混沌系统控制及参数辨识的若干问题研究
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摘要
混沌运动是自然界中客观存在的、最终有界的、有一定随机规则的、非常复杂的运动形式。它广泛地存在于自然界。近十几年来,混沌科学与其它科学互相渗透,无论是在生物学、物理学、心理学、数学、物理学、电子学、信息科学,还是在天文学、气象学、经济学,甚至在音乐、艺术等领域都得到了广泛的应用,混沌控制与混沌同步控制成为了非线性科学中的研究热点。但整个混沌控制及同步理论仍然不够完善,许多混沌控制及同步方法有待进一步发掘,如何设计简单实用的控制器需要进一步研究。而且现有的混沌控制及同步控制方法在面对实际混沌系统时,待解决问题还有许多,如往往不能得到混沌系统的全部状态变量、实际混沌系统的参数往往是部分未知或全部未知的、系统参数存在着扰动等。本文针对混沌系统的控制(包括同步控制)及其未知参数辨识中的一些问题进行了深入的研究,主要包括以下几个方面:
根据Lorenz系统族本身的参数结构,利用最佳Lyapunov函数思想,严格地证明了可以利用最简单的线性反馈控制,实现Lorenz系统族平衡位置的全局指数稳定、全局渐近稳定,;改进和推广了一些现有文献中仅仅限于局部稳定的结果,而且得到了比现有文献更尽可能简洁、更少保守较小的条件和尽可能好的结论。
根据微分方程稳定性理论和一类混沌系统的结构,得出了一类混沌系统PC同步和基于单变量控制同步较简单的充分条件;讨论了在驱动系统存在参数扰动和外界扰动的情况下(分参数已知和未知两种情形)同步控制器的设计,实现了存在扰动情况下一类混沌系统的鲁棒同步;在参数未知的情况下,研究了在一类混沌系统中,维数相同但结构不同混沌系统的同步问题,给出了相应的参数自适应律和控制器的设计方法。
研究了基于部分状态变量混沌系统的同步控制。根据微分方程的稳定性理论以及Liu混沌系统、广义Lorenz混沌系统(四维)和一个新超混沌系统的具体结构,利用驱动系统单个状态变量或部分状态变量设计了合适的控制器实现了分别实现了两个完全相同的Liu混沌系统、广义Lorenz混沌系统(四维)和一个新超混沌系统的同步。
研究了基于未知参数观测器混沌系统的参数辨识问题。根据状态观测器思想和微分方程的稳定性理论,将混沌系统的未知参数看成系统的状态变量,以关新平等人的工作为基础,提出了选择增益函数和构造相应的辅助函数的一般方法,对混沌系统的任一未知参数设计了未知参数观测器,并给出了相关的五个推论,较大地扩展了未知参数观测器的应用范围。
借鉴自适应控制同步参数辨识的思想,利用Lyapunov稳定性理论和LaSalle不变原理,提出了同步参数观测器的概念,给出了同步参数观测器中控制器的设计和参数自适应律设计的一般方法,在混沌系统模型已知的情况下,利用同步参数观测器对混沌系统的未知参数进行了辨识,在模型未知的情况下,利用同步参数观测器对混沌系统的模型结构和参数进行了辨识,并给出了相应的数值仿真。
讨论了基于极值点的混沌系统的参数辨识估计问题。在已知系统结构模型的情况下,根据混沌信号具有多极值点的特点,利用函数在极值点导数为零的知识和最小二乘法,提出了基于极值点的混沌系统参数估计方法,使用该方法对Lorenz混沌系统和Chen超混沌系统的参数进行了估计,以Lorenz混沌系统为例,并对该方法抗噪声干扰能力进行了测试。
最后对全文工作进行了总结,指出了下一步需要进行的工作。
Chaos is a very complex motion with definite stochastic rules and a final bound in nature. In recent years, chaos is widely applied to biology, psychology, mathematics, physics, electronics, information science, astronomy, aerography, economics, and even to music and art. The control and synchronization of chaos become a hot issue of study in nonlinear science. However, the theories of control and synchronization for chaos systems are not perfect enough. The methods for the control and synchronization of chaos systems need to be ulteriorly investigated, so do the designs of simple and effective controllers. Moreover, when the existing methods of control and synchronization are applied in real chaos systems, many problems remain solving. For example, sometimes the parameters of real chaos systems are partially or fully unknown and disturbed. In this thesis, some problems in control and parameter identification of chaos systems are studied thoroughly. The main work and research results are as follows:
According to the intrinsic parameter structure of Lorenz system family, by using optimal Lyapunov function, it is strictly proved that the simplest linear feedback control can realize the globally exponentially stable and asymptotically stable of the equilibrium point. It improves and extends the results in existing literature that is merely locally stable. Moreover, the conditions adopted in our theorems are more succinct, less conservative than those of previous results, and more general conclusions are obtained.
According to the stability theory of differential equation and the structure of a class of chaos systems, simple sufficient conditions are obtained for PC synchronization and single state-based synchronization. When there are disturbances from parameters and environment, the design of synchronization controllers are discussed and the robust synchronization of a class of chaos systems is realized under two cases whether the parameters of driving system are known or unknown. Moreover, the synchronization of two different chaos systems with the same dimensions in the class is studied and the adaptive law of parameters and controllers are designed.
The synchronization of chaos systems based on partial states is studied. According to the stability theory of differential equation and the structure of Liu chaos system, generalized Lorenz chaos system and a novel hyperchaos system, the synchronization of two identical Liu chaos systems, generalized Lorenz systems and novel hyperchaos systems are realized using single state or partial states.
Parameter identification of chaos system based on unknown parameter observer is discussed. According to observer idea and stability theory of differential equation, taking unknown parameters of chaotic system as state variables, general methods to choose an appropriate gain function and construct corresponding auxiliary function are proposed based on the work of Guan Xinping et al. The design of unknown parameter observer for any unknown parameter of chaos is achieved and five correlative corollaries are obtained. The application field of unknown parameter observer is extended.
According to the idea of adaptive synchronization-based parameter identification, Lyapunov stability theory and LaSalle invariable principle, the synchronization-based parameter observer is proposed and general methods for design of controller and parameter adaptive laws are given. When the model is known, the unknown parameters of chaos system are identified by means of synchronization-based parameter observer. When the model is unknown, the unknown model structure and parameters are identified. The corresponding simulations are given.
Parameter identification of continuous chaos system based on extremum point is discussed. When the system model is known, a new parameter estimation approach of continuous chaos system is proposed according to the characteristic that state variables of continuous chaos system present many extremum points, and the values and positions of these extremum points are stochastic, and the derivative of a variable at extremum equals zero. By means of this new method, unknown parameters of Lorenz system and hyperchaotic Chen system are estimated. Finally, anti-disturbance performance of this new method is tested.
Finally, a summary is given and some problems are pointed out for further research.
根据Lorenz系统族本身的参数结构,利用最佳Lyapunov函数思想,严格地证明了可以利用最简单的线性反馈控制,实现Lorenz系统族平衡位置的全局指数稳定、全局渐近稳定,;改进和推广了一些现有文献中仅仅限于局部稳定的结果,而且得到了比现有文献更尽可能简洁、更少保守较小的条件和尽可能好的结论。
根据微分方程稳定性理论和一类混沌系统的结构,得出了一类混沌系统PC同步和基于单变量控制同步较简单的充分条件;讨论了在驱动系统存在参数扰动和外界扰动的情况下(分参数已知和未知两种情形)同步控制器的设计,实现了存在扰动情况下一类混沌系统的鲁棒同步;在参数未知的情况下,研究了在一类混沌系统中,维数相同但结构不同混沌系统的同步问题,给出了相应的参数自适应律和控制器的设计方法。
研究了基于部分状态变量混沌系统的同步控制。根据微分方程的稳定性理论以及Liu混沌系统、广义Lorenz混沌系统(四维)和一个新超混沌系统的具体结构,利用驱动系统单个状态变量或部分状态变量设计了合适的控制器实现了分别实现了两个完全相同的Liu混沌系统、广义Lorenz混沌系统(四维)和一个新超混沌系统的同步。
研究了基于未知参数观测器混沌系统的参数辨识问题。根据状态观测器思想和微分方程的稳定性理论,将混沌系统的未知参数看成系统的状态变量,以关新平等人的工作为基础,提出了选择增益函数和构造相应的辅助函数的一般方法,对混沌系统的任一未知参数设计了未知参数观测器,并给出了相关的五个推论,较大地扩展了未知参数观测器的应用范围。
借鉴自适应控制同步参数辨识的思想,利用Lyapunov稳定性理论和LaSalle不变原理,提出了同步参数观测器的概念,给出了同步参数观测器中控制器的设计和参数自适应律设计的一般方法,在混沌系统模型已知的情况下,利用同步参数观测器对混沌系统的未知参数进行了辨识,在模型未知的情况下,利用同步参数观测器对混沌系统的模型结构和参数进行了辨识,并给出了相应的数值仿真。
讨论了基于极值点的混沌系统的参数辨识估计问题。在已知系统结构模型的情况下,根据混沌信号具有多极值点的特点,利用函数在极值点导数为零的知识和最小二乘法,提出了基于极值点的混沌系统参数估计方法,使用该方法对Lorenz混沌系统和Chen超混沌系统的参数进行了估计,以Lorenz混沌系统为例,并对该方法抗噪声干扰能力进行了测试。
最后对全文工作进行了总结,指出了下一步需要进行的工作。
Chaos is a very complex motion with definite stochastic rules and a final bound in nature. In recent years, chaos is widely applied to biology, psychology, mathematics, physics, electronics, information science, astronomy, aerography, economics, and even to music and art. The control and synchronization of chaos become a hot issue of study in nonlinear science. However, the theories of control and synchronization for chaos systems are not perfect enough. The methods for the control and synchronization of chaos systems need to be ulteriorly investigated, so do the designs of simple and effective controllers. Moreover, when the existing methods of control and synchronization are applied in real chaos systems, many problems remain solving. For example, sometimes the parameters of real chaos systems are partially or fully unknown and disturbed. In this thesis, some problems in control and parameter identification of chaos systems are studied thoroughly. The main work and research results are as follows:
According to the intrinsic parameter structure of Lorenz system family, by using optimal Lyapunov function, it is strictly proved that the simplest linear feedback control can realize the globally exponentially stable and asymptotically stable of the equilibrium point. It improves and extends the results in existing literature that is merely locally stable. Moreover, the conditions adopted in our theorems are more succinct, less conservative than those of previous results, and more general conclusions are obtained.
According to the stability theory of differential equation and the structure of a class of chaos systems, simple sufficient conditions are obtained for PC synchronization and single state-based synchronization. When there are disturbances from parameters and environment, the design of synchronization controllers are discussed and the robust synchronization of a class of chaos systems is realized under two cases whether the parameters of driving system are known or unknown. Moreover, the synchronization of two different chaos systems with the same dimensions in the class is studied and the adaptive law of parameters and controllers are designed.
The synchronization of chaos systems based on partial states is studied. According to the stability theory of differential equation and the structure of Liu chaos system, generalized Lorenz chaos system and a novel hyperchaos system, the synchronization of two identical Liu chaos systems, generalized Lorenz systems and novel hyperchaos systems are realized using single state or partial states.
Parameter identification of chaos system based on unknown parameter observer is discussed. According to observer idea and stability theory of differential equation, taking unknown parameters of chaotic system as state variables, general methods to choose an appropriate gain function and construct corresponding auxiliary function are proposed based on the work of Guan Xinping et al. The design of unknown parameter observer for any unknown parameter of chaos is achieved and five correlative corollaries are obtained. The application field of unknown parameter observer is extended.
According to the idea of adaptive synchronization-based parameter identification, Lyapunov stability theory and LaSalle invariable principle, the synchronization-based parameter observer is proposed and general methods for design of controller and parameter adaptive laws are given. When the model is known, the unknown parameters of chaos system are identified by means of synchronization-based parameter observer. When the model is unknown, the unknown model structure and parameters are identified. The corresponding simulations are given.
Parameter identification of continuous chaos system based on extremum point is discussed. When the system model is known, a new parameter estimation approach of continuous chaos system is proposed according to the characteristic that state variables of continuous chaos system present many extremum points, and the values and positions of these extremum points are stochastic, and the derivative of a variable at extremum equals zero. By means of this new method, unknown parameters of Lorenz system and hyperchaotic Chen system are estimated. Finally, anti-disturbance performance of this new method is tested.
Finally, a summary is given and some problems are pointed out for further research.
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