Liapunov函数的计算机构造及其应用

A New Method for Constructing Liapunovs Functions and Determining Stability with a Computer

本文提出了一种易于编程借助计算机实现的构造线性离散时间系统Ｌｉａｐｕｎｏｖ函数的方法。该方法迥避了直接求解离散Ｌｉａｐｕｎｏｖ矩阵方程，而是利用相应矩阵的负定性条件，通过逐次改变解矩阵中某些元素之值，构造出系统的Ｌｉａｐｕｎｏｖ函数。作为实例。

This paper proposed a new method for constructing Liapunovs function and determining the stability of discrete time systems. First, the system matrix and the Liapunov matrix equation are changed easily into the simpler forms with the method of similarity transformation. Secondly, in the case that the weight matrix D of Liapunovs function is the form of triple diagonal matrix, the matrix Q in the discrete Liapunov matrix equation F TDF D=Q and its order principal minors can be expressed by the elements of matrix F and D. Lastly, the required Liapunovs function can be constructed by choosing properly several differences in the matrix D under the negative definitive condition of matrix Q. The peculiarities of this method are: the solving directly Liapunov matrix equation is avoided. The all steps are easily in programming and are very applicable for the accomplishment with a computer. It is of universality and practicality in a certain sense.