复杂动态网络系统的同步控制研究
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摘要
将复杂系统抽象成相互作用的个体组成的复杂网络,是描述和理解复杂系统的重要方法,受到国内外学者的广泛关注.由此展开的复杂网络研究已渗透到物理学、生物科学、社会科学、计算机科学与工程等众多领域,涉及到其中的诸多研究内容.网络系统的同步就是其中一项代表性课题.同步是复杂网络上典型的集体行为和涌现现象.它普遍存在于各类复杂网络系统中,并关系着众多具体问题.研究网络系统上同步行为的控制问题对消除或增强同步影响,仿真、控制与设计复杂系统等具有积极的现实意义和理论价值.本文以连续时间耦合网络的同步控制展开研究,依次针对静态线性耦合网络、时变线性耦合网络以及更一般的不确定时滞耦合网络等设计有效的控制器,保证网络同步的实现.本文主要包括以下内容:
1.提出了一类随机伪分形演化网络,为本文随后章节的相关讨论提供了网络拓扑结构模型.该模型通过分形生长和边权竞争机制实现网络演化.利用连续域方法解析计算了网络的度分布、簇系数和平均路径长度,并执行数值仿真加以验证.结果表明这一类随机伪分形网络同时具有现实网络系统的两个普适特性—小世界效应和无标度特性,并且其幂律分布下的斜率可通过调节竞争参数α使之位于(2,3)之间,与绝大多数真实网络的分布斜率相符合.
2.设计了一种自适应牵制反馈控制手段用于同步加权耦合网络,并解决了实施牵制时的两个基本问题:牵制节点数目估计和牵制节点选择策略.考虑到一般的线性反馈牵制需要预估反馈增益,提出了利用局部余差自主调节反馈增益的自适应牵制控制手段,并给出了控制加权网络局部同步和全局同步的理论保证.特别地,证明了这种自适应牵制手段在理论分析研究时可等价为一般的线性反馈牵制,故可用线性牵制来研究自适应牵制的节点选择问题.以此为基础,利用矩阵论和线性矩阵不等式相关知识展开讨论.一方面,基于网络耦合矩阵特征值谱已知的情况,给出了牵制节点数目估计的必要条件和充分条件.另一方面,给出了判定是否为牵制节点的线性矩阵不等式判据,进而提出了基于节点度选择牵制节点的策略,并制定了可应用到一般网络拓扑结构的牵制节点搜索策略.
3.研究了一类快速切换耦合网络的同步控制问题.本文尝试用随机牵制控制策略来同步一类快速切换耦合网络模型,其中,网络节点是由位于平面上快速移动的智能体及其上的混沌振子构成,节点间的耦合关系则是通过智能体位置的邻近程度确定,网络的同步状态由振子来定义.以R(o丨¨)ssler振子为例,针对其两种典型耦合方式分别从理论和仿真实验上探讨了牵制控制同步的条件.研究结果表明,在具有无界同步化区域的网络中,牵制控制能力由牵制密度决定,与网络规模、牵制概率及节点密度无关;而具有有界同步化区域的网络实施牵制时对反馈增益、节点密度和牵制概率等有苛刻的要求,这导致牵制控制难以实施,需要新的控制手段如随机快速切换牵制网络节点来实现.
4.设计了不确定时滞耦合网络的同步控制器.针对一般时滞线性不确定耦合网络,考虑网络中的耦合关系有界且界值在已知和未知的情况下,分别引入线性反馈控制和自适应反馈控制手段来同步网络.针对时滞非线性耦合网络中的节点未知和存在不确定参数这两种情况,分别设计了自适应反馈控制律以保证网络同步.通过构造Lyapunov-Krasovskii范函给出理论证明,并用仿真示例进一步说明这两种自适应控制手段都对不确定参数如节点函数、耦合矩阵和时滞等表现出非常强的鲁棒性.
Representing a complex system as a network consisting of numerous coupled individuals is a key approach to describe and understand different behaviors of a complex system. Complex network has become highly active, with many new systems being proposed in application areas ranging from physics to biology to social science and computer science. Synchronization problem, as a common phenomenon of a population of dynamically interacting units, is one of the most demonstrating topics in complex networks. Ubiquitous in various kinds of complex networks, synchronization is closely related to multiple external flock behaviors. Therefore it is potentially of great significance to investigate synchronization control problem of dynamical systems on complex networks. The main objective of this dissertation is to investigate the synchronization control of a class of continuous-time coupled dynamical networks. We introduce several effective controllers for synchronization of distinct networks, which primarily encompass static coupled networks, time-varying networks and general uncertain delayed coupled networks. The main content of this dissertation are summarized as follows:
1. A class of random pseudofractal network models (RPNs) is proposed for modeling the network topology in the sequel. RPNs are generated and evolved according to the rules of fractal growth and competition. Aspects of degree distribution, clustering coefficient and average path length of the RPNs can be obtained by taking continuous approximation. In addition, analytical and numerical results indicate that the PRN model possess a power-law distribution with the exponents tuned by model parameters between 2 and 3, and small-world effect, consistent with most real networks.
2. An adaptive pinning control scheme is proposed for synchronizing coupled weighted dynamical networks. Number estimation and selection strategy of pinned nodes, as two fundamental pinning design problems, are solved. To avoid the prediction of linear feedback gains in pinning process, an adaptive pinning scheme are proposed, with the criteria of local and global synchronization being given by construction of Lyapunov functions. In particular, adaptive pinning and linear pinning, in the aspect of pinning strategy, are proven to be equivalent. As a result, number estimation and selection strategy of pinned nodes can be solved by using matrix theory and linear matrix inequality theory, which also set the foundation for subsequent discussion. Firstly, necessary and sufficient condition of the number of pinned nodes is given in form of eigenvalues of coupling matrix. Secondly, LMI-based criteria are given to determine the pinned nodes. As an extension to networks with general topology, several pinning and searching strategies are introduced respectively according to degrees of nodes.
3. Synchronization control of fast switching coupled network is studied, with the specific task of dictating multiple mobile agents in a plane towards a desired orbit. The problem in this dissertation is investigated by means of pinning, where each agent is associated with a chaotic oscillator coupled with those of neighboring ones. In addition, the pinning strategy is achieved by exerting common linear feedback on a small fraction of agents that are chosen randomly. We explore aspects of pinning probability, feedback gains and agent density under the assumption of fast switching constraint. Particularly, we show that in cases where network synchronization region is unbounded, synchronization of the dynamical network is exclusively determined by pinning density. However, such control strategy works little for or even weakens the synchronizability for networks with single bounded synchronization region.
4. Synchronization controllers for uncertain dynamical networks with delays are designed. We consider linearly-coupled networks with uncertain delay, typified by bounded entries of coupling matrix. Synchronization schemes based on linear feedback and adaptive control are demonstrated respectively. For a nonlinearly-coupled network with completely unknown or partially known nodes, two adaptive synchronization controllers are proposed respectively. Several criteria guaranteeing synchronization of these systems are established by employing the Lyapunov-Krasovskii functionals as well as numerical results for validation. In particular, we show that such controllers are highly robust against uncertain parameters, such as node functions, coupling matrix and time delays.
1.提出了一类随机伪分形演化网络,为本文随后章节的相关讨论提供了网络拓扑结构模型.该模型通过分形生长和边权竞争机制实现网络演化.利用连续域方法解析计算了网络的度分布、簇系数和平均路径长度,并执行数值仿真加以验证.结果表明这一类随机伪分形网络同时具有现实网络系统的两个普适特性—小世界效应和无标度特性,并且其幂律分布下的斜率可通过调节竞争参数α使之位于(2,3)之间,与绝大多数真实网络的分布斜率相符合.
2.设计了一种自适应牵制反馈控制手段用于同步加权耦合网络,并解决了实施牵制时的两个基本问题:牵制节点数目估计和牵制节点选择策略.考虑到一般的线性反馈牵制需要预估反馈增益,提出了利用局部余差自主调节反馈增益的自适应牵制控制手段,并给出了控制加权网络局部同步和全局同步的理论保证.特别地,证明了这种自适应牵制手段在理论分析研究时可等价为一般的线性反馈牵制,故可用线性牵制来研究自适应牵制的节点选择问题.以此为基础,利用矩阵论和线性矩阵不等式相关知识展开讨论.一方面,基于网络耦合矩阵特征值谱已知的情况,给出了牵制节点数目估计的必要条件和充分条件.另一方面,给出了判定是否为牵制节点的线性矩阵不等式判据,进而提出了基于节点度选择牵制节点的策略,并制定了可应用到一般网络拓扑结构的牵制节点搜索策略.
3.研究了一类快速切换耦合网络的同步控制问题.本文尝试用随机牵制控制策略来同步一类快速切换耦合网络模型,其中,网络节点是由位于平面上快速移动的智能体及其上的混沌振子构成,节点间的耦合关系则是通过智能体位置的邻近程度确定,网络的同步状态由振子来定义.以R(o丨¨)ssler振子为例,针对其两种典型耦合方式分别从理论和仿真实验上探讨了牵制控制同步的条件.研究结果表明,在具有无界同步化区域的网络中,牵制控制能力由牵制密度决定,与网络规模、牵制概率及节点密度无关;而具有有界同步化区域的网络实施牵制时对反馈增益、节点密度和牵制概率等有苛刻的要求,这导致牵制控制难以实施,需要新的控制手段如随机快速切换牵制网络节点来实现.
4.设计了不确定时滞耦合网络的同步控制器.针对一般时滞线性不确定耦合网络,考虑网络中的耦合关系有界且界值在已知和未知的情况下,分别引入线性反馈控制和自适应反馈控制手段来同步网络.针对时滞非线性耦合网络中的节点未知和存在不确定参数这两种情况,分别设计了自适应反馈控制律以保证网络同步.通过构造Lyapunov-Krasovskii范函给出理论证明,并用仿真示例进一步说明这两种自适应控制手段都对不确定参数如节点函数、耦合矩阵和时滞等表现出非常强的鲁棒性.
Representing a complex system as a network consisting of numerous coupled individuals is a key approach to describe and understand different behaviors of a complex system. Complex network has become highly active, with many new systems being proposed in application areas ranging from physics to biology to social science and computer science. Synchronization problem, as a common phenomenon of a population of dynamically interacting units, is one of the most demonstrating topics in complex networks. Ubiquitous in various kinds of complex networks, synchronization is closely related to multiple external flock behaviors. Therefore it is potentially of great significance to investigate synchronization control problem of dynamical systems on complex networks. The main objective of this dissertation is to investigate the synchronization control of a class of continuous-time coupled dynamical networks. We introduce several effective controllers for synchronization of distinct networks, which primarily encompass static coupled networks, time-varying networks and general uncertain delayed coupled networks. The main content of this dissertation are summarized as follows:
1. A class of random pseudofractal network models (RPNs) is proposed for modeling the network topology in the sequel. RPNs are generated and evolved according to the rules of fractal growth and competition. Aspects of degree distribution, clustering coefficient and average path length of the RPNs can be obtained by taking continuous approximation. In addition, analytical and numerical results indicate that the PRN model possess a power-law distribution with the exponents tuned by model parameters between 2 and 3, and small-world effect, consistent with most real networks.
2. An adaptive pinning control scheme is proposed for synchronizing coupled weighted dynamical networks. Number estimation and selection strategy of pinned nodes, as two fundamental pinning design problems, are solved. To avoid the prediction of linear feedback gains in pinning process, an adaptive pinning scheme are proposed, with the criteria of local and global synchronization being given by construction of Lyapunov functions. In particular, adaptive pinning and linear pinning, in the aspect of pinning strategy, are proven to be equivalent. As a result, number estimation and selection strategy of pinned nodes can be solved by using matrix theory and linear matrix inequality theory, which also set the foundation for subsequent discussion. Firstly, necessary and sufficient condition of the number of pinned nodes is given in form of eigenvalues of coupling matrix. Secondly, LMI-based criteria are given to determine the pinned nodes. As an extension to networks with general topology, several pinning and searching strategies are introduced respectively according to degrees of nodes.
3. Synchronization control of fast switching coupled network is studied, with the specific task of dictating multiple mobile agents in a plane towards a desired orbit. The problem in this dissertation is investigated by means of pinning, where each agent is associated with a chaotic oscillator coupled with those of neighboring ones. In addition, the pinning strategy is achieved by exerting common linear feedback on a small fraction of agents that are chosen randomly. We explore aspects of pinning probability, feedback gains and agent density under the assumption of fast switching constraint. Particularly, we show that in cases where network synchronization region is unbounded, synchronization of the dynamical network is exclusively determined by pinning density. However, such control strategy works little for or even weakens the synchronizability for networks with single bounded synchronization region.
4. Synchronization controllers for uncertain dynamical networks with delays are designed. We consider linearly-coupled networks with uncertain delay, typified by bounded entries of coupling matrix. Synchronization schemes based on linear feedback and adaptive control are demonstrated respectively. For a nonlinearly-coupled network with completely unknown or partially known nodes, two adaptive synchronization controllers are proposed respectively. Several criteria guaranteeing synchronization of these systems are established by employing the Lyapunov-Krasovskii functionals as well as numerical results for validation. In particular, we show that such controllers are highly robust against uncertain parameters, such as node functions, coupling matrix and time delays.
引文
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