Razumikhin-type ISS Lyapunov function and small gain theorem for discrete time time-delay systems with application to a biased min-consensus protocol

TL;DR

This paper develops a Razumikhin-type Lyapunov function-based small gain theorem for discrete time delay systems, providing necessary and sufficient conditions for stability, and applies it to analyze the robustness of a biased min-consensus protocol in networks.

Contribution

It introduces a novel Razumikhin-type Lyapunov function approach that simplifies stability analysis and offers necessary and sufficient conditions for exponential input-to-state stability in discrete delay systems.

Findings

Established necessary and sufficient stability conditions using Razumikhin functions.

Proved exponential input-to-state stability of the biased min-consensus protocol under network perturbations.

Validated theoretical results through simulations.

Abstract

This paper considers small gain theorems for the global asymptotic and exponential input-to-state stability for discrete time time-delay systems using Razumikhin-type Lyapunov function. Among other things, unlike the existing literature, it provides both necessary and sufficient conditions for exponential input-to-state stability in terms of the Razumikhin-type Lyapunov function and the small gain theorem. Previous necessary ad sufficient conditions were with the more computationally onerous, Krasovskii-type Lyapunov functions. The result finds application in the robust stability analysis of a graph-based distributed algorithm, namely, the biased min-consensus protocol, which can be used to compute the length of the shortest path from each node to its nearest source in a graph. We consider the biased min-consensus protocol under perturbations that are common in communication networks, including noise, delay and asynchronous communication. By converting such a perturbed protocol into a discrete time time-delay nonlinear system, we prove its exponential input-to-state stability under perturbations using our Razumikhin-type Lyapunov-based small gain theorem. Simulations are provided to verify the theoretical results.