不确定混沌系统的控制与同步研究
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摘要
混沌的控制、同步是混沌研究领域的重要分支。近半个世纪以来,随着非线性科学的蓬勃发展,混沌理论的研究和应用已成为复杂非线性科学研究中的主要方向。如何有效地控制混沌运动,如何使两个混沌(超混沌)系统达到同步,如何利用混沌同步方法实现复杂网络同步等问题是非常值得研究的课题。
本文首先介绍了混沌的一些基本概念,包括混沌理论形成与发展,混沌控制与同步研究的起源、现状和典型混沌系统模型,然后研究了混沌(超混沌)系统的控制与同步方法。最后,研究了复杂网络的自适应同步问题,设计自适应耦合强度算法。具体的研究工作概括如下：
1.研究了混沌模糊神经控制,针对一类未知混沌系统设计了一个自适应模糊神经控制器,该控制器由模糊神经控制器和鲁棒控制器构成。其中,模糊神经控制器主要包含一个模糊神经网络辨识器,通过调节模糊神经网络的参数实现对受控系统的估计。高斯函数是被广泛应用的径向基函数,因此,调整的参数包括后件部分的连接权,以及前件部分的高斯隶属度函数的均值和方差,应用BP算法实现了这些参数的在线调整。鲁棒控制器保证了受控系统的稳定性并达到了预期的控制性能。通过仿真结果验证了该方案的有效性。
2.研究了统一超混沌系统的参数已知、未知情况下同步问题。基于Lyapunov(?)急定性理论,首先以全状态混合投影自适应同步方法,设计了自适应控制器,通过稳定性理论推到证明了该控制器在参数已知情况下渐近稳定于不动点并实现统一超混沌系统的全状态混合投影同步。其次,使用主动控制同步方法,设计了同步控制器,实现了统一超混沌系统的完全同步。第三是在参数未知情况下的统一超混沌系统的自适应控制与反同步。设计了自适应控制器,证明了该控制器可使参数未知统一超混沌系统渐近稳定于不动点。设计了自适应反同步控制器,实现了参数未知统一超混沌系统的完全同步,数值仿真实验验证了所提出方案的有效性。
3.研究了存在参数扰动的超混沌系统的自适应同步问题。基于Lyapunov稳定性理论,设计了自适应控制器,从理论上证明了可以实现存在参数扰动下超混沌Qi系统的自同步,以及超混沌Qi系统和超混沌Lorenz系统间的异结构同步。最后数值仿真实验验证了所提出方案的有效性。
4.研究了规则耦合复杂网络的自适应同步,首先设计了自适应耦合强度算法,该算法能够有效地稳定和同步复杂网络,其次从数学的角度给出了该算法的严格证明。特别地指出该复杂网络的同步完全由耦合强度决定。最后,通过一个例子,利用LorenZ混沌系统作为复杂网络的节点,验证了所提算法的有效性。
Chaotic control and synchronization is important research branch of chaos. Nearly half a century, with the rapid development of nonlinear science, research and application of chaos theory has become the main direction of complex nonlinear scientific. Some subjects are very worthy of study, such as how to control the chaotic motion effectively, how to make two chaotic (hyperchaotic) systems to achieve synchronization, and how to utilize the chaotic synchronization method realizing complex network synchronization, etc.
The paper introduces some basic concepts of chaos, including the formation and development of chaos theory, the origin, present, and typical models of chaotic control and synchronization, and then proposed the chaotic (hyperchaotic) systems control and synchronization. Finally, the self-adaptive synchronization of complex network is studied and the Adaptive coupling strength algorithm is developed. Specific studies are summarized as follows:
1. The Fuzzy Neural Control of chaos is studied, and an adaptive fuzzy neural controller for a class of unknown chaotic system is developed, the controller consists of fuzzy neural controller and robust controller. The fuzzy neural controller consists mainly of a fuzzy neural network identifier, by adjusting the parameters of fuzzy neural network to achieve the estimates of the controlled system. Gaussian function is widely used radial basis function, therefore, the adjustment parameters include the connection weights after the piece parts, and the antecedent part of the Gaussian membership function of the mean and variance, the application of BP algorithm to adjust these parameters online. Robust controller guarantees the stability of the controlled system and achieves the desired control performance. Finally, the simulation results further demonstrate the effectiveness of the program.
2. The synchronization problems of a unified haperchaotic system over the known and unknown parameters are studied. Firstly, Based on Lyapunov stability theory, an adaptive controller is developed by the adaptive full state hybrid projective synchronization method, and theoretically proved the known parameters of the controller can achieve the unity of the whole state of super hybrid chaotic system map synchronization. Secondly, a synchronous controller is developed by the active control to achieve the unified hyperchaotic system synchronization. The third argument is the unknown chaotic system under a unified hyperchaotic adaptive control and anti-synchronization. Adaptive controller was designed to prove that the controller can make a unified hyperchaotic system with unknown parameters asymptotically stable in the fixed point. Adaptive anti-synchronization controller is designed to achieve a unified haperchaotic system with unknown parameters of the complete synchronization. Numerical simulation results further validate the effectiveness of the proposed scheme.
3. The adaptive synchronization of haperchaotic systems with parameter perturbations is studied. Based on Lyapunov stability theory, an adaptive controller is designed, which proved theoretically that the self-synchronization of the hyperchaotic Qi system can be achieved with parameter perturbations, and the different structure synchronization of the hyperchaotic Qi system and hyperchaotic Lorenz system. Finally, the simulation results further validate the effectiveness of the proposed scheme.
4. The rule coupling complex network self-adaptive synchronization is studied, firstly, the coupling strength algorithm is developed, which can stabilize effectively and synchronize the complex network, and secondly, the algorithm is strict proofed from the view of mathematics. Particularly, the complex network synchronization is determined by coupling strength completely. Finally, through an example, take the Lorenz system as the complex network nodes, further verified the effectiveness of the proposed algorithm.
本文首先介绍了混沌的一些基本概念,包括混沌理论形成与发展,混沌控制与同步研究的起源、现状和典型混沌系统模型,然后研究了混沌(超混沌)系统的控制与同步方法。最后,研究了复杂网络的自适应同步问题,设计自适应耦合强度算法。具体的研究工作概括如下：
1.研究了混沌模糊神经控制,针对一类未知混沌系统设计了一个自适应模糊神经控制器,该控制器由模糊神经控制器和鲁棒控制器构成。其中,模糊神经控制器主要包含一个模糊神经网络辨识器,通过调节模糊神经网络的参数实现对受控系统的估计。高斯函数是被广泛应用的径向基函数,因此,调整的参数包括后件部分的连接权,以及前件部分的高斯隶属度函数的均值和方差,应用BP算法实现了这些参数的在线调整。鲁棒控制器保证了受控系统的稳定性并达到了预期的控制性能。通过仿真结果验证了该方案的有效性。
2.研究了统一超混沌系统的参数已知、未知情况下同步问题。基于Lyapunov(?)急定性理论,首先以全状态混合投影自适应同步方法,设计了自适应控制器,通过稳定性理论推到证明了该控制器在参数已知情况下渐近稳定于不动点并实现统一超混沌系统的全状态混合投影同步。其次,使用主动控制同步方法,设计了同步控制器,实现了统一超混沌系统的完全同步。第三是在参数未知情况下的统一超混沌系统的自适应控制与反同步。设计了自适应控制器,证明了该控制器可使参数未知统一超混沌系统渐近稳定于不动点。设计了自适应反同步控制器,实现了参数未知统一超混沌系统的完全同步,数值仿真实验验证了所提出方案的有效性。
3.研究了存在参数扰动的超混沌系统的自适应同步问题。基于Lyapunov稳定性理论,设计了自适应控制器,从理论上证明了可以实现存在参数扰动下超混沌Qi系统的自同步,以及超混沌Qi系统和超混沌Lorenz系统间的异结构同步。最后数值仿真实验验证了所提出方案的有效性。
4.研究了规则耦合复杂网络的自适应同步,首先设计了自适应耦合强度算法,该算法能够有效地稳定和同步复杂网络,其次从数学的角度给出了该算法的严格证明。特别地指出该复杂网络的同步完全由耦合强度决定。最后,通过一个例子,利用LorenZ混沌系统作为复杂网络的节点,验证了所提算法的有效性。
Chaotic control and synchronization is important research branch of chaos. Nearly half a century, with the rapid development of nonlinear science, research and application of chaos theory has become the main direction of complex nonlinear scientific. Some subjects are very worthy of study, such as how to control the chaotic motion effectively, how to make two chaotic (hyperchaotic) systems to achieve synchronization, and how to utilize the chaotic synchronization method realizing complex network synchronization, etc.
The paper introduces some basic concepts of chaos, including the formation and development of chaos theory, the origin, present, and typical models of chaotic control and synchronization, and then proposed the chaotic (hyperchaotic) systems control and synchronization. Finally, the self-adaptive synchronization of complex network is studied and the Adaptive coupling strength algorithm is developed. Specific studies are summarized as follows:
1. The Fuzzy Neural Control of chaos is studied, and an adaptive fuzzy neural controller for a class of unknown chaotic system is developed, the controller consists of fuzzy neural controller and robust controller. The fuzzy neural controller consists mainly of a fuzzy neural network identifier, by adjusting the parameters of fuzzy neural network to achieve the estimates of the controlled system. Gaussian function is widely used radial basis function, therefore, the adjustment parameters include the connection weights after the piece parts, and the antecedent part of the Gaussian membership function of the mean and variance, the application of BP algorithm to adjust these parameters online. Robust controller guarantees the stability of the controlled system and achieves the desired control performance. Finally, the simulation results further demonstrate the effectiveness of the program.
2. The synchronization problems of a unified haperchaotic system over the known and unknown parameters are studied. Firstly, Based on Lyapunov stability theory, an adaptive controller is developed by the adaptive full state hybrid projective synchronization method, and theoretically proved the known parameters of the controller can achieve the unity of the whole state of super hybrid chaotic system map synchronization. Secondly, a synchronous controller is developed by the active control to achieve the unified hyperchaotic system synchronization. The third argument is the unknown chaotic system under a unified hyperchaotic adaptive control and anti-synchronization. Adaptive controller was designed to prove that the controller can make a unified hyperchaotic system with unknown parameters asymptotically stable in the fixed point. Adaptive anti-synchronization controller is designed to achieve a unified haperchaotic system with unknown parameters of the complete synchronization. Numerical simulation results further validate the effectiveness of the proposed scheme.
3. The adaptive synchronization of haperchaotic systems with parameter perturbations is studied. Based on Lyapunov stability theory, an adaptive controller is designed, which proved theoretically that the self-synchronization of the hyperchaotic Qi system can be achieved with parameter perturbations, and the different structure synchronization of the hyperchaotic Qi system and hyperchaotic Lorenz system. Finally, the simulation results further validate the effectiveness of the proposed scheme.
4. The rule coupling complex network self-adaptive synchronization is studied, firstly, the coupling strength algorithm is developed, which can stabilize effectively and synchronize the complex network, and secondly, the algorithm is strict proofed from the view of mathematics. Particularly, the complex network synchronization is determined by coupling strength completely. Finally, through an example, take the Lorenz system as the complex network nodes, further verified the effectiveness of the proposed algorithm.
引文
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