臨界系統之穩定化及非線性系統穩定區間之估測

Abstract

本篇論文之主要目的在探討臨界系統(Critical Systems)的穩定化與非線 性系統穩定區間(Domain of Attraction)之估測。首先考慮臨界系統之穩 定化問題，其目的在於設計一適當的迴授控制法則保證臨界系統之穩定， 近年來多位學者已利用``中央流型簡化法則" (Center Manifold ormula) 來作設計本篇論文中亦應用中央流型簡化法則來設計臨界系統之控制信號 ，以達成系統穩定化之目的。本論文中主要考慮系統之線性化模式具有一 簡單零特徵根 或一對純虛數的特徵根之臨界系統。對於系統穩定區間之 估測而言，我們針對一線性部份為可控制且非時變之非線性系統，提出一 個簡易的演算法則以求得此一系統的穩定區間。此演算法則主要應用系統 之李亞普諾夫矩陣方程式(Lyapunov Matrix Equation)之建立，進而利用 方程式之特性以估測系統之穩定區間。此外我們利用類似的建構法則估測 具有一簡單零特徵根之臨界系統的穩定區間。 Issues of the stabilization for critical system and the estimation of domain of attraction for autonomous nonlinear systems are presented in this thesis. The center manifold reduction is applied to design the stabilizing control laws for nonlinear critical systems, specifically, for the systems whose linearization possesses a simple zero eigenvalue or a pair of simple pure imaginary eigenvalues. The design involves the application of center manifold reduction, normal form transformation and stability criterion. In the topic of the domain of attraction, we present theoretical analysis and computational methods for estimating the attraction region of locally stable (or stabilizable) nonlinear systems. This is achieved by the construction of Lyapunov function for the nonlinear systems. To treat the high order term of nonlinear dynamics as perturbation of linear system, we propose methods to estimate the attraction region. Numerical results are given to demonstrate the applications. Furthermore, an algorithm is also proposed to estimate the domain of attraction for the critical systems whose linearized model possesses one simple zero eigenvalue.