A Geometric Study of the Phase Margin and its Implications in Design for Multivariable Control Systems

小行星9301196 在鲁棒控制中，具有给定结构的扰动集合可以看作是具有非平凡拓扑的可微流形。 这个流形可以分为稳定区域和不稳定区域，其中单位矩阵是稳定区域中的一个点，代表未扰动系统。 这种结构化扰动的稳定裕度可以定义为流形上从单位矩阵到不稳定区域的最小“距离”。 Bar-on和Jonckheere通过将扰动类限制在酉矩阵群中来定义多变量相位裕度。 它已被证明为二维的情况下，从几何上看，相位裕度可以表示为三个球从北极的不稳定区域的最小弧长。 建议的研究中心推广这些属性的n维情况下，特别关注：（1）的几何和拓扑性质的相位裕度;（2）的酉扰动的流形的稳定和不稳定的区域的划分;和（3）补偿器的设计，以提高相位裕度。 本研究的主要目标是发展一个透彻的了解相位裕度，并将这些抽象的概念与鲁棒控制理论和设计中的工程问题联系起来。 ***

9301196 Bar-on In robust control a collection of perturbations with a given structure can be viewed as a differentiable manifold with a nontrivial topology. This manifold can be divided into regions of stability and instability, where the identity matrix is a point in the region of stability and represents the unperturbed system. A stability margin for such a collection of structured perturbations can be defined as the smallest "distance" on the manifold from the identity matrix to the region of instability. Bar-on and Jonckheere have defined the multivariable phase margin by restricting the class of perturbations to the group of unitary matrices. It has been shown for the two-dimensional case that, when viewed geometrically, the phase margin can be represented as the minimum arc length on the three sphere from the north pole to the region of instability. The proposed research centers on generalizing these properties to the n-dimensional case with special focus on: (1) the geometric and topological properties of the phase margin; (2) the division of the manifold of unitary perturbations into stabilizing and destabilizing regions; and (3) the design of compensators to improve the phase margin. The main goal of this research is to develop a thorough understanding of the phase margin and to relate these abstract concepts back to engineering issues in robust control theory and design. ***