Conditions for synchronizability in arrays of coupled linear systems

TL;DR

This paper establishes new conditions under which arrays of coupled linear systems can be synchronized using linear feedback, highlighting differences between neutrally stable and critically unstable systems.

Contribution

It introduces sufficient conditions and an algorithm for designing feedback laws that achieve synchronization in coupled linear systems.

Findings

Neutrally stable systems with detectable outputs can synchronize with fixed connected graphs.

Detectability of outputs is insufficient for critically unstable systems to synchronize.

Full-state coupling guarantees synchronization for all connected configurations in unstable systems.

Abstract

Synchronization control in arrays of identical output-coupled continuous-time linear systems is studied. Sufficiency of new conditions for the existence of a synchronizing feedback law are analyzed. It is shown that for neutrally stable systems that are detectable form their outputs, a linear feedback law exists under which any number of coupled systems synchronize provided that the (directed, weighted) graph describing the interconnection is fixed and connected. An algorithm generating one such feedback law is presented. It is also shown that for critically unstable systems detectability is not sufficient, whereas full-state coupling is, for the existence of a linear feedback law that is synchronizing for all connected coupling configurations.