Research On Stability And Stabilization Of Two-time-scale Systems

In various fields of science and engineering, there are a number of physical systems which contain two significantly different dynamic modes:fast mode and slow mode. Two-time-scale systems are also called singularly perturbed systems, which are ideal models to describe such systems. The characteristic of two-time-scale systems is that it can depict and describe slow and fast features of dynamical systems by introducing singular perturbation parameters.In this thesis, the problems of stability and stabilization of two-time-scale systems are studied. The main results are stated as follows.1. The Gauss-Seidel iteration method is applied to investigate the exponential stability of a class of linear time-varying singularly perturbed systems. A simple algebraic criterion for exponential stability for all small enough values of singular perturbation parameter ε is obtained. Moreover, an upper bound on the values of ε which is used to ensure exponential stability of the system can be found. 2. The feedback stabilization of two-time-scale systems with time-varying time delay is studied. Two types of time delay are considered. Firstly, the two-time-scale systems are decomposed into fast mode and slow mode. A sufficient condition of stability for the slow subsystem is given. Secondly, the stability of the full-order systems with time delay is deduced from the stability of two reduced-order subsystems. Thirdly, using the parametrization design method, the state feedback control law of the full-order systems is designed based on the stabilization of two reduced-order subsystems, which ensure the full-order systems to be stable. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the derived results.