硕士学位论文
基于微分几何理论的力学系统
建模、分析及非线性控制
DIFFERENTIAL GEOMETRY AND MECHANICAL SYSTEMS MODELING ANALYSIS AND NONLINEAR CONTROL
李益群
鬼灵精哈尔滨工业大学
2013年6月
亦舒说国内图书分类号：O186.12 学校代码：10213 国际图书分类号：514 密级：公开
理学硕士学位论文
基于微分几何理论的力学系统
建模、分析及非线性控制
硕士研究生：李益群
导师：高广宏副教授
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申请学位：理学硕士
学科：应用数学
所在单位：数学系
答辩日期：2013年6月
授予学位单位：哈尔滨工业大学
Classified Index: O186.12
U.D.C:514
Disrtation for the Master Degree in Science
DIFFERENTIAL GEOMETRY AND MECHANICAL SYSTEMS MODELING ANALYSIS AND NONLINEAR CONTROL Candidate：Li Yiqun
Supervisor：Assoc Prof. Gao Guanghong Academic Degree Applied for：Master of Science
Speciality：Applied Mathematics
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Affiliation：Department of Mathematics
Date of Defence：June, 2013
Degree-Conferring-Institution：Harbin Institute of Technology
摘要
力学系统的非线性控制问题是非常有挑战性的跨学科课题。它与控制理论、几何力学、数值计算等学科有着密不可分的联系。国内外研究该内容的学者主要集中在数学、力学、控制、航空航天和机器人等专业领域，他们彼此研究的侧重点也有所不同。这篇文章系统地从建模、分析和控制律设计这三个方面介绍了微分几何（包括李群、李代数）等理论知识在非线性控制系统中的应用。
不同于通常牛顿力学与分析力学的方法，本文在力学系统建模的过程中引入微分流形的理论，得到了一类形式简介的几何模型。特别是对于刚体的位姿控制问题，通常的欧拉角，四元素等模型表示形式显得很复杂，这里我们用特殊正交群和特殊刚体群的形式来描述系统显得简洁而且保证了系统的几何结构。可以看到微分几何理论在非线性分析中的强大力量和潜力。
大雪纷飞的诗句本文将欠驱动航天器的无驱动力方向投影到球面上，从而将航天器的姿态稳定问题转化成球面上的轨迹规划问题。利用欧氏空间的测地线理论，构造了欠驱动航天器系统的反馈控制律。并用Lyapunov方法证明了它的指数收敛性。结合李群、李代数等知识设计了一类全驱动系统的控制律，并验证了它的几乎全局收敛性。通过理论分析和数值模拟，证明其在较大的初始误差下依然具有良好的控制效果。
名字特殊符号但是这些方法并不能应用于所有的系统，且对于以上控制方法的数值模拟我们采用的是常用的四阶龙格库塔方法，该方法具有物理耗散型，在长时间模拟的情况下，大大降低了仿真效果的可信性。是否能够给出一个保持系统拓扑结构的数值算法，这给我们提供了一个数值计算与几何控制交叉的研究方向。
关键词：微分几何；李群；非线性控制；力学系统
Abstract
Nonlinear control of mechanical systems is a very challenging discipline lies at the interction among many disciplines. It is cloly related with control theory, geometric mechanical and numerical analysis. Experts work on this discipline usually background in mathematics, mechanic, control theory, aerospace, robotics and so on. They also usually focus on different aspects. We mainly introduce how differential geometry (include Lie theory) is applied in nonlinear control system
s from modeling, analysis, and design.
Different from the classic Newton and Lagrange method for the modeling of mechanical systems, here differential manifold is ud in the modeling and finally we get a compact geometry model. The traditional model of kinematic equations and dynamic equations are always illustrate by Euler angle and unit quaternion which is complicate, here we u a class of lie group like SO(3) and SE(3) to describe the geometric structure of a body which makes the kinematic and dynamic equations looks much more tidier. So we can e the great potential of the u of differential geometry.接种hpv疫苗前后禁忌
Here we make a correspondence between the under-actuated direction of spacecraft and the point on a unit sphere and u the notion of geodesic in designing feedback control of an under-actuated spacecraft and prove the exponential convergence by Lyapunov method. We also give a feedback control law which can be ud in a class of systems and here theory of lie group is applied. We illustrate all the control law by numerical experiment.
Disappointedly, we can not u the theory we described above in all control systems. In addition, the method we ud in numerical analysis is 4-order R-K scheme which have dissipation, for the long time simulation the result is not trustable. Are there any other schemes that could hold the topologic
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structure of systems? We should consider this problem between numerical analysis and geometric control.
Keywords: differential geometry, lie group, nonlinear control, mechanical system
