Equivalent Conditions for the Synchronization of Identical Linear Systems over Arbitrary Interconnections

We propose necessary and sufficient conditions for the synchronization of $N$ identical single-input-single-output (SISO) systems, connected through a directed graph {without imposing any assumption on the graph interconnection}. We consider both the continuous-time and the discrete-time case, and we provide conditions that {are equivalent to} the uniform global exponential stability, {with guaranteed convergence rate,} of the closed {and unbounded} attractor that corresponds to the synchronization set.