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Alternative TitleResearch on Some Problems on Simultaneous Stabilization of Linear Systems and Solutions to a Class of Global Optimization Problems Based on BOTTEMA
Thesis Advisor郁文生
Degree Grantor中国科学院研究生院
Place of Conferral中国科学院自动化研究所
Degree Discipline控制理论与控制工程
Keyword线性系统 同时镇定 不等式型定理机器证明 比利时巧克力问题 法国香槟问题 复分析 全局优化 Bottema程序包 Linear Systems Simultaneous Stabilization Automated Inequality-type Theorem Proving Belgian Chocolate Problem French Champagne Problem Complex Analysis Global Optimization Bottema
Abstract线性系统的同时镇定问题是系统与控制理论中的基本问题,有着重要的理论意义和广泛的应用价值。为揭示同时镇定问题的复杂性,学者们提出了若干公开问题。这些问题表述非常简单,实际解决起来却异常困难,而对这些公开问题的探讨对于系统与控制理论的研究具有深远意义。 在本论文的第一部分,主要针对线性系统同时镇定中的若干问题进行系统地研究,具体内容如下: 首先,简要回顾了同时镇定问题的研究状况,并对若干公开问题给出基本的描述。 其次,借助杨路等人新近发展的不等式型定理机器证明的理论与方法,研究了线性系统同时镇定中著名的比利时巧克力问题和法国香槟问题。对比利时巧克力问题,系统地给出了当线性控制器阶次不超过四阶时解的分布,以及相应的控制器数值算例,改进了已有文献中的结果,并根据得到的结果,提出了限定控制器阶次条件下该问题解分布的两个猜想。针对法国香槟问题较为系统地给出了当控制器阶次不超过三阶时解的分布,以及相应的控制器数值算例,改进了已有文献中的结果。 另外,结合Nehari给出的复分析理论结果,讨论了Blondel关于三个线性对象同时镇定问题非有理可决定性讨论中一处重要定理证明存在的缺陷。 在本论文的第二部分中,讨论了一类阶次较高的三角函数全局优化问题。针对杨路提出的一个15次三角函数在一定约束下全局最大值的公开问题,基于理论上的推导,借助新近发展的不等式机器证明程序包BOTTEMA中全局优化的解决方法,完全地给出了问题的解答。类似的方法还可以使目前所能够解决的此类问题阶次进一步提高。
Other AbstractSimultaneous stabilization of linear systems is a fundamental issue in system and control theory, and is of theoretical significance as well as practical interest. Some open problems have been proposed in the literature as indications of the potential complexity of simultaneous stabilization questions. These problems are simply-stated but are very hard to solve. The system and control community will benefit from the resolutions to these difficult problems. The first portion of this dissertation is devoted to some problems associated with simultaneous stabilization of linear systems. This part goes as follows: Firstly, researches on simultaneous stabilization are reviewed and some open problems are introduced. Secondly, two open problems, namely Belgian chocolate problem and French champagne problem are considered. For the Belgian chocolate problem, based on the recent development in automated inequality-type theorem proving, the exact of bounds which guarantee the existence of stabilizing controllers with fixed order no more than four are determined. Meanwhile, some numerical examples are worked out and improve the relevant results in the literature. In addition, two conjectures concerning this problem are formulated. For the French champagne problem, the explicit bounds which guarantee the existence of stabilizing controllers with fixed order no more than three are presented. Some numerical examples improve the relevant results in the literature. Thirdly, according to the results of complex analysis presented by Nehari, a condition proposed by Blondel is analyzed, which concerns the simultaneous stabilization of three linear systems. It is shown that there is an error in the sufficiency part of its proof. In the second portion of this dissertation, some high-degree global optimization problems concerning a class of triangle functions are considered. Based on a theoretical analysis and the algorithms as well as functions in the generic program BOTTEMA, an open problem posed by Yang which concerns the global optimization of a 15-th degree triangle function is solved completely. Meanwhile, the degree of triangle functions in solvable cases of similar problems can be improved by our method.
Other Identifier200518014628031
Document Type学位论文
Recommended Citation
GB/T 7714
何冠男. 线性系统同时镇定若干问题的研究与基于BOTTEMA的一类全局优化问题的解[D]. 中国科学院自动化研究所. 中国科学院研究生院,2008.
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