混沌与超混沌系统生成及控制若干问题
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摘要
自从1963年Lorenz在数值实验中发现第一个混沌模型以来,混沌在许多科学领域得到了巨大的发展,并导致了其后关于复杂非线性现实世界的革命性的再思考。混沌与超混沌系统的生成与控制是混沌理论与应用研究的两个重要组成部分。
本文给出了几类具有新的动力学特性的混沌和超混沌系统,研究了永磁同步电动机系统的混沌控制,研究了一类混沌与超混沌系统的输入输出稳定性控制问题,对超混沌系统的广义追踪控制进行了进一步的研究。主要工作归纳如下：
1.基于Lorenz系统,构建了一个新的三维混沌系统。讨论了平衡点的性质,给出了系统的功率谱图、Poincare截面图,并利用分岔图和Lyapunov指数谱详细分析了各参数变化对系统动力学行为的影响。进一步的理论分析发现,交叉乘积项参数d和平方项参数e变化时,系统的Lyapunov指数谱保持恒定,且参数d具有全局非线性调幅功能,参数e具有局部非线性调幅功能。并且根据乘法器调幅模型,采用乘法运算来实现混沌信号(状态变量)对余弦周期信号的振幅调制,研究发现在参数d和e的变化(增加)过程中,调幅系数也随之变化(增加),且信息信号幅值取较大值时,调幅系数也较大,从而信噪比和功率利用率都较高。同时,基于拓扑马蹄理论,理论上证明了系统的混沌性,设计出的模拟电路进一步验证了混沌吸引子的存在性。
2.通过在一个三维混沌系统中加入状态反馈控制器,构造出一个新的四维超混沌系统。分析表明,随着不同参数变化该系统呈现周期、复杂周期、准周期、混沌及超混沌运动。同时设计出模拟电子电路对该超混沌系统进行了实验验证。
3.提出了一个新的超混沌系统,此系统具有一个平衡点,却表现出四翼超混沌行为。仔细分析了其动力学特性,包括平衡点稳定性、分叉图、李氏指数。同时设计出对应的模拟电路,实验验证了超混沌系统吸引子的存在性。
4.基于对称群理论,提出了一类环状Chua系统。详细分析了其形成机制和参数选取范围,给出了DSP实现。
5.基于LaSalle不变集定理,设计了一种自适应控制器,实现了永磁同步电动机的混沌控制,并对该控制方案的改进形式进行了研究。研究了分数阶永磁同步电动机的动力学行为,并给出了其混沌控制方案。
6.研究了一类存在不确定参数和外界噪声干扰的混沌与超混沌系统的输入输出稳定性控制问题。基于Lyapunov稳定性理论,设计了一个线性状态反馈控制器,该控制器对于任何给定的有界干扰,都能保证系统渐近稳定并获得有界的状态变量。通过求解线性矩阵不等式(linear matrix inequality, LMI),能够方便地获得控制器的控制强度矩阵。而且,基于动力学方程的非奇异坐标变换,得到了一个简化的线性状态反馈控制器。以控制Lorenz系统和超混沌Lorenz-Stenflo系统为例,数值验证了该控制方案的有效性。
7.对超混沌系统的广义追踪控制进行了研究。具体包括：(1)设计一种统一形式的非线性状态反馈控制器,实现一类五阶超混沌电路系统的状态变量与任意给定参考信号的追踪广义投影同步。通过取不同的比例因子和加速因子,可以快速获得与超混沌系统和多个不同维混沌系统之间的异结构广义投影同步、周期信号的广义投影同步,以及将五阶超混沌系统快速控制到周期态和期望的平衡点。(2)通过构造新的Lyapunov函数,设计一个自适应追踪控制器,实现了一个超混沌系统对各种不同参考信号的单变量追踪控制。(3)对一类存在未知参数和外界随机干扰的超混沌系统的函数投影同步和追踪控制进行了研究。通过设计适当的自适应鲁棒控制器,按一定比例函数将超混沌系统驱动到期望的任意参考信号,同时辨识出此超混沌系统的未知参数。
In1963, Lorenz discovered the first chaotic system when studying the atmospheric convection. The Lorenz system is the first chaotic model, which has become a touchstone for the developing of chaos theory, and led to further think about the complexity of nonlinear phenomena in the real world. In the pursuit of theory and application research on chaos, the generation and control for chaotic and hyperchaotic systems became two important research directions.
This paper presented several chaotic and hyperchaotic systems, discussed stable control for the permanent magnet synchronous motor, investigated the issue of input-to-state stable control for a class of chaotic and hyperchaotic systems with undeterministic parameters and external interference, and investigated the tracking control of several hyperchaotic systems.
The main research works of this dissertation are as follows.
1. Based on the construction pattern of Chen and Liu chaotic systems, a new chaotic system is proposed by developing Lorenz chaotic system. The essential features of chaotic system are analyzed via equilibrium and stability, continuous spectrum, Poincare mapping. The different dynamic behaviors of the system are analyzed especially when changing each system parameter. It's found that when the parameters d and e vary, the Lyapunov exponent spectrum keeps invariable, and there exist the functions of global nonlinear amplitude adjuster for d and partial nonlinear amplitude adjuster for e. Then, based on the model of multiplier modulation, we realize the amplitude modulation of the sinusoidal signal via the chaotic singals. Finally, by picking a suitable cross-section with respect to the attractor carefully, a topological horseshoe of the corresponding first-returned Poincare map is found, thus giving a rigorous confirmation of the existence of chaos in this system, and a practical circuit is designed to implement this new chaotic system, which confirms that the chaotic system can be achieved physically.
2. A four-dimensional hyperchaotic system is proposed by adding a state feedback controller to a chaotic system. Numerical simulations and theoretical analysis show that this four-dimensional system will take on periodic, complex periodic, quasi-periodic, chaotic and hyperchaotic dynamical behaviours as parameters vary. Moreover, an electronic circuit diagram is designed for demonstrating the existence of the hyperchaos.
3. A novel four-dimensional smooth autonomous system is proposed, which is special since it has only one equilibrium, but it can generate a four-wing chaotic or hyperchaotic attractor. By applying either analytical or numerical methods, some basic properties of the4D system, such as phase diagrams, bifurcation diagram and Lyapunov exponents are investigated to observe chaotic motions.
4. A ring-scroll Chua chaotic system is proposed by introducing a generalized ring transformation. Some basic dynamical properties of this generalized ring transformation are discussed. The parameter regions and the periodic orbits, which are embedded in Chua chaotic attractor mapping to those in ring-scroll Chua chaotic attractor, are investigated, too. Finally, the ring-scroll Chua chaotic attractor is physically implemented by using digital signal processors.
5. Based on the LaSalle's invariant set theorem, an adaptive controller is developed to acquire chaos control for the permanent magnet synchronous motor. And then an extended adaptive controller is developed by introducing a control strength factor. The chaotic behaviour of the fractional-order permanent magnet synchronous motor is investigated, and an adaptive controller is developed based on the stability theory for fractional systems.
6. The issue of input-to-state stable control for a class of chaotic and hyperchaotic systems with undeterministic parameters and external interference is investigated. Based on the Lyapunov stable theory, a linear state feedback controller is presented to guarantee the asymptotic stability and achieve the bounded state variable for any bounded disturbance. The control strength matrix can be obtained by solving the linear matrix inequality (LMI). Furthermore, with a nonsingular coordinate transformation of the dynamics equation, a simplified linear state feedback controller is obtained. Finally, the illustration is given by using two different chaotic and hyperchaotic systems with numerical simulations to verify the proposed ISS control scheme.
7. The generalized tracking control is investigated. First, a nonlinear controller is proposed to acquire generalized projective synchronization of full states of a fifth-order circuit's hyperchaotic system based on adjusting the scaling factor and the accelerated factor. It allows one to drive the fifth-order system to arbitrary periodic orbits or fixed point rapidly. Based on this, hyperchaotic systems, chaotic systems, periodic signal, and constant signal are taken as examples respectively. Second, based on constructing a new Lyapunov function, an adaptive controller is proposed to acquire adaptive tracking control for a hyperchaotic system. Finally, the function projective synchronization and tracking control of a class of hyperchaotic systems with unknown parameters and disturbance are investigated. Based on the Lyapunov's stability theory, a robust controller is designed to drive the hyperchaotic system to any desired reference signals up to the scaling function factor, and a parameter update law for identifying the unknown parameters of the hyperchaotic system is gained simultaneously.
本文给出了几类具有新的动力学特性的混沌和超混沌系统,研究了永磁同步电动机系统的混沌控制,研究了一类混沌与超混沌系统的输入输出稳定性控制问题,对超混沌系统的广义追踪控制进行了进一步的研究。主要工作归纳如下：
1.基于Lorenz系统,构建了一个新的三维混沌系统。讨论了平衡点的性质,给出了系统的功率谱图、Poincare截面图,并利用分岔图和Lyapunov指数谱详细分析了各参数变化对系统动力学行为的影响。进一步的理论分析发现,交叉乘积项参数d和平方项参数e变化时,系统的Lyapunov指数谱保持恒定,且参数d具有全局非线性调幅功能,参数e具有局部非线性调幅功能。并且根据乘法器调幅模型,采用乘法运算来实现混沌信号(状态变量)对余弦周期信号的振幅调制,研究发现在参数d和e的变化(增加)过程中,调幅系数也随之变化(增加),且信息信号幅值取较大值时,调幅系数也较大,从而信噪比和功率利用率都较高。同时,基于拓扑马蹄理论,理论上证明了系统的混沌性,设计出的模拟电路进一步验证了混沌吸引子的存在性。
2.通过在一个三维混沌系统中加入状态反馈控制器,构造出一个新的四维超混沌系统。分析表明,随着不同参数变化该系统呈现周期、复杂周期、准周期、混沌及超混沌运动。同时设计出模拟电子电路对该超混沌系统进行了实验验证。
3.提出了一个新的超混沌系统,此系统具有一个平衡点,却表现出四翼超混沌行为。仔细分析了其动力学特性,包括平衡点稳定性、分叉图、李氏指数。同时设计出对应的模拟电路,实验验证了超混沌系统吸引子的存在性。
4.基于对称群理论,提出了一类环状Chua系统。详细分析了其形成机制和参数选取范围,给出了DSP实现。
5.基于LaSalle不变集定理,设计了一种自适应控制器,实现了永磁同步电动机的混沌控制,并对该控制方案的改进形式进行了研究。研究了分数阶永磁同步电动机的动力学行为,并给出了其混沌控制方案。
6.研究了一类存在不确定参数和外界噪声干扰的混沌与超混沌系统的输入输出稳定性控制问题。基于Lyapunov稳定性理论,设计了一个线性状态反馈控制器,该控制器对于任何给定的有界干扰,都能保证系统渐近稳定并获得有界的状态变量。通过求解线性矩阵不等式(linear matrix inequality, LMI),能够方便地获得控制器的控制强度矩阵。而且,基于动力学方程的非奇异坐标变换,得到了一个简化的线性状态反馈控制器。以控制Lorenz系统和超混沌Lorenz-Stenflo系统为例,数值验证了该控制方案的有效性。
7.对超混沌系统的广义追踪控制进行了研究。具体包括：(1)设计一种统一形式的非线性状态反馈控制器,实现一类五阶超混沌电路系统的状态变量与任意给定参考信号的追踪广义投影同步。通过取不同的比例因子和加速因子,可以快速获得与超混沌系统和多个不同维混沌系统之间的异结构广义投影同步、周期信号的广义投影同步,以及将五阶超混沌系统快速控制到周期态和期望的平衡点。(2)通过构造新的Lyapunov函数,设计一个自适应追踪控制器,实现了一个超混沌系统对各种不同参考信号的单变量追踪控制。(3)对一类存在未知参数和外界随机干扰的超混沌系统的函数投影同步和追踪控制进行了研究。通过设计适当的自适应鲁棒控制器,按一定比例函数将超混沌系统驱动到期望的任意参考信号,同时辨识出此超混沌系统的未知参数。
In1963, Lorenz discovered the first chaotic system when studying the atmospheric convection. The Lorenz system is the first chaotic model, which has become a touchstone for the developing of chaos theory, and led to further think about the complexity of nonlinear phenomena in the real world. In the pursuit of theory and application research on chaos, the generation and control for chaotic and hyperchaotic systems became two important research directions.
This paper presented several chaotic and hyperchaotic systems, discussed stable control for the permanent magnet synchronous motor, investigated the issue of input-to-state stable control for a class of chaotic and hyperchaotic systems with undeterministic parameters and external interference, and investigated the tracking control of several hyperchaotic systems.
The main research works of this dissertation are as follows.
1. Based on the construction pattern of Chen and Liu chaotic systems, a new chaotic system is proposed by developing Lorenz chaotic system. The essential features of chaotic system are analyzed via equilibrium and stability, continuous spectrum, Poincare mapping. The different dynamic behaviors of the system are analyzed especially when changing each system parameter. It's found that when the parameters d and e vary, the Lyapunov exponent spectrum keeps invariable, and there exist the functions of global nonlinear amplitude adjuster for d and partial nonlinear amplitude adjuster for e. Then, based on the model of multiplier modulation, we realize the amplitude modulation of the sinusoidal signal via the chaotic singals. Finally, by picking a suitable cross-section with respect to the attractor carefully, a topological horseshoe of the corresponding first-returned Poincare map is found, thus giving a rigorous confirmation of the existence of chaos in this system, and a practical circuit is designed to implement this new chaotic system, which confirms that the chaotic system can be achieved physically.
2. A four-dimensional hyperchaotic system is proposed by adding a state feedback controller to a chaotic system. Numerical simulations and theoretical analysis show that this four-dimensional system will take on periodic, complex periodic, quasi-periodic, chaotic and hyperchaotic dynamical behaviours as parameters vary. Moreover, an electronic circuit diagram is designed for demonstrating the existence of the hyperchaos.
3. A novel four-dimensional smooth autonomous system is proposed, which is special since it has only one equilibrium, but it can generate a four-wing chaotic or hyperchaotic attractor. By applying either analytical or numerical methods, some basic properties of the4D system, such as phase diagrams, bifurcation diagram and Lyapunov exponents are investigated to observe chaotic motions.
4. A ring-scroll Chua chaotic system is proposed by introducing a generalized ring transformation. Some basic dynamical properties of this generalized ring transformation are discussed. The parameter regions and the periodic orbits, which are embedded in Chua chaotic attractor mapping to those in ring-scroll Chua chaotic attractor, are investigated, too. Finally, the ring-scroll Chua chaotic attractor is physically implemented by using digital signal processors.
5. Based on the LaSalle's invariant set theorem, an adaptive controller is developed to acquire chaos control for the permanent magnet synchronous motor. And then an extended adaptive controller is developed by introducing a control strength factor. The chaotic behaviour of the fractional-order permanent magnet synchronous motor is investigated, and an adaptive controller is developed based on the stability theory for fractional systems.
6. The issue of input-to-state stable control for a class of chaotic and hyperchaotic systems with undeterministic parameters and external interference is investigated. Based on the Lyapunov stable theory, a linear state feedback controller is presented to guarantee the asymptotic stability and achieve the bounded state variable for any bounded disturbance. The control strength matrix can be obtained by solving the linear matrix inequality (LMI). Furthermore, with a nonsingular coordinate transformation of the dynamics equation, a simplified linear state feedback controller is obtained. Finally, the illustration is given by using two different chaotic and hyperchaotic systems with numerical simulations to verify the proposed ISS control scheme.
7. The generalized tracking control is investigated. First, a nonlinear controller is proposed to acquire generalized projective synchronization of full states of a fifth-order circuit's hyperchaotic system based on adjusting the scaling factor and the accelerated factor. It allows one to drive the fifth-order system to arbitrary periodic orbits or fixed point rapidly. Based on this, hyperchaotic systems, chaotic systems, periodic signal, and constant signal are taken as examples respectively. Second, based on constructing a new Lyapunov function, an adaptive controller is proposed to acquire adaptive tracking control for a hyperchaotic system. Finally, the function projective synchronization and tracking control of a class of hyperchaotic systems with unknown parameters and disturbance are investigated. Based on the Lyapunov's stability theory, a robust controller is designed to drive the hyperchaotic system to any desired reference signals up to the scaling function factor, and a parameter update law for identifying the unknown parameters of the hyperchaotic system is gained simultaneously.
引文
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