Stability of continuous-time distributed consensus algorithms

TLDR

These systems naturally arise in distributed decision, coordination, rendezvous, and synchronization problems. The study aims to establish sufficient conditions for uniform exponential stability of continuous‑time linear time‑varying systems with Metzler matrices and zero row sums, ensuring convergence of all state components to a common value. The equilibrium set consists of states with identical components, and the analysis focuses on linear time‑varying systems with Metzler matrices and zero row sums. The sufficient conditions guarantee uniform exponential stability and convergence to a common value, and this result remains robust to arbitrary delays affecting only off‑diagonal terms.

Abstract

We study the stability properties of linear time-varying systems in continuous time whose system matrix is Metzler with zero row sums. This class of systems arises naturally in the context of distributed decision problems, coordination and rendezvous tasks and synchronization problems. The equilibrium set contains all states with identical state components. We present sufficient conditions guaranteeing uniform exponential stability of this equilibrium set, implying that all state components converge to a common value as time grows unbounded. Furthermore it is shown that this convergence result is robust with respect to an arbitrary delay, provided that the delay affects only the off-diagonal terms in the differential equation.