MSPA-ENG: Research in Nonlinear Control Systems Theory: Lyapunov Functions, Stabilization, and Engineering Applications II

该项目将基于李雅普诺夫函数、广义微分和输入-状态稳定框架的系统应用，为数学控制理论中的重要非线性动力系统发展高度创新的镇定理论。最终目标是对现有的各种时变反馈稳定器和Lyapunov函数的构造进行大胆和深远的统一，它们将适用于具有离散和连续时间尺度和输出的非常一般的系统，包括不能通过连续时不变反馈全局镇定的非完整系统。这样的系统在科学界无处不在。(A)将反馈延迟引入先验稳定闭环系统的效果量化，(B)不连续反馈分歧点的分析，(C)具有不确定模型参数的工程模型的跟踪问题，(D)具有测量不确定性的系统的Lyapunov函数和光滑排斥反馈稳定器的显式构造，以及(E)准时间最优反馈镇定。通过开发强大的新的工程和数学技术，这项工作将使使用单一、系统和用户友好的方法分析跨越许多学科的一大类重要的稳定性问题成为可能。数学控制理论和优化为许多现代技术提供了理论基础，包括航空、生物技术、通信网络、制造和气候变化模型。反馈稳定化被用于从机电工程到生物数学的许多领域。这一创新项目将力求突破，而不是渐进式改进。它的方法将给出一类重要的控制系统的反馈稳定器，这些系统超出了已知的连续反馈镇定技术的范围，但在工程上经常出现，例如具有时滞和多个时间尺度的系统，对于这些系统来说，只有部分信息可用。该项目将在几个具有强烈工程兴趣的应用中建议和探索创造性和原创性的反馈概念，例如恒化器(模拟竞争限制营养的微生物)和微型机电继电器(用于打开或关闭电路中的连接)。这项研究将通过支持研究助理来促进学习，研究助理将在控制系统实验室应用和验证方法。这项工作将在一个传统上吸引许多少数族裔的机构进行，并将特别努力从代表性不足的群体中招聘合格的研究助理。

This project will develop highly innovative stabilization theory for important classes of nonlinear dynamical systems in mathematical control theory, based on the systematic use of Lyapunov functions, generalized differentials, and the input-to-state stability framework. The ultimate goal is to provide a bold and far-reaching unification of the various existing constructions of time-varying feedback stabilizers and Lyapunov functions, which will apply to very general systems with mixtures of discrete and continuous time scales and outputs, including nonholonomic systems that cannot be globally stabilized by continuous time invariant feedbacks. Such systems are ubiquitous in science. Other topics to be pursued include (a) quantifying the effects of introducing feedback delays into a priori stable closed loop systems, (b) analysis of bifurcation points of discontinuous feedbacks, (c) tracking problems for engineering models with uncertain model parameters, (d) explicit constructions of Lyapunov functions and smooth repulsive feedback stabilizers for systems with measurement uncertainty, and (e) quasi time optimal feedback stabilization. By developing powerful new engineering and mathematical techniques, this work will make it possible to analyze a large class of important stability problems across many disciplines using a single, systematic and user-friendly approach.Mathematical control theory and optimization provide the theoretical foundations that undergird many modern technologies including aeronautics, biotechnology, communications networks, manufacturing, and models of climate change. Feedback stabilization is used in many of these areas, ranging from electromechanical engineering to biomathematics. This innovative project will strive for breakthroughs rather than incremental improvements. Its methods will lead to feedback stabilizers for significant classes of control systems that are beyond the scope of the known continuous feedback stabilization techniques but which commonly arise in engineering, such as systems with time delays and multiple time scales for which only partial information is available. This project will suggest and explore creative and original feedback concepts in several applications that are of compelling engineering interest, such as chemostats (which model microorganisms competing for limiting nutrients) and micro-electromechanical relays (which are used to open or close connections in electric circuits). The research will promote learning by supporting research assistants, who will apply and validate the methods in a control system laboratory. The work will be carried out at an institution that traditionally attracts many minorities, and special efforts will be made to recruit qualified research assistants from under-represented groups.