Sufficient conditions on observability grammian for synchronization in arrays of coupled time-varying linear systems

TL;DR

This paper establishes conditions based on the observability grammian that guarantee exponential or asymptotic synchronization in arrays of coupled, time-varying linear systems, applicable to both continuous and discrete time.

Contribution

It introduces new persistence of excitation conditions on the observability grammian ensuring synchronization, including a weaker condition for asymptotic synchronization, without assuming strong coupling.

Findings

Persistence of excitation guarantees exponential synchronization.

Weaker grammian conditions ensure asymptotic synchronization.

Bounded trajectories in integrator arrays with output matrices are shown.

Abstract

Synchronizability of stable, output-coupled, identical, time-varying linear systems is studied. It is shown that if the observability grammian satisfies a persistence of excitation condition, then there exists a bounded, time-varying linear feedback law that yields exponential synchronization for all fixed, asymmetrical interconnections with connected graphs. Also, a weaker condition on the grammian is given for asymptotic synchronization. No assumption is made on the strength of coupling. Moreover, related to the main problem, a particular array of output-coupled systems that is pertinent to much-studied consensus problems is investigated. In this array, the individual systems are integrators with identical, time-varying, symmetric positive semi-definite output matrices. Trajectories of this array are shown to stay bounded using a time-invariant, quadratic Lyapunov function. Also, sufficient conditions on output matrix for synchronization are provided. All of the results in the paper are generated for both continuous time and discrete time.