一类具混合时滞复杂网络的状态估计与同步分析
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摘要
本文研究一类含有混合时滞(mixed time delays)和随机参数的复杂网络的状态估计与同步分析问题。所考虑的复杂网络模型既含有离散时滞又含有无穷分布时滞,且离散时滞和无穷分布时滞均依赖于某一马尔科夫链(Markov chian),即网络系统时滞是马尔科夫模态依赖的。此外,还考虑了网络节点受维纳过程(Wiener process)的影响。论文首先研究上述模态依赖于马尔可夫链的网络系统的状态估计器的设计问题。通过构造网络的估计系统,选择合适的状态估计增益矩阵,获得网络系统的全阶状态估计器。进一步,通过构造新的Lyapunov-Krasovskii泛函和一些新的分析技巧,导出了网络系统节点在均方意义下同步的一些充分条件。所导出的状态估计器的存在性以及同步条件均能表示成线性矩阵不等式(LMI)的形式,从而可借助于Matlab LMI Toolbox有效地进行求解。最后,给出两个数值例子说明所提出理论的有效性和可应用性。值得指出的是:许多现存的结果是我们的特殊情形。
全文共由三个部分组成:第一部分,简要概述时滞的复杂网络研究的相关背景和意义,接着介绍时滞复杂网络动力学研究工作的进展,最后阐述了本文所要做的主要工作。第二部分,首先引入了所要考虑的复杂网络模型,通过构造新的Lyapunov-Krasovskii泛函和一些新的分析技巧,研究了复杂网络系统的状态估计问题,导出了系统状态估计器存在的LMI条件,数值例子说明我们提出方法的有效性。第三部分,进一步研究了复杂网络均方同步(或均方指数同步)问题,应用Lyapunov理论并利用随机分析的一些技巧,得到了系统同步(或指数同步)的一些充分性判据。一个数值例子被用来说明我们的方法。最后,我们给出简短的总结和展望。
This thesis is concerned with the state estimate and synchronization problem for a class of stochastic complex network with mixed time delays. The complex network under study is involved in both discrete time delay and unbounded distributed time delay, and both time delays are dependent on a Markov chian. i.e., the time delays are mode-dependent In addition, it is also assumed that the nodes of the network are disturbed by a wiener process. First, we cope with the state estimation for the complex network. By constructing the state estimation system and choosing an appropriate state estimation gain matrix, we obtain the desired estimator. Then, by constructing a novel Lyapunov-Krasovskii functional and employing some new analysis techniques, the sufficient conditions are derived to guarantee the mean-square synchronization between the nodes of the network. The obtained criteria, for either the existence of the state estimator or synchronization, can be expressed in the form of inequality (LMI), which can be easily checked by Matlab LMI Toolbox. Finally, two numerical examples are presented to demonstrate the effectiveness and applicability of the proposed methods. It is worth pointing out that our results include the existing ones as the special cases.
The thesis falls into three parts. The opening section gives an overview on the related background, and states the significance and the latest progress in the study of the complex networks. We conclude the chapter with the formulation of problems to be investigated in this thesis.
In Chapter 2, we begin with introducing the model of the complex network to be studied. A new Lyapunov-Krasovskii functional and some new techniques are employed to deal with the state estimation problem for the given network, the sufficient conditions are derived for the existence of the state estimator, which are in the form of LMI. And a numerical example is used to show our approach.
Chapter 3 focuses on the synchronization analysis for the given complex network. Based on Lyapunov theory, and by conducting stochastic analysis, we derive some easy-check criteria for the synchronization between the nodes of the network. An example is given for illustrative purpose. We conclude the chapter with some remarks .
全文共由三个部分组成:第一部分,简要概述时滞的复杂网络研究的相关背景和意义,接着介绍时滞复杂网络动力学研究工作的进展,最后阐述了本文所要做的主要工作。第二部分,首先引入了所要考虑的复杂网络模型,通过构造新的Lyapunov-Krasovskii泛函和一些新的分析技巧,研究了复杂网络系统的状态估计问题,导出了系统状态估计器存在的LMI条件,数值例子说明我们提出方法的有效性。第三部分,进一步研究了复杂网络均方同步(或均方指数同步)问题,应用Lyapunov理论并利用随机分析的一些技巧,得到了系统同步(或指数同步)的一些充分性判据。一个数值例子被用来说明我们的方法。最后,我们给出简短的总结和展望。
This thesis is concerned with the state estimate and synchronization problem for a class of stochastic complex network with mixed time delays. The complex network under study is involved in both discrete time delay and unbounded distributed time delay, and both time delays are dependent on a Markov chian. i.e., the time delays are mode-dependent In addition, it is also assumed that the nodes of the network are disturbed by a wiener process. First, we cope with the state estimation for the complex network. By constructing the state estimation system and choosing an appropriate state estimation gain matrix, we obtain the desired estimator. Then, by constructing a novel Lyapunov-Krasovskii functional and employing some new analysis techniques, the sufficient conditions are derived to guarantee the mean-square synchronization between the nodes of the network. The obtained criteria, for either the existence of the state estimator or synchronization, can be expressed in the form of inequality (LMI), which can be easily checked by Matlab LMI Toolbox. Finally, two numerical examples are presented to demonstrate the effectiveness and applicability of the proposed methods. It is worth pointing out that our results include the existing ones as the special cases.
The thesis falls into three parts. The opening section gives an overview on the related background, and states the significance and the latest progress in the study of the complex networks. We conclude the chapter with the formulation of problems to be investigated in this thesis.
In Chapter 2, we begin with introducing the model of the complex network to be studied. A new Lyapunov-Krasovskii functional and some new techniques are employed to deal with the state estimation problem for the given network, the sufficient conditions are derived for the existence of the state estimator, which are in the form of LMI. And a numerical example is used to show our approach.
Chapter 3 focuses on the synchronization analysis for the given complex network. Based on Lyapunov theory, and by conducting stochastic analysis, we derive some easy-check criteria for the synchronization between the nodes of the network. An example is given for illustrative purpose. We conclude the chapter with some remarks .
引文
[1] D. J. Watts, S. H. Strogatz, Collective dynamics of small–world networks, Nature, 393(1998), pp.440-442.
[2] R. Albert, H. Jeong, A. L. Barabási, Attack and Error Tolerance of Complex Networks, Nature, 406(2000), pp.378-382.
[3] X .F. Wang, Complex networks: Topology, dynamics and synchronization, Int. J. of Bifur. Chaos,12 (2002), pp.885-916.
[4] R. Albert, A. L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys., 74 (2002), pp.47-97.
[5] R. Albert, A. L. Barabási, Topology of evolving networks: local events and universality, Phys. Rev. Lett., 85(2000), pp.5234-5237.
[6] L. A. N. Amaral, et al, Virtual Round Table on ten leading questions for network research, Eur. Phys. J.B., 38(2004), pp.143-145.
[7] A. L. Barabási, R. Albert, Emergence of scaling in random networks, Science, 286(1999), pp.509-512.
[8] J. Camacho, R. Guimerá, L. A. N. Amaral, Robust patterns in food web structure, Phys. Rev. Lett., 88 (2002), pp.228102.
[9]方锦清,汪小帆,刘曾荣,略论复杂性问题和非线性复杂网络系统的研究,基础科学,科技导报, 2004年02期, 9-12.
[10] C.G. Li and G. Chen, A comprehensive weighted evolving network model, Physica A,343(2004), pp. 288-294.
[11] K. Klemm, V. M. Eguíluz, Highly clustered scale-free networks, Phy. Rev. E, 65(2002), pp. 036123.
[12] K.. Klemm, V. M. Eguíluz, Growing scale-free networks with small-world behavior, Phy. Rev. E, 65 (2002), pp.057102.
[13] X . Li and G. Chen, A local-world evolving network model, Physica A, 328(2003), pp. 274-286.
[14] Z. H. Liu, Y. C. Lai, N. Ye, Statistical properties and attack tolerance of growing networks with Algebraic preferential attachment, Phys. Rev. E, 66(2002), pp.036112.
[15] J.H. Lu, X.H. Yu and G. Chen, Chaos synchronization of general complex dynamical networks, Physica A, 334(2004), pp. 281-302.
[16] X.. F. Wang and G. Chen, Complex networks: small-world, scale-free, and beyond, IEEE Circ. Syst. Magazine, 3(2003), No. 1, pp. 6-20.
[17] X. F. Wang and G. Chen, Pinning control of scale-free dynamical networks, Physica A, 310(2002), pp.521-531.
[18] X.. F. Wang and G. Chen, Synchronization in small-world dynamical networks, Int. J. of Bifur. Chaos, 12(2002), pp.187-192.
[19] X. F. Wang and G. Chen, Synchronization in scale-free dynamical networks: Robustness and fragility, IEEE Trans. on Circ. Syst.–I, 49(2002), pp. 54-62.
[20] S. N. Dorogovtsev, J. F. F. Mendes, Evolution of networks, Adv. Phys., 51(2002), pp.1079-1187·
[21] M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45(2003), pp. 167-256.
[22] S. H. Strogatz, Exploring complex networks, Nature, 410(2001), pp. 268-276.
[23]吴彤,复杂网络研究及其意义,哲学研究,2004年第8期,pp.58-63.
[24] Y. Chen, G. Rangarajan and M. Ding, General stability analysis of synchronized dynamics in coupled systems, Phys Rev E 67 (2003), pp1–4( 026209).
[25] C.Li, W. Sun, D.Xu, Synchronization of complex dynamical network with nonlinear inner-coupling functions and time delays, Progress of Theoretical Physics, (2005), 114(4).
[26] J. Zhou and T. Chen, Synchronization in general complex delayed dynamical networks, IEEE Trans. Circuits Syst. I 53 (2) (2006), pp. 733–744.
[27] J. Cao, G. Chen, P. Li, Global synchronization in an array of delayed neural networks with hybrid coupling, IEEE Trans. Systems, Man, and Cybernetics-Part B: Cybernetics, 38:2(2008) 488-498
[28] W. Yu, J. Cao, J. Lu, Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM Journal on Applied Dynamical Systems, 7:1(2008) 108-133.
[29] R. Zhang, M. Hu and Z. Xu, Synchronization in complex networks with adaptive coupling, Physics Letters A, Volume 368, Issues 3-4, 20 August 2007, Pages 276-280.
[30] C. Li, G. Chen. Synchronization in general complex dynamical networks with coupling delays. Physica A. Volume 343, 15 November 2004, Pages 263-278
[31] Y. Liu, Z. Wang, J. Liang and X. Liu, Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time-delays, IEEE Transactions on Neural Networks, Vol. 20, No. 7, Jul. 2009, pp. 1102-1116.
[32] Y. Liu, Z. Wang, J. Liang and X. Liu, Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Transactions on Systems, Man, and Cybernetics - Part B, Vol. 38, No. 5, Oct. 2008, pp. 1314-1325.
[33] Y. Liu, Z. Wang and X. Liu, Exponential synchronization of complex networks with Markovian jump and mixed delays, Physics Letters A, Vol. 372, No. 22, May 2008, pp. 3986-3998.
[34] Q. K. Song, Design of controller on synchronization of chaotic neural networks with mixed time-varying delays. Neuro computing 72(13-15):(2009) 3288-3295.
[35] Q. K. Song: Synchronization analysis of coupled connected neural networks with mixed time delays. Neuro computing 72(16-18):(2009) 3907-3914.
[36] Baldi, P., & Atiya, A. F., How delays affect neural dynamics and learning. IEEE Transactions on Neural Networks, 5,(1994) 612–621.
[37] B.Y. Zhang, S.Y. Xu, Y. Zou, Improved stability criterion and its applications in delayed controller design for discrete-time systems. Automatica 44(11):(2008)2963-2967.
[38] S.Y. Xu, B. Song, J.W. Lu, James Lam: Robust stability of uncertain discrete-time singular fuzzy systems. Fuzzy Sets and Systems 158(20):(2007)2306-2316.
[39] H. Zhao, Global asymptotic stability of Hopfield neural network involving distributed delays, Neural Networks, 17,(2004)47–53.
[40] H. Zhao, Existence and global attractivity of almost periodic solution for cellular neural network withdistributed delays. Applied Mathematics and Computation, 154, (2004)683–695.
[41] J.D. Cao, F.G. Ren: Exponential Stability of Discrete-Time Genetic Regulatory Networks With Delays. IEEE Transactions on Neural Networks 19(3):(2008) 520-523.
[42] J.D. Cao, M.Xiao: Stability and Hopf Bifurcation in a Simplified BAM Neural Network With Two Time Delays. IEEE Transactions on Neural Networks 18(2):(2007)416-430.
[43] Arik, S., Stability analysis of delayed neural networks, IEEE Transactions on Circuits Systems-I, 47, (2000)1089–1092.
[44] X.. Mao, J. Lam, and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems & Control Letters 57 (2008), 927-935.
[45] X.. Mao,, Y. Shen, and C. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Processes and their Applications 118 (2008), 1385-1406.
[46] Z. Wang, Y. Liu, K. Fraser and X. Liu, Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays, Physics Letters A, Vol. 354, No. 4, Jun. 2006, pp. 288-297.
[47] Y. Liu, Z. Wang and X. Liu, On global exponential stability of generalized stochastic neural networks with mixed time delays, Neurocomputing, Vol. 70, No. 1-3, Dec. 2006, pp. 314-326.
[48] Z. Wang, Y. Liu, L. Yu and X. Liu, Exponential stability of delayed recurrent neural networks with Markovian jumping parameters, Physics Letters A, (ISSN 0375-9601) Vol. 356, No. 4-5, Aug. 2006, pp. 346-352.
[49] Z. Wang, Ho, D. W. C., & Liu, X, State estimation for delayed neural networks. IEEE Transactions on Neural Networks, 16(1), (2005)279–284.
[50] S.H. Strogatz and I. Stewart., Coupled oscillators and biological synchronization, Scientific American 269 (6), December, (1993) 102-109.
[51] C.M. Gray, Synchronous oscillations in neuronal systems: mechanisms and functions. J Comput Neurosci. Jun 1994;1(1-2):11–38.
[52] L. Glass, Synchronization and rhythmic processes in physiology J 1. Nature, 410.(6825):(2001)277-284.
[53] Néda Z, Ravasz E, Vicsek T, Brechet Y, Barabási A L, Physics of the rhythmic applause, Phys. Rev. E, (2000), 61(6):6987-6992.
[54] Boccaletti S, Kurths J, Osipov G, Valladares D L, Zhou C S, The synchronization of chaotic systems, Physics Reports, (2002), 366:1-101.
[55] X. F. Wang Complex networks: topology, dynamics and synchronization, International Journal of Bifurcation and Chaos, (2002), 12(5): 885-916.
[56] L. M. Pecora, T. L.Carroll., Master stability functions for synchronized coupled systems. Phys Rev Lett, (1998), 80(10):2109-2112.
[2] R. Albert, H. Jeong, A. L. Barabási, Attack and Error Tolerance of Complex Networks, Nature, 406(2000), pp.378-382.
[3] X .F. Wang, Complex networks: Topology, dynamics and synchronization, Int. J. of Bifur. Chaos,12 (2002), pp.885-916.
[4] R. Albert, A. L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys., 74 (2002), pp.47-97.
[5] R. Albert, A. L. Barabási, Topology of evolving networks: local events and universality, Phys. Rev. Lett., 85(2000), pp.5234-5237.
[6] L. A. N. Amaral, et al, Virtual Round Table on ten leading questions for network research, Eur. Phys. J.B., 38(2004), pp.143-145.
[7] A. L. Barabási, R. Albert, Emergence of scaling in random networks, Science, 286(1999), pp.509-512.
[8] J. Camacho, R. Guimerá, L. A. N. Amaral, Robust patterns in food web structure, Phys. Rev. Lett., 88 (2002), pp.228102.
[9]方锦清,汪小帆,刘曾荣,略论复杂性问题和非线性复杂网络系统的研究,基础科学,科技导报, 2004年02期, 9-12.
[10] C.G. Li and G. Chen, A comprehensive weighted evolving network model, Physica A,343(2004), pp. 288-294.
[11] K. Klemm, V. M. Eguíluz, Highly clustered scale-free networks, Phy. Rev. E, 65(2002), pp. 036123.
[12] K.. Klemm, V. M. Eguíluz, Growing scale-free networks with small-world behavior, Phy. Rev. E, 65 (2002), pp.057102.
[13] X . Li and G. Chen, A local-world evolving network model, Physica A, 328(2003), pp. 274-286.
[14] Z. H. Liu, Y. C. Lai, N. Ye, Statistical properties and attack tolerance of growing networks with Algebraic preferential attachment, Phys. Rev. E, 66(2002), pp.036112.
[15] J.H. Lu, X.H. Yu and G. Chen, Chaos synchronization of general complex dynamical networks, Physica A, 334(2004), pp. 281-302.
[16] X.. F. Wang and G. Chen, Complex networks: small-world, scale-free, and beyond, IEEE Circ. Syst. Magazine, 3(2003), No. 1, pp. 6-20.
[17] X. F. Wang and G. Chen, Pinning control of scale-free dynamical networks, Physica A, 310(2002), pp.521-531.
[18] X.. F. Wang and G. Chen, Synchronization in small-world dynamical networks, Int. J. of Bifur. Chaos, 12(2002), pp.187-192.
[19] X. F. Wang and G. Chen, Synchronization in scale-free dynamical networks: Robustness and fragility, IEEE Trans. on Circ. Syst.–I, 49(2002), pp. 54-62.
[20] S. N. Dorogovtsev, J. F. F. Mendes, Evolution of networks, Adv. Phys., 51(2002), pp.1079-1187·
[21] M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45(2003), pp. 167-256.
[22] S. H. Strogatz, Exploring complex networks, Nature, 410(2001), pp. 268-276.
[23]吴彤,复杂网络研究及其意义,哲学研究,2004年第8期,pp.58-63.
[24] Y. Chen, G. Rangarajan and M. Ding, General stability analysis of synchronized dynamics in coupled systems, Phys Rev E 67 (2003), pp1–4( 026209).
[25] C.Li, W. Sun, D.Xu, Synchronization of complex dynamical network with nonlinear inner-coupling functions and time delays, Progress of Theoretical Physics, (2005), 114(4).
[26] J. Zhou and T. Chen, Synchronization in general complex delayed dynamical networks, IEEE Trans. Circuits Syst. I 53 (2) (2006), pp. 733–744.
[27] J. Cao, G. Chen, P. Li, Global synchronization in an array of delayed neural networks with hybrid coupling, IEEE Trans. Systems, Man, and Cybernetics-Part B: Cybernetics, 38:2(2008) 488-498
[28] W. Yu, J. Cao, J. Lu, Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM Journal on Applied Dynamical Systems, 7:1(2008) 108-133.
[29] R. Zhang, M. Hu and Z. Xu, Synchronization in complex networks with adaptive coupling, Physics Letters A, Volume 368, Issues 3-4, 20 August 2007, Pages 276-280.
[30] C. Li, G. Chen. Synchronization in general complex dynamical networks with coupling delays. Physica A. Volume 343, 15 November 2004, Pages 263-278
[31] Y. Liu, Z. Wang, J. Liang and X. Liu, Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time-delays, IEEE Transactions on Neural Networks, Vol. 20, No. 7, Jul. 2009, pp. 1102-1116.
[32] Y. Liu, Z. Wang, J. Liang and X. Liu, Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Transactions on Systems, Man, and Cybernetics - Part B, Vol. 38, No. 5, Oct. 2008, pp. 1314-1325.
[33] Y. Liu, Z. Wang and X. Liu, Exponential synchronization of complex networks with Markovian jump and mixed delays, Physics Letters A, Vol. 372, No. 22, May 2008, pp. 3986-3998.
[34] Q. K. Song, Design of controller on synchronization of chaotic neural networks with mixed time-varying delays. Neuro computing 72(13-15):(2009) 3288-3295.
[35] Q. K. Song: Synchronization analysis of coupled connected neural networks with mixed time delays. Neuro computing 72(16-18):(2009) 3907-3914.
[36] Baldi, P., & Atiya, A. F., How delays affect neural dynamics and learning. IEEE Transactions on Neural Networks, 5,(1994) 612–621.
[37] B.Y. Zhang, S.Y. Xu, Y. Zou, Improved stability criterion and its applications in delayed controller design for discrete-time systems. Automatica 44(11):(2008)2963-2967.
[38] S.Y. Xu, B. Song, J.W. Lu, James Lam: Robust stability of uncertain discrete-time singular fuzzy systems. Fuzzy Sets and Systems 158(20):(2007)2306-2316.
[39] H. Zhao, Global asymptotic stability of Hopfield neural network involving distributed delays, Neural Networks, 17,(2004)47–53.
[40] H. Zhao, Existence and global attractivity of almost periodic solution for cellular neural network withdistributed delays. Applied Mathematics and Computation, 154, (2004)683–695.
[41] J.D. Cao, F.G. Ren: Exponential Stability of Discrete-Time Genetic Regulatory Networks With Delays. IEEE Transactions on Neural Networks 19(3):(2008) 520-523.
[42] J.D. Cao, M.Xiao: Stability and Hopf Bifurcation in a Simplified BAM Neural Network With Two Time Delays. IEEE Transactions on Neural Networks 18(2):(2007)416-430.
[43] Arik, S., Stability analysis of delayed neural networks, IEEE Transactions on Circuits Systems-I, 47, (2000)1089–1092.
[44] X.. Mao, J. Lam, and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems & Control Letters 57 (2008), 927-935.
[45] X.. Mao,, Y. Shen, and C. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Processes and their Applications 118 (2008), 1385-1406.
[46] Z. Wang, Y. Liu, K. Fraser and X. Liu, Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays, Physics Letters A, Vol. 354, No. 4, Jun. 2006, pp. 288-297.
[47] Y. Liu, Z. Wang and X. Liu, On global exponential stability of generalized stochastic neural networks with mixed time delays, Neurocomputing, Vol. 70, No. 1-3, Dec. 2006, pp. 314-326.
[48] Z. Wang, Y. Liu, L. Yu and X. Liu, Exponential stability of delayed recurrent neural networks with Markovian jumping parameters, Physics Letters A, (ISSN 0375-9601) Vol. 356, No. 4-5, Aug. 2006, pp. 346-352.
[49] Z. Wang, Ho, D. W. C., & Liu, X, State estimation for delayed neural networks. IEEE Transactions on Neural Networks, 16(1), (2005)279–284.
[50] S.H. Strogatz and I. Stewart., Coupled oscillators and biological synchronization, Scientific American 269 (6), December, (1993) 102-109.
[51] C.M. Gray, Synchronous oscillations in neuronal systems: mechanisms and functions. J Comput Neurosci. Jun 1994;1(1-2):11–38.
[52] L. Glass, Synchronization and rhythmic processes in physiology J 1. Nature, 410.(6825):(2001)277-284.
[53] Néda Z, Ravasz E, Vicsek T, Brechet Y, Barabási A L, Physics of the rhythmic applause, Phys. Rev. E, (2000), 61(6):6987-6992.
[54] Boccaletti S, Kurths J, Osipov G, Valladares D L, Zhou C S, The synchronization of chaotic systems, Physics Reports, (2002), 366:1-101.
[55] X. F. Wang Complex networks: topology, dynamics and synchronization, International Journal of Bifurcation and Chaos, (2002), 12(5): 885-916.
[56] L. M. Pecora, T. L.Carroll., Master stability functions for synchronized coupled systems. Phys Rev Lett, (1998), 80(10):2109-2112.
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