Algebraic Topology in Robust Control

鲁棒控制中的代数拓扑

• 批准号: 9113088
• 负责人: Edmond Jonckheere
• 金额: $ 10万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-08-15 至 1994-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9113088&HistoricalAwards=false
• 关键词: Algebraic; Topology; Robust; Control

The purpose of this proposal is to being together robust control and algebraic topology. It is argued that such robust control issues as structured singular values, multivariable phase margin, Kharitonov's theorem, etc. receive their natural and unifying formulation in the context of algebraic topology, as formulated by Poincare, Cartan, Eilenberg, and many others. The central mathematical issue is whether the mapping of the high dimensional manifold of structured uncertainties into the Nyquist template commutes with the boundary. While in the simple Kharitonov case it does, in the over-whelming majority of situations the Nyquist mapping does not commute with the boundary. However, the simplicial approximation theorem provides us with an approximate Nyquist map that does commute with the boundary. Fast implementation of the simplicial approximation theorem opens the road to a variety of fast, "simplicial" algorithms. Finally, performing some algebra on the simplicial approximation yield the "topography" of the stability boundary, that can be very complicated.

该方案的目的是将鲁棒控制和代数拓扑结合在一起。结构奇异值、多变量相位裕度、Kharitonov定理等鲁棒控制问题在Poincare、Cartan、Eilenberg等人提出的代数拓扑学中得到了自然而统一的表述。核心的数学问题是，结构不确定性的高维流形到奈奎斯特模板的映射是否与边界交换。而在简单的哈里托诺夫情形中，在大多数情况下，奈奎斯特映射并不与边界交换。然而，单纯逼近定理为我们提供了一个与边界可交换的近似奈奎斯特映射。单纯逼近定理的快速实现为各种快速的“单纯”算法开辟了道路。最后，对单纯近似进行一些代数运算，得到稳定边界的“地形”，这可能是非常复杂的。