Algebraic and Differential Topology in Robust Control

9300016乔克希尔提出的研究旨在更深入地理解不确定参数下反馈系统的闭环系统(鲁棒)稳定性问题。提出了问题的三个不同的攻击方向，它们都来自于代数和微分拓扑学--单纯逼近定理、Morse理论和障碍理论。由此，计算不确定空间中稳定性边界的新的“单纯”算法应运而生。这些算法的共同特点是它们是基于不确定性空间的单纯分解的思想。这种方法的一个副产品是阐明了分离边界的拓扑、稳定性的Betti数的界限，以及最重要的是，在已被宣布为“确定”的参数的扰动下的结构稳定性。

9300016 Jonckheere The proposed research is aimed at achieving a deeper understanding of the closed-loop (robust) stability issue in feedback systems under uncertain parameters. Three different directions of attack of the problem, all inspired from algebraic and differential topology, are proposed - the Simplicial Approximation Theorem, The Morse Theory, and Obstruction Theory. From there, new "simplicial" algorithms for computing the stability boundary in the space of uncertainties emerge. The general feature of these algorithms is that they are based on the idea of simplicial decomposition of the space of uncertainty. A byproduct of this approach is the elucidation of the topology of the separating boundary, bounds on the Betti numbers of the stability, and most importantly, its structural stability under perturbation of the parameters that have been declared "certain." ***