demonstrate our results through a numerical example on thepractical application in accelerating an agreement algorithm fornetworked~systems by use of a delayed feedback.

On the Positive Effect of Delay on the Rate of Convergence of a Class of Linear Time-Delayed Systems