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arXiv 2209.00345eess.SYcs.SY
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具有双不变度量的李群上一致性与同步性的统一框架

A Unified Framework for Consensus and Synchronization on Lie Groups admitting a Bi-Invariant Metric

Rama Seshan Chandrasekharan, Ravi N Banavar, Arun D Mahindrakar

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摘要：

对于在欧几里得空间上演化并通过连通图相互链接的有限数量的代理，基于代理间错误的拉普拉斯流确保了一阶和二阶动态的共识或同步。当这样的代理演变成一个圆圈（仓本振荡器），流，取决于正弦的代理间的误差角概括相同。在这项工作中，它表明，拉普拉斯流和仓本振荡器是一个更一般的理论的李群，承认双不变度量的共识的特殊情况。这样的理论不仅使这些共识和同步算法的推广李群，但也提供了洞察抽象的群论和微分几何性质，确保在欧几里德空间和圆收敛。

英文摘要：

For a finite number of agents evolving on a Euclidean space and linked to each other by a connected graph, the Laplacian flow that is based on the inter-agent errors, ensures consensus or synchronization for both first and second-order dynamics. When such agents evolve on a circle (the Kuramoto oscillator), the flow that depends on the sinusoid of the inter-agent error angles generalizes the same. In this work, it is shown that the Laplacian flow and the Kuramoto oscillator are special cases of a more general theory of consensus on Lie groups that admit bi-invariant metrics. Such a theory not only enables generalization of these consensus and synchronization algorithms to Lie groups but also provide insight on to the abstract group theoretic and differential geometric properties that ensures convergence in Euclidean space and the circle.

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