Analysis and Design of Control Systems Utilizing Data Structure and Algorithms

通过研究取得的主要成果概括如下.线性系统鲁棒控制器设计CAD系统的开发我们开发了一个基于数值集的线性系统鲁棒控制器设计CAD系统。对于具有完全可分解的传递函数或特征方程的系统，利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多项式函数值集的计算方法.分段线性李雅普诺夫函数的产生：提出了一类新的分段线性李雅普诺夫函数，并得到了稳定性结果。PLLF的候选者具有对应于与稳定区域相交的超平面的参数。稳定性条件的一组制定为线性规划问题（LP）的超平面插入的参数。如果计算出的最优值为负，我们构造一个 ...更多信息 PWLLF使用的解决方案。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，以在LP公式中引入更多的自由度，并达到预期的结果。由此产生的新LP的最优值总是小于或等于旧LP的最优值。我们还提出了一个计划，以产生超平面，使新的LP的最佳值小于旧的LP。此外，我们提出了一种快速的方法来计算新的LP的最优解。提出并实现了一种新的欧氏胞腔复形的数据结构。我们在程序中使用了这种数据结构，用于生成PLLF，并将用于使用PLLFs4的控制器设计。双线性优化问题的求解方法。使用PLLF设计控制器的主要问题是解决双线性优化问题，这是非凸问题，因此不容易解决。但我们认为这个问题比较容易解决，即双线性矩阵不等式问题。我们提出了一种使用Zoutendijk方法的方法。我们实现了这种方法，但需要进一步的研究，以提高收敛性的方法。少

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 have 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 Ly apunov functions.We proposed a new class of piecewise linear Lyapunov functions (PLLFs) and derived stability results. A candidate of PLLF 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 … More 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, we propose a fast method to compute the optimal solution 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 PLLFs and will be used for controller design using PLLFs4. Solution method of bilinear optimization problems.The main issue in designing controllers using PLLFs is to solve bilinear optimization problems, which is non-convex problems, and, hence, it is not so easy to solve. But we believe that the problem rather easy to solve that bilinear matrix inequality problems. We proposed a method using Zoutendijk method. We implement this method, but further research is needed to improve convergence of the method. Less