耦合动力系统的同步控制研究
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摘要
近几年耦合动力系统的集体行为受到了科学界与工程界的广泛关注.该方向的研究工作揭示了众多自然现象的本质,比如蝉的共鸣,疾病传播等.如果将每个人看作是一个节点并且直接认识的人之间存在一条连接边,人类社会就是一个由网络描述的耦合系统.将系统整体抽象成由许多独立运行却又互相影响的个体所组成的网络是分析系统性能的有效方法,学者们基于此方法研究了通讯系统、电力系统、交通系统、代谢系统、供应链系统等.耦合动力系统的复杂性不仅源于其中每个成员自身的动力学行为,还有它们交错在一起产生的复杂结构.
虽然耦合系统具有高度复杂性,但在一定条件下却能演化出自组织、涌现等集体行为.研究涌现的集体行为对认识和设计耦合动力系统具有理论意义和实际价值.同步是自然界中一类典型的集体行为,它被广泛地运用到经济、社会、物理、生物等领域.不同于孤立系统,耦合系统的动力学行为还受到耦合方式的影响.但是系统状态演化是否会影响系统结构?众所周知通过外部控制作用或者内部耦合作用可以达到耦合同步效果,但两者之间有无关系?在不同的耦合机制下如何设计高效的控制方法?此外由于规模庞大,复杂系统容易遭受随机扰动、内部组件失效、信息传递延迟等.这些因素会对系统同步性能造成什么影响?如何处理?本文针对上述问题研究了网络上动力系统的同步控制方法,以及系统结构与同步表现之间的相互影响.针对几类具有不同连接方式的耦合动力系统,综合运用稳定性理论、随机分析、线性矩阵不等式、代数图论等工具,建立了相应的同步规范并归纳了实现耦合同步的一般方法.本文的研究内容和创新成果包含以下几个方面:
考虑一类具有Markov切换结构和混合耦合时滞的强连接网络模型.综合运用代数图论和随机分析等工具,分别在全局控制和局部控制作用下研究了网络的H∞同步表现.该结论推广了确定系统的同步结果.
学习一类具有节点时滞和耦合时滞的网络模型,该网络具有社团结构.综合运用稳定性理论和矩阵理论等工具,建立了牵制控制策略下网络的聚类同步规范.基于谱分析理论设计算法定位受控对象,使用该方法可以在大规模网络中实现聚类同步.
分析一系列含有Markov切换参数和Wiener过程的模糊神经网络,耦合系统对应弱连接网络.综合运用随机分析和代数图论等工具,建立了耦合随机系统的同步标准.通过构造辅助矩阵定义了系统的同步误差,并在均方意义下证明了误差系统是指数稳定的.同时使用此辅助矩阵解释了耦合系统牵制同步与自同步之间的关系.
在没有外界控制输入的情况下,探索一系列随机神经网络如何在不连续耦合作用下达到同步效果,耦合系统对应弱连接网络.不连续耦合作用包括周期间隔耦合和脉冲耦合.综合运用随机分析和归纳法等工具,推导出均方意义下的同步标准.该结论推广了连续耦合系统的同步结果.
根据忆阻器的物理特性,建立具有状态切换参数的耦合忆阻神经网络模型.分别在耦合系统对应网络是不连接和弱连接的情况下研究其指数同步性.综合运用微分包含理论、集值映射理论和广义稳定性理论等工具,在不连续系统解的框架下推导出基于线性矩阵不等式的指数同步条件.对于第一种情况,由聚类算法定位受控对象并施加控制作用实现牵制同步;对于第二种情况,由弱连接耦合方式实现系统自同步.
最后对全文进行了总结并讨论了拟开展的研究工作.本文对耦合动力系统同步控制问题的研究不仅丰富了耦合动力系统的同步理论还为耦合动力系统的同步控制实现提供了可靠依据.通过数值仿真证明了理论结论的正确性和有效性.
In recent years, collective behaviors of coupled dynamical systems have gained in-creasing research interest in both science and engineering. The study contributes to revealthe nature of some interesting phenomena, such as resonances of cicadas, epidemic spread-ing, etc. The human society can be described by a network, in which each individuality isviewed as a node and there exists an edge between two persons if they know each other. Re-garding the whole system as a network composed of interactively dynamical individualitiesis an effective approach to study system properties, based on which great efforts have beenmade to study communication systems, power systems, transportation systems, metastasissystems, supply chain systems, etc. The complexities of coupled dynamical systems comefrom dynamics of each subsystem as well as their complicate connections.
Although processing high complexity, coupled dynamical systems have some collec-tive behaviors under suitable conditions, such as self organization and emergence. Inves-tigating these collective behaviors has both theoretical and practical significance in under-standing and designing complex systems. Synchronization is a typical collective behaviorin nature and has been extensively applied to economy, society, physics and biology, etc.Different from isolated system, the dynamics of coupled systems are also determined bycoupling pattern. However, does dynamic evolution have certain impact on system struc-ture? It is generally known that synchronization can be realized by external control or innercoupling. Is there any relation between them? How do we design effective control methodsfor coupled dynamical systems with different structures? Owing to extremely large size,coupled dynamical systems are often subject to stochastic disturbances, invalid componentsand information transfer delay, etc. How will these noises impact on synchronization perfor-mances? And how do we manage them? To solve above problems, this dissertation mainlyfocuses on synchronization control of coupled dynamical systems, as well as mutual effectsbetween coupling structure and synchronization performance. For some classes of coupleddynamical systems, the authors establish corresponding synchronization criteria by stabilitytheory, stochastic analysis, linear matrix inequality and algebraic graph theory, etc. Further-more, general methods that guarantee synchronization for coupled dynamical systems aresummarized. The main contributions of this dissertation are as follows:
A class of dynamical networks with Markovian switching structures and mixed cou-pling delays is considered, whose underlying graph is strongly connected. By algebraicgraph theory and stochastic analysis, the authors study the H∞synchronization perfor- mances of such stochastic system with global controllers and local controllers, respectively.The results include some existing results of H∞synchronization of coupled dynamical sys-tems without stochastic disturbances as special cases.
A class of dynamical networks with node delays and coupling delays is studied, whoseunderlying graph contains many communities. Through stability theory and matrix theory,the authors establish cluster synchronization conditions. The controlled objects are detectedby means of spectral analysis, based on which cluster synchronization can be realized inlarge scale network.
An array of fuzzy neural networks with Markovian switching parameters and Wienerprocess is analyzed, whose underlying network is weakly connected. Applying stochasticanalysis and algebraic graph theory, the authors develop self synchronization criteria forsuch coupled stochastic systems. Synchronization error is defined by some auxiliary matri-ces and error system is proved to be exponentially stable in the mean square. Furthermore,the relation between pinning synchronization and self synchronization of coupled dynamicalsystems is revealed by these auxiliary matrices.
Without control inputs, the authors explore how to synchronize coupled stochastic neu-ral networks by discontinuous coupling, whose underlying network is weakly connected.Two kinds of discontinuous coupling modes are involved, periodically intermittent couplingand impulsive coupling. The results include some existing results of stochastic synchroniza-tion of continuous coupled systems as special cases.
According to physical properties of memristors, the authors establish coupled memris-tive neural networks with state-dependent switching. Its exponential synchronization per-formances are analyzed from two aspects, disconnected underlying network and weaklyconnected one. By differential inclusion theory, set valued map theory and generalized sta-bility theory, linear matrix inequality-based synchronization conditions are obtained underthe framework of discontinuous system’s solutions. For the first case, controlled objects aredetected by aggregated approach and pinning synchronization is realized. For the secondcase, weakly connected coupling pattern leads to self synchronization.
Finally, conclusion is made and future work is presented. In this dissertation, theresearch on synchronization control of coupled dynamical systems not only enrichessynchronization theories, but also provides reliable bases for practical applications. Nu-merical simulations are given to verify the usefulness and effectiveness of theoretical results.
虽然耦合系统具有高度复杂性,但在一定条件下却能演化出自组织、涌现等集体行为.研究涌现的集体行为对认识和设计耦合动力系统具有理论意义和实际价值.同步是自然界中一类典型的集体行为,它被广泛地运用到经济、社会、物理、生物等领域.不同于孤立系统,耦合系统的动力学行为还受到耦合方式的影响.但是系统状态演化是否会影响系统结构?众所周知通过外部控制作用或者内部耦合作用可以达到耦合同步效果,但两者之间有无关系?在不同的耦合机制下如何设计高效的控制方法?此外由于规模庞大,复杂系统容易遭受随机扰动、内部组件失效、信息传递延迟等.这些因素会对系统同步性能造成什么影响?如何处理?本文针对上述问题研究了网络上动力系统的同步控制方法,以及系统结构与同步表现之间的相互影响.针对几类具有不同连接方式的耦合动力系统,综合运用稳定性理论、随机分析、线性矩阵不等式、代数图论等工具,建立了相应的同步规范并归纳了实现耦合同步的一般方法.本文的研究内容和创新成果包含以下几个方面:
考虑一类具有Markov切换结构和混合耦合时滞的强连接网络模型.综合运用代数图论和随机分析等工具,分别在全局控制和局部控制作用下研究了网络的H∞同步表现.该结论推广了确定系统的同步结果.
学习一类具有节点时滞和耦合时滞的网络模型,该网络具有社团结构.综合运用稳定性理论和矩阵理论等工具,建立了牵制控制策略下网络的聚类同步规范.基于谱分析理论设计算法定位受控对象,使用该方法可以在大规模网络中实现聚类同步.
分析一系列含有Markov切换参数和Wiener过程的模糊神经网络,耦合系统对应弱连接网络.综合运用随机分析和代数图论等工具,建立了耦合随机系统的同步标准.通过构造辅助矩阵定义了系统的同步误差,并在均方意义下证明了误差系统是指数稳定的.同时使用此辅助矩阵解释了耦合系统牵制同步与自同步之间的关系.
在没有外界控制输入的情况下,探索一系列随机神经网络如何在不连续耦合作用下达到同步效果,耦合系统对应弱连接网络.不连续耦合作用包括周期间隔耦合和脉冲耦合.综合运用随机分析和归纳法等工具,推导出均方意义下的同步标准.该结论推广了连续耦合系统的同步结果.
根据忆阻器的物理特性,建立具有状态切换参数的耦合忆阻神经网络模型.分别在耦合系统对应网络是不连接和弱连接的情况下研究其指数同步性.综合运用微分包含理论、集值映射理论和广义稳定性理论等工具,在不连续系统解的框架下推导出基于线性矩阵不等式的指数同步条件.对于第一种情况,由聚类算法定位受控对象并施加控制作用实现牵制同步;对于第二种情况,由弱连接耦合方式实现系统自同步.
最后对全文进行了总结并讨论了拟开展的研究工作.本文对耦合动力系统同步控制问题的研究不仅丰富了耦合动力系统的同步理论还为耦合动力系统的同步控制实现提供了可靠依据.通过数值仿真证明了理论结论的正确性和有效性.
In recent years, collective behaviors of coupled dynamical systems have gained in-creasing research interest in both science and engineering. The study contributes to revealthe nature of some interesting phenomena, such as resonances of cicadas, epidemic spread-ing, etc. The human society can be described by a network, in which each individuality isviewed as a node and there exists an edge between two persons if they know each other. Re-garding the whole system as a network composed of interactively dynamical individualitiesis an effective approach to study system properties, based on which great efforts have beenmade to study communication systems, power systems, transportation systems, metastasissystems, supply chain systems, etc. The complexities of coupled dynamical systems comefrom dynamics of each subsystem as well as their complicate connections.
Although processing high complexity, coupled dynamical systems have some collec-tive behaviors under suitable conditions, such as self organization and emergence. Inves-tigating these collective behaviors has both theoretical and practical significance in under-standing and designing complex systems. Synchronization is a typical collective behaviorin nature and has been extensively applied to economy, society, physics and biology, etc.Different from isolated system, the dynamics of coupled systems are also determined bycoupling pattern. However, does dynamic evolution have certain impact on system struc-ture? It is generally known that synchronization can be realized by external control or innercoupling. Is there any relation between them? How do we design effective control methodsfor coupled dynamical systems with different structures? Owing to extremely large size,coupled dynamical systems are often subject to stochastic disturbances, invalid componentsand information transfer delay, etc. How will these noises impact on synchronization perfor-mances? And how do we manage them? To solve above problems, this dissertation mainlyfocuses on synchronization control of coupled dynamical systems, as well as mutual effectsbetween coupling structure and synchronization performance. For some classes of coupleddynamical systems, the authors establish corresponding synchronization criteria by stabilitytheory, stochastic analysis, linear matrix inequality and algebraic graph theory, etc. Further-more, general methods that guarantee synchronization for coupled dynamical systems aresummarized. The main contributions of this dissertation are as follows:
A class of dynamical networks with Markovian switching structures and mixed cou-pling delays is considered, whose underlying graph is strongly connected. By algebraicgraph theory and stochastic analysis, the authors study the H∞synchronization perfor- mances of such stochastic system with global controllers and local controllers, respectively.The results include some existing results of H∞synchronization of coupled dynamical sys-tems without stochastic disturbances as special cases.
A class of dynamical networks with node delays and coupling delays is studied, whoseunderlying graph contains many communities. Through stability theory and matrix theory,the authors establish cluster synchronization conditions. The controlled objects are detectedby means of spectral analysis, based on which cluster synchronization can be realized inlarge scale network.
An array of fuzzy neural networks with Markovian switching parameters and Wienerprocess is analyzed, whose underlying network is weakly connected. Applying stochasticanalysis and algebraic graph theory, the authors develop self synchronization criteria forsuch coupled stochastic systems. Synchronization error is defined by some auxiliary matri-ces and error system is proved to be exponentially stable in the mean square. Furthermore,the relation between pinning synchronization and self synchronization of coupled dynamicalsystems is revealed by these auxiliary matrices.
Without control inputs, the authors explore how to synchronize coupled stochastic neu-ral networks by discontinuous coupling, whose underlying network is weakly connected.Two kinds of discontinuous coupling modes are involved, periodically intermittent couplingand impulsive coupling. The results include some existing results of stochastic synchroniza-tion of continuous coupled systems as special cases.
According to physical properties of memristors, the authors establish coupled memris-tive neural networks with state-dependent switching. Its exponential synchronization per-formances are analyzed from two aspects, disconnected underlying network and weaklyconnected one. By differential inclusion theory, set valued map theory and generalized sta-bility theory, linear matrix inequality-based synchronization conditions are obtained underthe framework of discontinuous system’s solutions. For the first case, controlled objects aredetected by aggregated approach and pinning synchronization is realized. For the secondcase, weakly connected coupling pattern leads to self synchronization.
Finally, conclusion is made and future work is presented. In this dissertation, theresearch on synchronization control of coupled dynamical systems not only enrichessynchronization theories, but also provides reliable bases for practical applications. Nu-merical simulations are given to verify the usefulness and effectiveness of theoretical results.
引文
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