摘要
耦合是指两个或两个以上的系统进行交互作用。把这些耦合的系统看成一个整体，若相互作用的系统本身是非线性的，或者耦合方式是非线性的，则称为非线性的耦合系统。本论文研究的主体内容是如何使非线性的耦合系统所有子系统的最终动力学行为趋于一致。以这一视角出发，针对不同的应用背景，本论文研究了动态神经网络对形状记忆合金的迟滞行为在线建模、自治的混沌系统与非自治的混沌系统同步和中立型时滞耦合复杂网络中联接点状态同步。
本论文的主要内容如下：
①动态神经网络与未知函数的耦合。在对基于智能材料的机械系统精确建模与控制的应用课题中，将形状记忆合金的高度非线性迟滞现象当作未知函数，利用迟滞行为的输入信号、输出信号与建模系统的交互耦合，使建模系统能够重构迟滞行为的输入信号与输出信号之间的函数关系，即建模系统体现的函数关系与迟滞行为函数关系要一致。将形状记忆合金的位移变化作为动态神经网络的输入，相应的预测电压变化作为动态神经网络的输出，从而实现动态神经网络对迟滞现象的逆函数观察。针对这个耦合系统，利用逆函数的观察结果，提出自适应逆控制策略，驱使形状记忆合金按照理想的轨迹运动，实现对形状记忆合金的精确控制。
②非自治混沌系统与自治混沌系统的耦合。在混沌同步的应用课题中，针对驱动系统的外部激励是一个相位与幅度都不确定的正弦类项，结合正弦类函数的特性，构造了一个比驱动系统更高阶的响应系统，实现了这两个系统的同步。与传统的响应系统的构造方式即和驱动系统同阶的构造方式相比，本论文的构造方式不用再假定响应系统与驱动系统的相位误差范围为已知，然后研究混沌同步的条件。仿真实验表明：对于带有任何相位和幅度激励项的驱动系统，本论文高阶的自治系统都能正确追踪驱动系统，最终两个系统都能在一定误差范围内保持一致。
③复杂系统中多个联节点状态之间的中立型时滞耦合。在复杂系统同步的应用课题中处理复杂网络联节点状态耦合方式时，不仅考虑了联节点过去状态对彼此的影响，还考虑联结过去状态的变化情况对彼此的影响，即考虑了联节点之间状态离散时滞耦合和中立型时滞耦合的情况。具体方法是先通过模型变换，把复杂网络模型的同步性问题转换成中立系统的稳定性问题。通过时滞分段方法构造了新颖的Lyapunov-Krasovskii泛函，得到了具有较少约束的渐近稳定性条件和指数稳定性条件，也即得到判定复杂网络内状态行为最终是否一致的条件。数值模拟结果表明本论文能证明的一致性条件的限制性比现有文献条件的限制性更宽松。
The nonlinear coupling system is defined as two or more than two nonlinearsubsystems coupling together or the coupling way among these nonlinearity. How tomake sure these systems reach consensus is the topic of this paper. In this field, westudied modeling shape memory effect by dynamical neural networks, chaossynchronization between non-autonomous master system and autonomous slave system,neutral coupling complex networks.
The main contents of this dissertation are as follows:
①Dynamical neural network (DNN) couples unknown function. In the project ofmodeling and controlling the smart materials and structure, regarding the shape memoryalloy’s hysteresis behavior as unknown function, coupling the dynamic neural networkand the unknown function, make sure that the two systems could reach consensus andthe dynamical neural network could accurately model the hysteresis behavior. That is,the consensus between the DNN and the unknown function could be reached withreasonable accuracy. When predicting voltage, the DNN acts as the inverse function ofthe shape memory effect. This kind of DNN is called IDNN. Based on the IDNN, weproposed an adaptive inverse control strategy which driven the shape memory alloyaccurately tracking the desired command.
②Time delay dynamic neural network (TDDNN) couples unknown function.Considering the past states’ effect on the hysteresis, a new TDDNN which introduces atime delay between the input and output response is proposed for modeling thehysteresis online. Experimental results demonstrated the time delay’s importance andthe effectiveness of the TDDNN. That is, the consensus between the TDDNN and theunknown function could be reached with reasonable accuracy.
③Autonomous slave system couples non-autonomous master system. In thefield of robust synchronization of chaotic systems with unknown phase term in thetriangular function of master chaos system, through using the properties of thetriangular function, a novel slave system whose dimension is larger than the mastersystem is proposed. Most literatures investigated the synchronization effects with theassumed phase difference and the assumed amplitude of the sinusoidal forcing term.With unknown phase difference and unknown amplitude of the sinusoidal forcing termin the master system, numerical simulations show that the effectiveness of the proposed and novel slave system. That is, the consensus between the master system and the novelslave system could be reached with reasonable accuracy.
④The neutral coupling among nodes in complex networks. The coupling waysnot only include the past states of one node’s effect on each other, but also include thederivative past states of one node’s effect on each other. In other words, the complexnetworks not only consider the delay coupling but also the neutral delay coupling.Using the transformation, we can transform the synchronization problem intostabilization problem. Based on these new complex models, we derive asymptotical andexponential criteria via delay fraction approach. Numerical examples are given toillustrate the effectiveness of our scheme, which shows that our results are better thanthat of the recent proposals.
引文
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