Mathematical Sciences: Minimal Order Dynamic Pole Assignment Problems

9224541 Wang本课题研究了带动态补偿的输出反馈极点配置问题。其目标是找出任意极点可配性所需的实际动态补偿器之间的最小阶数，并推导出动态补偿器空间的Rosenthal紧化度的公式。主要研究者在解决静态输出反馈极点配置问题中开发的涉及Schubert微积分的技术有望在最小阶动态补偿器问题的研究中发挥重要作用。线性系统的极点决定了它的振动频率和振幅，极点的位置决定了系统是否稳定。因此，在物理系统（如飞机或机械臂）的控制设计中，主要目标是使用输出反馈将系统的极点移动到其稳定区域，即振动的主要模式以规定的速率消失的区域。输出反馈通常涉及一个动态补偿器，补偿器的阶数越高，控制器需要移动极点就越复杂。因此，能够设计出最小阶动态补偿器是非常重要的。这个项目将运用代数几何的方法来解决这个问题。＊＊＊

9224541 Wang This project addresses the problem of pole assignment by output feedback with dynamic compensation. Its goals are to find the minimal order among real dynamic compensators needed for arbitrary pole assignability and to derive a formula for the degree of the Rosenthal compactification of dynamic compensator space. Techniques involving the Schubert calculus developed by the principal investigator in solving the static output feedback pole assignment problem are expected to be important in the study of the minimal order dynamic compensator problem. The poles of a linear system determine its frequencies and amplitudes of vibration, and their locations determine whether or not the system is stable. Therefore, in control design for a physical system such as an airplane or robot arm, a major objective is to move the poles of the plant into its stability region, i.e., a region where the principle modes of vibration die out at a prescribed rate, using output feedback. Output feedback usually involves a dynamic compensator and the higher the order of that compensator, the more complicated the controller needed to shift the poles becomes.Therefore, it is important to be able to design minimal order dynamic compensators. This project will brings the method of algebraic geometry to bear on this problem. ***