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

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，给出了不考虑区间参数变化的线性系统稳定的充分必要条件，并给出了非单调系统的单调化方法。考虑李代数结构的反馈设计：研究了轴对称欠驱动飞机和垂直起降飞机等欠驱动机械系统的反馈镇定和跟踪性能关系的改善。本文综述了具有约束的解析动力学的现代研究趋势，重点介绍了泊松几何、李代数及其在控制理论、不变性最优性和李代数中的应用。根据可达性代数、状态空间和输入空间的维度划分了最优奇异轨迹的存在域，给出了Bang-bang控制的最优性条件、最优奇异控制器的存在性条件和时间最优控制的逼近方法。其次，结合已有的研究结果，寻找统一理论[5]控制理论结构和开设导论课程的可能性：利用正则方程和输入、初始条件的变化，推导出性能指标或输出的变分表达式，发现该表达式非常便于概览控制理论的整体结构，具有开设控制理论导论课程的可能性。这是本研究在最后阶段达到的主要目的，虽然整个内容还没有准备好。

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.