互补状态约束系统与微分包含的等价性研究

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摘要

实际的控制系统通常都存在着状态约束条件，其中带有互补状态约束的控制

系统作为混杂系统的一个子类已引起人们越来越多的关注．

到目前为止，人们已经对互补控制系统做了大量的研究，并取得了丰硕的研

究成果．由于互补控制系统实际上是一类混杂系统，同时又具有互补控制系统自

身的特性，因此互补控制系统的理论与应用研究是系统科学与系统控制领域当前

研究的热点之一．

在带有互补状态约束的控制系统中，前人已经研究了状态约束函数为光滑函

数时互补控制系统与无状态约束的微分包含系统的等价性．本文针对互补控制系

统的状态约束函数为非光滑函数的情形，得出了其与无状态约束的微分包含系统

的等价性．同时，给出了与非光滑控制系统稳定性密切相关的集合近似法锥的简

化表示，将集合的近似法锥化成了有限点集的凸包．

首先基于凸分析理论，通过构造不同的切锥和法锥，得出了带有非光滑凸状

态约束的互补控制系统和无状态约束的微分包含系统的等价性．其次考虑互补控

制系统的状态约束函数为次可微、拟凸函数的情形，利用非光滑分析理论将次可

微拟凸函数的次微分化成有限点集的凸包，从而进一步得到互补控制系统与无状

态约束的微分包含系统的等价性．由此，就可以对互补控制系统进行更加广泛、

深入的研究了．

最后研究了与互补控制系统稳定性密切相关的集合近似法锥的简化表示问

题．主要根据非光滑分析的理论，考虑了比函数上图更一般的集合近似法锥的表

达式．首先经过理论推导，得出了近似法锥与方向导数间的关系，进而得到近似

法锥的简化表示．

关键词：互补控制系统微分包含非光滑分析近似法锥

ABSTRACT

Recently, complementary problem has been applied to many practical

departments, such as mechanics, engineering, economics, traffics and so on. Along

with the development of the control theory, the researchers introduce complementary

conditions into the control theory and define a class of complementarity systems. The

complementarity system is a subclass of hybrid control systems, it has attracted much

more attentions of people. Studies of the complementarity system make the control

theory get farther developments.

By far, people have conducted much research work in complementarity systems,

and make a series of satisfied achievements. Owing to a complementarity system is

actually a class of hybrid systems, at the same time, it has its own characteristics, at

present, the topic of complementarity systems’ theories and applications is one of the

hot spots of studies in the field of system science and control theory.

This paper aims at the equivalence between the complementarity system and a

differential inclusion. It also gives the simple representation of the proximal normal

cone which is close to nonsmooth control theories, and we get some meaningful

results.

Firstly, we introduce the background and significance of researching, and give a

presentation of research status of complementarity systems. We describe the

complementary problem concisely and also provide the general model of the

complementarity system as well as concrete complementarity systems. At the same

time the nonsmooth analysis theory and some preliminary knowledge are presented.

Secondly, we discuss the equivalence between control systems with

complementarity constraints and differential inclusions. We mainly use convex

analysis to get the equivalence between the complementarity system and a differential

inclusion by constructing different tangent cone and normal cone, the functions related

to complementarity constraints are nonsmooth and convex. Also, based on nonsmooth

analysis, we give the equivalence of the systems again, and the functions related to

complementarity constraints are subdifferentiable pseudoconvex at this time.

Thirdly, the problem of the simple representation of the proximal normal cone is

considered. On certain condition, the relation among the proximal normal cone and the

directional derivative is found. It benefits to the advanced research of the

complementarity system.

At last, we sum up this paper, at the same time, we conceive the future researches

about complementarity systems and the proximal normal cone.

Key Words: complementarity systems, differential inclusion,

nonsmooth analysis, proximal normal cone

中文摘要

ABSTRACT

第一章绪论 ................................................................................................................. 1

§1.1 研究背景及意义 ................................................................................................1

§1.2 国内外研究现状 ................................................................................................2

§1.3 研究内容和章节结构 ........................................................................................3

第二章预备知识 ......................................................................................................... 5

§2.1 非光滑分析理论 ................................................................................................5

§2.2 互补控制系统综述 ..........................................................................................10

§2.2.1 互补问题的简要介绍 ............................................................................... 10

§2.2.2 互补控制系统模型 ................................................................................... 11

§2.2.3 例子 .......................................................................................................... 12

§2.3 微分包含简介 ..................................................................................................15

第三章凸互补状态约束系统与微分包含的等价性 ............................................... 17

§3.1 引言 ..................................................................................................................17

§3.2 问题的提出 ......................................................................................................17

§3.3 系统的等价性 ..................................................................................................18

§3.3.1 含有线性等式约束的情况 ....................................................................... 18

§3.3.2 非光滑凸函数约束情形 ........................................................................... 19

§3.4 例子 ..................................................................................................................21

§3.5 小结 ..................................................................................................................22

第四章拟凸互补状态约束系统与微分包含的等价性 ........................................... 23

§4.1 引言 ..................................................................................................................23

§4.2 问题的描述 ......................................................................................................23

§4.3 系统的等价性 ..................................................................................................25

§4.4 例子 ..................................................................................................................28

§4.5 小结 ..................................................................................................................28

第五章近似法锥的简化表示 ................................................................................... 31

§5.1 引言 ..................................................................................................................31

§5.2 相关理论 ..........................................................................................................31

§5.3 近似法锥和方向导数的关系 ..........................................................................32

§5.4 近似法锥的拟微分表示 ..................................................................................34

§5.5 例子 ..................................................................................................................35

§5.6 小结 ..................................................................................................................36

第六章结论与展望 ................................................................................................... 39

§6.1 结论 ..................................................................................................................39

§6.2 未来工作设想 ..................................................................................................39

参考文献 ..................................................................................................................... 41

在读期间公开发表的论文和承担科研项目及取得成果 ......................................... 45

致谢 ............................................................................................................................. 47

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摘要：

摘要实际的控制系统通常都存在着状态约束条件，其中带有互补状态约束的控制系统作为混杂系统的一个子类已引起人们越来越多的关注．到目前为止，人们已经对互补控制系统做了大量的研究，并取得了丰硕的研究成果．由于互补控制系统实际上是一类混杂系统，同时又具有互补控制系统自身的特性，因此互补控制系统的理论与应用研究是系统科学与系统控制领域当前研究的热点之一．在带有互补状态约束的控制系统中，前人已经研究了状态约束函数为光滑函数时互补控制系统与无状态约束的微分包含系统的等价性．本文针对互补控制系统的状态约束函数为非光滑函数的情形，得出了其与无状态约束的微分包含系统的等价性．同时，给出了与非光滑控制系统稳定性密切相...

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