Global Coordinated Stabilization of Multiple Simple Mechanical Control Systems on a Class of Lie Groups

This article investigates the problem of global coordinated stabilization for a multiagent system, which consists of a general class of mechanical systems that evolve on compact connected Lie groups. First, a distributed synergistic hybrid controller is synthesized by leveraging a coadjoint incidence matrix to assign to neighboring agents a hybrid feedback that is based on their relative position. It leads to robust and global asymptotic stability in tree networks. Second, the existence of synergistic potential functions (SPFs)—the key ingredient for deriving the hybrid feedback—is established on compact Lie groups. Moreover, a direct extension shows that there exists an SPF on the noncompact special Euclidean group <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{SE}(n)$</tex-math></inline-formula>. In contrast with some existing results, the proposed controller removes the invariance condition on the hybrid feedback, and the existence of the SPF ensures that our controller is applicable to many Lie groups of practical interests. Finally, a worked example of multiple planar rigid-body systems on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{SE} (2)$</tex-math></inline-formula> is used to illustrate our approach.