Computer Aided Design of Robust Control Systems Computational Geometry Approach

主要研究结果总结如下：1、研究结果：线性系统鲁棒控制器设计CAD系统的开发。我们开发了一个CAD系统用于鲁棒控制器设计的线性系统使用值集。对于具有完全可分解传递函数和/或特征方程的系统，使用多边形区间算法计算值集是非常有用的。提出并实现了一种不需要完全可分解的多项式函数值集的计算方法。分段线性李雅普诺夫函数的生成。提出了一类新的分段线性Lyapunov函数（pwllf），并给出了稳定性结果。一个候选的PWLLF具有与超平面相交的稳定域对应的参数。将稳定性条件集表述为关于超平面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。得到的新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，还提出了一种快速求解新LP.3最优解的方法。欧几里得细胞复合体的数据结构。提出并实现了欧几里得细胞复合体的数据结构。我们在程序中使用这种数据结构来生成pwllf，并将使用pwllf进行控制器设计。

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.