Mathematical Sciences: Transformations of Linear Systems: Classification and Orbit Closure

9404641 O‘Halloran反馈和等价性是线性微分方程组或差分方程组的变换，可被视为对簇的代数群作用。对于给定的代数群作用，两个基本问题是分类(确定两个元素是否在同一轨道上的方法)和轨道闭合(对轨道拓扑闭合中的元素的描述)。本研究的主要目的是利用代数几何、代数几何软件和线性代数的方法，解决状态空间系统输出反馈作用的分类问题。对于完全反馈群在广义状态空间系统空间上的作用，建立了一种分类，解决了这种情况下的轨道闭合问题。从飞机控制系统到恒温器调节加热系统，许多自动化系统背后的原理是反馈控制，其中关于设备及其环境的当前信息被用作输入，以实现必要的调整。“状态反馈”指的是完全信息可用的情况；当只有部分信息可用时，根据“输出反馈”进行调整。当反馈的重复导致系统的演变，其中反馈输入发挥着越来越大的作用时，就会出现“高增益反馈”。这项研究的重点是反馈控制系统的数学模型，其中将进行两个主要项目。第一个是确定通过应用输出反馈从给定的控制系统中可以获得哪些数学模型，从而反映当利用输出反馈时在物理系统中可能出现的情况的范围。第二个项目解决了当应用高增益状态反馈时，可以从给定的控制系统数学模型获得的控制系统的数学模型的辨识问题。***

9404641 O'Halloran Feedback and equivalence are transformations of linear systems of differential or difference equations which may be viewed as algebraic group actions on varieties. For a given algebraic group action, two basic questions are classification (a method for determining whether or not two elements are in the same orbit) and orbit closure (a description of the elements in the topological closure of an orbit). The main purpose of this research is to solve the classification problem for the output feedback action on the space of state space systems, using methods from algebraic geometry, algebraic geometry software, and linear algebra. For the action of the full feedback group on the space of generalized state space systems, a classification has been established; this project will solve the orbit closure problem in this setting. The principle behind many automated systems, from aircraft control systems to thermostat regulated heating systems, is that of control by feedback, in which current information about the device and its environment is used as input in order to effect necessary adjustments. "State feedback" refers to the situation in which full information is available; when only partial information is available, adjustments are made based on "output feedback." "High gain feedback" occurs when reiteration of feedback causes an evolution of the system in which feedback input is playing a progressively larger role. This research focuses on mathematical models of feedback control systems, in which two major projects will be pursued. The first is to determine which mathematical models of control systems can be obtained from a given one by applying output feedback, thus reflecting the range of situations that could arise in a physical system when output feedback is utilized. The second project addresses the identification of those mathematical models of control systems which can be obtained from a given one when high gain state feedb ack is applied. ***