随机与不确定因素下复杂网络控制与同步研究
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摘要
复杂动力学网络的控制与同步研究是目前国内外关于复杂网络研究的热点问题之一。在实际的复杂动力学网络中,由于结构的建模误差、环境温度和湿度等外部条件的变化、通信流量的随机性等因素的影响,复杂动力学网络中的随机和不确定因素是客观存在的,而且有时是无法忽略的。因此,建立保守性小的随机或不确定复杂动力学网络模型并根据模型的变化改变分析综合所采用的方法及工具是有必要的。鉴于此,本文致力于研究随机和不确定因素下的复杂动力学网络的控制与同步问题,这对理解和应用复杂网络有重要的意义。
论文建立了带马尔可夫模式依赖时延的不确定无向离散复杂动力学网络模型,运用奇异系统并结合变量变换的方法,基于线形矩阵不等式,得到了带马尔可夫模式依赖时延的无向离散网络随机稳定的充分条件。
针对一种连续时间、参数和时延服从马尔可夫切换的复杂动力学网络模型,论文借助奇异系统的方法,构造了新的Lyapunov泛函结合耦合的线性矩阵不等式,得到了该复杂动力学网络随机稳定的充分条件。论文提出了另一类连续马尔可夫切换时延复杂动力学网络模型,即参数、时延和内部耦合矩阵都服从马尔可夫切换的情况,运用Kronecker积的性质结合构造的Lyapunov泛函,得到了连续时间马尔可夫切换时延网络指数均方同步的充分条件。
论文还研究了随机非线性耦合时变时延动力学网络,其中,耦合项为一个布朗运动形式的随机扰动项。基于Lyapunov稳定性定理、It微分方程、不变微分方程的LaSalle不变原理,运用自适应反馈控制从理论上研究并得到了这类网络均方同步的充分条件。
Control and Synchronization of complex dynamical networks with stochastic and uncertain factors is one of the most famous problems in the world. In the real world, because of existence of the structure error, the influence caused by the change of the environment temperature and humidity and the random factor in information traffic, the stochastic and uncertain factors of complex dynamical network exist, and can not ignored sometimes. Therefore, it is necessary to build stochastic or uncertain complex dynamical network model with low conservative property and change the analysis methods and tools according to the change of the model. Accordingly, the control and synchronization of complex dynamic networks with stochastic and uncertain factors are analyzed, which is meaningful for the understanding and application of complex networks.
Based on liner matrix inequality, undirected discrete Markovian time-dependent delay dynamical network model has been researched by using descriptor system approach combining with variable transformation method. The sufficient condition for the stability of undirected discrete Markovian time-dependent delay network can be obtained.
A continuous Markov jump delay network dynamical model is introduced, which consists of Markov jump mode-dependent delays and parameters. The stability and synchronization have been analyzed by using descriptor system approach and the sufficient condition of the stability of continues-time Markov jump delay network can be obtained. Another continuous Markov jump delay network dynamical model is presented, namely, parameters, delays and inside coupling matrix are all subjected to Markov jump. Combining the property of Kronecker product with the Lyapunov functional, the sufficient condition of exponential mean square synchronization of continues-time Markov jump delay dynamical network can be obtained.
The synchronization problem of time-varying delay network with stochastic nonlinear coupling has been investigated by using adaptive feedback control strategy. In this model, the random coupling factor is a random disturbance in the form of Brown motion. Based on Lyapunov stability theorem, It differential equation and LaSalle invariable differential equation theory, the sufficient condition for the synchronization of this network has been obtained by using adaptive feedback control.
论文建立了带马尔可夫模式依赖时延的不确定无向离散复杂动力学网络模型,运用奇异系统并结合变量变换的方法,基于线形矩阵不等式,得到了带马尔可夫模式依赖时延的无向离散网络随机稳定的充分条件。
针对一种连续时间、参数和时延服从马尔可夫切换的复杂动力学网络模型,论文借助奇异系统的方法,构造了新的Lyapunov泛函结合耦合的线性矩阵不等式,得到了该复杂动力学网络随机稳定的充分条件。论文提出了另一类连续马尔可夫切换时延复杂动力学网络模型,即参数、时延和内部耦合矩阵都服从马尔可夫切换的情况,运用Kronecker积的性质结合构造的Lyapunov泛函,得到了连续时间马尔可夫切换时延网络指数均方同步的充分条件。
论文还研究了随机非线性耦合时变时延动力学网络,其中,耦合项为一个布朗运动形式的随机扰动项。基于Lyapunov稳定性定理、It微分方程、不变微分方程的LaSalle不变原理,运用自适应反馈控制从理论上研究并得到了这类网络均方同步的充分条件。
Control and Synchronization of complex dynamical networks with stochastic and uncertain factors is one of the most famous problems in the world. In the real world, because of existence of the structure error, the influence caused by the change of the environment temperature and humidity and the random factor in information traffic, the stochastic and uncertain factors of complex dynamical network exist, and can not ignored sometimes. Therefore, it is necessary to build stochastic or uncertain complex dynamical network model with low conservative property and change the analysis methods and tools according to the change of the model. Accordingly, the control and synchronization of complex dynamic networks with stochastic and uncertain factors are analyzed, which is meaningful for the understanding and application of complex networks.
Based on liner matrix inequality, undirected discrete Markovian time-dependent delay dynamical network model has been researched by using descriptor system approach combining with variable transformation method. The sufficient condition for the stability of undirected discrete Markovian time-dependent delay network can be obtained.
A continuous Markov jump delay network dynamical model is introduced, which consists of Markov jump mode-dependent delays and parameters. The stability and synchronization have been analyzed by using descriptor system approach and the sufficient condition of the stability of continues-time Markov jump delay network can be obtained. Another continuous Markov jump delay network dynamical model is presented, namely, parameters, delays and inside coupling matrix are all subjected to Markov jump. Combining the property of Kronecker product with the Lyapunov functional, the sufficient condition of exponential mean square synchronization of continues-time Markov jump delay dynamical network can be obtained.
The synchronization problem of time-varying delay network with stochastic nonlinear coupling has been investigated by using adaptive feedback control strategy. In this model, the random coupling factor is a random disturbance in the form of Brown motion. Based on Lyapunov stability theorem, It differential equation and LaSalle invariable differential equation theory, the sufficient condition for the synchronization of this network has been obtained by using adaptive feedback control.
引文
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[8] Xiang J., Chen G.. On the V-stability of complex dynamical networks. Automatica, 2007, 43:1049~1057
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[10] Wu X., Wang J., and Lu J.-a. et al. Hyperchaotic behavior in a non-autonomous unified chaotic system with continuous periodic switch. Chaos, Solit. Fract., 2007, 32: 1485~1490
[11] Lu J.-a., Wu X.. A unified chaotic system with continuous periodic switch. Chaos, Solit. Fract., 2004, 20: 245~251
[12] Mansouri B., Manamanni N., and Guelton K. et al. Robust pole placement controller design in LMI region for uncertain and disturbed switched systems. Nonlinear Anal. Hybrid Syst., 2008, 2: 1136~1143
[13] Lin L.. Stabilization analysis for economic compartmental switched systems based on quadratic Lyapunov function. Nonlinear Anal. Hybrid Syst., 2008, 2: 1187~1197
[14] Moulay E., Bourdais R., W. Perruquetti, Stabilization of nonlinear switched systems using control Lyapunov functions. Nonlinear Anal. Hybrid Syst., 2007, 1: 482~490
[15] Liberzon D., Morse A. S.. Basic problems in stability and design of switchedsystems. IEEE Control Systems Magazine, 1999, 19(1): 59~70
[16] Agrachev A. A., Liberzon D.. Lie-algebraic stability criteria for switched systems. SIAM J. Control Opt., 2002, 40:253~269
[17] Morse A. S.. Supervisory control of families of linear set-point controllers. Part 1: exact matching, IEEE Tran. On Automatic Control, 1996,41(10): 1413~1431
[18] Hespanha J. P., Morse A. S.. Stability of switched systems with average dwell-time. Proceedings of the 38th IEEE Conference on Decision and Control, 1999: 2655~2660
[19] Guan Z. H., Zhang H., Stabilization of complex network with hybrid impulsive and switching control. Chaos Solitons & Fractals, 2008, 37: 1372~1382
[20] Wang Y. W., Yang M., Wang O. and Hua et al, Robust stabilization of complex switched networks with parametric uncertainties and delays via impulsive control. IEEE Trans. Circuits Syst. I, in press
[21]程代展.《切换系统》进展.控制理论与应用学报20周年纪念及学术交流会,2004,广州.
[22] Wang Z., Liu Y., Yu L.. Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Physics Letters A, 2006, 356(4): 346~352
[23] Gelenbe E.. Stability of the random neural network model. Neural Computation, 1990, 2(2): 239~247
[24] Mori T, Kokame H, Stability of x&(t )= Ax(t)+Bx(t?τ). IEEE Trans. on Autom. Contr., 1989, 34(4): 460~462
[25] Brierley S. D., Chiasson J. N., Lee E. B. et al. On stability independent of delay. IEEE Trans. on Autom. Contr., 1982, 27(1): 252~254
[26] Zhang Jiye, Yang Yiren. Global stability analysis of bidirectional associative memory neural networks with time-delay. Int. Journal. of Cir. Theo. and App.,2001, 29(2): 185~196
[27] Chen T. P., Rong L.B..Delay-independent stability analysis of Cohen-Grossberg neural networks. Phys. L. A, 2003, 317(5): 436~449
[28] Hua C. C., Long C. N., Guan X. P.. New results on stability analysis of neural networks with time-varying delays. Phys. L. A, 2005, 352(4): 335~340
[29] Niculescu S. I. On delay-dependent stability under model transformations of some neutral linear systems. International Journal of Control, 2001, 74(6): 1447~1455
[30] Kolmanovskii V., Richard J. P.. Stability of some linear systems with delays. IEEETrans. on Autom. Contr., 1999, 44(5): 984~989
[31] Fridman E.. New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Systems and Control Letters, 2001, 43(2): 309~319
[32] Cao J. D., Huang D. S., and QuY.. Global robust stability of delayed recurrent neural networks Chaos, Solitons & Fractals, 2005, 23(1): 221~229
[33] Wang Zidong, Lauria Stanislao, and Jian’an Fang et al. Exponential stability of uncertain stochastic neural networks with mixed time-delays. Chaos, Solitons & Fractals, 2007, 32(1): 62~72
[34] Wonham W. M.. Optimal stationary control of a linear system with state-dependent noise. SIAM J. Contr., 1967, 5: 486~500
[35] Willems J. L., Willems J. C.. Robust stabilization of uncertain systems. SIAM J. Contr. Opt., 1983, 21: 352~374
[36] Bertsekas D. P., Rhodes I. B.. Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems. IEEE Trans. Autom. Contr., 1973, 18: 117~124
[37] Chang S., Peng T.. Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans. Automat. Contr., 1972, 17: 474~483
[38] Bermstein D. S., Greeley S.W.. Robust output-feedback stabilization: deterministic and stochastic perspectives, 1986, 17: 1818~1826
[39] Gutman S.. Uncertain dynamical systems a Lyapunov minimax approach. IEEE Trans. Automat. Contr., 1979, 24: 437-443
[40] Carlucci D., Donati F.. Control of norm uncertain systems. IEEE Trans. Automat. Contr., 1975, 20: 792~795
[41] Blythe S., Mao X. and Liao X., Stability of stochastic delay neural networks. J Franklin Inst., 2001, 338: 481~495
[42]廖晓昕,毛学荣.随机时滞Hopfield神经网络的均方指数稳定性.华中理工大学学报,1997, 25(6),7~10
[43] Pecora L.M., Carroll T. L.. Synchronization in Chaotic Systems. Phys. Rev. L., 1990, 19: 821~824
[44] Strogatz S. H., Sync.. The emerging science of spontaneous order. Theia Hyperion, New York, 2003
[45] Chen G., Duan Z.. Network synchronizability analysis: a graph-theoretic approach, Chaos, 2008, 18(3): 037102/1~037102/8
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