LAUSR

Abstract A graph elegantly describes the connection topology in a networked system of autonomous agents and, dynamic stability is an essential consideration in the successful design of formations. In this paper, we explore the graph Laplacian and its role in the stability of relative error dynamics of formations. We then use the graph Laplacian to investigate the dynamic stability in the presence of reciprocal connections between agents. Restoring instabilities can be challenging in multi-agent systems; we introduce the adaptive-key-component controller as a practical means to restore stability in graph-based networks. Throughout the paper, we use several examples to illustrate stability issues that can arise from connection geometries.