Characterizations on Stability Properties for Nonlinear Systems

[72620]本提案概述了一个研究计划，旨在了解系统在外部输入（如干扰和跟踪信号）方面的行为。具体来说，将研究两种类型的稳定性特性：投入产出稳定性和增量稳定性。粗略地说，输入输出稳定性要求在外部输入信号很小的情况下，输出信号最终变小，特别是在没有输入作用于系统的情况下，输出信号收敛于零。增量稳定性特性要求具有附近输入的附近轨迹保持在附近，并最终彼此收敛。这些性质是线性系统研究中一些常用性质的非线性推广。为了更好地理解非线性情况下的性质，探索它们的等效特征是至关重要的，其中类李雅普诺夫函数提供了最内在的洞察力。本课题的主要目的是研究非线性稳定性性质的等效表征和发展类李雅普诺夫函数理论。系统和控制理论中的一个主要问题是研究系统对干扰的敏感性，更一般地说，是研究输出对执行器和测量误差、跟踪信号的大小等的依赖性。对于线性系统，频域方法在这类问题的研究中取得了巨大的成功。然而对于非线性系统，情况要复杂得多。在非线性情况下的一个关键问题是确定最需要的特征，并将它们表述成合适的数学概念。在过去10年左右的时间里，输入状态稳定性的概念被提出，并迅速成为许多现代非线性反馈分析和设计所依赖的基本概念。在这个项目中，它被提议研究输入到状态稳定性概念的各种扩展。目的是为非线性系统类稳定性质的研究奠定理论基础，从而为非线性系统设计提供一套完善的理论工具。

0072620WangThis proposal outlines a research program aimed at understanding the behavior of systems with regard to external inputs such as disturbances and track signals. Specifically, two types of stability properties will be examined: input-to-output stability and incremental stability. Roughly speaking, input-to-output stability requires that the output signals eventually become small if the external input signals are small, and in particular, the output signals should converge to zero when there is no input acting on the system. Incremental stability properties require nearby trajectories with nearby inputs stay nearby, and eventually converge to each other. These properties are nonlinear generalizations of some frequently used ones in the study of linear systems. To understand the properties in the nonlinear case well, it is crucial to explore their equivalent characterizations, among which the Lyapunov-like functions provide the most intrinsic insight. The main objective of the project is to investigate the equivalent characterizations and to develop the theory of Lyapunov-like functions for the nonlinear stability properties.One of the main issues in systems and control theory concerns with the study of system sensitivity to disturbances, and more generally, of the dependence of outputs on actuator and measurement errors, magnitudes of tracking signals, and the like. For linear systems, the frequency domain approach has led great success in the study of such problems. Yet for nonlinear systems, the situation is much more complicated. A key issue in the nonlinear case is to identify the most desired features and to formulate them into suitable mathematical notions. During the last 10 years or so, the notion of input-to-state stability was formulated and it quickly became a fundamental concept upon which much of modern nonlinear feedback analysis and design rest. In this project it is proposed to investigate a wide variety of extensions of the notion of input-to-state stability. The goal is to build a theoretical foundation for the study of stability-like properties for nonlinear systems, thereby providing a set of sound theoretical tools for nonlinear systems design.