无平衡点分数阶混沌系统的同步研究
发布时间：2019-04-22 09:06
【摘要】：分数阶系统是由微分阶次为任意实数甚至为任意复数的微分方程所描述的动力学系统。能更准确、更有效地描述实际系统中出现的复杂现象,现已被广泛应用于诸多高精尖领域。混沌系统是一类特殊的非线性系统。自二十世纪初学者们提出混沌系统以来,由于它具有的随机性和对初始条件的极端敏感性等特点,混沌系统的研究一直处于前沿课题。对于特殊的无平衡点的分数阶混沌系统来说,常规判定不能证明其混沌特性。对于这样的系统,本文的研究将着重以下几个方面:首先,研究了分数阶微积分的基本定义、性质以及它的稳定性理论。结合自然界和社会科学中普遍存在的混沌现象,介绍了整数阶混沌系统同步的基本方法。对一个新的无平衡点分数阶混沌系统进行了分析,利用分数阶微分变换方法,得到了它的解序列。研究了无平衡点分数阶混沌系统的Kaplan-Yorke维数和耗散性,然后基于系统的离散映射,根据QR分解方法得到最大Lyapunov特征指数,从而判断系统是否保持混沌。最后给出了一种全状态自适应控制方法,使系统的状态可以追踪期望轨迹。最后通过数值仿真,验证了该控制方法的有效性。其次,研究了一类参数不确定的无平衡点分数阶混沌系统的同步问题。根据自适应控制理论,将自适应控制法引入到混沌同步中。把它应用到一类新的无平衡点分数阶混沌系统中,设计了性能可靠的同步控制器和参数自适应律,实现了上述提到的无平衡点分数阶混沌系统与其自结构的同步。同时运用主动控制方法构造主动控制器对系统进行了同步研究。把误差系统中无益的非线性项通过设计控制器进行抵消,提出了一类简化分数阶混沌系统的同步控制器设计方法。与上述所研究的自适应法进行数值仿真对比,既验证了两种同步方法的有效性也体现出了两种方法所存在的差异。经过对比,可以得出改进的自适应控制法具有更快的控制速度和更高的控制效率,而主动控制方法具有更简单的计算过程。最后针对无平衡点的分数阶混沌系统,令其为驱动系统,令一个新的四维分数阶混沌系统为响应系统,采用自适应控制方法进行异结构同步研究。设计了合适的同步控制器和可以很好辨识参数的自适应律,实现了无平衡点分数阶混沌系统与新的四维分数阶混沌系统之间的异结构同步,并进行仿真,验证了该方法的有效性。
[Abstract]:Fractional-order systems are dynamical systems described by differential equations with differential orders of arbitrary real numbers or even arbitrary complex numbers. It can more accurately and effectively describe the complex phenomena in the actual system, and has been widely used in many high-precision fields. Chaotic system is a kind of special nonlinear system. Since chaos system was put forward by scholars in the early 20th century, because of its randomness and extreme sensitivity to initial conditions, the study of chaotic system has been in the forefront of the subject. For a special fractional chaotic system with no equilibrium point, the conventional decision can not prove its chaotic characteristics. For such a system, this paper will focus on the following aspects: firstly, the basic definition, properties and stability theory of fractional calculus are studied. Based on the common chaotic phenomena in nature and social science, this paper introduces the basic methods of synchronization of integer-order chaotic systems. In this paper, a new fractional chaotic system with no equilibrium point is analyzed, and its solution sequence is obtained by means of fractional differential transformation. The Kaplan-Yorke dimension and dissipation of fractional order chaotic systems without equilibrium point are studied. Then based on the discrete mapping of the system the maximum Lyapunov characteristic exponent is obtained according to the QR decomposition method so as to judge whether the system is chaotic or not. Finally, a full-state adaptive control method is presented to make the state of the system track the desired trajectory. Finally, the effectiveness of the control method is verified by numerical simulation. Secondly, the synchronization problem of a class of unequilibrium fractional chaotic systems with uncertain parameters is studied. According to the adaptive control theory, the adaptive control method is introduced into chaos synchronization. It is applied to a new class of fractional chaotic systems without equilibrium point, and the synchronization controller and parameter adaptive law with reliable performance are designed. The synchronization between the above mentioned fractional order chaotic system without equilibrium point and its self-structure is realized. At the same time, the active control method is used to construct the active controller to study the synchronization of the system. A synchronization controller design method for a class of simplified fractional chaotic systems is proposed by canceling the non-linear terms of the error system by designing a controller. Compared with the self-adaptive method mentioned above, the effectiveness of the two methods is verified and the difference between the two methods is demonstrated. Through comparison, it can be concluded that the improved adaptive control method has faster control speed and higher control efficiency, while the active control method has a simpler calculation process. Finally, for the fractional chaotic system without equilibrium point, it is a driving system, and a new four-dimensional fractional chaotic system is a response system. An adaptive control method is used to study the heterogeneous structure synchronization. A suitable synchronization controller and an adaptive law which can identify the parameters well are designed, and the heterogeneous synchronization between the unequilibrium fractional order chaotic system and the new four dimensional fractional chaotic system is realized, and the simulation is carried out. The validity of the method is verified.
【学位授予单位】：东北石油大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O415.5;TP273
，
本文编号：2462706
[Abstract]:Fractional-order systems are dynamical systems described by differential equations with differential orders of arbitrary real numbers or even arbitrary complex numbers. It can more accurately and effectively describe the complex phenomena in the actual system, and has been widely used in many high-precision fields. Chaotic system is a kind of special nonlinear system. Since chaos system was put forward by scholars in the early 20th century, because of its randomness and extreme sensitivity to initial conditions, the study of chaotic system has been in the forefront of the subject. For a special fractional chaotic system with no equilibrium point, the conventional decision can not prove its chaotic characteristics. For such a system, this paper will focus on the following aspects: firstly, the basic definition, properties and stability theory of fractional calculus are studied. Based on the common chaotic phenomena in nature and social science, this paper introduces the basic methods of synchronization of integer-order chaotic systems. In this paper, a new fractional chaotic system with no equilibrium point is analyzed, and its solution sequence is obtained by means of fractional differential transformation. The Kaplan-Yorke dimension and dissipation of fractional order chaotic systems without equilibrium point are studied. Then based on the discrete mapping of the system the maximum Lyapunov characteristic exponent is obtained according to the QR decomposition method so as to judge whether the system is chaotic or not. Finally, a full-state adaptive control method is presented to make the state of the system track the desired trajectory. Finally, the effectiveness of the control method is verified by numerical simulation. Secondly, the synchronization problem of a class of unequilibrium fractional chaotic systems with uncertain parameters is studied. According to the adaptive control theory, the adaptive control method is introduced into chaos synchronization. It is applied to a new class of fractional chaotic systems without equilibrium point, and the synchronization controller and parameter adaptive law with reliable performance are designed. The synchronization between the above mentioned fractional order chaotic system without equilibrium point and its self-structure is realized. At the same time, the active control method is used to construct the active controller to study the synchronization of the system. A synchronization controller design method for a class of simplified fractional chaotic systems is proposed by canceling the non-linear terms of the error system by designing a controller. Compared with the self-adaptive method mentioned above, the effectiveness of the two methods is verified and the difference between the two methods is demonstrated. Through comparison, it can be concluded that the improved adaptive control method has faster control speed and higher control efficiency, while the active control method has a simpler calculation process. Finally, for the fractional chaotic system without equilibrium point, it is a driving system, and a new four-dimensional fractional chaotic system is a response system. An adaptive control method is used to study the heterogeneous structure synchronization. A suitable synchronization controller and an adaptive law which can identify the parameters well are designed, and the heterogeneous synchronization between the unequilibrium fractional order chaotic system and the new four dimensional fractional chaotic system is realized, and the simulation is carried out. The validity of the method is verified.
【学位授予单位】：东北石油大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O415.5;TP273
，
本文编号：2462706
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