非线性混沌系统分析和控制问题的研究
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摘要
混沌学是一门新兴的非线性科学,它的研究热潮始于二十世纪七十年代,但是其渊源可以追溯到十九世纪三十年代。最近几十年来,在国内外众多学者的不懈努力下,混沌理论得到了迅速的发展,其研究内容也越来越广泛、深入。同时随着各学科间的相互交叉与渗透,混沌理论在许多领域得到了有效的应用。混沌研究对于现代科学的影响不仅限于自然科学,而是几乎覆盖了所有学科领域。那么从控制论的角度出发,自然考虑到混沌系统的控制问题,其中包括混沌的控制(混沌的抑制),混沌的反控制(混沌的产生和加强)以及混沌同步的控制等。
与此同时,分数阶系统的研究在近年来逐渐成为了热点。然而分数阶微积分的概念并不是一个全新的概念,它产生于三百多年前,可以说与整数阶微积分有着同样的历史渊源。分数阶微积分实际上是研究和应用任意阶微分和积分的理论,它是整数阶微积分的自然延伸。上个世纪后期,分数阶微积分的应用研究取得的一些成果,引起了不同学科领域学者们的广泛关注,使其应用领域有了显著地增加。现在人们已经认识到现实系统中许多是分数阶的,但迄今为止,所有的控制系统均是整数阶描述的,这实际上是忽略了系统的真实性。之所以将其考虑为整数阶,是因为系统本身的复杂性和缺乏有效的数学工具。随着分数阶微积分理论的发展,近些年来将分数阶微积分理论应用于控制理论和控制实践的研究已经开始,并不断取得进展。分数阶微积分的发展,为以整数阶微积分理论为基础的控制理论和控制工程提供了一个新的发展空间。其中,分数阶与混沌系统的结合,由于它们自身特性上的联系,如:分维,自相似等等,也变得越来越紧密。但是由于混沌系统和分数阶系统理论都还不成熟,很多问题还有待进一步地研究和探讨。
基于以上的阐述,论文的主要研究工作和创新包括以下两个方面:一方面是对整数阶非线性混沌系统的控制问题进行研究;另一方面是探讨控制和分析分数阶非线性混沌系统。具体可以分为:
-利用微扰判据的Melnikov方法讨论周期扰动参数引导Lorenz混沌系统进入低周期轨道的参数条件。
-提出了利用奇异单输入状态反馈混沌化系统的方法;并在Marroto定理基础上,从理论上证明经过反馈以后的闭环系统确实存在有Li-Yorke意义上的混沌。
-首次针对一类分数阶混沌系统,提出了利用状态的PIα调节器实现控制混沌的方法。同时从理论上证明了一个非线性分数阶系统的稳定性定理;在这个基础上,证明了所提出控制混沌方法的有效性。
-在整数阶系统分析基础上,将谐波平衡原则进行推广,用以分析非线性分数阶系统的混沌参数域。
-提出了两种方法来控制一类非同元次分数阶混沌系统的同步,并且根据不同方法的特性,分别在理论上给予证明。
Chaology is a rising nonlinear science. Its research upsurge started from 1970s’, butits origin could be traced back to 1830s’. In recent years, chaos theory is developed rapidlyunder the incessant efforts of scholars over the world, and the research areas of chaologyare also expanded thoroughly. With the intercrossing and penetration of different knowl-edge, chaos theory has been applied in many other fields effectively. The in?uences of chaosresearch towards modern sciences are not bounded in natural science, but cover almost allfields of science. Well, from the point of cybernetic, it yields a natural yet nontrivial ques-tion whether one can control chaos, including chaos control (chaos inhibition) anti-chaosControl(chaos producing and enhancing), and chaos synchronization.
Meanwhile, in recent years, the fractional-order systems become a hot research topic.The concepts of fractional derivatives and integral were proposed three hundred years ago.Fractional calculus is the theory on the research and application of derivatives and integralof arbitrary order. It is a natural extension of the classical mathematics. In the last three orfour decades, many researchers have made a great effort to apply this knowledge in practiceand in different research fields. Now, the application domains of fractional calculus have in-creased significantly. Real systems in general are fractional-order systems, although in sometypes of systems the order is very close to an integer order. The control systems used so farwere all considered as integer-order systems, regardless of the reality, the reason is the highercomplexity and the absence of adequate mathematical tool. Since major advances have beenmade in fractional calculus in the last few years, the knowledge of fractional calculus be-gins to be applied in control theory and control engineering. It provides new landscape forcontrol theory and control engineering based on integer-order differential equations. Amongthose, the research of the fractional-order systems and chaos systems become increasinglyclose, because of their own nature, such as the fractal dimension, self-similarity and so on.However, as the theory of chaotic systems and fractional-order systems being in developing,many issues still need further study and discussion.
Based on above, the research and contributions in this dissertation are divided into twoaspects: one is about controlling chaos in the integer-order nonlinear chaos systems; theother is study and discussion the analysis and controlling the fractional-order nonlinear chaossystems. Specifically,
- By using Melnikov method of perturbation criteria, the conditions of periodicparamtric perturbations to control chaos in Lorenz system are discussed.
- The single input state feedback approach for chaotifying a stable system is presented..Based on the Marotto theorem, it is proven theoretically that the closed-loop systemis chaotic in the sense of Li and Yorke.
- The PI~αregulator of system states to control a kind of fractional-order chaos systemsis first introduced in this dissertation. At same time, a stability theorem of nonlinearfractional-order differential equations is proven theoretically. Then, according to that,a new criterion is derived for designing the controller gains for stabilization this kindof fractional-order chaos systems.
- Based on the integer-order system, the harmonic balance principle is used to analyzethe parameter domain of nonlinear fractional-order system.
- Twoapproachestosynchronizeakindofincommensuratefractional-orderchaoticsys-tem are addressed in this thesis. According to the different characteristic, the differentprocesses of theoretical proof are presented respectively.
与此同时,分数阶系统的研究在近年来逐渐成为了热点。然而分数阶微积分的概念并不是一个全新的概念,它产生于三百多年前,可以说与整数阶微积分有着同样的历史渊源。分数阶微积分实际上是研究和应用任意阶微分和积分的理论,它是整数阶微积分的自然延伸。上个世纪后期,分数阶微积分的应用研究取得的一些成果,引起了不同学科领域学者们的广泛关注,使其应用领域有了显著地增加。现在人们已经认识到现实系统中许多是分数阶的,但迄今为止,所有的控制系统均是整数阶描述的,这实际上是忽略了系统的真实性。之所以将其考虑为整数阶,是因为系统本身的复杂性和缺乏有效的数学工具。随着分数阶微积分理论的发展,近些年来将分数阶微积分理论应用于控制理论和控制实践的研究已经开始,并不断取得进展。分数阶微积分的发展,为以整数阶微积分理论为基础的控制理论和控制工程提供了一个新的发展空间。其中,分数阶与混沌系统的结合,由于它们自身特性上的联系,如:分维,自相似等等,也变得越来越紧密。但是由于混沌系统和分数阶系统理论都还不成熟,很多问题还有待进一步地研究和探讨。
基于以上的阐述,论文的主要研究工作和创新包括以下两个方面:一方面是对整数阶非线性混沌系统的控制问题进行研究;另一方面是探讨控制和分析分数阶非线性混沌系统。具体可以分为:
-利用微扰判据的Melnikov方法讨论周期扰动参数引导Lorenz混沌系统进入低周期轨道的参数条件。
-提出了利用奇异单输入状态反馈混沌化系统的方法;并在Marroto定理基础上,从理论上证明经过反馈以后的闭环系统确实存在有Li-Yorke意义上的混沌。
-首次针对一类分数阶混沌系统,提出了利用状态的PIα调节器实现控制混沌的方法。同时从理论上证明了一个非线性分数阶系统的稳定性定理;在这个基础上,证明了所提出控制混沌方法的有效性。
-在整数阶系统分析基础上,将谐波平衡原则进行推广,用以分析非线性分数阶系统的混沌参数域。
-提出了两种方法来控制一类非同元次分数阶混沌系统的同步,并且根据不同方法的特性,分别在理论上给予证明。
Chaology is a rising nonlinear science. Its research upsurge started from 1970s’, butits origin could be traced back to 1830s’. In recent years, chaos theory is developed rapidlyunder the incessant efforts of scholars over the world, and the research areas of chaologyare also expanded thoroughly. With the intercrossing and penetration of different knowl-edge, chaos theory has been applied in many other fields effectively. The in?uences of chaosresearch towards modern sciences are not bounded in natural science, but cover almost allfields of science. Well, from the point of cybernetic, it yields a natural yet nontrivial ques-tion whether one can control chaos, including chaos control (chaos inhibition) anti-chaosControl(chaos producing and enhancing), and chaos synchronization.
Meanwhile, in recent years, the fractional-order systems become a hot research topic.The concepts of fractional derivatives and integral were proposed three hundred years ago.Fractional calculus is the theory on the research and application of derivatives and integralof arbitrary order. It is a natural extension of the classical mathematics. In the last three orfour decades, many researchers have made a great effort to apply this knowledge in practiceand in different research fields. Now, the application domains of fractional calculus have in-creased significantly. Real systems in general are fractional-order systems, although in sometypes of systems the order is very close to an integer order. The control systems used so farwere all considered as integer-order systems, regardless of the reality, the reason is the highercomplexity and the absence of adequate mathematical tool. Since major advances have beenmade in fractional calculus in the last few years, the knowledge of fractional calculus be-gins to be applied in control theory and control engineering. It provides new landscape forcontrol theory and control engineering based on integer-order differential equations. Amongthose, the research of the fractional-order systems and chaos systems become increasinglyclose, because of their own nature, such as the fractal dimension, self-similarity and so on.However, as the theory of chaotic systems and fractional-order systems being in developing,many issues still need further study and discussion.
Based on above, the research and contributions in this dissertation are divided into twoaspects: one is about controlling chaos in the integer-order nonlinear chaos systems; theother is study and discussion the analysis and controlling the fractional-order nonlinear chaossystems. Specifically,
- By using Melnikov method of perturbation criteria, the conditions of periodicparamtric perturbations to control chaos in Lorenz system are discussed.
- The single input state feedback approach for chaotifying a stable system is presented..Based on the Marotto theorem, it is proven theoretically that the closed-loop systemis chaotic in the sense of Li and Yorke.
- The PI~αregulator of system states to control a kind of fractional-order chaos systemsis first introduced in this dissertation. At same time, a stability theorem of nonlinearfractional-order differential equations is proven theoretically. Then, according to that,a new criterion is derived for designing the controller gains for stabilization this kindof fractional-order chaos systems.
- Based on the integer-order system, the harmonic balance principle is used to analyzethe parameter domain of nonlinear fractional-order system.
- Twoapproachestosynchronizeakindofincommensuratefractional-orderchaoticsys-tem are addressed in this thesis. According to the different characteristic, the differentprocesses of theoretical proof are presented respectively.
引文
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