Analysis and Synthesis of Control Systems via Randomized Algorithms

随机算法是近年来被公认的控制系统设计的有效工具。事实上，在这一研究领域，越来越多的复杂问题出现，概率方法给出了有效的算法来解决这些问题。例如，鲁棒最优控制通常被重新定义为全局优化，其中问题的大小由于系统中包含的不确定性而相当大。因此，通过确定性算法在合理的时间内解决它是非常困难的，这被称为维数灾难。另一方面，基于随机抽样的算法的不确定性，使我们能够获得一个概率的解决方案，没有沉重的计算工作量，虽然一定的风险水平应该是accepted.In本研究项目中，控制系统的分析和综合，通过随机算法进行了进一步的研究，这导致了一个新的框架控制系统理论。特别是以下重要的结果获得。(i)控制系统分析与综合的随机化算法：为保成本调节器、固定阶控制器和切换系统的概率设计开发了特定的随机化算法。(ii)随机化算法在控制中的特征：研究了多约束和/或非凸约束，特别是由凸约束逻辑和描述的艾德约束的随机化算法的基本过程。(iii)相关应用：该方法进一步应用于鲁棒辨识，其中的隶属度集的大小估计概率存在的干扰和参数的不确定性。

Randomized algorithms are recently recognized as viable tools for control systems design. In fact, in this research field, more and more complex problems arise, and probabilistic approach gives efficient algorithms to solve these problems. For example, robust optimal control is recast as a global optimization in general, where the size of the problem is quite large due to uncertainty contained in systems. Thus, to solve it within a reasonable time is very difficult via deterministic algorithms, which is known as the curse of dimensionality. On the other hand, algorithms based on random sampling of uncertainties enables us to obtain a probabilistic solution, without heavy computational effort, though a certain risk-level should be accepted.In this research project, analysis and synthesis of control systems via randomized algorithms are further investigated, which leads to a novel framework of control systems theory. In particular the following important results are obtained.(i) Randomized Algorithms for Analysis and Synthesis of Control Systems: Specific randomized algorithms are developed for probabilistic design of guaranteed cost regulator, fixed order controller, and switched systems.(ii) Characterizations of Randomized Algorithms in Control: Fundamental procedures of randomized algorithms are investigated for typical control problems with multiple constraints and/or nonconvex constraints, in particular, constraints described ed by a logical sum of convex constraints.(iii) Related Applications: The approach is further applied to a robust identification, where the size of the membership set is estimated probabilistically in the presence of disturbance and parameter uncertainty.