分数阶混沌稳定性理论及同步方法研究
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摘要
混沌现象普遍存在于自然科学和社会科学中。混沌现象通常是有害的。在控制领域，混沌现象会降低设备的控制性能、加剧设备的损坏。在测试技术领域，混沌现象会使测量的结果混有混沌噪声，因而必须研究如何去除混沌噪声提高测量精度。本文重点研究了利用混沌同步降低混沌噪声的方法和更具有普遍性的分数阶混沌同步方法。
分数阶混沌系统稳定性理论是正在发展中的研究领域，其稳定性理论尚不如整数阶稳定性理论成熟，分数阶混沌同步方法发展受得了制约，针对目前研究现状，本文主要做了以下工作：
1、建立了真分数阶系统和假分数阶系统的概念，提出了对分数阶系统进行分段研究的思路。
2、根据分数阶线性系统的稳定性理论，基于“如果分数阶非线性系统在变量的变化域内都满足分数阶系统稳定性理论则该分数阶系统一定稳定”的思想，提出并证明了分数阶非线性系统稳定的三个判据：1)其整数阶系统稳定则对应的真分数阶系统也稳定的理论；2)h函数稳定理论，3)Lyapunov方程的稳定理论。
3、对整数阶Chua's系统和Lorenz系统的混沌特性进行了分析。引入“中间过程”理论，建立了分数阶Chua's系统和分数阶Lorenz系统的物理模型。并分析了分数阶Chua's系统和分数阶Lorenz系统混沌特性与其对应的分数阶阶次的关系。
4、提出了矩阵配置法的控制器设计方法，利用该方法不仅实现了参数未知的分数阶Lorenz混沌系统自适应同步及未知参数辩识，还实现了分数阶超混沌CYQY系统与参数未知的分数阶超混沌Lorenz系统异结构同步：将Backstepping控制器设计方法拓展到了分数阶系统，实现了分数阶Newton-Leipnik混沌系统的同步；将线性反馈方法运用于分数阶系统中，设计自适应规则，实现了参数未知的分数阶L(u|¨)混沌系统同步，较好地解决了反馈系数难以确定的问题。
5、设计了分数阶蔡氏无量纲状态方程的电路。首先对分数阶蔡氏无量纲状态方程各变量进行比例压缩变换、分数阶微分等效转换；其次根据变换后的方程设计出各模块电路，再将各模块按方程中各状态变量的对应关系联结起来。整个电路只由反相加法器、分数阶微分等效电路、反相器、控制器几大模块构成，电路结构对称。电路实验结果与计算机模拟结果完全符合，实现了分数阶蔡氏电路同步。
Chaos universally exists in natural sciences and social sciences. Chaos is usually harmful. In controllingarea, it will cut down the controllability of the equipment controlled and exacerbate the damage toequipment. In testing area, the measuring results will be mixed by chaotic noise, and then it necessary toremove chaotic noise to ensure measure precision. Hence, it is necessary to study how to remove chaoticnoise to improve measure precision. This paper puts emphasis on studying the approach of removing noiseby synchronizing chaotic systems and the universally approach of synchronizing fractional chaotic system.
As the stable theory for fractional chaotic systems is a developing area and imperfect, it hinders thedevelopment of the approach synchronizing and controlling fractional chaotic system. The maincontribution of this paper is about fractional system as follows:
1.Fractional systems are distinguished into proper fraction systems and improper fraction systems andthe idea that different types of systems are studied respectively is proposed.
2.Three stable theories about proper fractional nonlinear system are proposed and proved: 1)the theorem that a fractional system with order not more than 1 is stable if integer order formof the fractional system is stable; 2) h function stable theorem; 3) Lyapunov function stabletheorem.
3.Integer order Chua's chaotic system and integer order Lorenz chaotic are analyzed in this paper.Fractional physical models of Chua's chaotic system and Lorenz chaotic system are respectivelyintroduced according to "middle processing" theorem. The relations between fractional chaotic system andfractional order are respectively analyzed.
4.1) Configurating special matrix approach is proposed. Synchronizing not only fractional chaoticLorenz system with unknown parameters but also fractional chaotic CYQY system withfractional chaotic Lorenz system with unknown parameters is realized based on this approach.2) Backstepping approach is extended from integer order system to fractional order system. Synchronizingfractional order chaotic Newton-Leipnik system is realized by designing controllerbased on backstepping approach. 3) Linear feedback approach is used to synchronize fractionalchaotic system with unknown parameters and unknown parameter update rule is also designed. Theproblem how to fix on feedback coefficient is also resolved。
5.Fractional order chaotic Chua's circuit is designed in this paper. First, viriable-scale reduction, fractional to integral are made on the Chua's non-dimension state equation. Then each module circuit basedon the state equation is designed and linked with the other circuits by corresponding relationships betweenstate variables. Then each module circuit based on the state equation is designed and linked with the othercircuits by corresponding relationships between state variables. The circuit consists of inverted adder,fractional differential, inverter and controller, and has symmetric structure. Hardware experiment results arein good agreement with the computer simulations and synchronizing fractional chaotic Chua's system isrealized with circuit.
分数阶混沌系统稳定性理论是正在发展中的研究领域，其稳定性理论尚不如整数阶稳定性理论成熟，分数阶混沌同步方法发展受得了制约，针对目前研究现状，本文主要做了以下工作：
1、建立了真分数阶系统和假分数阶系统的概念，提出了对分数阶系统进行分段研究的思路。
2、根据分数阶线性系统的稳定性理论，基于“如果分数阶非线性系统在变量的变化域内都满足分数阶系统稳定性理论则该分数阶系统一定稳定”的思想，提出并证明了分数阶非线性系统稳定的三个判据：1)其整数阶系统稳定则对应的真分数阶系统也稳定的理论；2)h函数稳定理论，3)Lyapunov方程的稳定理论。
3、对整数阶Chua's系统和Lorenz系统的混沌特性进行了分析。引入“中间过程”理论，建立了分数阶Chua's系统和分数阶Lorenz系统的物理模型。并分析了分数阶Chua's系统和分数阶Lorenz系统混沌特性与其对应的分数阶阶次的关系。
4、提出了矩阵配置法的控制器设计方法，利用该方法不仅实现了参数未知的分数阶Lorenz混沌系统自适应同步及未知参数辩识，还实现了分数阶超混沌CYQY系统与参数未知的分数阶超混沌Lorenz系统异结构同步：将Backstepping控制器设计方法拓展到了分数阶系统，实现了分数阶Newton-Leipnik混沌系统的同步；将线性反馈方法运用于分数阶系统中，设计自适应规则，实现了参数未知的分数阶L(u|¨)混沌系统同步，较好地解决了反馈系数难以确定的问题。
5、设计了分数阶蔡氏无量纲状态方程的电路。首先对分数阶蔡氏无量纲状态方程各变量进行比例压缩变换、分数阶微分等效转换；其次根据变换后的方程设计出各模块电路，再将各模块按方程中各状态变量的对应关系联结起来。整个电路只由反相加法器、分数阶微分等效电路、反相器、控制器几大模块构成，电路结构对称。电路实验结果与计算机模拟结果完全符合，实现了分数阶蔡氏电路同步。
Chaos universally exists in natural sciences and social sciences. Chaos is usually harmful. In controllingarea, it will cut down the controllability of the equipment controlled and exacerbate the damage toequipment. In testing area, the measuring results will be mixed by chaotic noise, and then it necessary toremove chaotic noise to ensure measure precision. Hence, it is necessary to study how to remove chaoticnoise to improve measure precision. This paper puts emphasis on studying the approach of removing noiseby synchronizing chaotic systems and the universally approach of synchronizing fractional chaotic system.
As the stable theory for fractional chaotic systems is a developing area and imperfect, it hinders thedevelopment of the approach synchronizing and controlling fractional chaotic system. The maincontribution of this paper is about fractional system as follows:
1.Fractional systems are distinguished into proper fraction systems and improper fraction systems andthe idea that different types of systems are studied respectively is proposed.
2.Three stable theories about proper fractional nonlinear system are proposed and proved: 1)the theorem that a fractional system with order not more than 1 is stable if integer order formof the fractional system is stable; 2) h function stable theorem; 3) Lyapunov function stabletheorem.
3.Integer order Chua's chaotic system and integer order Lorenz chaotic are analyzed in this paper.Fractional physical models of Chua's chaotic system and Lorenz chaotic system are respectivelyintroduced according to "middle processing" theorem. The relations between fractional chaotic system andfractional order are respectively analyzed.
4.1) Configurating special matrix approach is proposed. Synchronizing not only fractional chaoticLorenz system with unknown parameters but also fractional chaotic CYQY system withfractional chaotic Lorenz system with unknown parameters is realized based on this approach.2) Backstepping approach is extended from integer order system to fractional order system. Synchronizingfractional order chaotic Newton-Leipnik system is realized by designing controllerbased on backstepping approach. 3) Linear feedback approach is used to synchronize fractionalchaotic system with unknown parameters and unknown parameter update rule is also designed. Theproblem how to fix on feedback coefficient is also resolved。
5.Fractional order chaotic Chua's circuit is designed in this paper. First, viriable-scale reduction, fractional to integral are made on the Chua's non-dimension state equation. Then each module circuit basedon the state equation is designed and linked with the other circuits by corresponding relationships betweenstate variables. Then each module circuit based on the state equation is designed and linked with the othercircuits by corresponding relationships between state variables. The circuit consists of inverted adder,fractional differential, inverter and controller, and has symmetric structure. Hardware experiment results arein good agreement with the computer simulations and synchronizing fractional chaotic Chua's system isrealized with circuit.
引文
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