Propagation Stability Concepts for Network Synchronization Processes

TL;DR

This paper introduces a new concept of disturbance propagation stability for network synchronization, linking it to frequency response and gain conditions, with extensions to subnetworks and planar systems.

Contribution

It develops a novel stability notion for network processes, characterizes it via frequency response, and simplifies it for SISO subsystems, extending to subnetwork robustness.

Findings

Propagation stability is equivalent to a local closed-loop gain being less than one.

For SISO subsystems, stability can be checked using Nyquist plots.

The concept extends to subnetworks, showing robustness to disturbances.

Abstract

A notion of disturbance propagation stability is defined for dynamical network processes, in terms of decrescence of an input-output energy metric along cutsets away from the disturbance source. A characterization of the disturbance propagation notion is developed for a canonical model for synchronization of linearly-coupled homogeneous subsystems. Specifically, propagation stability is equivalenced with the frequency response of a certain local closed-loop model, which is defined from the subsystem model and local network connections, being sub-unity gain. For the case where the subsystem is single-input single-output (SISO), a further simplification in terms of the subsystem's open loop Nyquist plot is obtained. An extension of the disturbance propagation stability concept toward imperviousness of subnetworks to disturbances is briefly developed, and an example focused on networks with planar subsystems is considered.