Analysis and Design of Control Systems Using Piecewise Linear Lyapunov Functions

主要研究结果总结如下：1、研究结果：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Lyapunov函数（pwllf），并给出了稳定性结果。一个PWLLF候选函数的参数对应于定义在超平面划分区域内的分段线性函数。将稳定性条件集表述为分段线性函数插入参数的线性规划问题（LP）。如果计算出的最优值为负，则利用该解构造一个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候选者，从而在LP公式中引入更多的自由度，从而得到期望的结果。我们得到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。这个条件是对先前结果的改进。通过采用这种方法，我们可以为一些无法生成PLLF的系统生成PLLF。稳定区域估计的扩大。在这个研究项目中，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估计或设计控制器使封闭系统具有更大的稳定区域是非常重要的问题。我们提出了一种实现这一目标的方法。非线性伺服系统的设计。提出了一种基于PLLFs的非线性伺服系统设计方法。该方法降低了以往结果的保守性。为了改善系统的暂态响应特性，提出了一种基于参考调速器和线性二次调速器理论的方案。双线性优化问题的快速求解方法。当我们使用PLLFs设计控制器时，我们需要解决双线性优化问题，而不是凸问题。为了解决双线性优化问题，我们应用了Zoutendijk方法、基于Imanishi进化理论的遗传算法以及一些实例的概率方法。每种方法都有优点和缺点。这个问题需要进一步的研究。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