Algebraic Topology in Robust Control

这次求婚的目的是为了 鲁棒控制和代数拓扑。 有人认为 结构奇异值等鲁棒控制问题， 多变量相位裕度、Kharitonov定理等。 他们的自然和统一的配方， 代数拓扑学的背景，由 庞加莱，嘉当，艾伦伯格，还有许多其他人。 的 数学的核心问题是， 高维结构流形 不确定性进入奈奎斯特模板与 边界 在简单的Kharitonov案件中， 在绝大多数情况下， 奈奎斯特映射与边界不对易。 然而，单纯逼近定理 为我们提供了一个近似的奈奎斯特图， 与边界交换。 的快速实现 单纯逼近定理为 各种快速的“单纯”算法。 最后， 对单纯形进行代数运算 近似产生的“地形”的稳定性 边界，这可能非常复杂。

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.