Algebraic Topology in Robust Control

该方案的目的是将鲁棒控制和代数拓扑结合起来。本文认为，结构奇异值、多变量相裕度、Kharitonov定理等鲁棒控制问题在代数拓扑的背景下得到了庞加莱、Cartan、Eilenberg等人表述的自然而统一的表述。中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。虽然在简单的Kharitonov情况下确实如此，但在绝大多数情况下，奈奎斯特映射并不与边界交换。然而，简单逼近定理为我们提供了一个与边界交换的近似奈奎斯特映射。简单近似定理的快速实现为各种快速的“简单”算法开辟了道路。最后，在简单近似上进行一些代数运算，得到稳定性边界的“地形”，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.