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Study on the construction of control theory by the notion of invariance and the development of its introductory course

用不变性概念构建控制理论的研究及其入门课程的开发

基本信息

批准号：
13650479
负责人：
SHIMA Masasuke
金额：
$ 1.66万
依托单位：
Hokkaido University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

Firstly, the role of invariance in the control theory is studied [1]Invariance with respect to the variation of parameters : Using Frazer-Duncan theorem and monotonicity conditions, we derived a necessary and sufficient condition for the stability of linear systems without regard to the variation of interval parameters, and presented a monotonization method for the system which is not monotone. [2]Feedback design considering structures of Lie algebra : Feedback stabilization and improvement of tracking properties ties are studied for the under actuated mechanical systems such as axi-symmetric underactuated aircraft and PVTOL aircraft [3]Invriant formulation of design theory using invariance condition : Modem trends of studies of analytical dynamics with constraints are surveyed with focus to Poisson geometry, Lie algebroid and applications to control theory [4]Invariance optimality and Lie algebra : The existence region of optimal singular trajectories is classified by the dimensions of accessibility algebra, state space and input space A condition of optimality of Bang-bang control, conditions for the existence of optimal singular controllers and an approximation method of time optimal control are presented. Secondly, considering the shove results, we looked for the possibility of unified theory [5]Structure of control theory and development of an introductory course : Using the canonical equations and variations of inputs and initial conditions, we derived a variational expression of performance criterion or output and found that the expression is very convenient to survey the whole structure of control theory and has the possibility of constructing an introductory course to control theory. This is the main objective of this study attained in the last stage, though the whole contents are not prepared yet.

首先，研究了不变性在控制理论中的作用[1]关于参数变化的不变性：利用Frazer-Duncan定理和单调性条件，我们得到了线性系统在不考虑区间参数变化的情况下稳定的充要条件，并给出了系统非单调的单调化方法。[2]考虑李代数结构的反馈设计：针对欠驱动机械系统，如轴对称欠驱动飞机和PVTOL飞机，研究了反馈镇定和跟踪性能的改善[3]利用不变条件提出了设计理论：综述了带约束分析动力学研究的现代趋势，重点介绍了泊松几何、李代数体及其在控制理论中的应用[4]不变性最优性和李代数：根据可达代数、状态空间和输入空间的维度对最优奇异轨迹的存在区域进行了分类。给出了Bang-bang控制最优性的条件、最优奇异控制器存在的条件和时间最优控制的一种近似方法。其次，考虑到前人的研究结果，我们寻找了控制理论的统一理论[5]结构和开发导论课程的可能性：利用正则方程以及输入和初始条件的变化，我们导出了性能指标或输出的变分表达式，发现该表达式非常方便地考察了控制理论的整体结构，并且具有构建控制理论导论课程的可能性。这是本研究在最后一个阶段达到的主要目标，虽然整个内容还没有准备好。