Leader-Follower Matrix-Weighted Consensus with Time-Delays

This paper investigates the convergence conditions of leader-follower matrix-weighted consensus networks in the presence of constant time delays. Several delayed consensus algorithms for networks of single-integrator agents, relying solely on relative state information, are analyzed. Convergence to a matrix-weighted leader-follower consensus is established using eigenvalue-based criteria and the Lyapunov-Krasovskii method. It is shown that, in most cases, when the leaders have non-identical initial states, the followers asymptotically converge to distinct clusters, which may lie outside the convex hull of the leaders' initial states. The theoretical results are supported with numerical simulations.