Effects of adding arcs on the consensus convergence rate of leader-follower multi-agent systems

For a first-order leader-follower multi-agent system (MAS) with a directed graph as its interaction topology, the consensus convergence rate is determined by the algebraic connectivity (the smallest real part of the nonzero eigenvalues of the Laplacian matrix). Adding arcs to the followers is an effective approach to improve the consensus convergence rate of a leader-follower MAS. In this paper, the effects of adding arcs to the followers on the algebraic connectivity are investigated, when the followers’ interaction topology is a strongly connected directed graph. Our results include: (1) If arcs are added to the followers, then the algebraic connectivity increases if and only if the sum of entries of the Fiedler vector, corresponding to all the tails, is smaller than that of the heads; (2) For the case when a fixed number of arcs with a common head are added, the smaller the sum of entries of the Fiedler vector is, corresponding to all the tails, the larger the algebraic connectivity will be; (3) Each entry of the Fiedler vector, corresponding to the informed agents, is greater than that of the other types of followers; (4) If the Laplacian matrix of a leader-follower interaction topology can be divided into equal row sum blocks by layers, then the entries of the Fiedler vector in each layer are the same and the entries increase by layers. Thus the effects of adding arcs on algebraic connectivity can be determined by the layers of heads and tails of the arcs. Finally, the theoretical results are illustrated by numerical experiments.