On leader-follower consensus with switching topologies : an analysis inspired by pigeon hierarchies

Inspired by the interesting findings regarding the hierarchy and coordination of pigeons in Nagy et al. [Nature, 2010], we revisit the discrete-time leader-follower consensus problem with switching topologies. The purpose of this paper is to investigate the impact of hierarchical topologies and followers' self-loops on the convergence performance of leader-follower consensus, including the convergence rate and robustness to switching topologies. We first study the fixed topology case, and show that the followers converge to the leader's state in finite time if and only if each follower has no self-loops and the topology is hierarchical. However, we show via counterexample, that leader-follower consensus may not be achieved when some followers have no self-loops and the topology is switching; even if each interaction graph has a spanning tree rooted at the leader. With the aid of binary relation theory, we further develop a new approach to present a novel sufficient condition for leader-follower consensus with switching topologies and with some followers having no self-loops. We prove that when no followers have self-loops and the same hierarchical organization is kept under the switching topologies, then the fastest rate of convergence in leader-follower consensus can be achieved; even in the presence of complex dynamic topologies. This is consistent with the natural phenomena found in pigeons by Nagy et al.