发表机构
Cornell University; The University of Sydney（康奈尔大学; 悉尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出弱不变性这一放宽的对称性概念，将非对称残差由另一控制系统捕获，证明其推广经典对称性与群仿射系统，并可用于降维，以重力下飞行器为例展示九维弱对称性。
AI 中文摘要
对称性（或不变性）是一种强大的结构性质，能够为估计和控制提供高效、有效的解决方案。然而，经典不变性对系统动力学所施加的约束使得对称性成为一种非常刚性的性质，这种性质可能被外力破坏，或者仅限于整个系统中的某一部分。为了寻求更大的灵活性，本工作引入了一种新颖的、放宽的对称性概念，称为“弱不变性”，其中动力学的非对称部分（“残差”）可以完全由另一个在对称群上演化的控制系统来捕获。弱不变系统在严格意义上比经典不变系统更具一般性，但它们仍然享有许多类似的有利性质。特别地，我们证明了任何弱不变系统都允许一个级联分解，其中被驱动的子系统是群仿射的，这表明弱对称性不仅推广了经典对称性，还推广了（迄今为止不同的）群仿射系统类。我们还表明，具有自主残差的弱对称性可以从系统的误差动力学中分解出来，从而与经典对称性相比，能够实现更大程度的维度缩减。最后，我们研究了重力影响下的飞行器示例，为此我们提出了一个九维的弱对称性（严格包含该系统熟悉的四维经典对称性）。因此，弱不变性推广了经典对称性，同时保留了关键的结构性质，从而为更灵活的对称性信息控制方法奠定了基础。
英文摘要
Symmetry (or invariance) is a powerful structural property that enables efficient, effective solutions for estimation and control. However, the constraints imposed on a system's dynamics by classical invariance make symmetry a very rigid property, which may be broken by external forces or confined to only a portion of the overall system. Seeking greater flexibility, this work introduces a novel relaxed notion of symmetry, termed ``weak invariance'', in which the non-symmetric part of the dynamics (the ``residual'') can be captured entirely by another control system evolving on the symmetry group. Weakly invariant systems are strictly more general than classical invariant systems, but they nonetheless enjoy many similar favorable properties. In particular, we prove that any weakly invariant system admits a cascade decomposition in which the driven subsystem is group affine, showing that weak symmetry generalizes not only classical symmetry, but also the (thus far distinct) class of group affine systems. We also show that a weak symmetry with autonomous residual can be factored out of the system's error dynamics, enabling yet a greater reduction of dimensionality as compared to classical symmetries. Finally, we study the example of an aerial vehicle under the influence of gravity, for which we propose a nine-dimensional weak symmetry (strictly containing the system's familiar four-dimensional classical symmetry). Weak invariance thus generalizes classical symmetry while also preserving key structural properties, thereby laying a foundation for more flexible methods of symmetry-informed control.
Comments15 pages, 5 figures
