分数阶混沌系统及其同步控制研究
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摘要
分数阶微积分理论的研究已有300多年的历史,但长久以来的研究者主要集中在数学领域里。直到1983年Mandelbort首次指出自然界以及许多科学技术领域中存在大量分维数的事实,且在整数与分数部分之间存在自相似现象以后,作为分形几何和分维数的动力学基础,分数阶微积分才重新获得了新的发展并成为当前国际上的一个热点研究课题。分数阶动力学系统的混沌控制与混沌同步已成为非线性领域研究的重点,但分数阶混沌控制与同步的策略和方法还较少,尚处在研究初期。由于分数阶混沌系统在保密通信等领域中拥有潜在的应用前景,在这一领域将会有更大的发展空间。
本文利用计算机数值模拟的方法研究了分数阶混沌系统及其同步控制问题。基本内容包括以下几个方面:(1)介绍了混沌的发展历程、混沌的特性及通向混沌的道路。研究了国内外分数阶混沌系统及其同步发展的现状,总结了所取得的成果及存在的不足。(2)介绍了分数阶混沌系统的数值模拟方法。(3)基于分数阶系统的非线性观测器理论和稳定性理论的结合,推导出一类分数阶混沌系统的状态观测器同步设计方法,得到了此类状态不能全部测量的分数阶混沌系统的同步方案。(4)对一个新的分数阶混沌系统的动态行为进行研究。基于分数阶稳定性理论,推导出该分数阶系统在对称阶和非对称阶时产生混沌的必要条件;最后分别运用改进的非线性观测器控制方法和主动反馈控制方法实现了新的分数阶混沌系统的自同步。(5)在对称阶和非对称阶两种情况下,分别对一个新的分数阶超混沌系统的运行状态进行讨论,然后运用主动反馈控制分别实现了该分数阶超混系统的自同步和该系统与分数阶Chen超混沌系统的异结构同步。(6)首先对整数阶简单互联电力系统的混沌和分叉的研究现状作简要介绍;然后基于分数阶理论,对分数阶互联电力系统模型进行仿真分析;最后,分别运用非线性反馈控制和主动反馈控制方法,实现了分数阶互联电力系统混沌振荡的同步控制。
The study on fractional order calculus theory has a history more than 300 years, however, the scholars studying on fractional order calculus theory mainly exist in the field of math during the past long period. In1983, Mandelbort initially suggested that there are plenty of fractal dimension phenomena in the field of nature and the field of science and technology, and there are self-similarities between integer and fraction. Then, as the dynamics foundation of geometry and fractal dimension, fractional order calculus gained new development and become a hot issue of the international. The synchronization control of fractional order system is the focus of recent researches, but chaotic synchronization methods are limited. It is believed that the chaos synchronization will play an important role in fields such as secure communication in the future.
In this paper, fractional order chaotic systems and their synchronization are studied by numerical simulation. The main contents can be divided into six parts: The first part, the development process of chaos, the characteristics of chaos and the road leading to chaos are introduced. The development of fractional order chaos systems and their synchronization are investigated at home and abroad, the results achieved and the existing shortcomings are summarized. In the second part, the numerical simulation methods for fractional order chaos system are given. In the thirdly part, based on the nonlinear observer and stability theory of fractional order systems, a class of fractional synchronization of chaotic systems state observer are designed, has been such a state can not be fully measured fractional order chaotic systems synchronization program. In the fourth part, a new fractional order chaos system is invesgated. Based on the stability theory of fractional order systems, we analyse stability condition for chaos to exist in the commensurate and incommensurate new fractional order system respectively. At last, based on active control method and observe-based control method, two identical new fractional order chaotic systems achieve synchronization respectively. In the fifth part, a new fractional order hyperchaos system is studied. We analyse the dynamics behavior of the commensurate and incommensurate new fractional order hyperchaotic system respectively, followed by the use of active feedback control, two identical new fractional order hyperchaotic systems and nonidentical new fractional order hyperchaotic systems achieve synchronization. In the sixth part, the model of integer-order simple interconnected power system and the bifurcation of the chaotic status are introducted, and then based on fractional order theory, the model of fractional order interconnected power system is analysed by numerical simulation. Finally, based on nonlinear feedback control and active feedback control method, we realize the synchronization control for the fractional order interconnected power system chaotic oscillation respectively.
本文利用计算机数值模拟的方法研究了分数阶混沌系统及其同步控制问题。基本内容包括以下几个方面:(1)介绍了混沌的发展历程、混沌的特性及通向混沌的道路。研究了国内外分数阶混沌系统及其同步发展的现状,总结了所取得的成果及存在的不足。(2)介绍了分数阶混沌系统的数值模拟方法。(3)基于分数阶系统的非线性观测器理论和稳定性理论的结合,推导出一类分数阶混沌系统的状态观测器同步设计方法,得到了此类状态不能全部测量的分数阶混沌系统的同步方案。(4)对一个新的分数阶混沌系统的动态行为进行研究。基于分数阶稳定性理论,推导出该分数阶系统在对称阶和非对称阶时产生混沌的必要条件;最后分别运用改进的非线性观测器控制方法和主动反馈控制方法实现了新的分数阶混沌系统的自同步。(5)在对称阶和非对称阶两种情况下,分别对一个新的分数阶超混沌系统的运行状态进行讨论,然后运用主动反馈控制分别实现了该分数阶超混系统的自同步和该系统与分数阶Chen超混沌系统的异结构同步。(6)首先对整数阶简单互联电力系统的混沌和分叉的研究现状作简要介绍;然后基于分数阶理论,对分数阶互联电力系统模型进行仿真分析;最后,分别运用非线性反馈控制和主动反馈控制方法,实现了分数阶互联电力系统混沌振荡的同步控制。
The study on fractional order calculus theory has a history more than 300 years, however, the scholars studying on fractional order calculus theory mainly exist in the field of math during the past long period. In1983, Mandelbort initially suggested that there are plenty of fractal dimension phenomena in the field of nature and the field of science and technology, and there are self-similarities between integer and fraction. Then, as the dynamics foundation of geometry and fractal dimension, fractional order calculus gained new development and become a hot issue of the international. The synchronization control of fractional order system is the focus of recent researches, but chaotic synchronization methods are limited. It is believed that the chaos synchronization will play an important role in fields such as secure communication in the future.
In this paper, fractional order chaotic systems and their synchronization are studied by numerical simulation. The main contents can be divided into six parts: The first part, the development process of chaos, the characteristics of chaos and the road leading to chaos are introduced. The development of fractional order chaos systems and their synchronization are investigated at home and abroad, the results achieved and the existing shortcomings are summarized. In the second part, the numerical simulation methods for fractional order chaos system are given. In the thirdly part, based on the nonlinear observer and stability theory of fractional order systems, a class of fractional synchronization of chaotic systems state observer are designed, has been such a state can not be fully measured fractional order chaotic systems synchronization program. In the fourth part, a new fractional order chaos system is invesgated. Based on the stability theory of fractional order systems, we analyse stability condition for chaos to exist in the commensurate and incommensurate new fractional order system respectively. At last, based on active control method and observe-based control method, two identical new fractional order chaotic systems achieve synchronization respectively. In the fifth part, a new fractional order hyperchaos system is studied. We analyse the dynamics behavior of the commensurate and incommensurate new fractional order hyperchaotic system respectively, followed by the use of active feedback control, two identical new fractional order hyperchaotic systems and nonidentical new fractional order hyperchaotic systems achieve synchronization. In the sixth part, the model of integer-order simple interconnected power system and the bifurcation of the chaotic status are introducted, and then based on fractional order theory, the model of fractional order interconnected power system is analysed by numerical simulation. Finally, based on nonlinear feedback control and active feedback control method, we realize the synchronization control for the fractional order interconnected power system chaotic oscillation respectively.
引文
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