Leader–Follower Formation of Second-Order Agents via Delayed Relative Displacement Feedback

This letter investigates a leader-follower formation of a group of second-order agents, considering an undirected and connected topology. It is assumed that no velocity information is available, and that a uniform delay affects the processing of the displacement information. To address these issues, a delayed relative displacement feedback is introduced. A necessary and sufficient condition for stability is derived in terms of a delay threshold, given that the delay-free controller is stable. Moreover, a stability region in the complex plane for the eigenvalues of the Laplacian interaction matrix is introduced. The results are illustrated through numerical examples.