Robust Synchronization of Directed Networks with Coprime Factor Perturbed Agent Dynamics

This paper deals with robust synchronization of directed and undirected multi-agent networks with uncertain agent dynamics. Given a network with identical nominal dynamics, we allow uncertainty in the form of coprime factor perturbations of the transfer matrix of the agent dynamics. These perturbations are assumed to be stable and have H∞norm that is bounded by an a priori given desired tolerance. We derive state space equations for dynamic observer based protocols that achieve robust synchronization in the presence of such uncertainty. We show that this robust synchronization of the network by the dynamic protocol is equivalent to robust stabilization of a single linear system by all controllers from a given set of feedback controllers. The synchronizing protocols are expressed in terms of real symmetric solutions to algebraic Riccati equations related to the nominal agent dynamics, and contain a weighting factor depending on the eigenvalues of the graph Laplacian. We obtain an achievable interval, i.e. an interval such that for each value of the tolerance contained in this interval there exists a robustly synchronizing protocol. For undirected networks, the supremum of this interval is proportional to the square root of the quotient of the smallest and the largest eigenvalue of the graph Laplacian.