从一般组合学视角看编队控制：边动力学与有向感知
Formation control from the generic combinatorial viewpoint: edge dynamics and directed sensing
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中文总结 AI 辅助
本文提出基于距离的编队控制几何框架，通过边动力学与谱条件判定局部指数稳定性，并给出有向感知下的充要条件及半定规划综合方法。
中文摘要 AI 辅助
我们建立了一个基于距离的编队控制的几何框架，该框架将智能体间距离的演化与其通过相容节点运动实现的过程分离开来，从而将稳定性问题归结到边空间上。我们证明了边动力学的局部指数收敛蕴含编队的局部指数收敛，并且稳定性由线性边算子的谱性质所保证。我们引入了一组一般谱性质的分层结构——弱容许性、容许性和强容许性——它们为局部指数稳定性提供了必要条件。专门针对有向感知情形，我们得到了在任意目标处局部稳定的一个充要谱条件，以及一个二次充分性证书。这些条件揭示出稳定性同时依赖于图的定向和目标几何，并表明持久性既不是局部收敛的必要条件也不是充分条件。我们证明了每个一般刚性图都容许一个容许定向，并且对于无环定向，我们给出了容许性的精确组合刻画。最后，该二次证书导出了一个用于综合稳定化边增益的半定规划。
英文摘要
We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties --- weak admissibility, admissibility, and strong admissibility --- that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.
发表机构
- School of Mathematics and Statistics, University of St Andrews（圣安德鲁斯大学数学与统计学院）
- Stephen B. Klein Faculty of Aerospace Engineering, Technion-Israel Institute of Technology（以色列理工学院斯蒂芬·B·克莱恩航空航天工程学院）
- Numerical Analysis and Applied Mathematics (NUMA) unit at KU Leuven（荷语天主教鲁汶大学数值分析与应用数学（NUMA）单元）
- School of Mathematical Sciences, Lancaster University（兰卡斯特大学数学科学学院）
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