OPERATION ON SETS AND IT'S APPLICATIONS TO COMPUTER AIDED DESIGN OF ROBUST CONTROL SYSTEMS

由于系统参数的变化和不确定性，数学模型Including给出了一个不完善的描述真实的系统。因此，鲁棒性是对控制系统最基本的要求之一。本课题的主要研究成果如下：1.多边形间隔算术。为了处理不确定性，我们定义了由所有凸多边形组成的集合上的运算（加法、乘法和反转），我们称之为多边形区间运算。给出了多边形区间算术的几个重要性质。我们还提出了一个有效的算法来计算凸矩阵的加法.鲁棒稳定性分析（1）反馈系统的稳定性可以通过考察反馈微分矩阵在各个频率上的行列式来分析。计算时采用了一种基于映射定理的方法，但计算量很大。提出了一种基于多边形区间算法和映射定理的行列式计算方法。（2）非线性系统的稳定性可以用李雅普诺夫函数来检验。提出了利用计算几何技术构造Liapunov函数计算凸壳的方法. RSRD（Robust Sequential Return Difference）方法提出了鲁棒控制系统设计的RSRD方法，该方法采用多边形区间算法，使得各回路控制器的设计可以“独立”进行，并保证了系统的完整性. CAD（计算机辅助设计）系统.基于RSRD方法，我们开发了一个用于设计鲁棒控制系统的CAD系统。我们还实现了一个程序来计算多输入多输出系统的稳定裕度。

Because of variation and uncertainty of system parameters, mathematical models Invariably give an imperfect description of real systems. Therefore, robustness is one of the most fundamental requirements for control systems. Main results of this research project may be summarized as follows:1. Polygon Interval Arithmetic. To treat uncertainty, we defined operations (addition, multiplication, and reversion) on sets consisting of all the convex polygons, which we call polygon interval arithmetic. We derived several important properties of polygon interval arithmetic. We also proposed an efficient algorithm to calculate addition of convex polygons.2. Robust Stability Analysis. (1)Stability of feedback systems can be analyzed by examining determinants of return differential matrix at every frequencies. A method based on the mapping theorem was used to calculate it, but it is very much time consuming. We proposed to use a method based on both polygon interval arithmetic and the mapping theorem to calculate determinants. (2)Stability of (nonlinear) systems can be examined by using Liapunov functions. We proposed a method to construct a Liapunov function via computational geometric technique to calculate convex hulls.3. RSRD(Robust Sequential Return Difference) method. We proposed RSRD method to design robust control systems, which uses polygon interval arithmetic, and which makes possible to design controllers of each loops "independently", and to guarantee the integrity.4. CAD(Computer Aided Design) System. We developed a CAD system to design robust control systems based on RSRD method. We also implemented a program to calculate stability margin of multi-input multi-output systems.