【作者】 张丽丽；

【导师】 王银河；

【作者基本信息】 广东工业大学 ， 控制理论与控制工程， 2014， 博士

【摘要】 从工程角度讲,复杂动态网络稳定是能够有效应用的前提,而现实中的网络并不都能通过自身实现稳定,因此,非常有必要探讨复杂动态网络的稳定控制问题。还应该注意到,同步现象无论在自然界还是人工网络中都是非常常见的,而且一些同步现象对我们日常生活是非常有用的。近十几年来,同步作为复杂动态网络的一种非常重要的集体行为已经得到了国内外工程界和科学界人员的广泛关注。物以类聚,人以群分。现实中有一类复杂动态网络,其节点的内部动态具有某种特殊的相似性,例如,同型号发电机组成的大规模电能网络,结构相同的传感器组成的无线传感网络等都是节点内部动态具有相似性特点的动态网络。这种特殊的相似性对于网络的稳定性和节点间的同步行为有直接的影响。从理论分析的角度上看,这种相似性有助于简化网络稳定和同步化控制器的设计。值得指出的是,本文中的相似节点并不要求节点具有相同的状态维数,更不要求节点完全相同,具有不同维数相似节点的复杂动态网络能更好地描述现实网络。同时还要看到,现实网络尤其是生物网络和工程网络中节点之间存在时变外部耦合关系,而且经常存在时滞,其内部耦合函数通常也是非线性的。鉴于以上考虑,本文主要研究多种耦合情况下的具有相似节点的复杂动态网络的稳定和同步化控制策略,所用数学分析工具主要是常微分方程的Lyapunov稳定性理论、时滞系统的Lyapunov-Krasovskii稳定性定理以及非自治系统的Barbalat’s引理等。本文的研究工作主要分为以下几个方面：1.简单介绍复杂网络及其特性、研究意义与发展简史；简单介绍复杂动态网络的研究内容、分类：简单介绍复杂动态网络稳定和同步的定义及它们的研究意义和研究内容,并提出本文需要探讨的问题；2.分别针对仿射线性和非线性动力系统形式的孤立节点,给出了节点相似性的定义；3.探讨了两类具有相似节点的耗散复杂动态网络的镇定问题。首先设计了具有非线性相似节点和非线性常时滞耦合的时不变耗散复杂动态网络的渐近稳定控制器；其次,针对具有非线性相似节点和非线性内部耦合的时变耗散复杂动态网络,根据该网络外部耦合系数的公共界已知和未知两种情况分别设计了其渐近稳定控制器,其中当该公共界未知时,本文引入了自适应控制策略,而且只有一个自适应律。本文提出的时变耗散复杂动态网络的稳定控制方案不需要依赖于网络节点间的外部耦合信息,具有一般性和普适性。数值仿真验证了上述两类网络镇定方案的有效性。4.提出了三类具有不同维数相似节点的无时滞耦合复杂动态网络的同步控制器设计方案,包括：(1)提出了具有线性相似节点和非线性耦合的时不变耗散复杂动态网络的分散动力补偿渐近同步控制器；(2)分别设计了两类具有不同维数非线性相似节点和非线性耦合的时不变耗散复杂动态网络的分散动力补偿渐近同步控制器；(3)为更好地描述现实网络,提出了具有非线性相似节点和非线性耦合的时变非耗散复杂动态网络模型,同时基于网络中各节点的轨迹提出了网络指数同步和渐近同步的定义,并在此定义的意义下,利用网络外部耦合系数的已知或未知公共界设计了网络的同步控制器,其中,当此公共界已知时,本文提出的分散状态反馈控制器能够实现网络的指数同步,而当此公共界未知时,本文提出的自适应分散状态反馈控制器能够实现网络的渐近同步,两种情况下网络都能很快达到同步状态。各部分的数值仿真都验证了相应理论结果的有效性。5.设计了具有不同维数非线性相似节点和非线性时滞耦合的复杂动态网络的同步控制方案,具体内容为：(1)提出了具有不同维数非线性相似节点和非线性常时滞耦合的时不变耗散复杂动态网络的分散动力补偿时滞独立渐近同步控制器；(2)为更好地描述现实网络,本文首先提出了具有不同维数非线性相似节点和非线性时变时滞耦合的时变非耗散复杂动态网络模型,此网络模型具有一般性,而且网络中各节点的时变时滞不同,然后根据网络外部耦合系数公共界已知和未知两种情况,分别设计了其分散状态反馈时滞独立指数同步控制器和分散自适应状态反馈时滞独立渐近同步控制器；(3)考虑到时滞独立同步控制器的局限性,本文还根据外部耦合系数公共界已知和未知两种情况,分别设计了具有不同维数非线性相似节点和非线性的不同时变时滞耦合的时变非耗散复杂动态网络的分散状态反馈时滞依赖指数同步控制器和分散自适应状态反馈时滞依赖渐近同步控制器。各部分分别通过数值实例验证了理论结果的有效性。本论文受到国家自然科学基金(61273219)、广东省自然科学基金(S2013010015768)、高等学校博士学科点专项科研基金(20134420110003)等资助。更多还原

【Abstract】 From the engineering view of point, the stabilization property of complex dynamical networks is prerequisite towards effective applications. But the realist networks do not always realize stabilization by themselves. Thus, it is very necessary to investigate the stabilization problem for complex dynamical networks via control strategies. It should also be noticed that synchronization phenomena are ubiquitous in natural and artificial networks, and some synchronous phenomena are very useful for our life. In the past ten or more years, as an important collective behaviors of complex dynamical networks, synchronization has attracted extensive attention from various fields of science and engineering throughout the world.Like to like. In real world, there exists a class of complex dynamical networks whose nodes possess some special similarities in their inner dynamics. For example, the large-scale electric power networks with identical generator units, the wireless sensor networks composed of the same structure sensors and so on are the networks whose nodes possess some inner similarities. These special similarities have direct effect on the stability and synchronization of the networks. From the view of theoretical analysis, these similarities will help to simplify the stabilization and synchronization controllers. It is worth pointing out that similar nodes may be with different state dimensions and may be different from each other.In fact, the complex dynamical network with different-dimensional similar nodes can be more coincident with the real-world networks. It should also be noted that time-varying coupling coefficients, coupling delays or nonlinear coupling functions exist in most realist networks, especially in biological networks and engineering networks.Inspired by the above discussions, this dissertation investigates the stabilization and synchronization problems for the complex dynamical networks with similar nodes and different kinds of outer and inner couplings. Lyapunov stability theory of ordinary differential equations, Lyapunov-Krasovskii stability theorem of delayed systems and Barbalat’s lemma of non-autonomous systems are the main mathematical tools used in the theoretical analysis of this dissertation.The main research work and results are listed in the followings.1. Complex networks and their characters, research meaning and development overview are briefly introduced. Besides, the research contents and categories of complex dynamical networks are also briefly introduced. Furthermore, the definitions, research meaning and research contents of stabilization and synchronization for the complex dynamical networks are introduced, and some necessary topics to be studied in this dissertation are presented.2. The definitions of similar nodes are proposed for the affine linear and nonlinear isolated nodes, respectively.3. Stabilization control problems for two classes of dissipatively coupled complex dynamical networks with similar nodes are investigated, respectively. Firstly, the asymptotic stabilization controllers are synthesized for the complex dynamical networks with nonlinear similar nodes and nonlinear delayed coupling functions. Then, for the time-varying and dissipatively coupled complex dynamical network model with nonlinear similar nodes and nonlinear coupling functions, two kinds of asymptotic stabilization controllers are designed according to its certain or uncertain common bound of the outer coupling coefficients, respectively. Especially, when the common bound is unknown, adaptive control mechanism is introduced and there is only one adaptive law needed. The stabilization strategies proposed here do not depend on the outer coupling information of the network, so they are general. Numerical simulations have verified the effectiveness of the theoretical results mentioned above.4. Synchronization controllers are proposed for three classes of non-delayed coupling complex dynamical networks with different-dimensional similar nodes, which are listed as follows.(1) Decentralized dynamical compensation asymptotic synchronization controllers are proposed for the complex dynamical networks with linear similar nodes and nonlinearly coupled functions.(2) Two kinds of decentralized dynamical compensation asymptotic synchronization controllers are synthesized for the complex dynamical networks with nonlinear similar nodes and nonlinearly coupling functions.(3) To well describe the realist networks, a non-dissipatively coupled time-varying complex dynamical network model with nonlinear similar nodes and nonlinearly coupled functions is introduced. For this kind of networks, the definitions of exponential synchronization and asymptotic synchronization are proposed based on the nodes’ trajectories in the network. Furthermore, synchronization control strategies are proposed for this kind of networks according to the known or unknown common bound of the outer coupling coefficients, respectively. When the common bound is known, the decentralized state-feedback controllers synthesized here can guarantee the network realizing exponential synchronization. And when the common bound is unknown, the adaptive decentralized state-feedback controllers designed here can make the network achieve asymptotic synchronization. Corresponding simulation examples have verified the effectiveness of the theoretical results mentioned above.5. Synchronization controllers are designed for the delayed coupling complex dynamical networks with different-dimensional nonlinear similar nodes, which are listed as follows.(1) Decentralized dynamical compensation and delay-independent synchronization controllers are synthesized for the time-invariant complex dynamical networks with nonlinear similar nodes and constant coupling delays, which can guarantee the networks achieving asymptotic synchronization.(2) To well describe the real-world networks, a general non-dissipatively coupled time-varying complex dynamical network model is introduced here, in which, different time-varying coupling delays for different nodes are considered. Furthermore, decentralized state-feedback delay-independent exponential synchronization controllers or adaptive asymptotic synchronization controllers are proposed for the networks with the known or unknown common bound of their coupling coefficients, respectively.(3) Considering the limitation of the delay-independent synchronization controllers, delay-dependent exponential or adaptive asymptotic synchronization controllers are also designed for the non-dissipatively coupled time-varying complex dynamical networks with nonlinear different-dimensional similar nodes and different time-varying coupling delays according to their certain or uncertain common bound, respectively. Corresponding numerical simulations have verified the effectiveness of the theoretical results mentioned above.This dissertation was supported by the National Science Foundation of China (61273219), the National Science Foundation of Guangdong Province of China (S2013010015768) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20134420110003).更多还原