切换广义系统的一类统一Lyapunov函数初探

Investigation of a Common Lyapunov Function for Switched Descriptor Systems

切换系统作为混杂系统中的重要一类,已经进行了深入广泛的研究.本文通过构造一类统一Lyapunov函数的方法,研究切换线性广义系统的稳定性.首先构造了i∈{1,2}时的一个Lyapunov函数;并证明此Lyapunov函数是切换广义系统的一个统一Lyapunov函数;其次利用相似的构造方法推广到i∈{1,2,…,N}的情况,其中A_i,i∈{1,2,…,N}是状态矩阵且是可交换的;最后,给出了在任意切换律下使得该切换广义系统渐近稳定的充分条件.

The scope of control theory is being enlarged to include hybrid dynamical systems. One of the main features of such hybrid dynamical systems is the systematic application of the idea of switched sys-tems. In this paper,the existence of a common Lyapunov function for a class of switched descriptor sys-tems was studied. Stability of switchedlinear singular systems under arbitrary switching signals was investi-gated. We first present work on i∈（1,2）,and a Lyapunov function is constructed and proved to be a common Lyapunov function of the system,then we extend to the case of i∈（1,2,…,N） using the similar method. The matrices Ai,i∈（1,2,…,N） are commute pairwise. Finally,a sufficient condition assures the system asymptotically stable under arbitrary switching strategy.