统一模型混沌系统控制、同步及其应用
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摘要
混沌运动是一种确定性的非线性运动,它的运动轨迹非常复杂但又不完全随机。在许多情况下都可以观察到混沌运动的存在。由于混沌信号固有的连续宽带和似噪声等特性,为保密通信提供了高度保密的手段,混沌控制、同步及在保密通信中的应用吸引了众多学者的注意。但是,由于混沌系统所固有的系统输出对状态初值的敏感性以及混沌系统和混沌现象的复杂性和奇异性,使得混沌控制理论的研究更具有挑战性,也使得这一领域的研究和发展成为当代非线性科学的研究热点。
本论文采用自适应理论、无源控制、变结构控制等方法,研究基于统一模型的混沌动力学系统的控制、同步及其在保密通信系统中的应用。主要研究内容如下：
1.构造基于发送系统和响应系统的误差系统,将系统同步控制的问题转化为误差系统渐近稳定性分析问题。在统一混沌系统的非线性函数满足Lipschitz条件的基础上,利用无源性网络理论,分析并推导了基于统一混沌模型的误差同步系统等效为无源系统的条件,实现了将最小相位混沌误差同步系统等效为无源系统,即构造混沌系统的控制器,将混沌同步误差系统配置为无源系统,实现了基于统一模型的混沌同步控制。
2.基于最速下降法,讨论了统一混沌系统的不确定参数自适应辨识方法,构造不确定混沌系统的同步控制器,将含有不确定参数的混沌同步误差系统等效为无源系统,实现混沌同步误差系统的稳定性控制,进而实现参数不确定混沌系统同步。
3.给出了混沌时滞同步的定义,讨论了含有不确定性的混沌时滞同步控制器设计。基于最速下降法,讨论了不确定参数统一混沌系统的时滞同步控制,将含有不确定参数的混沌时滞同步误差系统等效为无源系统,实现参数不确定混沌系统时滞同步及其广义同步。
4.研究了分数阶统一混沌系统的渐近稳定性。对分数阶统一混沌系统进行了深入分析,并针对系统的不稳定平衡点,借鉴整数阶非线性无源系统理论,设计了系统的等效无源控制器,给出了实现分数阶系统稳定的控制参数取值范围,实现了系统的渐近稳定控制。利用拉普拉斯变换,采用将分数阶系统近似等效为整数阶系统的方法,建立分数阶系统的等效整数阶状态空间方程,从而得到系统的状态输出。
5.针对分数阶统一混沌系统的不稳定平衡点,利用分数阶微积分的定义,构造系统的李亚普诺夫函数及滑动模态,设计了系统的变结构控制器,实现了系统的渐近稳定控制。
Chaotic motion is a complex nonlinear motion, whose trajectory of the orbits in the phase plane is very complex but not stochastic. We can observe the chaotic phenomenon in a lot of systems. The dynamic properties of chaos signal such as ergodicity, aperiodic, uncorrelated, broad band and noise-like have been proved to be useful for communication and picture encrypt systems and in describing and diagnosing nonlinear dynamic systems. Chaos plays an important role in dynamical systems and is applied in many fields such as physics, chemistry, economics and so on. The sensitivity to changes in initial conditions, a kind of intrinsically character, and complexity of chaotic dynamical system make the research on the chaotic control theory to be more challenging. Research on chaotic control, synchronization and its application becomes a new researching focus in nonlinear science fields.
The main content of this paper contains the analysis and control of unified chaotic dynamical system. The control theory of chaotic control, synchronization system mainly contains passive control, adaptive state feedback control, sliding-mode synchronous control, and so on. In this dissertation, the main contributions are as follows:
1. Based on the nonlinear function of unified chaotic system satisfies Lipschitz condition, the synchronization error system for the drive system and the response system is given and the problem to make system synchronize can be transformed to ensure the global approximate stability of the synchronization system. An adaptive passivity method is adopted to design statement feedback controller to ensure the global stability of the synchronization system. With this method, the unified system can be synchronized with different initial conditions. The simulation results show the validity of the method.
2. The passive synchronization method of unified chaotic system with uncertain parameters is studied. Based on the method of steepest descent, the uncertain parameters can be identified adaptively. Through making the unified chaotic system be a passive system, the synchronization controller is designed with the uncertain parameters, and the synchronization error system of unified chaotic system can be stabilized. The simulation results show that the control method is effective for unified chaotic system synchronization.
3. A synchronization control approach for unified chaotic system with uncertainties and time delay is presented using passive control theory. The stability of the synchronization error system is studied so that the unified chaotic system can be synchronized with uncertainties and channel time delay. Because the chaotic trajectory is limitary and the nonlinear function of the unified chaotic system satisfies Lipschitz condition, the error system transformed by synchronization error system can be globally asymptotically stabilized by state feedback, therefore it implements the synchronization control of unified chaotic system with uncertainties and time delay. Simulation results show that the adaptive passive synchronization scheme is efficient, and the generalized synchronization can also be realized.
4. The equivalent passive control theory for fractional order unified chaotic system is studied. The chaotic behaviors in the fractional order unified system are numerically investigated. On the unstable equilibrium point, it is used the "equivalent passivity" method of nonlinear control design to derive the nonlinear controller. How to choose the control parameter is also studied so that the controller is to stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction. With Laplace transform theory, the equivalent integer order state equation from fractional order nonlinear system is gotten, and the statement output can be solved.
5. The stability of fractional order unified chaotic system is studied with sliding mode control theory. The sliding manifold is constructed by the definition of fractional order derivative and integral for fractional order unified chaotic system. It is proved that the sliding manifold exists. In order to improve the convergence rate, the controller includes two parts:continuous controller and switching controller. According to the boundness of chaotic attractor, the chosen values of the controlling parameters are decided by parameters inequality. Simulation results are obtained to verify the effectiveness of this method.
本论文采用自适应理论、无源控制、变结构控制等方法,研究基于统一模型的混沌动力学系统的控制、同步及其在保密通信系统中的应用。主要研究内容如下：
1.构造基于发送系统和响应系统的误差系统,将系统同步控制的问题转化为误差系统渐近稳定性分析问题。在统一混沌系统的非线性函数满足Lipschitz条件的基础上,利用无源性网络理论,分析并推导了基于统一混沌模型的误差同步系统等效为无源系统的条件,实现了将最小相位混沌误差同步系统等效为无源系统,即构造混沌系统的控制器,将混沌同步误差系统配置为无源系统,实现了基于统一模型的混沌同步控制。
2.基于最速下降法,讨论了统一混沌系统的不确定参数自适应辨识方法,构造不确定混沌系统的同步控制器,将含有不确定参数的混沌同步误差系统等效为无源系统,实现混沌同步误差系统的稳定性控制,进而实现参数不确定混沌系统同步。
3.给出了混沌时滞同步的定义,讨论了含有不确定性的混沌时滞同步控制器设计。基于最速下降法,讨论了不确定参数统一混沌系统的时滞同步控制,将含有不确定参数的混沌时滞同步误差系统等效为无源系统,实现参数不确定混沌系统时滞同步及其广义同步。
4.研究了分数阶统一混沌系统的渐近稳定性。对分数阶统一混沌系统进行了深入分析,并针对系统的不稳定平衡点,借鉴整数阶非线性无源系统理论,设计了系统的等效无源控制器,给出了实现分数阶系统稳定的控制参数取值范围,实现了系统的渐近稳定控制。利用拉普拉斯变换,采用将分数阶系统近似等效为整数阶系统的方法,建立分数阶系统的等效整数阶状态空间方程,从而得到系统的状态输出。
5.针对分数阶统一混沌系统的不稳定平衡点,利用分数阶微积分的定义,构造系统的李亚普诺夫函数及滑动模态,设计了系统的变结构控制器,实现了系统的渐近稳定控制。
Chaotic motion is a complex nonlinear motion, whose trajectory of the orbits in the phase plane is very complex but not stochastic. We can observe the chaotic phenomenon in a lot of systems. The dynamic properties of chaos signal such as ergodicity, aperiodic, uncorrelated, broad band and noise-like have been proved to be useful for communication and picture encrypt systems and in describing and diagnosing nonlinear dynamic systems. Chaos plays an important role in dynamical systems and is applied in many fields such as physics, chemistry, economics and so on. The sensitivity to changes in initial conditions, a kind of intrinsically character, and complexity of chaotic dynamical system make the research on the chaotic control theory to be more challenging. Research on chaotic control, synchronization and its application becomes a new researching focus in nonlinear science fields.
The main content of this paper contains the analysis and control of unified chaotic dynamical system. The control theory of chaotic control, synchronization system mainly contains passive control, adaptive state feedback control, sliding-mode synchronous control, and so on. In this dissertation, the main contributions are as follows:
1. Based on the nonlinear function of unified chaotic system satisfies Lipschitz condition, the synchronization error system for the drive system and the response system is given and the problem to make system synchronize can be transformed to ensure the global approximate stability of the synchronization system. An adaptive passivity method is adopted to design statement feedback controller to ensure the global stability of the synchronization system. With this method, the unified system can be synchronized with different initial conditions. The simulation results show the validity of the method.
2. The passive synchronization method of unified chaotic system with uncertain parameters is studied. Based on the method of steepest descent, the uncertain parameters can be identified adaptively. Through making the unified chaotic system be a passive system, the synchronization controller is designed with the uncertain parameters, and the synchronization error system of unified chaotic system can be stabilized. The simulation results show that the control method is effective for unified chaotic system synchronization.
3. A synchronization control approach for unified chaotic system with uncertainties and time delay is presented using passive control theory. The stability of the synchronization error system is studied so that the unified chaotic system can be synchronized with uncertainties and channel time delay. Because the chaotic trajectory is limitary and the nonlinear function of the unified chaotic system satisfies Lipschitz condition, the error system transformed by synchronization error system can be globally asymptotically stabilized by state feedback, therefore it implements the synchronization control of unified chaotic system with uncertainties and time delay. Simulation results show that the adaptive passive synchronization scheme is efficient, and the generalized synchronization can also be realized.
4. The equivalent passive control theory for fractional order unified chaotic system is studied. The chaotic behaviors in the fractional order unified system are numerically investigated. On the unstable equilibrium point, it is used the "equivalent passivity" method of nonlinear control design to derive the nonlinear controller. How to choose the control parameter is also studied so that the controller is to stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction. With Laplace transform theory, the equivalent integer order state equation from fractional order nonlinear system is gotten, and the statement output can be solved.
5. The stability of fractional order unified chaotic system is studied with sliding mode control theory. The sliding manifold is constructed by the definition of fractional order derivative and integral for fractional order unified chaotic system. It is proved that the sliding manifold exists. In order to improve the convergence rate, the controller includes two parts:continuous controller and switching controller. According to the boundness of chaotic attractor, the chosen values of the controlling parameters are decided by parameters inequality. Simulation results are obtained to verify the effectiveness of this method.
引文
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