Mathematical Sciences: Nonlinear Differential Equations, Vector Field Approximations and Control

数学科学：非线性微分方程、矢量场逼近和控制

• 批准号: 9301039
• 负责人: Henry Hermes
• 金额: $ 7.54万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-04-01 至 1996-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9301039&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Differential; Equations

The focus of this research is the development of a coherent program to determine the stabilizability of nonlinear control systems. Understanding stabilizability is important; almost all issues in both systems theory and systems practice eventually reduce to some version of the stabilization problem. The importance of linear systems theory rests more on the ease of testing a linear systems for stabilization than on its linearity per se. Earlier work concentrated on the question of short time local controllability, a necessary condition for stabilizability. viewed as stabilization by open loop controls while the latter is stabilization by closed loop controls. As a result of the prior work, one now has computational criteria for deciding STLC. Work has turned to approximations of nonlinear systems by simpler ones such as nilpotent systems and systems which are homogeneous with respect to a dilation. This work holds promise for a relatively comprehensive theory of nonlinear stabilizability. In the present project, a new concept of vector fields which are stable with respect to measurement are introduced. The concept is closely related to the shadowing ideas of nonlinear dynamics. It has been shown already that if Brockett's locally onto conditions not satisfied then a nonlinear system is not locally stabilizable to a vector field which is stable with respect to measurement. Work will also be done using homogeneous approximations and homogeneous Lagrangians to construct dynamic asymptotically stabilizing feedbacks for systems which don't satisfy Brockett's locally onto condition using approximate feedback linearization and optimal control. The combination of mathematical power and engineering applications has long made control theory one of the healthiest sources of new mathematical themes. The bridges built between mathematicians and engineers have provided for dynamic interchanges leading to the advancement of both fields. This project takes up the problem of what one can do when some of the traditional tests for stabilizing state feedback control are not present. In many important problems, including mechanical systems with nonholonomic constraints and the inverted pendulum in a gravity free environment, efforts are to be made to construct discontinuous state feedback controls.

这项研究的重点是开发一个连贯的程序来确定非线性控制系统的可镇定性。理解稳定性是很重要的；系统理论和系统实践中的几乎所有问题最终都会归结为某种形式的稳定性问题。线性系统理论的重要性更多地取决于线性系统稳定性测试的容易程度，而不是其线性本身。早期的工作集中在短期局部能控性问题上，这是可镇定的必要条件。被认为是开环控制的稳定，而后者是闭环控制的稳定。作为先前工作的结果，现在已经有了确定STLC的计算标准。工作已经转向用更简单的系统来逼近非线性系统，例如幂零系统和关于伸缩是齐次的系统。这项工作为建立一个相对全面的非线性镇定理论奠定了基础。在本项目中，引入了关于测量稳定的矢量场的新概念。这一概念与非线性动力学的跟踪思想密切相关。已经证明，如果Brockett的局部条件不满足，则非线性系统不能局部镇定到关于测量稳定的矢量场。对于不满足Brockett局部条件的系统，还将利用齐次逼近和齐次拉格朗日函数，利用近似反馈线性化和最优控制来构造动态渐近镇定反馈。长期以来，数学能力和工程应用的结合使控制理论成为新数学主题最健康的来源之一。在数学家和工程师之间建立的桥梁提供了动态的交流，导致了这两个领域的进步。这个项目解决了当一些传统的稳定状态反馈控制的测试不存在时，人们可以做什么的问题。在许多重要的问题中，包括具有非完整约束的机械系统和无重力环境中的倒立摆，需要努力构造不连续的状态反馈控制。