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

9301196在鲁棒控制中，具有给定结构的扰动集合可以被视为具有非平凡拓扑的可微流形。这个流形可以分为稳定和不稳定两个区域，其中单位矩阵是稳定区域中的一个点，表示未受扰动的系统。这种结构扰动集合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”。Baron和Jonck heere定义了多变量相位裕度，他们把摄动的类别限制在酉阵的群上。对于二维情况，从几何上看，相位裕度可以表示为从北极到不稳定区域的三个球面上的最小弧长。所提出的研究重点是将这些性质推广到n维情形，特别关注：(1)相位裕度的几何和拓扑性质；(2)么正扰动流形的稳定区和失稳区的划分；以及(3)提高相位裕度的补偿器的设计。这项研究的主要目标是深入了解阶段裕度，并将这些抽象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. ***