含有微分代数子系统的混杂系统稳定性研究

针对目前广泛存在的微分代数混杂系统(DAHS)的一般模型,提出了包括稳定性和(大范围)渐近稳定性概念的稳定性理论框架.利用单李雅普诺夫(Lyapunov)函数和多李雅普诺夫函数工具,得出了在任意切换及慢切换条件下判断DAHS的稳定性及渐近稳定性的充分条件.尤其对仅包含线性微分代数子系统的混杂系统,还具体给出了其切换序列的选择办法,以使整个系统达到大范围渐近稳定.在一个包括两个线性微分代数子系统的DAHS实例中,用相关定理得出了切换序列的选择办法,使得整个系统稳定.

Study on stability of hybrid systems composed of differential-algebraic subsystems

For the actuality that Differential-Algebraic Hybrid System (DAHS), whose subsystem is modeled by differential-algebraic equations, are seldom focused on by researchers at present but are important to direct the correlative research, a framework of stability theory including the definition of stability and (globally) asymptotic stability was put forward based on the model of DAHS. And the sufficient conditions to test for stability and asymptotic stability were achieved in the case of arbitrary switching and slow switching by Single Lyapunov Function (SLF) and Multiple Lyapunov Functions (MLF). For DAHS composed of only linear differential-algebraic subsystems, the switching law was presented to stabilize the whole system globally. The simulation results of a numerical example, in which the hybrid system composes of two linear differential-algebraic subsystems,are also investigated to show the feasibility and the effectiveness of the switching law. Analysis of power system showed that its voltage stability has the typical characteristics of hybrid system. However, the power subsystems cannot be described by pure Ordinary Differential Equation (ODE) but by Differential Algebraic Equation (DAE). Therefore, voltage stability boils down to the stability problem of Differential Algebraic Hybrid System (DAHS). In this paper, the power system model was used to degine the general form of DAHS ,then the stability theory framework was presented for the first time. The results of stable and asymptotically stable voltage were educed using Lyapunov's direct method. In the end, the simulation results showed that the stability theory results presented in this paper, can be successfully used for analysis of voltage stability. Then a novel tool is presented for research of voltage stability. The subsystem of hybrid system was modeled by differential-algebraic equations and pure differential equations. This class of hybrid systems, which was defined as differential-algebraic hybrid system (DAHS), was seldom researched at present. A framework of stability theory including the definition of stability and (globally) asymptotic stability was put forward based on the model of DAHS. And the sufficient conditions to test for stability and asymptotic stability were achieved in the case of arbitrary switching and slow switching by Single Lyapunov Function (SLF) and Multiple Lyapunov Functions (MLF). For DAHS composed of only linear differential-algebraic subsystems, the switching law governing the stability of the whole system globally was presented. The simulation results of a numerical example, in which the hybrid system composed of two linear differential-algebraic subsystems, were also investigated to show the feasibility and the effectiveness of the switching law.