Mathematical Science: Nonlinear Control Theory: Topics Related to Stabilizability, Observability and Realizability

9403924王/林在这项研究中将涉及非线性控制理论的两个主要领域。第一个与控制系统的稳定性和可镇定性有关，第二个与可观性和可达性有关。在稳定性领域，研究计划将依赖于三个密切相关的主题的整合，最终目标是了解干扰对控制设计的影响。第一个主题通过分析可达集的结构及其延拓来分析一般动力系统的稳定性，从而得到一般动力系统的逆李雅普诺夫函数。第二个主题涉及所谓的输入到状态的稳定性，第三个主题涉及设计显式稳定反馈律的通用公式的发展。这项调查的第二个主要领域的主要目标是分析系统的不同表示之间的关系。注意力将集中在澄清在微分代数和微分几何框架内发展起来的与可观测性有关的不同概念和结果之间的联系。还需要研究的是状态空间系统允许积分输入/输出方程的条件。粗略地说，控制理论关注的是改变动力系统的行为，以满足预期的性能目标。控制系统的分析和设计在制造自动化和许多其他应用中起着基础性的作用。本研究所涉及的主要问题是受干扰的控制系统的稳定性问题，即如何确保感兴趣的量最终以适当的方式运行而不考虑作用于系统的扰动的具体性质的问题，以及系统的识别问题，即如何根据对其运行的观察获得的数据来获得系统或“对象”的模型的问题。***

9403924 Wang/Lin Two major areas within nonlinear control theory will be pursued in this investigation. The first is related to stability and stabilizability of control systems, and the second to observability and reachability. Within the area of stability, the research program will rely upon the integration of three closely related topics with the ultimate goal of developing an understanding of the effect of disturbances on control design. The first topic focuses on the analysis of the stability of general dynamical systems through an analysis of the structure of reachable sets and their prolongations, so that a converse Lyapunov function for general dynamical systems can be developed. The second topic concerns so-called input-to-state stability, and the third topic concerns the development of universal formulae for designing explicit stabilizing feedback laws. The main goal in the second major area of this investigation is to analyze the relationship between different representations of systems. Attention will be focused on clarifying the connections between different notions and results related to observability which have been developed within the frameworks of differential algebra and differential geometry. Also to be investigated are conditions under which a state space system admits integral input/output equations. Roughly speaking, control theory concerns itself with changing the behavior of dynamical systems so as to meet desired performance objectives. The analysis and design of control systems play fundamental roles in manufacturing automation and many other applications. The main issues addressed in this research deal with the problem of stability of control systems subject to disturbances, that is, with the question of how to assure that the quantities of interest ultimately behave in an appropriate manner regardless of the specific nature of the disturbances acting on the system, and with identification of systems, that is, with the questi on of how to obtain models of the system, or "plant", on the basis of data obtained from observations of its operation. ***