Characterizations on Stability Properties for Nonlinear Systems

Florida Atlantic University, Boca Raton FL

Investigators

Abstract

0072620 Wang This 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.