Analysis and Design of Control Systems Using Piecewise Linear Lyapunov Functions

本文的主要研究成果如下：1.分段线性李雅普诺夫函数类的推广提出了一类新的分段线性李雅普诺夫函数（PWLLF），并得到了稳定性结果。PWLLF的候选者具有对应于在由超平面划分的区域中定义的分段线性函数的参数。稳定性条件的设置制定为线性规划问题（LP）的参数插入分段线性函数。如果计算出的最优值为负，我们使用该解构造一个PWLLF。当LP的最优值为非负时，我们通过添加适当的超平面来修改PWLLF候选，以在LP公式中引入更多的自由度，并达到预期的结果。我们导出了一个新的生成超平面的条件，使得新LP的最优值小于旧LP的最优值。这个条件是对以前结果的改进。通过收养这个我 ...更多信息 2.稳定域的估计方法在本研究项目中，我们关注的是半全局稳定性，而不是全局稳定性。在这方面，计算更大的稳定区域的估计或设计控制器，使闭环系统具有大的稳定区域是非常重要的问题。3.非线性伺服系统的设计提出了一种基于PLLF的非线性伺服系统设计方法。这种方法减少了以往结果的保守性。为了改善系统的瞬态响应特性，本文提出了一种基于参考调速器和线性二次型调节器理论的控制器设计方案。4.双线性优化问题的快速求解方法在利用PLLF设计控制器时，需要求解双线性优化问题，而这类问题不是凸问题。为了解决双线性优化问题，我们应用Zoutendijk的方法，遗传算法的基础上Imanishi的进化理论和概率的方法为一些例子。每种方法都有其优点和缺点。这一问题需要进一步研究。少

The main results obtained through the research are summarized as follows.1.Generalization of the class of piecewise linear Lyapunov functions.We proposed a new class of piecewise linear Lyapunov functions (PWLLFs) and derived stability results. A candidate of PWLLF has parameters corresponding to piecewise linear function defined in the region divided by hyperplanes. The set of stability conditions are formulated as Linear Programming Problem (LP) in terms of parameters inserted by piecewise linear functions. If the computed optimal value is negative, we construct a PWLLF using the solution. When the optimal value of the LP is nonnegative, we modify the PWLLF candidate by adding appropriate hyperplanes to introduce more freedom in the LP formulation and arrive at the desired result. We derived a new condition for generating hyperplanes such that the optimal value of the new LPs is less than that of the old LP. This condition is an improvement of the previous result. By adopting this me … More thod, we can generate a PLLF for some systems, for which we could not generate a PLLF.2.Enlargement of estimates of stability regions.In this research project, we are interested in semi-global stability rather than the global stability. In this respect, it is very important issues to compute larger estimates of stability regions or to design controller so that the closed system has large stability region. We proposed a method to achieve this.3.Design of nonlinear servo systems.We proposed a design method of nonlinear servo systems by using PLLFs. This method reduces conservativeness included in previous results. To improve the transient response characteristic, we proposed a scheme, which adopts idea based on the reference governor and the linear quadratic regulator theory.4.Fast solving methods for bilinear optimization problems.When we design controller using PLLFs, we need to solve bilinear optimization problems, which are not convex problems. To solve bilinear optimization problems, we applied the Zoutendijk's method, a genetic algorithm based on Imanishi's evolution theory and the probabilistic approach for some examples. Each method has both the advantage and the disadvantage. Further research on this issue is needed. Less