Computer Aided Design of Robust Control Systems Using Polygon Interval Arithmetic

主要研究结果总结如下：1、研究结果：没有完全可分解表达式的函数的值集估计。考虑了当给定解析函数f(p)没有完全可分解表达式，且p的每个参数p1都属于一个多边形时，计算值集的良好估计问题，并证明了f的值集的边界包含在多边形的边缘图像中。利用这一结果，我们提出了两种计算值集估计的方法。一类是由p的仿射函数之比给出的函数，另一类是由p的多仿射函数给出的函数。鲁棒控制系统CAD系统的开发。设计了满足鲁棒性要求的PID控制器、超前补偿器、滞后补偿器和二自由度PID控制器。该系统利用NPIA计算被控对象的值集，利用集（并集、交集集和差集）上的操作计算控制器参数的可行域。由于它计算可行域，即使优化问题不是凸的，它也能给出全局最优解。非线性系统的稳定性分析。本文不仅分析了非线性不确定系统的稳定性，而且分析了线性不确定系统的稳定性。利用NPIA计算了线性部分对应的传递函数的值集，并用波波夫准则分析了稳定性。

The main results obtained through the research are summarized as follows.1. Value sets estimate for functions which have no totally decomposable expression.We consider a problem to compute good estimate of value sets when a given analytic function f(p) has no totally decomposable expression, where each parameter p1 of p belongs to a polygon, and showed that the boundary of the value set of f is included by the image of edges of polygons. Using this result, we proposed two methods to compute estimates of value sets. One is for the functions which are given by the ratio of affine functions of p. The other is for the functions which are multi-affine functions of p.2. Development of CAD system for robust control systems.We developed a CAD system which design PID controllers, lead compensators, lag compensators and 2 degree of freedom PID controllers, which satisfy several robust performances. This system compute value sets of plants using NPIA and feasible regions of parameters of controllers using operations on sets (union, intersection and difference set). Since it computes feasible regions, it gives global optimum solution even if optimization problems are not convex.3. Stability analysis of nonlinear systems.We analyze stability of Lur'e systems which have uncertainties in not only for the nonlinear part but also the linear part. Value sets of transfer function corresponding to the linear part are computed using NPIA and stability is analyzed applying Popov criterion.