Computer Aided Design of Robust Control Systems Computational Geometry Approach

本研究取得的主要成果如下：1.研究成果。线性系统鲁棒控制器设计CAD系统的开发我们开发了一个利用值集进行线性系统鲁棒控制器设计的CAD系统。对于具有完全可分解的传递函数和/或特征方程的系统，使用多边形区间算法的方法对于计算值集是非常有用的。提出并实现了一种计算不需要完全分解的多项式函数的值集的方法。提出了一类新的分段线性Lyapunov函数(PWLLF)，并得到了稳定性结果。PWLLF的一个候选者具有对应于超平面相交稳定区域的参数。根据超平面插入的参数，将稳定性条件集表示为线性规划问题(LP)。如果计算的最优值为负，则使用该解来构造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，从而在LP公式中引入更多的自由度，从而达到预期的结果。由此产生的新LP的最优值总是小于或等于旧LP的最优值。我们还提出了一种生成超平面的方案，使得新LP的最优值小于旧LP的最优值。此外，还提出了一种快速计算新LP的最优解的方法。欧氏单元复合体的数据结构。我们提出并实现了欧氏单元复合体的数据结构。我们在生成PWLLF的程序中使用了这种数据结构，并将用于使用PWLLF的控制器设计。

The main results obtained through the research are summarized as follows.1. Development of CAD system for robust controllers design of linear systems.We developed a CAD system for robust controllers design of linear systems using value sets. For systems which has completely decomposable transfer functions and/or characteristic equations, the method using polygon interval arithmetic is very useful to compute value sets. We proposed and implemented a method computing value sets of multi-polynomial functions, which need not be completely decomposable.2. Generation 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 hyperplanes intersecting stability region. The set of stability conditions are formulated as Linear Programming Problem (LP) in terms of parameters inserted by the hyperplanes. 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. The optimal value of the resulting new LP is always less than or equal to that of the old LP.We also proposed a scheme to generate hyperplanes such that the optimal value of the new LPs is less than that of the old LP.Moreover, a fast method to compute the optimal solutoion of the new LP.3. Data structure for Euclidean cell complex.We proposed and implemented data structure for Euclidean cell complex. We used this data structure in programs for generation of PWLLFs and will be used for controller design using PWLLFs.