摘要
伴随着分数阶理论的发展与混沌系统同步的广泛应用,分数阶混沌系统的同步问题引起了越来越多学者的关注。由于混沌系统对参数扰动与外部干扰特别敏感,同时分数阶系统的稳定性与整数阶系统稳定性又存在很大的区别。所以,分数阶混沌系统的鲁棒同步问题一直是一个棘手的问题。
本文围绕分数阶系统的稳定性与分数阶混沌系统的鲁棒同步方法进行了深入研究。主要内容包括:
1.针对具有外部扰动的不确定分数阶混沌系统,基于终端滑模控制理论设计了一种鲁棒同步控制方案。其原理是基于分数阶Lyapunov稳定性理论对分数阶混沌系统引入了鲁棒滑模控制律,确保了滑模运动在有限时间内发生。对具有外部扰动的不确定分数阶Lorenz混沌系统与分数阶Chen's混沌系统进行了鲁棒同步仿真,结果验证了方法的有效性。特别指出,该分数阶终端滑模控制器针对一大类具有外部扰动的不确定异结构分数阶混沌系统都是有效的。
2.针对参数不确定与外部扰动整体有界的分数阶混沌系统基于模型逼近方法提出了一种新的鲁棒修正投影方案。通过针对具有扰动的不确定分数阶Lorenz混沌系统与分数阶Chen's混沌系统的完全同步与修正投影同步两个仿真实例来验证了所设计的基于模型逼近的同步控制器的鲁棒性。从该控制器的设计过程可以得出,该控制方案针对一大类包含外部扰动的不确定异结构分数阶混沌系统均可以实现鲁棒同步。
3.通过分析时变分数阶系统的稳定性,给出了一种微分阶次0<α<1时的时变分数阶系统的稳定性判定定理。基于此定理对分数阶Lu混沌系统的同步进行了仿真研究,仿真结果验证了所提出的时变分数阶系统稳定性定理在实际控制器设计过程中是有效的。
4.首先给出了永磁同步电机(PMSM)混沌系统的分数阶数学模型;然后基于分数阶Lyapunov矩阵微分方程稳定性理论,给出了分数阶Lyapunov鲁棒稳定性定理和推论,理论的提出使时域内分数阶系统的稳定性判定更加便捷。此后,应用此定理和推论分别设计了不同的控制器来实现分数阶PMSM混沌系统的混沌控制与同步,数值仿真曲线表明了控制与同步方案的有效性。
综上,本文针对分数阶系统的稳定性与分数阶混沌系统的鲁棒同步问题的创新性成果主要有以下几点:
1.基于终端滑模控制方法,设计分数阶滑模控制实现了具有扰动的不确定异结构分数阶混沌系统鲁棒同步。
2.针对具有扰动的不确定异结构分数阶混沌系统提出了一种基于模型逼近误差上界的鲁棒同步方法。
3.提出了时变分数阶系统的稳定性定理并将其成功应用到分数阶混沌系统的同步。
4.给出了线性分数阶系统的稳定性定理并将其成功应用于分数阶PMSM混沌系统的同步。
The robust synchronization problem of fractional-order chaotic system, which is pre-sented along with the development of fractional-order theory and extensive use of chaotic synchronization, has attracted more and more attention of scholars. But at the same time, it is also a thorny problem in that chaotic system is particularly sensitive to parameter perturbation and external disturbance, and there is significant difference between the sta-bility of fractional-order chaotic system and integer order chaotic system.
This dissertation focuses on the stability of fractional-order system and the robust synchronization method of fractional-order chaotic systems. Main contributions are given as follows:
1. A robust synchronization control scheme for uncertain fractional-order chaotic sys-tems with external disturbances is designed based on the terminal sliding mode control theory. On the basis of fractional-order Lyapunov stability theory, a robust sliding mode control scheme, which is used to ensure the sliding mode motion occurs in limited time, is introduced to the fractional-order chaotic system. The proposed control scheme is applied to synchronize the fractional order Lorenz chaotic system and fractional-order Chen's cha-otic system with uncertainty and external disturbance parameters, simulation results show the applicability and efficiency of the proposed scheme. It should be pointed out in partic-ular that the introduced fractional-order terminal sliding mode controller is applicable for a large class of different uncertain fractional-order chaotic systems under external disturb-ances.
2. Based on model approximation method, a new robust modified projective synchro-nization control scheme is presented for the fractional-order chaotic system which is global bounded for parameter uncertainty and external disturbances. Then the proposed method is used to the complete synchronization and revised projection synchronization of frac-tional-order Lorenz and Chen's chaotic systems with parameter uncertainty and external disturbances, simulation results show the robustness of the designed controller. It can also be concluded from the scheme's design process that the proposed scheme is effective for a large class of different uncertain fractional-order chaotic systems under external disturb-ances.
3. By analyzing the stability of a time-varying fractional-order system, a stability the-orem is proposed for a time varying fractional-order system with order0<α<1. Then the presented stability theorem is used in the synchronization of the fractional-order Lii chaotic system, simulation results verify again that it is also effective when used in the process of controller design.
4. First, a fractional-order mathematical model of permanent magnet synchronous motor (PMSM) is given. Then a fractional-order Lyapunov robust stability theorem and deduction, which makes it is more convenient to judge the stability of fractional-order system in time domain, are derived based on the fractional-order Lyapunov matrix differ-ential equation stability theory. Finally, different controllers which are designed respec-tively according to the proposed theorem and deduction are used to achieve the chaotic control and synchronization of the fractional-order PMSM system, numerical simulation curves show the effectiveness of the method.
In conclusion, the main innovative achievements for fractional-order systems stability and fractional-order chaotic system robust synchronization in this paper are as follows:
1. Based on the terminal sliding mode control scheme, fractional-order sliding mode surface and controller are designed to realize the robust synchronization of different un-certain fractional-order chaotic systems under external disturbances.
2. Based on the upper bound of model approximation error method, a robust synchronization scheme is proposed for different uncertain fractional-order chaotic sys-tems under external disturbances.
3. A stability theorem for time-varying fractional-order system is proposed and is successfully applied to the synchronization of fractional-order chaotic systems.
4. A stability theorem for linear fractional-order system is proposed and is successfully applied to the synchronization of fractional-order PMSM chaotic system.
引文
[1]陈关荣,汪小帆.动力系统的混沌化:理论、方法与应用[M].上海:上海交通大学出版社,2006.
[2]格雷席克,胡凯.混沌及其秩序:走近复杂体系[M].百家出版社,2001.
[3]孙光辉.分数阶混沌系统的控制及同步研究[D].哈尔滨:哈尔滨工业大学,2010.
[4]Lorenz E N. Deterministic nonperiodic flow [J]. J Atmos Sci,1963,20(2):130-141.
[5]张静.混沌同步控制中若干问题的研究[D].大连:大连理工大学,2008.
[6]陈关荣,吕金虎.Lorenz系统族的动力学分析,控制与同步[M].北京:科学出版社,2003.
[7]Ott E, Grebogi C, Yorke J A. Controlling chaos [J]. Phys Rev Lett 1990,64(11): 1196-1199.
[8]Pecora L M, Carroll T L. Driving systems with chaotic signals [J]. Phys Rev A,1991, 44(4):2374-2383.
[9]Carroll T L, Pecora L M. Synchronizing chaotic circuits [J]. Circuits and Systems, IEEE Transactions on,1991,38(4):453-456.
[10]Zhang Q L, Lu L. Synchronization of Spatiotemporal Chaos in a Class of Complex Dynamical Networks [J]. Chin Phys B,2011,20(1):010510.
[11]Wei W, Li D H, Wang J. Synchronization of hyperchaotic Chen systems:A class of the adaptive control approach [J]. Chin Phys B,2010,19(4):040507.
[12]Pourgholi M, Majd V J. A Novel Robust Proportional-integral (PI) Adaptive Observer Design for Chaos Synchronization [J]. Chin Phys B,2011,20(12):120503.
[13]Hu A H, Xu Z Y, Guo L X. Holder Continuity of Generalized Chaos Synchroniza-tion in Complex Networks [J]. Chin Phys B,2011,20(9):090511.
[14]Vinagre B M, Chen Y. Fractional calculus applications in automatic control and robotics [A]. Lecture Notes Prepared for The Tutorial Workshop [C],2002(2).
[15]Torvik P J, Bagley R L. On the appearance of the fractional derivative in the behavior of real materials [J]. J Appl Mech 1984,51(2):294-298.
[16]Podlubny I. Fractional-order systems and PIλDμ-controllers [J]. IEEE Trans Autom Control 1999,44(1):208-214.
[17]李翊神.非线性科学选讲[M].中国科学技术大学出版社,1994.
[18]刘曾荣.非线性科学浅论[J].自然杂志,1996,18(4):229-232.
[19]李响.混沌系统的几类同步问题的研究[D].无锡:江南大学,2009.
[20]Li T Y, Yorke J A. Period three implies chaos [J]. The American Mathematical Monthly,1975,82(10):985-992.
[21]Day R H. The emergence of chaos from classical economic growth [J]. The Quarterly Journal of Economics,1983,98(2):201-213.
[22]Devaney R L. An introduction to chaotic dynamical systems [M]. New York: Addison-Wesley Publishing Company,1989.
[23]王兴元.混沌系统的同步及在保密通信中的应用[M].北京:科学出版社,2012.
[24]高远,方锦清.超混沌电路的广义同步[J].电子与信息学报,2002,24(6):855-859.
[25]Yang T, Chua L O. Generalized synchronization of chaos via linear transformations [J]. Int J Bifurcation Chaos 1999,9:215-220.
[26]闵富红,王执铨.两个四维混沌系统广义投影同步[J].物理学报,2007,56(11):6238-6244.
[27]张若洵.混沌和分数阶混沌系统同步研究及电路仿真[D].石家庄:河北师范大学,2008.
[28]Pecora L M, Carroll T L. Synchronization in chaotic systems [J]. Phys Rev Lett 1990, 64(8):821-824.
[29]Oppenheim A V, Wornell G W, Isabelle S H, et.al. Signal processing in the context of chaotic signals [A]. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing [C], New York, USA, March 23-26,1992: 117-120.
[30]Corron N J, Hahs D W. A new approach to communications using chaotic signals [J]. Circuits and Systems I:Fundamental Theory and Applications, IEEE Transactions on,1997,44(5):373-382.
[31]Kevin M. Steps toward unmasking secure communications [J]. Int J Bifurcation Chaos 1994,4(4):959-977.
[32]Kocarev L, Parlitz U. General approach for chaotic synchronization with applications to communication [J]. Phys Rev Lett 1995,74(25):5028-5031.
[33]Pyragas K. Continuous control of chaos by self-controlling feedback [J]. Phys Lett A 1992,170(6):421-428.
[34]Zou Y L, Zhu J, Chen G, et.al. Synchronization of hyperchaotic oscillators via single unidirectional chaotic-coupling [J]. Chaos, Solitons Fractals 2005,25(5):1245-1253.
[35]Yassen M. Controlling chaos and synchronization for new chaotic system using linear feedback control [J]. Chaos, Solitons Fractals 2005,26(3):913-920.
[36]Bu S, Wang S, Ye H. An algorithm based on variable feedback to synchronize chaotic and hyperchaotic systems [J]. Physica D 2002,164(1):45-52.
[37]Li D, Lu J, Wu X. Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems [J]. Chaos, Solitons Fractals 2005,23(1):79-85.
[38]Park J H. Stability criterion for synchronization of linearly coupled unified chaotic systems [J]. Chaos, Solitons Fractals 2005,23(4):1319-1325.
[39]Yan J, Li C. On synchronization of three chaotic systems [J]. Chaos, Solitons Fractals 2005,23(5):1683-1688.
[40]Zhou T, Lii J, Chen G, et.al. Synchronization stability of three chaotic systems with linear coupling [J]. Phys Lett A 2002,301(3):231-240.
[41]Yu Y, Zhang S. Global synchronization of three coupled chaotic systems with ring connection [J]. Chaos, Solitons Fractals 2005,24(5):1233-1242.
[42]Wu X, Guan Z H, Wu Z. Adaptive synchronization between two different hyperchaotic systems [J]. Nonlinear Analysis:Theory, Methods & Applications, 2008,68(5):1346-1351.
[43]May R M. Simple mathematical models with very complicated dynamics [J]. Nature, 1976,261(5560):459-467.
[44]Yassen M. Adaptive synchronization of two different uncertain chaotic systems [J]. Phys Lett A 2005,337(4):335-341.
[45]Yan J J, Yang Y S, Chiang T Y, et.al. Robust synchronization of unified chaotic systems via sliding mode control [J]. Chaos, Solitons Fractals 2007,34(3):947-954.
[46]Chen F, Zhang W. LMI criteria for robust chaos synchronization of a class of chaotic systems [J]. Nonlinear Analysis:Theory, Methods & Applications,2007,67(12): 3384-3393.
[47]Andrievsky B. Adaptive synchronization methods for signal transmission on chaotic carriers [J]. Math Comput Simulat,2002,58(4):285-293.
[48]Yan J J, Lin J S, Liao T L. Synchronization of a modified Chua's circuit system via adaptive sliding mode control [J]. Chaos, Solitons Fractals 2008,36(1):45-52.
[49]Sinha S, Henrichs J, Ravindra B. A general approach in the design of active controllers for nonlinear systems exhibiting chaos [J]. Int J Bifurcation Chaos 2000, 10(01):165-178.
[50]薛月菊,冯汝鹏.连续时间耦合系统中时空混沌的自适应模糊控制[J].物理学报,2001,50(3):440-444.
[51]Di Bernardo M. An adaptive approach to the control and synchronization of continuous-time chaotic systems [J]. Int J Bifurcation Chaos 1996,6(3):557-568.
[52]Li Z, Chen G, Shi S, et.al. Robust adaptive tracking control for a class of uncertain chaotic systems [J]. Phys Lett A 2003,310(1):40-43.
[53]胡建兵,韩焱,赵灵冬.自适应同步参数未知的异结构分数阶超混沌系统[J].物理学报,2009,58(3):1441-1445.
[54]Tavazoei M S, Haeri M. Synchronization of chaotic fractional-order systems via active sliding mode controller [J]. Physica a-Statistical Mechanics and Its Applications,2008,387(1):57-70.
[55]Kiani-B A, Fallahi K, Pariz N, et.al. A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter [J]. Communications in Nonlinear Science and Numerical Simulation,2009,14(3):863-879.
[56]Xu J Q. Adaptive synchronization of the fractional-order unified chaotic system with uncertain parameters [A]. Proceedings of the 30th Chinese Control Conference [C], Yantai, China, July 22-24, IEEE Computer Society,2011:2423-2428.
[57]Zhou P, Ding R. Adaptive function projective synchronization between different fractional-order chaotic systems [J]. Indian J Phys 2012,86(6):497-501.
[58]Yang C C. One input control for exponential synchronization in generalized Lorenz systems with uncertain parameters [J]. J Franklin Inst 2012,349(1):349-365.
[59]Zhang R X, Yang S P. Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations [J]. Nonlinear Dyn 2012,69(3):983-992.
[60]Xiang W, Chen F Q. Robust synchronization of a class of chaotic systems with disturbance estimation [J]. Communications in Nonlinear Science and Numerical Simulation,2011,16(8):2970-2977.
[61]邢生焰.分数阶不确定性系统鲁棒控制研究[D].上海:上海交通大学,2009.
[62]张隆阁.一类参数不确定混沌系统的分数阶自适应同步[J].中国科技信息,2009,15:57-58.
[63]孙宁,张化光,王智良.不确定分数阶混沌系统的滑模投影同步[J].浙江大学学报(工学版),2010,44(7):1288-1291.
[64]许建强.参数不确定分数阶统一混沌系统的自适应同步[A].第三十届中国控制会议[C],烟台,2011,1312-1317.
[65]王振滨.分数阶线性系统及其应用[D].上海:上海交通大学,2004.
[66]Nigmatullin R R, Mehaute A L. Is there geometrical/physical meaning of the fractional integral with complex exponent? [J]. J Non-Cryst Solids 2005,351(33): 2888-2899.
[67]Mandelbrot B B, Van Ness J W. Fractional Brownian motions, fractional noises and applications [J]. SIAM Rev 1968,10(4):422-437.
[68]Yao K, Su W, Zhou S. On the connection between the order of fractional calculus and the dimensions of a fractal function [J]. Chaos, Solitons Fractals 2005,23(2): 621-629.
[69]Yao K, Zhang X. Research announcements:On the Fractional Calculus of a Type of Weierstrass Functions [J]. Adv Math,2002,31(5):483-484.
[70]Oldham K B, Oldhamm K B, Spainer J. The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitary Order [M]. New York: Academic press,1974.
[71]Miller K S, Ross B. An introduction to the fractional calculus and fractional differential equations [M]. New York:Wiley-Interscience Publication,1993.
[72]Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives:Theory and Applications [M]. Switzerland:Gordon and Breach Science Publishers.1993: 48,49,66,67.
[73]Das S. Functional Fractional Calculus for System Identification and Controls [M]. Springer,2011.
[74]Podlubny I. Fractional differential equation [M]. New York:Academic press.1999: 41-48,71.
[75]谭琳.函数札记[M].杭州:浙江大学出版社,1997:1-21.
[76]常福宣,吴吉春,薛禹群等.考虑时空相关的分数阶对流-弥散方程及其解[J].水动力学研究与进展,A辑,2005,20(2):233-240.
[77]常福宣,陈进,黄薇.反常扩散与分数阶对流-扩散方程[J].物理学报,2005,54(3):1113-1117.
[78]常福宣,吴吉春,薛禹群等.多孔介质溶质运移问题中的分数弥散[J].水动力学研究与进展:A辑,2005,20(1):50-55.
[79]常福宣,吴吉春,戴水汉.多孔介质溶质运移的分数弥散过程与L(e)vy分布[J].南京大学学报(自然科学版),2004,40(3):287-291.
[80]段广仁.线性系统理论[M].第二版.哈尔滨:哈尔滨工业大学出版社,2004.
[81]Matignon D. Stability properties for generalized fractional differential systems [J]. Fractional Differential Systems:Models, Methods and Applications,1998,5:145-158.
[82]Lan Y H, Huang H X, Zhou Y. Observer-based robust control of a (1≤α< 2) fractional-order uncertain systems:a linear matrix inequality approach [J]. IET Control Theory Appl 2012,6(2):229-234.
[83]Li Y, Chen Y Q, Podlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability [J]. Computers and Mathematics with Applications,2010,59(5):1810-1821.
[84]Mittag-Leffler G. Sur la nouvelle fonction Ea(x) [J]. CR Acad Sci Paris,1903,137: 554-558.
[85]Mittag-Leffler G. Sopra la funzione Ea(x) [J]. Rend Accad Lincei, ser,1904,5(13): 3-5.
[86]Mittag-Leffler G. Sur la representation analytique d'une function monogene (cinquieme note) [J]. Acta Math,1905,29:101-181.
[87]Erdelyi A, Magnus W, Oberhettinger F, et.al. Higher transcendental functions, Vol. 3 [M]. McGraw-Hill, New York,1955.
[88]陈文,孙洪广.分数阶微分方程的数值算法:现状和问题[J].计算机辅助工程,2010,19(2):1-2.
[89]林然,刘发旺.分数阶常微分方程初值问题的高阶近似[J].厦门大学学报:自然科学版,2004,43(1):25-30.
[90]薛定宇,陈阳泉.控制数学问题的MATLAB求解[M].北京:清华大学出版社,2007.
[91]刘金琨.滑模变结构控制MATLAB仿真[M].北京:清华大学出版社,2005.
[92]徐世许.不确定系统的终端滑模变结构控制[D].上海:复旦大学,2012.
[93]陈义.不确定性非线性系统控制方法研究[D].西安:西北工业大学,2006.
[94]马新,平静水,陈宝国.终端滑模控制器设计在永磁同步电动机中的应用[J].通化师范学院学报,2011,32(10):11-13.
[95]张晓鹏.混沌同步的若干问题研究[D].大连:大连理工大学,2011.
[96]Zak M. Terminal attractors for addressable memory in neural networks [J]. Phys Lett A 1988,133(1):18-22.
[97]Man Z H, Paplinski A P, Wu H R. A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators [J]. IEEE Trans Autom Control 1994,39(12): 2464-2469.
[98]Man Z H, Yu X H. Terminal sliding mode control of MIMO linear systems [J]. Circuits and Systems I:Fundamental Theory and Applications, IEEE Transactions on,1997,44(11):1065-1070.
[99]Iglesias W J, Xu L, Saito O. Fast convergence for tracking problem of nth degree system by sliding mode control [A]. Proceedings of the 35th Conference on Decision and Control [C], Kobe, Japan, December 11-13, IEEE,1996 (4):4607-4612.
[100]Li C P, Yan J P. The synchronization of three fractional differential systems [J]. Chaos, Solitons Fractals 2007,32(2):751-757.
[101]Caponetto R, Dongola G, Fortuna L. Fractional order systems:modeling and control applications [M]. Hackensack:World Scientific Pub Co Inc,2010.
[102]Ye M-Y, Wang X-D, Zhang H-R. Chaotic time series forecasting using online least squares support vector machine regression [J]. Acta Photonica Sinica,2005,34(6): 2568-2573.
[103]Li C G, Chen G R. Chaos in the fractional order Chen system and its control [J]. Chaos, Solitons Fractals 2004,22(3):549-554.
[104]Chen G R, Ueta T. Yet another chaotic attractor [J]. Int J Bifurcation Chaos Appl Sci Eng 1999,9(7):1465-1466.
[105]Yang Q G, Zeng C B. Chaos in fractional conjugate Lorenz system and its scaling attractors [J]. Communications in Nonlinear Science and Numerical Simulation, 2010,15(12):4041-4051.
[106]Sang J Y, Yang J, Yue L J. Complete Synchronization of Double-delayed Rossler Systems with Uncertain Parameters [J]. Chin Phys B,2011,20(8):080507.
[107]Ye M Y. Control of chaotic system based on least squares support vector machine modeling [J]. Acta Phys Sin,2005,54(1):30-34.
[108]Lyapunov A M. The general problem of stability of motions [D]. Fizmatgiz: Moscow,1950.
[109]廖晓昕.稳定性的理论、方法和应用[M].第二版.武汉:华中科技大学出版社,2010.
[110]王振滨,曹广益,朱新坚.分数阶线性定常系统的稳定性及其判据[J].控制理论与应用,2004,21(6):922-926.
[111]王振滨,曹广益,朱新坚.分数阶线性系统的内部和外部稳定性研究[J].控制与决策,2004,19(10):1171-1174.
[112]汪纪锋.分数阶系统控制性能分析[M].北京:电子工业出版社,2010.
[113]高哲,廖晓钟.一种线性分数阶系统稳定性的频域判别准则[J].自动化学报,2011,37(11):1387-1394.
[114]Chen Y Q, Moore K L. Analytical stability bound for a class of delayed fractional-order dynamic systems [J]. Nonlinear Dyn 2002,29(1):191-200.
[115]张凤荣.分数阶微分系统的稳定性分析[D].上海:上海大学,2012.
[116]李丽香,彭海朋,罗群等.一种分数阶非线性系统稳定性判定定理的问题及分析[J].物理学报,2013,62(2):20502.
[117]Zhu Z W, Leung H. Adaptive identification of nonlinear systems with application to chaotic communications [J]. IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications,2000,47(7):1072-1080.
[118]Kuroe Y, Hayashi S. Analysis of bifurcation in power electronic induction motor drive systems [M]. Power Electronics Specialists Conference. Milwaukee, WI IEEE Conference Publishing Services.1989:923-930.
[119]Slemon G R. Electrical machines for variable-frequency drives [J]. Proc IEEE 1994, 82(8):1123-1139.
[120]Choi H H, Jung J W. Discrete-time fuzzy speed regulator design for PM synchronous motor [J]. IEEE Trans Ind Electron 2013,60(2):600-607.
[121]Jing Z J, Yu C, Chen G R. Complex dynamics in a permanent-magnet synchronous motor model [J]. Chaos Soliton Fract,2004,22(4):831-848.
[122]Li Z, Park J B, Joo Y H, et.al. Bifurcations and chaos in a permanent-magnet synchronous motor [J]. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,2002,49(3):383-387.
[123]Gao Y, Fan J W, Luo W G, et al. Chaos in the fractional order permanent magnet synchronous motor and its control [J]. Journal of Wuhan University of Technology, 2012,34(7):134-140.
[124]Li C L, Yu S M, Luo X S. Fractional-order permanent magnet synchronous motor and its adaptive chaotic control [J]. Chin Phys B,2012,21(10):100506.
[125]Xue D, Zhao C, Chen Y. Fractional order PID control of A DC-motor with elastic shaft:a case study [A]. Proceedings of the American Control Conference,2006 [C], Minneapolis, Minnesota, USA,14-16 June,2006:3182-3187.
[126]Watts D J, Strogatz S H. Collective dynamics of small-world'networks [J]. Nature, 1998,393(6684):440-442.
[127]Barabasi A L, Albert R. Emergence of scaling in random networks [J]. Science,1999, 286(5439):509-512.
[128]Lii J H, Chen G R. A time-varying complex dynamical network model and its controlled synchronization criteria [J]. IEEE Trans Autom Control 2005,50(6):841-846.
[129]Liu H, Lu J A, Lu J H, et.al. Structure identification of uncertain general complex dynamical networks with time delay [J]. Automatica,2009,45(8):1799-1807.
[130]Duan Z S, Chen G R. Does the eigenratio λ2/λN represent the synchronizability of a complex network? [J]. Chin Phys B,2012,21(8):080506.
[131]Lii J H, Yu X H, Chen G R, et.al. Characterizing the synchronizability of small-world dynamical networks [J]. Circuits and Systems I:Regular Papers, IEEE Transactions on,2004,51(4):787-796.
[132]Zhou J, Lu J A, Lu J H. Adaptive synchronization of an uncertain complex dynamical network [J]. IEEE Trans Autom Control 2006,51(4):652-656.
[133]He D R, Liu Z H, Wang B H. Complex Systems and Complex Networks [M]. Beijing; Higher Education Press.2009:1-4.