基于复杂网络理论的复杂系统同步控制研究
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摘要
复杂网络的理论研究经过近十年的发展,已经取得了令人瞩目的一些成果。复杂网络的复杂结构和动力学行为的多样性使得复杂系统的研究更具意义,也更具挑战性,复杂网络的研究也被认为是21世纪科学技术前沿战略性研究课题之一。其中,关于复杂网络系统动力学行为中同步的研究也已成为控制工程界的重要研究课题。本文在阅读现有的复杂网络有关文献的基础上,对这一研究课题进行了全面的综述,并从复杂网络理论的角度对复杂系统的同步控制问题进行了较为深入的研究。论文的主要研究内容及创新点包括如下七个方面:
1.从复杂网络理论这一新的角度对大规模工程系统进行讨论,并分别构建了几类常见的大规模工程系统的复杂网络模型。随后,利用复杂网络理论分析了工程系统的网络拓扑特性,并在此基础上,将复杂网络系统同步方面的研究与大型工程系统的应用相结合,根据复杂网络同步理论对工程系统的动态性能进行探讨,从而为此类问题的解决提供了新的思路。
2.针对星形耦合网络系统同步问题进行了研究。首先讨论了此类系统同步问题,发现系统达到同步与网络的耦合强度无关。本章提出了复杂网络系统可反馈同步化的概念,并研究了星形耦合网络系统的可反馈同步化问题,得到了此类系统可反馈同步化的判据。其次,讨论了采用牵制控制策略实现低阶星形网络系统的同步问题,并以充分利用网络的结构信息为基础,分别给出局部控制律和全局控制律的设计方法。最后,将之推广到高阶系统,研究了高阶非线性星形网络系统的牵制控制问题,得到了比较具体的结论,具有实际应用价值。
3.针对最近邻耦合网络系统的同步控制进行了讨论。首先研究了此类系统达到同步的条件,发现只要节点本身的动力学行为稳定,则此类系统能够达到同步。根据第三章提出的可反馈同步化的概念,研究了最近邻耦合网络系统可反馈同步化的问题,并得到相应的判据。其次,本文利用控制作用衰减率概念,研究了最近邻耦合网络系统的牵制控制能力,并对此类网络分析了不同的牵制控制策略的有效性。随后,在此基础上,采用牵制控制策略实现此类系统的同步,并分别给出了系统局部控制律和全局控制律的设计方法。此方法充分利用了网络结构信息,从而减少了控制律设计的保守性。
4.讨论了一类广义时滞复杂动态网络系统的同步稳定性和牵制控制问题。针对此类系统,本文首先分别给出了保守性小的连续时间时滞复杂动态网络系统和离散时间时滞复杂动态网络系统的同步稳定性条件。其次,通过对部分节点施加牵制控制作用的方法,设计分散反馈控制器,分别使得连续时间时滞复杂动态网络系统和离散时间时滞复杂动态网络系统达到同步,从而保证了整个系统的同步稳定性。随后,在此基础上,将控制器的设计问题转化为求解线性矩阵不等式(LMI)组合的凸优化问题。该问题便于利用现有的优化软件求解,也大大降低了问题求解的复杂性。
5.针对由动力学行为不同的节点构成的异质复杂网络系统进行研究。大规模复杂系统往往根据工艺、空间或时间的不同划分为不同的子系统。根据各子系统的特点,可分别采用最适合的方式来建立模型,并据此形成由不同类型的子系统模型构成的关联系统。本文以复杂网络理论为基础,通过构建此类关联系统的网络模型,研究了使此类新颖的异质复杂动态网络系统稳定的牵制控制问题。该类复杂网络系统分别由不同模型描述的节点相互关联而成,利用牵制控制使得该类系统达到稳定。根据Lyapunov稳定性定理,分别独立地求解了对应于各节点及其关联拓扑的线性矩阵不等式(LMI),从而判断出异质复杂网络系统牵制控制的稳定性。上述方法将原问题转化为多个低维线性矩阵不等式的并行求解,大大减少计算的复杂性。
6.利用网络结构优化的方法,研究了如何改善复杂网络系统的同步性和一致性的问题。本文首先给出了加权网络熵的定义,将之用来测量加权网络的均匀性。其次,分析了网络均匀性、一致性和同步性三者之间的关系。随后,基于此加权网络熵指标和网络的统计特性,提出了一种改善复杂网络系统一致性和同步性的结构优化方法,即通过尽可能少地增加连线来最大程度地增大复杂网络系统的一致性和同步性。该方法也为此类问题的解决提供了一个新的途径。
7.基于复杂动态网络系统同步控制理论,研究了分布式多移动机器人的队形保持和跟踪控制问题。首先,为了估算外部向量场变化情况下的所有移动机器人的测量的平均值,每个移动机器人需测量自身运动轨迹的向量场局部值,并将之与其他邻近机器人共享。根据共享的信息,移动机器人以同步协商的合作方式控制自身运动轨迹,同时维持一定的队形。其次,研究了跟随领航者的队形控制方法。针对多机器人之间形成的两种队形(最近邻耦合网络和星形网络)进行分析,并提出了具体的移动机器人动态控制律的设计方法,可较方便地对系统的极点进行配置。最后,探讨了复杂系统同步理论在多移动机器人中的应用。
The recent decade has witnessed the great development of researches on the study on complex networks. Diversity of the structure and the dynamical behaviors of complex networks make the research of them more meaningful and more challenging. The research on the synchronization has been an important object in control engineering field. Based on the former research, the thesis gives a survey of complex networks systems on the basis of related references and deals with some issues of synchronization control of complex networks systems on a theoretical and practical background. The main contents contain the following seven parts:
1. From the new view of complex networks large-scale engineering systems are discussed. And the models of complex networks are built for these engineering systems. Then, the properties of the networks topology of the systems are analyzed based on complex networks theory. Combining the research of the synchronization of complex networks with large-scale engineering systems, we discuss the dynamical properties of the systems. So a novel way for solving this kind of problems for dealing with large-scale systems is presented.
2. Focusing on star-shaped complex networks systems, conditions of achieving synchronization for this kind of systems are firstly discussed in the thesis. And it is found that the conditions are not related with the coupled strength of the networks. Then the concept of capable feedback synchronization (CFS) is presented. The problem of CFS of star-shaped complex networks is investigated and the criterion of CFS is obtained. Secondly, pinning control strategy is designed to make lower-order star-shaped complex networks systems synchronized based on the structural information of the networks. The strategies of global pinning control and local pinning control are presented in this thesis. At the same time, the pinning strategy of high-order star-shaped complex networks systems is also designed. Finally, concrete conclusions are arrived which are more suitable to the application.
3. For the nearest-neighbor coupled complex networks systems, conditions of achieving synchronization for this kind of systems are firstly discussed in this thesis and it is found the system can achieve synchronization if the dynamics of nodes are stable. Based on the definition of CFS, the problem of CFS for this kind of systems is investigated. Secondly, the concept of attenuation rate of control ability is presented. The capacity of pinning control for the nearest-neighbor coupled complex networks systems is studied. In the following, the validity of different control strategies is compared in the given coupled complex networks. Finally, the strategies of global pinning control and local pinning control are achieved for this type of complex networks systems.
4. The problems of synchronization and pinning control for general time-delay dynamical systems are investigated. In this thesis less conservative criterions of continuous-time complex dynamical systems with time-delay and discrete-time complex dynamical systems with time-delay are obtained. We apply pinning control strategies to a small fraction of nodes. Then the decentralized feedback controllers are respectively designed to make both continuous-time complex dynamical systems with time-delay and discrete-time complex dynamical systems with time-delay synchronized. At the same time, the stability of synchronization for the two kinds of systems is guaranteed. Moreover, the problems of designing controllers are converted into solving optimal problems of a series of linear matrix inequalities. The solution can be made use of available software, which reduces the complexity.
5. Heterogeneous complex dynamical networks are investigated, in which the dynamics of nodes are different. Complex large scale systems possess the characteristics of high dimensionality, large number of variables and strong nonlinearity. One of the effective approaches for the complex system modeling is using one hybrid interconnected model to describe the dynamics of each subsystem. This thesis focuses on pinning control of a kind of heterogeneous complex network systems, whose dynamics of nodes described by different models. Through constructing a set of independent lower-dimensional linear matrix inequalities, and solving them in parallel, a novel stability analysis method is proposed. Thus the computational complexity is greatly reduced. Computer simulation is conducted to validate the effectiveness and efficiency of the proposed method.
6. The problem of improving the consensus and synchronizability for weighted networks of dynamic systems via adding links among them is investigated in this paper. A weighted network entropy is introduced to measure the homogeneity of the weighted networks. The relationship between the homogeneity and the consensus and synchronizability is analyzed. A solution scheme is proposed to improve the consensus and synchronizability of the weighted networks through maximizing their homogeneity via adding as few links as possible. Weighted networks created from the Barrat model are tested by using the proposed scheme, and two commonly used link-adding approaches are selected to compare with this scheme. Computer simulation results show that the proposed link-adding scheme can enhance the consensus and the synchronizability of the weighted networks effectively and efficiently.
7. Based on the theory of synchronization in complex networks, formation and tracking control of distributed multi mobile-robots are investigated under a changing external vector field. Each mobile robot measures the local value of the field along its trajectory and occasionally shares relevant information with other agents, in order to estimate the spatial average obtained from measurements across all mobile robots. Using shared information, mobile robots control their trajectories in a cooperative manner, with the goals of maintaining a desired formation about the average. Then two kind of communication networks including star-shaped networks and the nearest-neighbor networks formed between mobile robots is studied, based on which control strategies are given for mobile robots. Finally, applications of synchronization to mobile robots are further analyzed.
1.从复杂网络理论这一新的角度对大规模工程系统进行讨论,并分别构建了几类常见的大规模工程系统的复杂网络模型。随后,利用复杂网络理论分析了工程系统的网络拓扑特性,并在此基础上,将复杂网络系统同步方面的研究与大型工程系统的应用相结合,根据复杂网络同步理论对工程系统的动态性能进行探讨,从而为此类问题的解决提供了新的思路。
2.针对星形耦合网络系统同步问题进行了研究。首先讨论了此类系统同步问题,发现系统达到同步与网络的耦合强度无关。本章提出了复杂网络系统可反馈同步化的概念,并研究了星形耦合网络系统的可反馈同步化问题,得到了此类系统可反馈同步化的判据。其次,讨论了采用牵制控制策略实现低阶星形网络系统的同步问题,并以充分利用网络的结构信息为基础,分别给出局部控制律和全局控制律的设计方法。最后,将之推广到高阶系统,研究了高阶非线性星形网络系统的牵制控制问题,得到了比较具体的结论,具有实际应用价值。
3.针对最近邻耦合网络系统的同步控制进行了讨论。首先研究了此类系统达到同步的条件,发现只要节点本身的动力学行为稳定,则此类系统能够达到同步。根据第三章提出的可反馈同步化的概念,研究了最近邻耦合网络系统可反馈同步化的问题,并得到相应的判据。其次,本文利用控制作用衰减率概念,研究了最近邻耦合网络系统的牵制控制能力,并对此类网络分析了不同的牵制控制策略的有效性。随后,在此基础上,采用牵制控制策略实现此类系统的同步,并分别给出了系统局部控制律和全局控制律的设计方法。此方法充分利用了网络结构信息,从而减少了控制律设计的保守性。
4.讨论了一类广义时滞复杂动态网络系统的同步稳定性和牵制控制问题。针对此类系统,本文首先分别给出了保守性小的连续时间时滞复杂动态网络系统和离散时间时滞复杂动态网络系统的同步稳定性条件。其次,通过对部分节点施加牵制控制作用的方法,设计分散反馈控制器,分别使得连续时间时滞复杂动态网络系统和离散时间时滞复杂动态网络系统达到同步,从而保证了整个系统的同步稳定性。随后,在此基础上,将控制器的设计问题转化为求解线性矩阵不等式(LMI)组合的凸优化问题。该问题便于利用现有的优化软件求解,也大大降低了问题求解的复杂性。
5.针对由动力学行为不同的节点构成的异质复杂网络系统进行研究。大规模复杂系统往往根据工艺、空间或时间的不同划分为不同的子系统。根据各子系统的特点,可分别采用最适合的方式来建立模型,并据此形成由不同类型的子系统模型构成的关联系统。本文以复杂网络理论为基础,通过构建此类关联系统的网络模型,研究了使此类新颖的异质复杂动态网络系统稳定的牵制控制问题。该类复杂网络系统分别由不同模型描述的节点相互关联而成,利用牵制控制使得该类系统达到稳定。根据Lyapunov稳定性定理,分别独立地求解了对应于各节点及其关联拓扑的线性矩阵不等式(LMI),从而判断出异质复杂网络系统牵制控制的稳定性。上述方法将原问题转化为多个低维线性矩阵不等式的并行求解,大大减少计算的复杂性。
6.利用网络结构优化的方法,研究了如何改善复杂网络系统的同步性和一致性的问题。本文首先给出了加权网络熵的定义,将之用来测量加权网络的均匀性。其次,分析了网络均匀性、一致性和同步性三者之间的关系。随后,基于此加权网络熵指标和网络的统计特性,提出了一种改善复杂网络系统一致性和同步性的结构优化方法,即通过尽可能少地增加连线来最大程度地增大复杂网络系统的一致性和同步性。该方法也为此类问题的解决提供了一个新的途径。
7.基于复杂动态网络系统同步控制理论,研究了分布式多移动机器人的队形保持和跟踪控制问题。首先,为了估算外部向量场变化情况下的所有移动机器人的测量的平均值,每个移动机器人需测量自身运动轨迹的向量场局部值,并将之与其他邻近机器人共享。根据共享的信息,移动机器人以同步协商的合作方式控制自身运动轨迹,同时维持一定的队形。其次,研究了跟随领航者的队形控制方法。针对多机器人之间形成的两种队形(最近邻耦合网络和星形网络)进行分析,并提出了具体的移动机器人动态控制律的设计方法,可较方便地对系统的极点进行配置。最后,探讨了复杂系统同步理论在多移动机器人中的应用。
The recent decade has witnessed the great development of researches on the study on complex networks. Diversity of the structure and the dynamical behaviors of complex networks make the research of them more meaningful and more challenging. The research on the synchronization has been an important object in control engineering field. Based on the former research, the thesis gives a survey of complex networks systems on the basis of related references and deals with some issues of synchronization control of complex networks systems on a theoretical and practical background. The main contents contain the following seven parts:
1. From the new view of complex networks large-scale engineering systems are discussed. And the models of complex networks are built for these engineering systems. Then, the properties of the networks topology of the systems are analyzed based on complex networks theory. Combining the research of the synchronization of complex networks with large-scale engineering systems, we discuss the dynamical properties of the systems. So a novel way for solving this kind of problems for dealing with large-scale systems is presented.
2. Focusing on star-shaped complex networks systems, conditions of achieving synchronization for this kind of systems are firstly discussed in the thesis. And it is found that the conditions are not related with the coupled strength of the networks. Then the concept of capable feedback synchronization (CFS) is presented. The problem of CFS of star-shaped complex networks is investigated and the criterion of CFS is obtained. Secondly, pinning control strategy is designed to make lower-order star-shaped complex networks systems synchronized based on the structural information of the networks. The strategies of global pinning control and local pinning control are presented in this thesis. At the same time, the pinning strategy of high-order star-shaped complex networks systems is also designed. Finally, concrete conclusions are arrived which are more suitable to the application.
3. For the nearest-neighbor coupled complex networks systems, conditions of achieving synchronization for this kind of systems are firstly discussed in this thesis and it is found the system can achieve synchronization if the dynamics of nodes are stable. Based on the definition of CFS, the problem of CFS for this kind of systems is investigated. Secondly, the concept of attenuation rate of control ability is presented. The capacity of pinning control for the nearest-neighbor coupled complex networks systems is studied. In the following, the validity of different control strategies is compared in the given coupled complex networks. Finally, the strategies of global pinning control and local pinning control are achieved for this type of complex networks systems.
4. The problems of synchronization and pinning control for general time-delay dynamical systems are investigated. In this thesis less conservative criterions of continuous-time complex dynamical systems with time-delay and discrete-time complex dynamical systems with time-delay are obtained. We apply pinning control strategies to a small fraction of nodes. Then the decentralized feedback controllers are respectively designed to make both continuous-time complex dynamical systems with time-delay and discrete-time complex dynamical systems with time-delay synchronized. At the same time, the stability of synchronization for the two kinds of systems is guaranteed. Moreover, the problems of designing controllers are converted into solving optimal problems of a series of linear matrix inequalities. The solution can be made use of available software, which reduces the complexity.
5. Heterogeneous complex dynamical networks are investigated, in which the dynamics of nodes are different. Complex large scale systems possess the characteristics of high dimensionality, large number of variables and strong nonlinearity. One of the effective approaches for the complex system modeling is using one hybrid interconnected model to describe the dynamics of each subsystem. This thesis focuses on pinning control of a kind of heterogeneous complex network systems, whose dynamics of nodes described by different models. Through constructing a set of independent lower-dimensional linear matrix inequalities, and solving them in parallel, a novel stability analysis method is proposed. Thus the computational complexity is greatly reduced. Computer simulation is conducted to validate the effectiveness and efficiency of the proposed method.
6. The problem of improving the consensus and synchronizability for weighted networks of dynamic systems via adding links among them is investigated in this paper. A weighted network entropy is introduced to measure the homogeneity of the weighted networks. The relationship between the homogeneity and the consensus and synchronizability is analyzed. A solution scheme is proposed to improve the consensus and synchronizability of the weighted networks through maximizing their homogeneity via adding as few links as possible. Weighted networks created from the Barrat model are tested by using the proposed scheme, and two commonly used link-adding approaches are selected to compare with this scheme. Computer simulation results show that the proposed link-adding scheme can enhance the consensus and the synchronizability of the weighted networks effectively and efficiently.
7. Based on the theory of synchronization in complex networks, formation and tracking control of distributed multi mobile-robots are investigated under a changing external vector field. Each mobile robot measures the local value of the field along its trajectory and occasionally shares relevant information with other agents, in order to estimate the spatial average obtained from measurements across all mobile robots. Using shared information, mobile robots control their trajectories in a cooperative manner, with the goals of maintaining a desired formation about the average. Then two kind of communication networks including star-shaped networks and the nearest-neighbor networks formed between mobile robots is studied, based on which control strategies are given for mobile robots. Finally, applications of synchronization to mobile robots are further analyzed.
引文
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