Geometry and Algebra of Nonlinear Control Systems

0072369 Kawski这个项目建立在时序代数的形式主义基础上，研究非线性控制系统的几何基础。在几何上，时间乘积编码了受控非线性动力学组件的相互作用。 在代数上，它是有效的级数展开和形式解算法的基础结构。特别关注的领域是系统的互连，以及对控制表现出完全非线性依赖的系统。在任何一种情况下，具体的目标是开发规范形式，并找到有效的系列（或产品）表示（使用代数组合学的工具）。可能的应用包括路径规划，反馈稳定，最优控制，甚至数值积分的算法。一个补充的第二个线程开发用于研究（模拟和实验）和通信的交互式可视化工具。这项研究的动机和驱动力是希望了解从分子水平到天文尺度的任何尺度上的动力系统相互作用的基本，共同的原则。可能结果的多样性是渗透在我们环境中的非线性效应的结果：当组合两个子系统时，结果通常不同于部分之和。本研究的重点是进一步发展捕捉这种非线性相互作用的代数，其中a+B的效果不仅仅是a和B的效果之和。时序代数提供了一种形式化的语言来模拟这样的交互作用，即使是将片段放在一起的顺序也很重要，其中a*B通常不同于B*a。控制理论的目标是利用相互作用的微妙性质来塑造复杂的系统-通常只通过对部件如何相互作用的微小干扰。虽然这项数学研究适用于几乎任何类型的动力系统，甚至包括社会，医疗和金融环境，但该项目将重点关注机械系统（如具有多个移动部件的大型卫星），以努力展示一般原理。该项目的一个重要组成部分是开发交互式可视化工具。这些用于研究，并用于沟通的几何非线性相互作用，以展示如何深刻理解的基本数学结构导致控制复杂系统的有效手段。这种可视化的努力也为本科生提供了一个丰富的环境，以连接先进的理论研究。它可能被证明是一种有效的手段，以扩大管道带来新的人才进入数学研究。

0072369KawskiThis project builds on the formalism of chronological algebras to study the geometric foundations of nonlinear control systems. Geometrically, the chronological product encodes the interaction of the components of controlled nonlinear dynamics. Algebraically it is the fundamental structure that underlies effective series expansion and formal solution algorithms. Areas of special focus are interconnections of systems, and systems that exhibit full nonlinear dependence on the control. In either case, specific objectives are to develop normal forms and to find effective series (or product) representations (using tools from algebraic combinatorics). Possible applications include algorithms for path planning, feedback stabilization, optimal control and even numerical integration. A complementary second thread develops interactive visualization tools for research (simulation and experimentation) and for communication. This research is motivated and driven by the desire to understand the fundamental, common principles that govern the interactions of dynamical systems on any scale, from molecular levels to astronomical dimensions. The diversity of possible outcomes is a consequence of nonlinear effects that permeate our environment: When combining two subsystems, the result is generally different from just the sum of the parts.This research focuses on further developing the algebra which captures such nonlinear interaction, where the effect of a+b is more than just the sum of the effects of a and b. The chronological algebra provides the formal language to model such interactions where even the order in which pieces are put together matters, where a*b is generally different from b*a. The control perspective is distinguished from the basic study of dynamical systems as it aims beyond just descriptive understanding: The objective of control theory is to exploit the subtle nature of the interactions in order to shape complex systems -- often via only minute interference with how the parts interact. While this mathematical research applies to virtually any kind of dynamical system, including even social, medical and financial environments, this project will focus on mechanical systems (like large satellites with several moving parts) in efforts to demonstrate the general principles. An important component of this project is the development of interactive visualization tools. These are used for research, and for communicating the geometry of the nonlinear interactions, to demonstrate how the profound understanding of the fundamental mathematical structures leads to effective means of controlling complex systems. This visualization effort also provides a rich environment for undergraduate students to connect with advanced theoretical research. It may prove to be an effective means to expand the pipeline bringing new talents into mathematical research.