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

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.