  This study deals with analysis and control of cooperative synchronization for
identical agents interacting on a directed graph topology. The agents are
considered to have general continuous linear time-invariant dynamics with
homogeneous communication and/or control delays. An LMI approach based on a
Lyapunov-Krasovskii functional is proposed, together with the synchronizing
region concept, which decouples the single-agent dynamics from the detailed
graph topology. Moreover, the conventional notion of synchronizing region is
here extended by an LMI relaxation utilizing quasi-convex characteristic of the
problem. This leads to less conservative results for the region of graph matrix
eigenvalues in the complex domain, where the synchronization is guaranteed. The
proposed method to calculate the allowable delay bound for synchronization is
also less conservative as compared to the results from the literature.
Furthermore, two designs for distributed state-feedback control are suggested.
The precise delay value and the detailed graph topology need not be known for
their application; it suffices only to know the upper bound on the delay and
the approximate region where the Laplacian eigenvalues lie. Specific
improvements over the results existing in the literature are demonstrated by a
numerical example, which validates the proposed approaches.
