磁浮列车悬浮系统的Hopf分岔及滑模控制研究
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摘要
车轨振动问题影响工程造价和车辆运行的舒适度,对车轨耦合振动进行深入研究是当前悬浮控制领域的热点和难点.悬浮控制是磁悬浮列车的核心和关键技术.选取合适的控制算法,对悬浮控制起着至关重要的作用.
本文对磁浮列车控制系统的数学模型进行动力学行为和控制理论两大方面的研究：动力学行为研究主要是通过分析时滞状态反馈控制的磁悬浮系统的稳定性和Hopf分岔,从理论上找出车轨振动的原因,为以后抑制车轨耦合振动提供参考；控制理论研究主要是采用滑模控制算法确定状态反馈控制输入,使磁浮列车悬浮系统快速稳定.
第三章以时滞作为分岔参数讨论了时滞状态反馈控制的磁浮列车悬浮系统模型的动力学行为.前两节运用中心流形和正规型理论的方法,分别研究了时滞状态反馈控制的刚性轨道磁悬浮系统和弹性轨道磁悬浮系统的稳定性和Hopf分岔,计算出正规型方程并判断分岔的方向和极限环的稳定性.中心流形的方法虽然经典但是计算繁琐,与之相比,摄动法快速简便.在第四节中采用了Pseudo-Oscillator分析的摄动方法来计算Hopf分岔引起的周期解的振幅近似表达式,并用数值仿真验证了其有效性.
时滞是不可避免的,在时滞状态反馈控制的磁浮列车系统中,如何选取适当的控制参数也十分重要.在第四章讨论时滞和速度反馈控制增益的变化关系对系统稳定性的影响.并在时滞固定的情况下,以系统反馈控制参数作为分岔参数,采用多尺度的方法分析磁浮列车模型的Hopf分岔性质和计算周期解的近似表达式,得出速度反馈控制参数取值在一定范围内时滞反馈控制的磁悬浮系统保持稳定,变化其控制参数会产生两个极限环,并可能产生双Hopf分岔.接着分析了双Hopf分岔存在时所产生周期解的存在性和稳定性,并发现在特定条件下,系统会发生二次分岔.
然后选取适当的控制策略,由简到繁地对列车系统进行悬浮控制.在第五章运用滑模控制算法来确定控制输入,保证磁悬浮系统趋于稳定.本章分为两个方面：线性滑模控制和非线性滑模控制.首先采用与系统线性二次型调节器结合的方法对磁浮列车的线性系统进行滑模控制；接着采用修正的正切函数对磁浮列车的非线性系统进行自适应滑模控制,并通过数值仿真来验证其有效性.
The vehicle-guideway vibration greatly influences the project price and ride comfort. It is currently attractive and difficult to study the vibration of the vehicle-guideway in the field of suspension control. Maglev suspension control is the core and key technologies. The appropriate control algorithm plays a vital role in maglev suspension control.
This paper studies two aspects of the mathematical model of the maglev train control system:one is dynamic behavior and the other is control theory. The dynamic behavior study includes analyzing the stability and Hopf bifurcation of the delay feedback control maglev system to find the reason of vibration of the vehicle-guideway. And control theory research includes calculating the control input and stabilizing the suspension system of magnetic lev-itation train using the sliding mode control algorithms.
We discuss the dynamical behavior of the delay state feedback maglev suspension sys-tem by choosing the time delay as the bifurcation parameter in Chapter three. Using the center manifold and normal form theory, in the first two sections we analyze the stability and Hopf bifurcation of rigid guideway and flexible guideway magnetic levitation systems with the delayed state feedback control, and determine the nature of Hopf bifurcation and the sta-bility of the limit cycle by calculating the normal equation. Comparing with the method of center manifold, which is classic but needs tedious calculation, the method of perturbation is more quickly and easier. In the forth Section, using the Pseudo-Oscillator analysis, we calculate the approximate expression of the amplitude of the periodic solutions, and verify its validity by numerical simulations.
Delays are inevitable. Selecting appropriate control parameters in the delay state feed-back control of the maglev train system is also important. In the fourth Chapter, we discuss the relationship of time delay and velocity feedback control gain which influence the stabil-ity of maglev system. When time delay is fixed, choosing the velocity feedback control gain as the bifurcation parameter, we analyze the quality of the Hopf bifurcation and calculate the approximate expression of periodic solutions of the maglev system by using the method of multi-scale. The delay maglev system is stable when the velocity feedback control gain within a certain range, and would have two limit cycles even exist a double Hopf bifurcation by changing the value of the feedback gain. And then we analyze the existence and the sta-bility of the periodic solutions when double Hopf bifurcation occurs. In certain conditions, the maglev system would occur second bifurcation.
Then we select the appropriate control strategy, from simple to complex, to control the suspension system of maglev train. In the fifth Chapter, using the sliding mode control algorithm, the control input is determined to ensure the maglev system stable. This Chapter includes two aspects:linear and nonlinear sliding mode control. First of all, combining with linear quadratic regulator, the sliding mode control is applied to the linear maglev train system; then we apply the modified tangent function to the adaptive sliding mode control of nonlinear system of maglev train, and verify its effectiveness through numerical simulations.
本文对磁浮列车控制系统的数学模型进行动力学行为和控制理论两大方面的研究：动力学行为研究主要是通过分析时滞状态反馈控制的磁悬浮系统的稳定性和Hopf分岔,从理论上找出车轨振动的原因,为以后抑制车轨耦合振动提供参考；控制理论研究主要是采用滑模控制算法确定状态反馈控制输入,使磁浮列车悬浮系统快速稳定.
第三章以时滞作为分岔参数讨论了时滞状态反馈控制的磁浮列车悬浮系统模型的动力学行为.前两节运用中心流形和正规型理论的方法,分别研究了时滞状态反馈控制的刚性轨道磁悬浮系统和弹性轨道磁悬浮系统的稳定性和Hopf分岔,计算出正规型方程并判断分岔的方向和极限环的稳定性.中心流形的方法虽然经典但是计算繁琐,与之相比,摄动法快速简便.在第四节中采用了Pseudo-Oscillator分析的摄动方法来计算Hopf分岔引起的周期解的振幅近似表达式,并用数值仿真验证了其有效性.
时滞是不可避免的,在时滞状态反馈控制的磁浮列车系统中,如何选取适当的控制参数也十分重要.在第四章讨论时滞和速度反馈控制增益的变化关系对系统稳定性的影响.并在时滞固定的情况下,以系统反馈控制参数作为分岔参数,采用多尺度的方法分析磁浮列车模型的Hopf分岔性质和计算周期解的近似表达式,得出速度反馈控制参数取值在一定范围内时滞反馈控制的磁悬浮系统保持稳定,变化其控制参数会产生两个极限环,并可能产生双Hopf分岔.接着分析了双Hopf分岔存在时所产生周期解的存在性和稳定性,并发现在特定条件下,系统会发生二次分岔.
然后选取适当的控制策略,由简到繁地对列车系统进行悬浮控制.在第五章运用滑模控制算法来确定控制输入,保证磁悬浮系统趋于稳定.本章分为两个方面：线性滑模控制和非线性滑模控制.首先采用与系统线性二次型调节器结合的方法对磁浮列车的线性系统进行滑模控制；接着采用修正的正切函数对磁浮列车的非线性系统进行自适应滑模控制,并通过数值仿真来验证其有效性.
The vehicle-guideway vibration greatly influences the project price and ride comfort. It is currently attractive and difficult to study the vibration of the vehicle-guideway in the field of suspension control. Maglev suspension control is the core and key technologies. The appropriate control algorithm plays a vital role in maglev suspension control.
This paper studies two aspects of the mathematical model of the maglev train control system:one is dynamic behavior and the other is control theory. The dynamic behavior study includes analyzing the stability and Hopf bifurcation of the delay feedback control maglev system to find the reason of vibration of the vehicle-guideway. And control theory research includes calculating the control input and stabilizing the suspension system of magnetic lev-itation train using the sliding mode control algorithms.
We discuss the dynamical behavior of the delay state feedback maglev suspension sys-tem by choosing the time delay as the bifurcation parameter in Chapter three. Using the center manifold and normal form theory, in the first two sections we analyze the stability and Hopf bifurcation of rigid guideway and flexible guideway magnetic levitation systems with the delayed state feedback control, and determine the nature of Hopf bifurcation and the sta-bility of the limit cycle by calculating the normal equation. Comparing with the method of center manifold, which is classic but needs tedious calculation, the method of perturbation is more quickly and easier. In the forth Section, using the Pseudo-Oscillator analysis, we calculate the approximate expression of the amplitude of the periodic solutions, and verify its validity by numerical simulations.
Delays are inevitable. Selecting appropriate control parameters in the delay state feed-back control of the maglev train system is also important. In the fourth Chapter, we discuss the relationship of time delay and velocity feedback control gain which influence the stabil-ity of maglev system. When time delay is fixed, choosing the velocity feedback control gain as the bifurcation parameter, we analyze the quality of the Hopf bifurcation and calculate the approximate expression of periodic solutions of the maglev system by using the method of multi-scale. The delay maglev system is stable when the velocity feedback control gain within a certain range, and would have two limit cycles even exist a double Hopf bifurcation by changing the value of the feedback gain. And then we analyze the existence and the sta-bility of the periodic solutions when double Hopf bifurcation occurs. In certain conditions, the maglev system would occur second bifurcation.
Then we select the appropriate control strategy, from simple to complex, to control the suspension system of maglev train. In the fifth Chapter, using the sliding mode control algorithm, the control input is determined to ensure the maglev system stable. This Chapter includes two aspects:linear and nonlinear sliding mode control. First of all, combining with linear quadratic regulator, the sliding mode control is applied to the linear maglev train system; then we apply the modified tangent function to the adaptive sliding mode control of nonlinear system of maglev train, and verify its effectiveness through numerical simulations.
引文
[1]Glatzel K, Khurdok G, Rogg D. The development of the magnetically suspended transporta-tion system in the federal republic of Germany. IEEE Transactions on Vehicular Technology, 1980,29(1):3-17
[2]Nakashima H, Seki A. The status of the technical development for the Yamanashi Maglev test line. MAGLEV Proceeding,1995,31-35
[3]Yan L. Development of the maglev transportation in China. MAGLEV Proceeding,2004, 1-12
[4]Yoshihide Y, Masaaki F, Masao T, et al. The first HSST maglev commercial train in Japan. MAGLEV Proceeding,2004,76-85
[5]Cho H, Sung H, Kim B, et al. Maglev program in Korea. MAGLEV Proceeding,2002, 01103-01107
[6]Lee H W, Kim K C, Lee J. Review of maglev train technologies. IEEE Transactions on Magnetics,2006,42(7):1917-1925
[7]魏庆朝.磁悬浮铁路系统与技术.北京：中国科学出版社,2003
[8]Liu Y, Sun G, Wei R. The developmental status and future prospects of maglev technology. MAGLEV Proceeding,2006,59-64
[9]张颖,陈慧星,吴志添,李云钢.电磁永磁混合磁悬浮列车的磁铁结构优化设计.机车电传动,2008,5：30-32
[10]洪华杰.EMS型低速磁浮车轨耦合振动研究.国防科技大学：博士学位论文,2005
[11]刘恒坤.常导高速磁浮列车双转向架搭接结构的悬浮控制问题研究.国防科技大学：博士学位论文,2005
[12]施晓红.常导高速磁浮列车车轨耦合非线性动力学问题研究.国防科技大学：博士学位论文,2005
[13]王洪波.EMS型低速磁浮列车／轨道系统的动力相互作用问题研究.国防科技大学：博士学位论文,2007
[14]Zhang L L, Huang L H, Zhang Z Z. Stability and Hopf bifurcation of the maglev system with delayed position and speed feedback control. Nonlinear Dynamics,2009,59(1):197-207
[15]Attilio M. Vibration control for parametrically exicited Lineard systems. Int. J. Nonlinear Mech.,2006,41:146-155
[16]Attilio M. Vibration control for primary resonance of the Van der Pol oscillator by a time-delay state feedback. Int. J.Nonlinear Mech.,2003,38:123-131
[17]Attilio M. Vibration control for the primary resonance of a cantilever by time-delay state feedback. J. Sound and Vibration,2003,259:241-251
[18]Giorgio B, Leonid F, Alessandro P, et al. Modern sliding mode control theory. Berlin:Spinker, 2008
[19]Suplina V, Shaked U. Robust Hoo output-feedback control of systems with time-delay. Sys-tems and Control Letters,2008,57(3):193-199
[20]Tomohisa H, Wassim M, Konstantin Y. Neural network hybrid adaptive control for nonlinear uncertain impulsive dynamical systems. Nonlinear Analysis:Hybrid Systems,2008,2:862-874
[21]Taher N. A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem. Applied Energy,2010,87(1):327-339
[22]Hassan I, Mohamed A. Variable structure control of a magnetic levitation system. American Control Conference,2001,3725-3730
[23]郝柏林.分岔与奇异性.上海：上海科技教育出版社,1995
[24]Iooss G, Joseph D D. Elements of applied bifurcation theory. New York:Spinger-Verlag, 1990
[25]Hassard B D, Kazarinoff N D, Wan Y H. Theory and applications of Hopf bifurcation. Cam-bridge:Cambridge University Press,1981
[26]Hale J. Theory of functional differential equations. New York:Springer-Verlag,1977
[27]Kubicek M, Marek M. Computational methods in bifurcation theory and diddipative structure. Berlin:Springer-Verlag,1983
[28]张锁春.现代震荡反应的数学理论和数值方法.郑州：河南教育出版社,1991
[29]朱正佑,程昌钧.分叉问题的数值计算方法.兰州：兰州大学出版社,1989
[30]柳贵东,佘龙华.Hopf分叉的劳斯判据及磁悬浮系统的振动分析.振动、测试与诊断,2003,23(4)：276-278
[31]施晓红,佘龙华.非线性磁悬浮控制系统的周期运动稳定性研究.动力学与控制学报,2005,3(3)：52-55
[32]施晓红,佘龙华,常文森.EMS磁浮列车车／轨耦合系统的分岔现象.力学学报,2004,36(5)：634-640
[33]刘金琨.滑模变结构MATLAB仿真.北京：清华大学出版社,2005
[34]Utkin V I. Sliding modes in control and optimization. Berlin:Springer-Verlag,1992
[35]Utkin V I, Parnakh A. Sliding modes and their application in variable structure systems. Mir: Moscow,1978
[36]Utkin V I. Sliding mode control design principles and applications to electric drives. IEEE Transactions on Industrial Electronics,1993,40(1):23-36
[37]Slotine J J, Sastry S S. Tracking control of nonlinear systems using sliding sufaces with application to robot manipulator. International Journal of Control,1983,38(2):465-492
[38]Chung S C, Lin C L. A tranformed Lure problem for sliding mode control and chattering reduction. IEEE Transactions on Automatic Control,1999,44(3):563-568
[39]Xu J X, Lee T H, Pan Y J. On the sliding mode control for DC servo mechanisms in the presence of unmodeled dynamics. Mechatronics,2003,13:755-770
[40]Kachroo P, Tomizuka M. Chattering reduction and error convergence in the sliding mode control of a class of nonlinear systems. IEEE Transactions on Automatic Control,1996,41(7): 1063-1068
[41]Kang B P, Ju J L. Sliding mode controller with filtered signal for robot manipulators using virtual plant/controller. Mechatronics,1997,7(3):277-286
[42]Yanada H, Ohnishi H. Frequency-shaped sliding mode control of an eletrohydraulic servo-moto. Journal of systems and Control and Dynamics,1999,213(1):441-448
[43]Ha Q P, Nguyen Q H, Rye H F, et al. Fuzzy sliding-mode controllers with applications. IEEE Transactions on Industrial Electronics,2001,48(1):38-41
[44]张天平,冯纯伯.基于模糊逻辑的连续滑模控制.控制与决策,1995,10(6)：503-507
[45]孙益标,郭庆鼎,孙艳娜.基于模糊自学习的交流直线伺服系统滑模变结构控制.电工技术学报,2001,16(1)：52-56
[46]Hwang C L. Sliding mode control using time-varying switching gain and boundary layer for eletrohydraulic position and differential pressure control. IEE Proceedings Control Theory and Applications,1996,143(4):325-332
[47]Wong L J, Leung F H, Tam P K. A chattering elimination algorithm for sliding mode control of uncertain non-linear systems. Mechatronics,1998,8:765-775
[48]林岩,毛剑琴,操云甫.鲁棒低增益变结构模型参考自适应控制.自动化学报,2001,27(5)：665-670
[49]Chen M S, Hwang Y R, Tomizuka M. A state-dependent boundary layer design for sliding mode control. IEEE Iransactions on Automatic Control,2002,47(10):1677-1681
[50]Vicente P V, Gerd H. Chattering-free sliding mode control for a class of nonlinear mechnical systems. International Journal of Robust and Nonlinear Control,2001,11:1161-1178
[51]Bartolini G, Ferrara A, Usani E. Chattering avoidance by second-order sliding mode control. IEEE Transactions on Automatic Control,1998,43(2):241-246
[52]Ertugrul M, Kaynak O. Neuro sliding mode control of robotic manipulators. Mechatronics, 2000,10(1):239-263
[53]Lin F J, Chou W D. An induction motor servo drive using sliding-mode controller with genetic algorithm. Electric Power Systems Research,2003,64(2):93-108
[54]Edwards C. A pratical method for the design of sliding mode controllers using matrix inequal-ities. Automatica,2004,40(10):1761-1769
[55]Bekiroglu N, Bozma H I, Istefanopulos Y. Model reference adaptive approach to sliding mode chontrol. American Control Conference,1995,1:1028-1032
[56]Song J B, Ishida Y. A robust sliding mode control for pneumatic servo systems. International Journal Engineering Science,1997,35(8):711-723
[57]盛蓉蓉,吴云飞.磁浮列车的模糊滑模控制器.黑龙江科技学院学报,2005,15(5)：305-308
[58]段吉安,陆新江,李群明.电磁悬浮平台系统的滑模控制研究.系统仿真学报,2005,17(8)：1966-1969
[59]郑永斌.EMS型中低速磁浮列车悬浮模块的解耦控制研究.国防科技大学：硕士学位论文,2006
[60]Yang Z J, Tateishi M. Adaptive robust nonlinear control of a magnetic levitation system. Automatica,2001,37(7):1125-1131
[61]刘德生,李杰,周丹峰.EMS型磁悬浮列车模块悬浮系统的模型参考自适应控制.微计算机信息,2006,9(1)：113-115
[62]常蕾,游小杰,杨中平.电磁永磁混合悬浮系统的滑模控制器设计.微计算机信息,2009,25(4)：24-26
[63]马忠宝,孙荣斌.磁悬浮列车电磁悬浮系统的自适应模糊滑模控制.机车电传动,2007,1：29-32
[64]吴祥明.磁浮列车.上海：上海科学技术出版社,2003
[65]张志洲.高速磁浮列车单铁悬浮系统的容错控制问题研究.国防科技大学：硕士学位论文,2006
[66]She L H, Wang H, Zou D S, et al. Hopf bifurcation of maglev system with coupled elastic guideway. The 20th Conference on Magnetically Levirated Systems and Linear Drives,2008: 1-7
[67]刘恒坤,常文森,施晓红.磁悬浮系统车轨耦合振动研究.计算机仿真,2006,23(9)：256-258
[68]Masada E. Development of maglev and linear drive technology for transportation in Japan. 14th Int. Conf. On Maglev Systems,1995:26-29
[69]张继业,曾京.Hopf分岔的代数判据及其在车辆动力学中的应用.力学学报,2000,32(5)：596-605
[70]龙驭球,包世华.结构力学.北京：高等教育出版社,1996
[71]宋一凡.公路桥梁动力学.北京：人民交通出版社,1999
[72]张丙飞.应用于磁浮列车车轨耦合振动控制的状态反馈组合方法研究.国防科学技术大学：硕士学位论文,2007
[73]王浩.应用Hopf分岔理论研究磁浮列车车轨弹性耦合振动控制.国防科学技术大学：硕士学位论文,2008
[74]胡寿松.自动控制原理.北京：国防工业出版社,2000
[75]曹建福,韩崇昭,方洋旺.非线性系统理论及应用.西安：西安交通大学出版社,2002
[76]马知恩,周义仓.常微分方程定性与稳定性方法.北京：科学出版社,2001
[77]李云.非线性动力系统的现代数学方法及其应用.北京：人民交通出版社,1998
[78]Nayfeh A H. Order reduction of retarded nonlinear systems-the method of multiple scales versus center-manifold. Nonlinear Dynamics,2008,51:483-500
[79]Das S L, Chatterjee A. Multiple scales without center manifold reduction for delay differential equations near Hopf bifurcations. Nonlinear Dynamics,2002,30:323-335
[80]Nayfeh A H. Perturbation methods. Shanghai:Shanghai Publishing House of Science and Technology,1984
[81]Casal A, Freedman M. A Poincare-Lindstedt approach to bifurcation problems for differential-delay equations. IEEE Trans.Autom.Control,1980,25:967-973
[82]Rand R, Verdugo A. Hopf bifurcation formula for first order differential-delay equations. Commun.Nonlinear Sci.Numer.Simul.,2007,12:859-864
[83]Nayfeh A H, Chin C M, Pratt J R. Perturbation mathods in nonlinear dynamics:applications to machining dynamics. ASME J. Manuf.Sci.Technol.,1996,119:485-493
[84]MacDonald N. Harmonic balance in delay-differential equations. J. Sound Vib.,1995,186: 649-656
[85]Wang Z H, Hu H Y. Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation. Int.J.Bifurc.Chaos,2007,17(8):2805-2814
[86]张芷芬.微分方程定性理论.北京：科学出版社,2003
[87]黄立宏,李雪梅.细胞神经网络动力学.北京：科学出版社,2006
[88]张恭庆,林源渠.泛函分析讲义.北京：北京大学出版社,1987
[89]钱祥征,戴斌祥,刘开宇.非线性常微分方程理论方法应用.长沙：湖南大学出版社,2006
[90]Stepan G. Retarded dynamical systems:stability and characteristic functions. Lonton:Long-man,1989
[91]Balachandran B, Kalmar-Nagy, Gilsinn D E. Delay differential equations:recent advances and new directions. New York:Springer,2008
[92]Nayfeh A H. Method of normal forms. New York:Wiley-Interscience,1993
[93]Zheng Y G, Wang Z H. Stability and Hopf bifurcation of an optoelectronic time-delay feed-back system. Nonlinear Dyn.,2009,57(1):125-134
[94]Masoud Z N, Nayfeh A H, Almousa A. Delayed position-feedback controller for the reduction of payload pendulations of rotary cranes. J. Vib. Control,2003,9(1):257-277
[95]Hale J K, Lunel S M V. Introduction to functional differential equations. New York:Springer Verlag,1993
[96]Takens F. Unfoldings of certain singularities of vector fields:generalized Hopf bifurcations. JDE,1973,14:476-493
[97]Golubitsky M, Langford W F. Classification and unfoldings of degenerate Hopf bifurcation. JDE,1981,41:525-546
[98]Engelborghs K, Luzyanina T, Samaey G. DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Department of Computer Science, K.U. Leuven,Belgium,2001
[99]Leblanc V G, Langford W F. Classification and unfoldings of 1:2 resonant Hopf bifurcation. Arch. Rational Mech. Anal.,1996,136:305-357
[100]Guckenheimer J, Holems P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York:Spring-Verlag,1983
[101]Campbell S A, Stone E. Analysis of the chatter instability in a nonlinear model for drilling. ASME J. Comp. Nonl. Dyn.,2006,1:294-306
[102]Campbell S A, Crawford S, Morris K. Friction and the inverted stabilization problem. ASME J. Dyn. Sys. Meas. Cont.,2008,130(5):054502-054508
[103]Husain A R, Ahmad M N, Mohd A H, et al. Sliding mode control with linear quadratic hyperplane design:an application to an active magnetic bearing system. The 5th Student Conference on Research and Development,2007,11-12:588-593
[104]Sivrioglu S, Nonami K. Sliding mode control with time-varying hyperplane for AMB systems. IEEE/ASME Trans. on Mechatronics,1998,3(1):51-59
[105]Hung J Y, Gao W B, Hung J C. Variable structure control:a survey. IEEE Trans. Ind. Electron, 1993,40(1):2-22
[106]Spurgeon S, Edwards C. Sliding mode control:theory and applications. London:Taylor and Francis,1998
[107]Slotine J J, Li W. Applied nonlinear control. Englewood Cliffs:Prentice-Hall,1991
[108]Huang Y J, Kuo T C. Robust position control of DC servomechanism with output measurement noise. Electrical Engineering,2006,88:223-338
[109]Furuta K. VSS type self-tuning control. IEEE Transactions on Industrial Electronics,1993, 40:37-44
[110]Lee P M, Oh J H. Improvements on VSS type self-tuning control for a tracking controller. IEEE Transactions on Industrial Electronics,1998,45:319-325
[111]Chen C T, Chang W D. A feedforward neural network with function shape autotuning Neural Networks,1996,9(4):627-641
[112]Chang W D, Hwang R C, Hsieh J G. Application of an auto-tuning neuron to sliding mode control IEEE Transactions on Systems, Man, and Cybernetics-Part C,2002,32:517-529
[113]Kuo T C, Ying J H, Chang S H. Sliding mode control with self-tuning law for uncertain nonlinear systems. ISA Transactions,2008,47:171-178
[114]Rong X, Umit O. Sliding mode control of a class of underactuated systems. Automatica, 2008,44:233-241
[115]Cui X, Shin K G. Direct control and coordination using neural networks. IEEE Transactions on Systems, Man, and Cybernetics,1993,23(3):686-697
[116]Richard C D, Rober H B. Modern control systems. London:Addison Wesley Longman,2008
[2]Nakashima H, Seki A. The status of the technical development for the Yamanashi Maglev test line. MAGLEV Proceeding,1995,31-35
[3]Yan L. Development of the maglev transportation in China. MAGLEV Proceeding,2004, 1-12
[4]Yoshihide Y, Masaaki F, Masao T, et al. The first HSST maglev commercial train in Japan. MAGLEV Proceeding,2004,76-85
[5]Cho H, Sung H, Kim B, et al. Maglev program in Korea. MAGLEV Proceeding,2002, 01103-01107
[6]Lee H W, Kim K C, Lee J. Review of maglev train technologies. IEEE Transactions on Magnetics,2006,42(7):1917-1925
[7]魏庆朝.磁悬浮铁路系统与技术.北京：中国科学出版社,2003
[8]Liu Y, Sun G, Wei R. The developmental status and future prospects of maglev technology. MAGLEV Proceeding,2006,59-64
[9]张颖,陈慧星,吴志添,李云钢.电磁永磁混合磁悬浮列车的磁铁结构优化设计.机车电传动,2008,5：30-32
[10]洪华杰.EMS型低速磁浮车轨耦合振动研究.国防科技大学：博士学位论文,2005
[11]刘恒坤.常导高速磁浮列车双转向架搭接结构的悬浮控制问题研究.国防科技大学：博士学位论文,2005
[12]施晓红.常导高速磁浮列车车轨耦合非线性动力学问题研究.国防科技大学：博士学位论文,2005
[13]王洪波.EMS型低速磁浮列车／轨道系统的动力相互作用问题研究.国防科技大学：博士学位论文,2007
[14]Zhang L L, Huang L H, Zhang Z Z. Stability and Hopf bifurcation of the maglev system with delayed position and speed feedback control. Nonlinear Dynamics,2009,59(1):197-207
[15]Attilio M. Vibration control for parametrically exicited Lineard systems. Int. J. Nonlinear Mech.,2006,41:146-155
[16]Attilio M. Vibration control for primary resonance of the Van der Pol oscillator by a time-delay state feedback. Int. J.Nonlinear Mech.,2003,38:123-131
[17]Attilio M. Vibration control for the primary resonance of a cantilever by time-delay state feedback. J. Sound and Vibration,2003,259:241-251
[18]Giorgio B, Leonid F, Alessandro P, et al. Modern sliding mode control theory. Berlin:Spinker, 2008
[19]Suplina V, Shaked U. Robust Hoo output-feedback control of systems with time-delay. Sys-tems and Control Letters,2008,57(3):193-199
[20]Tomohisa H, Wassim M, Konstantin Y. Neural network hybrid adaptive control for nonlinear uncertain impulsive dynamical systems. Nonlinear Analysis:Hybrid Systems,2008,2:862-874
[21]Taher N. A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem. Applied Energy,2010,87(1):327-339
[22]Hassan I, Mohamed A. Variable structure control of a magnetic levitation system. American Control Conference,2001,3725-3730
[23]郝柏林.分岔与奇异性.上海：上海科技教育出版社,1995
[24]Iooss G, Joseph D D. Elements of applied bifurcation theory. New York:Spinger-Verlag, 1990
[25]Hassard B D, Kazarinoff N D, Wan Y H. Theory and applications of Hopf bifurcation. Cam-bridge:Cambridge University Press,1981
[26]Hale J. Theory of functional differential equations. New York:Springer-Verlag,1977
[27]Kubicek M, Marek M. Computational methods in bifurcation theory and diddipative structure. Berlin:Springer-Verlag,1983
[28]张锁春.现代震荡反应的数学理论和数值方法.郑州：河南教育出版社,1991
[29]朱正佑,程昌钧.分叉问题的数值计算方法.兰州：兰州大学出版社,1989
[30]柳贵东,佘龙华.Hopf分叉的劳斯判据及磁悬浮系统的振动分析.振动、测试与诊断,2003,23(4)：276-278
[31]施晓红,佘龙华.非线性磁悬浮控制系统的周期运动稳定性研究.动力学与控制学报,2005,3(3)：52-55
[32]施晓红,佘龙华,常文森.EMS磁浮列车车／轨耦合系统的分岔现象.力学学报,2004,36(5)：634-640
[33]刘金琨.滑模变结构MATLAB仿真.北京：清华大学出版社,2005
[34]Utkin V I. Sliding modes in control and optimization. Berlin:Springer-Verlag,1992
[35]Utkin V I, Parnakh A. Sliding modes and their application in variable structure systems. Mir: Moscow,1978
[36]Utkin V I. Sliding mode control design principles and applications to electric drives. IEEE Transactions on Industrial Electronics,1993,40(1):23-36
[37]Slotine J J, Sastry S S. Tracking control of nonlinear systems using sliding sufaces with application to robot manipulator. International Journal of Control,1983,38(2):465-492
[38]Chung S C, Lin C L. A tranformed Lure problem for sliding mode control and chattering reduction. IEEE Transactions on Automatic Control,1999,44(3):563-568
[39]Xu J X, Lee T H, Pan Y J. On the sliding mode control for DC servo mechanisms in the presence of unmodeled dynamics. Mechatronics,2003,13:755-770
[40]Kachroo P, Tomizuka M. Chattering reduction and error convergence in the sliding mode control of a class of nonlinear systems. IEEE Transactions on Automatic Control,1996,41(7): 1063-1068
[41]Kang B P, Ju J L. Sliding mode controller with filtered signal for robot manipulators using virtual plant/controller. Mechatronics,1997,7(3):277-286
[42]Yanada H, Ohnishi H. Frequency-shaped sliding mode control of an eletrohydraulic servo-moto. Journal of systems and Control and Dynamics,1999,213(1):441-448
[43]Ha Q P, Nguyen Q H, Rye H F, et al. Fuzzy sliding-mode controllers with applications. IEEE Transactions on Industrial Electronics,2001,48(1):38-41
[44]张天平,冯纯伯.基于模糊逻辑的连续滑模控制.控制与决策,1995,10(6)：503-507
[45]孙益标,郭庆鼎,孙艳娜.基于模糊自学习的交流直线伺服系统滑模变结构控制.电工技术学报,2001,16(1)：52-56
[46]Hwang C L. Sliding mode control using time-varying switching gain and boundary layer for eletrohydraulic position and differential pressure control. IEE Proceedings Control Theory and Applications,1996,143(4):325-332
[47]Wong L J, Leung F H, Tam P K. A chattering elimination algorithm for sliding mode control of uncertain non-linear systems. Mechatronics,1998,8:765-775
[48]林岩,毛剑琴,操云甫.鲁棒低增益变结构模型参考自适应控制.自动化学报,2001,27(5)：665-670
[49]Chen M S, Hwang Y R, Tomizuka M. A state-dependent boundary layer design for sliding mode control. IEEE Iransactions on Automatic Control,2002,47(10):1677-1681
[50]Vicente P V, Gerd H. Chattering-free sliding mode control for a class of nonlinear mechnical systems. International Journal of Robust and Nonlinear Control,2001,11:1161-1178
[51]Bartolini G, Ferrara A, Usani E. Chattering avoidance by second-order sliding mode control. IEEE Transactions on Automatic Control,1998,43(2):241-246
[52]Ertugrul M, Kaynak O. Neuro sliding mode control of robotic manipulators. Mechatronics, 2000,10(1):239-263
[53]Lin F J, Chou W D. An induction motor servo drive using sliding-mode controller with genetic algorithm. Electric Power Systems Research,2003,64(2):93-108
[54]Edwards C. A pratical method for the design of sliding mode controllers using matrix inequal-ities. Automatica,2004,40(10):1761-1769
[55]Bekiroglu N, Bozma H I, Istefanopulos Y. Model reference adaptive approach to sliding mode chontrol. American Control Conference,1995,1:1028-1032
[56]Song J B, Ishida Y. A robust sliding mode control for pneumatic servo systems. International Journal Engineering Science,1997,35(8):711-723
[57]盛蓉蓉,吴云飞.磁浮列车的模糊滑模控制器.黑龙江科技学院学报,2005,15(5)：305-308
[58]段吉安,陆新江,李群明.电磁悬浮平台系统的滑模控制研究.系统仿真学报,2005,17(8)：1966-1969
[59]郑永斌.EMS型中低速磁浮列车悬浮模块的解耦控制研究.国防科技大学：硕士学位论文,2006
[60]Yang Z J, Tateishi M. Adaptive robust nonlinear control of a magnetic levitation system. Automatica,2001,37(7):1125-1131
[61]刘德生,李杰,周丹峰.EMS型磁悬浮列车模块悬浮系统的模型参考自适应控制.微计算机信息,2006,9(1)：113-115
[62]常蕾,游小杰,杨中平.电磁永磁混合悬浮系统的滑模控制器设计.微计算机信息,2009,25(4)：24-26
[63]马忠宝,孙荣斌.磁悬浮列车电磁悬浮系统的自适应模糊滑模控制.机车电传动,2007,1：29-32
[64]吴祥明.磁浮列车.上海：上海科学技术出版社,2003
[65]张志洲.高速磁浮列车单铁悬浮系统的容错控制问题研究.国防科技大学：硕士学位论文,2006
[66]She L H, Wang H, Zou D S, et al. Hopf bifurcation of maglev system with coupled elastic guideway. The 20th Conference on Magnetically Levirated Systems and Linear Drives,2008: 1-7
[67]刘恒坤,常文森,施晓红.磁悬浮系统车轨耦合振动研究.计算机仿真,2006,23(9)：256-258
[68]Masada E. Development of maglev and linear drive technology for transportation in Japan. 14th Int. Conf. On Maglev Systems,1995:26-29
[69]张继业,曾京.Hopf分岔的代数判据及其在车辆动力学中的应用.力学学报,2000,32(5)：596-605
[70]龙驭球,包世华.结构力学.北京：高等教育出版社,1996
[71]宋一凡.公路桥梁动力学.北京：人民交通出版社,1999
[72]张丙飞.应用于磁浮列车车轨耦合振动控制的状态反馈组合方法研究.国防科学技术大学：硕士学位论文,2007
[73]王浩.应用Hopf分岔理论研究磁浮列车车轨弹性耦合振动控制.国防科学技术大学：硕士学位论文,2008
[74]胡寿松.自动控制原理.北京：国防工业出版社,2000
[75]曹建福,韩崇昭,方洋旺.非线性系统理论及应用.西安：西安交通大学出版社,2002
[76]马知恩,周义仓.常微分方程定性与稳定性方法.北京：科学出版社,2001
[77]李云.非线性动力系统的现代数学方法及其应用.北京：人民交通出版社,1998
[78]Nayfeh A H. Order reduction of retarded nonlinear systems-the method of multiple scales versus center-manifold. Nonlinear Dynamics,2008,51:483-500
[79]Das S L, Chatterjee A. Multiple scales without center manifold reduction for delay differential equations near Hopf bifurcations. Nonlinear Dynamics,2002,30:323-335
[80]Nayfeh A H. Perturbation methods. Shanghai:Shanghai Publishing House of Science and Technology,1984
[81]Casal A, Freedman M. A Poincare-Lindstedt approach to bifurcation problems for differential-delay equations. IEEE Trans.Autom.Control,1980,25:967-973
[82]Rand R, Verdugo A. Hopf bifurcation formula for first order differential-delay equations. Commun.Nonlinear Sci.Numer.Simul.,2007,12:859-864
[83]Nayfeh A H, Chin C M, Pratt J R. Perturbation mathods in nonlinear dynamics:applications to machining dynamics. ASME J. Manuf.Sci.Technol.,1996,119:485-493
[84]MacDonald N. Harmonic balance in delay-differential equations. J. Sound Vib.,1995,186: 649-656
[85]Wang Z H, Hu H Y. Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation. Int.J.Bifurc.Chaos,2007,17(8):2805-2814
[86]张芷芬.微分方程定性理论.北京：科学出版社,2003
[87]黄立宏,李雪梅.细胞神经网络动力学.北京：科学出版社,2006
[88]张恭庆,林源渠.泛函分析讲义.北京：北京大学出版社,1987
[89]钱祥征,戴斌祥,刘开宇.非线性常微分方程理论方法应用.长沙：湖南大学出版社,2006
[90]Stepan G. Retarded dynamical systems:stability and characteristic functions. Lonton:Long-man,1989
[91]Balachandran B, Kalmar-Nagy, Gilsinn D E. Delay differential equations:recent advances and new directions. New York:Springer,2008
[92]Nayfeh A H. Method of normal forms. New York:Wiley-Interscience,1993
[93]Zheng Y G, Wang Z H. Stability and Hopf bifurcation of an optoelectronic time-delay feed-back system. Nonlinear Dyn.,2009,57(1):125-134
[94]Masoud Z N, Nayfeh A H, Almousa A. Delayed position-feedback controller for the reduction of payload pendulations of rotary cranes. J. Vib. Control,2003,9(1):257-277
[95]Hale J K, Lunel S M V. Introduction to functional differential equations. New York:Springer Verlag,1993
[96]Takens F. Unfoldings of certain singularities of vector fields:generalized Hopf bifurcations. JDE,1973,14:476-493
[97]Golubitsky M, Langford W F. Classification and unfoldings of degenerate Hopf bifurcation. JDE,1981,41:525-546
[98]Engelborghs K, Luzyanina T, Samaey G. DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Department of Computer Science, K.U. Leuven,Belgium,2001
[99]Leblanc V G, Langford W F. Classification and unfoldings of 1:2 resonant Hopf bifurcation. Arch. Rational Mech. Anal.,1996,136:305-357
[100]Guckenheimer J, Holems P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York:Spring-Verlag,1983
[101]Campbell S A, Stone E. Analysis of the chatter instability in a nonlinear model for drilling. ASME J. Comp. Nonl. Dyn.,2006,1:294-306
[102]Campbell S A, Crawford S, Morris K. Friction and the inverted stabilization problem. ASME J. Dyn. Sys. Meas. Cont.,2008,130(5):054502-054508
[103]Husain A R, Ahmad M N, Mohd A H, et al. Sliding mode control with linear quadratic hyperplane design:an application to an active magnetic bearing system. The 5th Student Conference on Research and Development,2007,11-12:588-593
[104]Sivrioglu S, Nonami K. Sliding mode control with time-varying hyperplane for AMB systems. IEEE/ASME Trans. on Mechatronics,1998,3(1):51-59
[105]Hung J Y, Gao W B, Hung J C. Variable structure control:a survey. IEEE Trans. Ind. Electron, 1993,40(1):2-22
[106]Spurgeon S, Edwards C. Sliding mode control:theory and applications. London:Taylor and Francis,1998
[107]Slotine J J, Li W. Applied nonlinear control. Englewood Cliffs:Prentice-Hall,1991
[108]Huang Y J, Kuo T C. Robust position control of DC servomechanism with output measurement noise. Electrical Engineering,2006,88:223-338
[109]Furuta K. VSS type self-tuning control. IEEE Transactions on Industrial Electronics,1993, 40:37-44
[110]Lee P M, Oh J H. Improvements on VSS type self-tuning control for a tracking controller. IEEE Transactions on Industrial Electronics,1998,45:319-325
[111]Chen C T, Chang W D. A feedforward neural network with function shape autotuning Neural Networks,1996,9(4):627-641
[112]Chang W D, Hwang R C, Hsieh J G. Application of an auto-tuning neuron to sliding mode control IEEE Transactions on Systems, Man, and Cybernetics-Part C,2002,32:517-529
[113]Kuo T C, Ying J H, Chang S H. Sliding mode control with self-tuning law for uncertain nonlinear systems. ISA Transactions,2008,47:171-178
[114]Rong X, Umit O. Sliding mode control of a class of underactuated systems. Automatica, 2008,44:233-241
[115]Cui X, Shin K G. Direct control and coordination using neural networks. IEEE Transactions on Systems, Man, and Cybernetics,1993,23(3):686-697
[116]Richard C D, Rober H B. Modern control systems. London:Addison Wesley Longman,2008