Computer Generation of Lyapunov Functions and Its Applications

1.改进多面体Lyapunov函数的算法：我们研究了几种改进多面体Lyapunov函数的方法，其中最有用的方法是在现有的多面体上添加由广义扇形条件产生的矢量场的极值点。此外，利用实Jordan标准形，我们提出了一种生成有用的初始多面体的方法。动态凸壳算法：提出了一种新的动态凸壳算法，该算法比Under-Beyond算法需要更少的内存和计算时间。不连续系统：我们推导了可能存在滑模的不连续系统的稳定性结果。所得结果与连续系统的结果基本一致。此外，利用多面体Lyapunov函数，提出了一种变结构控制系统的设计方法。该方法可以消除抖振。应用：我们将多面体Lyapunov函数应用于组合系统的稳定性分析。我们还将其应用于模糊控制系统的稳定性分析，结果表明，通过应用广义扇区条件，我们可以用比传统的Lyapunov函数如二次Lyapunov函数等更少的保守条件来保证系统的稳定性。

1. Algorithm for lmproving polytope lyapunov Function : We examined several method to improve polytope Lyapunov functions, and we conclude that the most useful method is that adding vectors which are extreme points of vector field generated by the generalized sector conditions to the present polytope. Moreover, by using real Jordan canonical form, we proposed a method to generate useful an initial polytope.2. Dynamic Convex Hull Algorithm : We proposed a new dynamic convex hull algorithm which requires less memories and computing time than the Beneath-Beyond method.3. Discontinuous Systems : We derived a stability result for discontinuous systems which may have sliding modes. The result is almost same with that for continuous systems. Moreover, by using a polytope Lyapunov function, we proposed a design method for variable structure control systems. This method can eliminate chattering.4. Applications : We applied the polytope Lyapunov function to the stability analysis of composite systems. We also applied it to the stability analysis of fussy control systems, and showed that, by applying generalized sector conditions, we can guarantee the stability with less conservative conditions than the traditional Lyapunov functions such as quadratic Lyapunov functions and so on.