Consensus of Second-Order Multi-Agent Systems Without a Spanning Tree: A Sequence-Based Topology-Dependent Method

This paper investigates the consensus of second-order multi-agent systems under switched topologies. Previous studies indicate that a consensus cannot be reached if the topology is fixed and has no spanning tree, but it is possible to reach a consensus for the multi-agent systems under switched topologies even if every topology has no spanning tree. However, in this paper, we show that some second-order multi-agent systems cannot reach a consensus even if the union of the directed interaction graphs has a spanning tree, and even if the union has a spanning tree frequently enough. It is often complex to judge whether the second-order multi-agent systems can reach a consensus or not under switched topologies if every topology has no spanning tree. This paper proposes a sequence-based topology-dependent method to determine whether a consensus can be reached in this circumstance. Our results are supported by examples and counterexamples.