标题： 不变控制系统的商 显示英文标题

标题： Quotients of Invariant Control Systems

摘要： 在以前的工作中表明，如果流形$M$上的控制系统$\mathcal{V}\subset TM$拥有控制对称性群$G$，则它通常拥有群商（或对称性约化）$\mathcal{V}/G$，这些商是静态反馈线性化的（SFL）。 反过来，这可以用于系统地构造控制系统$\mathcal{V}$的动态反馈线性化；或者当不存在动态反馈线性化时，构造部分反馈线性化。 由于这些及相关应用，本文对控制系统的对称性约化进行了详细研究。 我们表明，在控制系统的对称性约化中涉及的一个关键性质是对称群作用的横截性。 通过推广这一概念，我们分析了不变分布，特别是控制系统的几何结构如何由于对称性约化而改变。 这为理解SFL商的意外频繁出现提供了重要信息。 具体而言，我们详细说明了商对象的各种规范子丛的可积性属性和秩与给定分布的差异。 因此，我们能够根据 $\mathcal{V}$的几何性质和作用 $G$对 $G$-不变控制系统的 $\mathcal{V}$商进行分类。 此外，我们证明了静态反馈线性化在对称约简下是保持的，控制系统的静态反馈线性化的著名 Sluis-Gardner-Shadwick (S-G-S) 检验也被扩展到 Goursat 纤丛。 还给出了一种广义的 S-G-S 检验来确定 $\mathcal{V}/G$是否为 Goursat 纤丛，该检验基于 $G$的李代数。 最后，我们将所有结果应用于著名的 PVTOL 控制系统，并证明它可被视为某个 9 维李群上的不变控制系统。 显示英文摘要

摘要： In previous work it was shown that if a control system $\mathcal{V}\subset TM$ on manifold $M$ has a control symmetry group $G$ then it very often has group quotients (or symmetry reductions) $\mathcal{V}/G$ which are static feedback linearizable (SFL). This, in turn, can be applied to systematically construct dynamic feedback linearizations of $\mathcal{V}$; or to construct partial feedback linearizations, when no dynamic feedback linearization exists. Because of these and related applications, this paper makes a detailed study of symmetry reduction for control systems. We show that a key property involved in the symmetry reduction of control systems is that of transversality of Lie group actions. Generalizing this notion, we provide an analysis of how the geometry of an invariant distribution, and particularly a control system, is altered as a consequence of symmetry reduction. This provides important information toward understanding the unexpectedly frequent occurrence of SFL quotients. Specifically, we detail how the integrability properties and ranks of various canonical sub-bundles of the quotient object differ from those of the given distribution. As a consequence we are able to classify the SFL quotients of $G$-invariant control systems $\mathcal{V}$ based upon the geometric properties of $\mathcal{V}$ and the action $G$. Additionally, we prove that static feedback linearizability is preserved by symmetry reduction and the well-known Sluis-Gardner-Shadwick (S-G-S) test for the static feedback linearization of control systems is extended to Goursat bundles. A generalized S-G-S test to identify if $\mathcal{V}/G$ is a Goursat bundle is also given, based on the Lie algebra of $G$. Finally, we apply all our results to the well-known PVTOL control system and demonstrate that it may be viewed as an invariant control system on a certain 9-dimensional Lie group.