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

## TL;DR

This paper investigates how time delay can unexpectedly enhance the stability margin and convergence rate of linear systems, providing exact conditions and delays for optimal performance using Lambert W functions.

## Contribution

It introduces a precise analytical framework using Lambert W functions to identify when and how delay improves convergence in linear systems.

## Key findings

- Delay can increase the convergence rate in certain linear systems.
- Exact delay ranges for enhanced convergence are derived.
- Maximum convergence rate and optimal delay are explicitly calculated.

## Abstract

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