切换线性时滞系统的稳定性

Stability for Switched Linear Systems with Time-delay

研究了具有时滞的线性切换系统的稳定性问题。根据李雅普诺夫第二方法,找到适合讨论本文线性时滞系统的李雅普诺夫函数,然后对李雅普诺夫函数求导后的部分等式的处理采用了矩阵分解的方法,把矩阵分解为2个任意矩阵相乘的形式,这里主要采用的是矩阵的满秩分解,这样做的目的是便于运算,从而得到系统稳定的充分条件。同时给出了切换线性时滞系统切换律的设计,文中的判定条件与时滞无关,但切换律与时滞有关。利用MATLAB中线性矩阵不等式LMI工具箱求解,得到使得文中所讨论切换线性时滞系统稳定的解矩阵。最后通过数值算例验证了本文结果的可行性、正确性和有效性。

In this paper, the stability problem for a class of switched linear systems with time-delay is considered. Based on the Lyapunov second method, construct a suitable Lyapunov function for the discussion switched linear systems with time-delay, then Lyapunov function derivative part of the discussion systems are treated by using matrix decomposition method, where the matrix is decomposed into multipling by the form of two arbitrary matrices, and here the full rank matrix decomposition is mainly used aim-ing is to facilitate the operation, thus a asymptotic stability sufficient condition for switched linear systems is obtained. In the mean while, the switching laws which is designed for the systems is given, the sufficient condition is independent with time delays, but the switching laws is depend on it. At last empolying linear matrix inequality （LMI） toolbox in this work, the solution matrice for the switched linear systems in the paper is also obtained. Numerical examples are illustrated to show the feasibility, accuracy and effectiveness of our results.