Continuous-time quantized consensus: convergence of Krasowskii solutions

TL;DR

This paper analyzes the convergence behavior of continuous-time quantized consensus algorithms in directed networks with time-varying topology, using Krasovskii solutions to handle discontinuities and establishing conditions for finite-time consensus.

Contribution

It introduces a framework for analyzing Krasovskii solutions in quantized consensus, providing convergence conditions and bounds on convergence time for time-varying directed networks.

Findings

Krasovskii solutions reach consensus if the limit graph has a globally reachable node.

Convergence time is bounded when the topology is time-invariant.

Solutions can have very large convergence times on discontinuity surfaces.

Abstract

This note studies a network of agents having continuous-time dynamics with quantized interactions and time-varying directed topology. Due to the discontinuity of the dynamics, solutions of the resulting ODE system are intended in the sense of Krasovskii. A limit connectivity graph is defined, which encodes persistent interactions between nodes: if such graph has a globally reachable node, Krasovskii solutions reach consensus (up to the quantizer precision) after a finite time. Under the additional assumption of a time-invariant topology, the convergence time is upper bounded by a quantity which depends on the network size and the quantizer precision. It is observed that the convergence time can be very large for solutions which stay on a discontinuity surface.