基于分数阶微积分理论的滑模变结构控制
|
摘要
分数阶微积分理论主要是对任意实数阶或复数阶微积分的理论及其应用的研究,是经典整数阶微积分理论的推广。分数阶微积分控制理论是一个新兴的课题,最近三十多年发展迅速,目前已成为科研工作者研究的热点。
本论文主要研究分数阶滑模变结构控制方法。论文首先介绍了有关滑模变结构控制方法以及分数阶微积分理论的基础知识。通过基础分数阶微分方程的介绍,引入分数阶指数方程的解曲线。从解曲线的形状特点以及其随参数变化规律出发,将分数阶微积分引入到传统的整数阶滑模变结构指数控制律当中,延拓了传统的整数阶滑模变结构控制律,并通过能达性条件证明该分数阶滑模变结构控制律的有效性。将分数阶滑模变结构指数控制律应用于两混沌系统的同步问题当中。设计分数阶指数控制律使两混沌系统达到同步,实验结果证明了分数阶指数控制律较整数阶指数控制律的优越性。最后,论文对分数阶天棚阻尼半主动悬架设计了基于分数阶积分补偿的滑模观测器,基于分数阶积分补偿的滑模变结构控制方法是对传统滑模变结构指数控制律方法的另一种延拓,此种方法合理性与有效性是比较直观的。通过分数阶积分补偿方法与整数阶积分补偿方法的仿真比较,再次显示分数阶微积分理论的优越性。
Fractional calculus is a theory on the research and application of derivatives and integral of arbitrary real or complex order. It is a natural extension of the classical integer order calculus. Control theory of fractional calculus is a novel topic. It has developed rapidly on the recent 30 years and become the focus of so many researchers.
In this paper, fractional sliding mode control method will be researched. Firstly, a brief introduction is given about the sliding mode control and the basic knowledge and properties of fractional calculus. Then the fundamental linear fractional-order differential equation will be introduced, and fractional-order exponential differential equation curves are got and studied by changing the parameters regularly. According to the variations of curves with the change of parameters regularly, the fractional exponential curve and the integral exponential curve is compared and it supplies the basis of theory of next studies. The fractional calculus is introduced in the classical integer-order exponential sliding mode control law, called fractional-order exponential sliding mode control law. This law is proved the generalization of the classical sliding mode control law and can lead the state of systems into the sliding mode. Then it will be used in designing the control law in the synchronization of two chaos systems. By SIMULINK, the result of the fractional-order law method and the integer-order law method will be compared and the fractional-order law method is proved much more better. At the end of this paper, another fractional-order sliding mode control law is studied. It is called sliding mode control law based on the fractional compensation. People can much more easily deduce that the state of the system will reach the sliding mode by this method. It is another way of generalizing the classic sliding mode control law. It is used in designing the observer of the fractional skyhook damping semi-active suspension. The designed fractional-order observer improves the performance of the optimized fractional-order skyhook damping semi-active suspension. So one can conclude that the fractional-compensation sliding mode control method is better than integral-compensation one.
本论文主要研究分数阶滑模变结构控制方法。论文首先介绍了有关滑模变结构控制方法以及分数阶微积分理论的基础知识。通过基础分数阶微分方程的介绍,引入分数阶指数方程的解曲线。从解曲线的形状特点以及其随参数变化规律出发,将分数阶微积分引入到传统的整数阶滑模变结构指数控制律当中,延拓了传统的整数阶滑模变结构控制律,并通过能达性条件证明该分数阶滑模变结构控制律的有效性。将分数阶滑模变结构指数控制律应用于两混沌系统的同步问题当中。设计分数阶指数控制律使两混沌系统达到同步,实验结果证明了分数阶指数控制律较整数阶指数控制律的优越性。最后,论文对分数阶天棚阻尼半主动悬架设计了基于分数阶积分补偿的滑模观测器,基于分数阶积分补偿的滑模变结构控制方法是对传统滑模变结构指数控制律方法的另一种延拓,此种方法合理性与有效性是比较直观的。通过分数阶积分补偿方法与整数阶积分补偿方法的仿真比较,再次显示分数阶微积分理论的优越性。
Fractional calculus is a theory on the research and application of derivatives and integral of arbitrary real or complex order. It is a natural extension of the classical integer order calculus. Control theory of fractional calculus is a novel topic. It has developed rapidly on the recent 30 years and become the focus of so many researchers.
In this paper, fractional sliding mode control method will be researched. Firstly, a brief introduction is given about the sliding mode control and the basic knowledge and properties of fractional calculus. Then the fundamental linear fractional-order differential equation will be introduced, and fractional-order exponential differential equation curves are got and studied by changing the parameters regularly. According to the variations of curves with the change of parameters regularly, the fractional exponential curve and the integral exponential curve is compared and it supplies the basis of theory of next studies. The fractional calculus is introduced in the classical integer-order exponential sliding mode control law, called fractional-order exponential sliding mode control law. This law is proved the generalization of the classical sliding mode control law and can lead the state of systems into the sliding mode. Then it will be used in designing the control law in the synchronization of two chaos systems. By SIMULINK, the result of the fractional-order law method and the integer-order law method will be compared and the fractional-order law method is proved much more better. At the end of this paper, another fractional-order sliding mode control law is studied. It is called sliding mode control law based on the fractional compensation. People can much more easily deduce that the state of the system will reach the sliding mode by this method. It is another way of generalizing the classic sliding mode control law. It is used in designing the observer of the fractional skyhook damping semi-active suspension. The designed fractional-order observer improves the performance of the optimized fractional-order skyhook damping semi-active suspension. So one can conclude that the fractional-compensation sliding mode control method is better than integral-compensation one.
引文
[1]刘金琨,滑模变结构控制MATLAB仿真.清华大学出版社.2005.10
[2] Lacroix S F, Traitfédu Calcul Différentiel et du Calcul Intégral[M], Courcier, Paris, t. 3, 1819; pp. 409-410
[3] R.C.Koeller, Polynomial operators, stieltjes convolution, and fractional calculus in hereditary mechanics, Acta Mechanica,1986(58):251-264
[4] H.H.Sun, A.A.Abdelwahab and B.Onaral, Linear approximation of transfer function with a pole of fractional order, IEEE transactions on Automatic Control,1984(29):441-444
[5] O.Heaviside, Electromagetic Theory, Chelsea: New York,1971
[6] B.Ross.Fractional Calculus and its Applications.Berlin:Springer Verlag,1975
[7] K.B.Oldham, J.Spanier. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press,1974
[8] B.B.Mandelbort.The Fractal Geometry of Nature.New York:W.H.Freeman and Company,1982
[9]高安秀树.分数维.北京:地震出版社,1989
[10]王东生,曹磊.混沌、分形及应用.合肥:中国科学技术出版社,1995
[11] S.G.Samko,A.A.Kilbas,O.I.Marichev. Integral and Derivatives of Fractional Order and Some of Their Applications. Minsk: Naukai Tekhnika,1987
[12] S.G.Samko,A.A.Kilbas,O.I.Marichev. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach,1993
[13] K.S.Miller,B.Boss. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley-Interscience Publication,1993
[14] I.Podlubny. Fractional Differential Equations. San Diego: Academic Press,1999
[15] R.R.Nishimoto. Fractional Calculus(Integration and Differentiation of Arbitrary Order)Vol 2.Koriyama:Descartes Press Cor.,1987
[16]曾庆山,曹广益,王振滨.分数阶PIλDμ控制器的仿真研究.系统仿真学报.Vol.16 No.3.2004
[17]刘进英,李文.分数阶PIλDμ控制器的设计.自动控制与检测
[18] Podlubny I.Fractional differential equations[M].New York:Academic Press,1999
[19] Oldham K B,Spanier J. The Fractional Calculus[M].New York:Academic,1974
[20] Yongguang Yu,Han-Xiong Li.The synchronization of fractional-order Rossler hyperchaotic systems.Physica A 387(2008)1393-1403
[21] Chuanguang Li,Guanrong Chen.Chaos and hyperchaos in the fractional-order Rossler equations.Physica A341 (2004)55-61
[22]陈宁,台永鹏,陈南.分数微积分理论在非线性车辆悬架滑模控制中的应用,动力学与控制学报.Vol.7 No.3 Sep.2009
[23] Schuster H.G.Deterministic chaos: an introduction. Weinheim:Physik-Verlag,1984.
[24] Hirsch M.W., Smale S.Differential equations: dynamical systems and linear algebra. New York: Academic Press, 1974.
[25] Bode HW, Network Analysis and Feedback Amplifier Design[M], Van Nostrand, New York,1945
[26] Tustin A, Allanson JT, Layton JM,et al, The design of systems for automatic control of the position of massive object[R]. Proceedings of Institution of Electrical Engineers, 105, Part C, Suppl. 1958,No.1: 1-57
[27] Carlson Gordon E., C. A. Halijak (1961), Simulation of the fractional derivative operator s1/2 and the fractional integral operator 1/s1/2[J], in: Proc. Of the CSSCM, Kansas State U. B., 1961,45(7):1-22
[28] Carlson GE and. Halijak, CA. Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process[J], IEEE Transactions on Circuit Theory,1964,11(2): 210–213.
[29] Oustaloup A. La commande CRONE. Editions Hermès, Paris, (1991).
[30] Oustaloup A. La dérivation non entière: théorie,synthèse et applications[M]. Hermès, Paris. 1995.
[31] Oustaloup, A, Moreau, X, Nouillant, M, The CRONE suspension [J], Control Engineering Practice,1996. Vol 4(8):1101-1108
[32] Moreau X, Nouillant C, Oustaloup A, Effect Of The Crone Suspension Control System On Braking[C], In: Workshop IFAC On Advances in Automotive Control, Karlsruhe, Germany,2001 : 63-68.
[33] Pommier V,Sabatier J, Lanusse P,et al, Crone Control Of A Nonlinear Hydraulic Actuator[J], Control Engineering Practice 2002,10 :391–402 .
[34] Altet O, Moreau X, Moze M, and et al. Principles And Synthesis Of Hydractive Crone Suspension[J], Nonlinear Dynamics 38: 435–459, 2004.
[35] Sabatier J, Oustaloup A, Iturricha A.G, and Lanusse P, CRONE Control: Principles and Extension to Time-Variant Plants with Asymptotically Constant Coefficients[J], Nonlinear Dynamics 2002,29: 363–385.
[36] Matignon D, Stability result on fractional differential equations with applications to control processing[J], in: IMACS-SMC Proceedings, Lille, France, 1996:963-968
[37] Matignon D. (1998), Stability properties for generalized fractional differential systems[C],in: ESIAM Proceedings on Fractional Differential Systems 5, 1998: 145-158
[38] Podlubny I, Fractional-order systems and PIlDm-controllers[J], IEEE Transactions on Automatic Control 1999,44(1) :208–214
[39] Moze, M., J. Sabatier, A. Oustaloup, LMI Tools for Stability Analysis of Fractional Systems, ASME 2005 International Design Engineering Technical Conferences.
[40] Moze M, Sabatier J, and Oustaloup A. LMI Characterization Of Fractional Systems Stability[C], Sabatier J.et al (editors) Advances in Fractional Calculus: Theoretical Developments and Applicationsin Physics and Engineering, Springer Netherlands,2007: 419–434.
[41]陈宁,基于分数阶微积分理论的车辆动力学,东南大学博士学位论文.2009.15
[42] S. Saha Ray, R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method[J], Applied Mathematics and Computation 2005,167 : 561–571
[43] Varsha Daftardar-Gejji, Hossein Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations[J], J. Math. Anal. Appl. 301 (2005) 508–518
[44] Junsheng D, Jianye A, Mingyu X. Solution of System of Fractional Differential Equations by Adomian Decomposition Method[J], Appl. Math. J. Chinese Univ. Ser. B,2007, 22(1): 7-12
[45] Gaul L, Klein P, Kempfle S Impulse response function of an oscillator with fractional derivative in damping description[J]. Mech. Res. Commun. 1989,16(5):4447–4472.
[46] Machado JAT . Discrete-time fractional-order controllers[J]. FCAA J. 2001,4:47–66.
[47] Diethelm K, An algorithm for the numerical solution of differential equations of fractional order[J], Electron. Trans. Numer. Anal. 1997,5: 1–6.
[48] Luise Blank. Numerical Treatment of Differential Equations of Fractional Order[R], Manchester Center for Computational Mathematics Numerical Analysis Report No. 287,1996.
[49] Diethelm K, Ford NJ, Freed AD, et al. Algorithms For The Fractional Calculus: A Selection Of Numerical Methods[J], Comput. Methods Appl. Mech. Engrg. 2005,194: 743–773
[50] Diethelm K, Neville JF, Alan DF. A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations[J], Nonlinear Dynamics 2002,29: 3–22.
[51] Chien-Cheng T, Design of variable and adaptive fractional order FIR differentiators, Signal Processing, 2006,86:2554-2566,
[52] Chien-Cheng T, Design of FIR and IIR fractional order Simpson digital integrators, Signal Processing, 2007,87: 1045-1057,
[53] Barbosa RS,Tenreiro Machado JA, Silvaa MF. Time domain design of fractional differintegrators using least-squares[J], Signal Processing, Volume 86, Issue 10, October 2006, Pages 2567-2581
[54] Diethelm K, Ford NJ, Freed AD and et al. Algorithms for the fractional calculus: A selection of numerical methods[J], Computer Methods in Applied Mechanics and Engineering, 2005,194: 743-773
[55] Metzler R, Gl?ckle WG, Nonnenmacher TF. Fractional Model Equation for Anomalous Diffusion[J]. Physica A, 1994, 211(1):13–24.
[56] Wyss, W. The Fractional Diffusion Equation[J]. J. Math. Phys., 1986, 27(11): 1986
[57] Schnieder WR, Wyss W. Fractional Diffusion and Wave Equations[J]. J. Math. Phys.,1989,30(1): 134–144.
[58] Almeida LB. The fractional Fourier transform and time-frequency representations[J], Signal Processing, IEEE Transactions on1994,42: 3084-3091
[59] Levernhe, F. Montseny, G. Audounet, J. Markovian diffusive representation of 1/fαnoisesand application to fractional stochastic differential models[J], Signal Processing, IEEE Transactions on,2001,49:414-423
[60] B. Mandelbrot, Some noises with 1/ f spectrum, a bridge between direct current and white noise, IEEE Transactions on Information Theory, 1967(13): 289-298
[61] Ichise M, Nagayanagi Y, Kojima T. An Analog Simulation of Non-Integer Order Transfer Functions for Analysis of Electrode Processes[J]. J. Electroanal. Chem. Interfacial Electrochem., 1971,33(2): 253–265.
[62] Li CP, Peng CJ. Chaos in chen's system with a fractional order[J]. Chaos Solitons and Fractals, 2004,22:443-450.
[63] Tavazoei MS, Haeri M. Chaos control via a simple fractional-order controller[J], Physics Letters A, 2008, 372: 798-807.
[64]吴峥茂.非线性混沌系统分析和控制问题的研究[D],上海交通大学博士学位论文,2007
[65]林孔容.关于分数阶导数的几种不同定义的分析与比较.闽江学院学报.Vol.24 No.5 Oct.2003
[66] Zeng Qing-shan, Cao Guang-yi,Zhu Xin-jian.External stability of fractional-order control systems.Journal of HarbinInstitute of Technology,13(1):32-36,2006
[67] A.V.Hold.Chaos.Manchester University Press,1986
[68]陈奉苏.混沌学及其应用.中国电力出版社,1998
[69] B.L.Hao.Elementary symbolic dynamics and chaos in dissipative systems.Utopia Press,1989.
[70]吴祥兴,陈忠.混沌学导论.上海科学技术文献出版社,1997
[71]非线性动力学,刘秉正,彭建华.高等教育出版社,2004
[72] Schuster H.G.Deterministic chaos:an introduction.Weinheim:Physik-Verlag,1984.
[73] Hirsch M.W.,Smale S.Differential equations:dynamical systems and linear algebra.New York: Academic Press,1974.
[74] Li C G,Chen G R.Chaos and hyperchaos in the fractional-order Ro¨ssler equations.Physica A, 341:55–61,2004.
[75] Gao Xin,Yu Jue-Bang.Chaos and chaotic control in a fractional-order electronic oscillator.Chinese physics, 14(5): 908–913,2005.
[76] Lu J G,Chen G R.A note on the fractional-order chen system.Chaos,Solitons and Fractals 27:685–688,2006.
[77] Chunguang Li,Guanrong Chen, Chaos and hyperchaos in the fractional-order Rossler equations.Physica A 341 (2004) 55-61
[78] Jun Guo Lu.Chaotic dynamics and synchronization of fractional-order arneodo’s systems.Chaos,Solitons and Fractals, 26:1125–1133,2005.
[79] Hao Zhu,Shangbo Zhou,Jun Zhang.Chaos and synchronization of the fractional-order Chua’s system.Chaos,Solitons and Fractals 39 (2009) 1595-1603
[80] Lorenzo, C. F. and Hartley, T. T., Dynamics and Control of Initialized Fractional-Order Systems,2002(09):201-233
[81] Lorenzo, C. F. and Hartley, T. T.,‘Generalized functions for the fractional calculus’, NASA TP-1999-209424, 1999.
[82] Lorenzo, C. F. and Hartley, T. T.,‘R-function relationships for application in the fractional calculus’, NASA TM-2000-210361, 2000.
[83] T. T. Hartley, and C. F. Lorenzo,“Insights into the Initialization of Fractional Order Operators via emi-Infinite Lines”, NASA TM–1998-208407, Dec. 1998
[84] C. F. Lorenzo, and T. T. Hartley,“Initialization, Conceptualization, and Application in the Generalized Fractional Calculus”NASA TP-1998-208415, Dec. 1998
[85]刘宗华.混沌动力学基础及应用.高等教育出版社,2006
[86]尹万建.汽车空气弹簧悬架系统的非线性动力学行为研究[D].北京交通大学博士学位论文.2007
[87]翁建生.基于磁流变阻尼器的车辆悬架系统半主动控制[D].南京航空航天大学博士学位论文,2002
[88]于志生主编,汽车理论,机械工业出版社,1981
[89]台永鹏.基于分数阶微积分的车辆半主动悬架研究[M].南京林业大学硕士学位论文.2009
[90]郑玲,邓兆祥,李以农.汽车半主动悬架的滑模变结构控制.振动工程学报.Vol 16 No.4 Dec 2003
[91]段虎明,石峰,谢飞,张开斌.路面不平度研究综述.振动与冲击.Vol 28 No. 9 2009
[92]刘京津.基于滑模观测器的故障诊断技术及其在飞控系统中的应用研究.南京航空航天大学硕士论文.2008
[93]申忠宇,顾幸生,赵瑾.基于奇异值分解的鲁棒滑模观测器设计与仿真.系统仿真学报.Vol.21 No.3 Feb,2009
[94]李会艳,王江,胡龙根.基于滑模观测器的故障诊断及在主动悬架系统中的应用.上海海运学院学报.Vol.22 No.3Sep.2001
[95] Edwards C. Spurgeon S K. On the development of discontinuous observers [J]. International Journal of Control.1994,59(5):1211-1229
[96] Tan P C, Edwards C. An LMI approach for designing sliding mode observers[A]. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney[C]. Australia: IEEE,2000.2587-2592
[97] D.Karnopp, M.J.Crosby, R.A.Harwood.“Vibration control Using Semi-active force generators”. Transactions of ASME(1974):619-626
[98] Ning Chen, Yongpeng Tai, Application of Fractional Order Skyhook Damping Control on Semi-Active Suspension. 2008 International Pre-Olympic Workshop on Modelling and Simulation,Nanjin,China,2008.
[2] Lacroix S F, Traitfédu Calcul Différentiel et du Calcul Intégral[M], Courcier, Paris, t. 3, 1819; pp. 409-410
[3] R.C.Koeller, Polynomial operators, stieltjes convolution, and fractional calculus in hereditary mechanics, Acta Mechanica,1986(58):251-264
[4] H.H.Sun, A.A.Abdelwahab and B.Onaral, Linear approximation of transfer function with a pole of fractional order, IEEE transactions on Automatic Control,1984(29):441-444
[5] O.Heaviside, Electromagetic Theory, Chelsea: New York,1971
[6] B.Ross.Fractional Calculus and its Applications.Berlin:Springer Verlag,1975
[7] K.B.Oldham, J.Spanier. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press,1974
[8] B.B.Mandelbort.The Fractal Geometry of Nature.New York:W.H.Freeman and Company,1982
[9]高安秀树.分数维.北京:地震出版社,1989
[10]王东生,曹磊.混沌、分形及应用.合肥:中国科学技术出版社,1995
[11] S.G.Samko,A.A.Kilbas,O.I.Marichev. Integral and Derivatives of Fractional Order and Some of Their Applications. Minsk: Naukai Tekhnika,1987
[12] S.G.Samko,A.A.Kilbas,O.I.Marichev. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach,1993
[13] K.S.Miller,B.Boss. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley-Interscience Publication,1993
[14] I.Podlubny. Fractional Differential Equations. San Diego: Academic Press,1999
[15] R.R.Nishimoto. Fractional Calculus(Integration and Differentiation of Arbitrary Order)Vol 2.Koriyama:Descartes Press Cor.,1987
[16]曾庆山,曹广益,王振滨.分数阶PIλDμ控制器的仿真研究.系统仿真学报.Vol.16 No.3.2004
[17]刘进英,李文.分数阶PIλDμ控制器的设计.自动控制与检测
[18] Podlubny I.Fractional differential equations[M].New York:Academic Press,1999
[19] Oldham K B,Spanier J. The Fractional Calculus[M].New York:Academic,1974
[20] Yongguang Yu,Han-Xiong Li.The synchronization of fractional-order Rossler hyperchaotic systems.Physica A 387(2008)1393-1403
[21] Chuanguang Li,Guanrong Chen.Chaos and hyperchaos in the fractional-order Rossler equations.Physica A341 (2004)55-61
[22]陈宁,台永鹏,陈南.分数微积分理论在非线性车辆悬架滑模控制中的应用,动力学与控制学报.Vol.7 No.3 Sep.2009
[23] Schuster H.G.Deterministic chaos: an introduction. Weinheim:Physik-Verlag,1984.
[24] Hirsch M.W., Smale S.Differential equations: dynamical systems and linear algebra. New York: Academic Press, 1974.
[25] Bode HW, Network Analysis and Feedback Amplifier Design[M], Van Nostrand, New York,1945
[26] Tustin A, Allanson JT, Layton JM,et al, The design of systems for automatic control of the position of massive object[R]. Proceedings of Institution of Electrical Engineers, 105, Part C, Suppl. 1958,No.1: 1-57
[27] Carlson Gordon E., C. A. Halijak (1961), Simulation of the fractional derivative operator s1/2 and the fractional integral operator 1/s1/2[J], in: Proc. Of the CSSCM, Kansas State U. B., 1961,45(7):1-22
[28] Carlson GE and. Halijak, CA. Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process[J], IEEE Transactions on Circuit Theory,1964,11(2): 210–213.
[29] Oustaloup A. La commande CRONE. Editions Hermès, Paris, (1991).
[30] Oustaloup A. La dérivation non entière: théorie,synthèse et applications[M]. Hermès, Paris. 1995.
[31] Oustaloup, A, Moreau, X, Nouillant, M, The CRONE suspension [J], Control Engineering Practice,1996. Vol 4(8):1101-1108
[32] Moreau X, Nouillant C, Oustaloup A, Effect Of The Crone Suspension Control System On Braking[C], In: Workshop IFAC On Advances in Automotive Control, Karlsruhe, Germany,2001 : 63-68.
[33] Pommier V,Sabatier J, Lanusse P,et al, Crone Control Of A Nonlinear Hydraulic Actuator[J], Control Engineering Practice 2002,10 :391–402 .
[34] Altet O, Moreau X, Moze M, and et al. Principles And Synthesis Of Hydractive Crone Suspension[J], Nonlinear Dynamics 38: 435–459, 2004.
[35] Sabatier J, Oustaloup A, Iturricha A.G, and Lanusse P, CRONE Control: Principles and Extension to Time-Variant Plants with Asymptotically Constant Coefficients[J], Nonlinear Dynamics 2002,29: 363–385.
[36] Matignon D, Stability result on fractional differential equations with applications to control processing[J], in: IMACS-SMC Proceedings, Lille, France, 1996:963-968
[37] Matignon D. (1998), Stability properties for generalized fractional differential systems[C],in: ESIAM Proceedings on Fractional Differential Systems 5, 1998: 145-158
[38] Podlubny I, Fractional-order systems and PIlDm-controllers[J], IEEE Transactions on Automatic Control 1999,44(1) :208–214
[39] Moze, M., J. Sabatier, A. Oustaloup, LMI Tools for Stability Analysis of Fractional Systems, ASME 2005 International Design Engineering Technical Conferences.
[40] Moze M, Sabatier J, and Oustaloup A. LMI Characterization Of Fractional Systems Stability[C], Sabatier J.et al (editors) Advances in Fractional Calculus: Theoretical Developments and Applicationsin Physics and Engineering, Springer Netherlands,2007: 419–434.
[41]陈宁,基于分数阶微积分理论的车辆动力学,东南大学博士学位论文.2009.15
[42] S. Saha Ray, R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method[J], Applied Mathematics and Computation 2005,167 : 561–571
[43] Varsha Daftardar-Gejji, Hossein Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations[J], J. Math. Anal. Appl. 301 (2005) 508–518
[44] Junsheng D, Jianye A, Mingyu X. Solution of System of Fractional Differential Equations by Adomian Decomposition Method[J], Appl. Math. J. Chinese Univ. Ser. B,2007, 22(1): 7-12
[45] Gaul L, Klein P, Kempfle S Impulse response function of an oscillator with fractional derivative in damping description[J]. Mech. Res. Commun. 1989,16(5):4447–4472.
[46] Machado JAT . Discrete-time fractional-order controllers[J]. FCAA J. 2001,4:47–66.
[47] Diethelm K, An algorithm for the numerical solution of differential equations of fractional order[J], Electron. Trans. Numer. Anal. 1997,5: 1–6.
[48] Luise Blank. Numerical Treatment of Differential Equations of Fractional Order[R], Manchester Center for Computational Mathematics Numerical Analysis Report No. 287,1996.
[49] Diethelm K, Ford NJ, Freed AD, et al. Algorithms For The Fractional Calculus: A Selection Of Numerical Methods[J], Comput. Methods Appl. Mech. Engrg. 2005,194: 743–773
[50] Diethelm K, Neville JF, Alan DF. A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations[J], Nonlinear Dynamics 2002,29: 3–22.
[51] Chien-Cheng T, Design of variable and adaptive fractional order FIR differentiators, Signal Processing, 2006,86:2554-2566,
[52] Chien-Cheng T, Design of FIR and IIR fractional order Simpson digital integrators, Signal Processing, 2007,87: 1045-1057,
[53] Barbosa RS,Tenreiro Machado JA, Silvaa MF. Time domain design of fractional differintegrators using least-squares[J], Signal Processing, Volume 86, Issue 10, October 2006, Pages 2567-2581
[54] Diethelm K, Ford NJ, Freed AD and et al. Algorithms for the fractional calculus: A selection of numerical methods[J], Computer Methods in Applied Mechanics and Engineering, 2005,194: 743-773
[55] Metzler R, Gl?ckle WG, Nonnenmacher TF. Fractional Model Equation for Anomalous Diffusion[J]. Physica A, 1994, 211(1):13–24.
[56] Wyss, W. The Fractional Diffusion Equation[J]. J. Math. Phys., 1986, 27(11): 1986
[57] Schnieder WR, Wyss W. Fractional Diffusion and Wave Equations[J]. J. Math. Phys.,1989,30(1): 134–144.
[58] Almeida LB. The fractional Fourier transform and time-frequency representations[J], Signal Processing, IEEE Transactions on1994,42: 3084-3091
[59] Levernhe, F. Montseny, G. Audounet, J. Markovian diffusive representation of 1/fαnoisesand application to fractional stochastic differential models[J], Signal Processing, IEEE Transactions on,2001,49:414-423
[60] B. Mandelbrot, Some noises with 1/ f spectrum, a bridge between direct current and white noise, IEEE Transactions on Information Theory, 1967(13): 289-298
[61] Ichise M, Nagayanagi Y, Kojima T. An Analog Simulation of Non-Integer Order Transfer Functions for Analysis of Electrode Processes[J]. J. Electroanal. Chem. Interfacial Electrochem., 1971,33(2): 253–265.
[62] Li CP, Peng CJ. Chaos in chen's system with a fractional order[J]. Chaos Solitons and Fractals, 2004,22:443-450.
[63] Tavazoei MS, Haeri M. Chaos control via a simple fractional-order controller[J], Physics Letters A, 2008, 372: 798-807.
[64]吴峥茂.非线性混沌系统分析和控制问题的研究[D],上海交通大学博士学位论文,2007
[65]林孔容.关于分数阶导数的几种不同定义的分析与比较.闽江学院学报.Vol.24 No.5 Oct.2003
[66] Zeng Qing-shan, Cao Guang-yi,Zhu Xin-jian.External stability of fractional-order control systems.Journal of HarbinInstitute of Technology,13(1):32-36,2006
[67] A.V.Hold.Chaos.Manchester University Press,1986
[68]陈奉苏.混沌学及其应用.中国电力出版社,1998
[69] B.L.Hao.Elementary symbolic dynamics and chaos in dissipative systems.Utopia Press,1989.
[70]吴祥兴,陈忠.混沌学导论.上海科学技术文献出版社,1997
[71]非线性动力学,刘秉正,彭建华.高等教育出版社,2004
[72] Schuster H.G.Deterministic chaos:an introduction.Weinheim:Physik-Verlag,1984.
[73] Hirsch M.W.,Smale S.Differential equations:dynamical systems and linear algebra.New York: Academic Press,1974.
[74] Li C G,Chen G R.Chaos and hyperchaos in the fractional-order Ro¨ssler equations.Physica A, 341:55–61,2004.
[75] Gao Xin,Yu Jue-Bang.Chaos and chaotic control in a fractional-order electronic oscillator.Chinese physics, 14(5): 908–913,2005.
[76] Lu J G,Chen G R.A note on the fractional-order chen system.Chaos,Solitons and Fractals 27:685–688,2006.
[77] Chunguang Li,Guanrong Chen, Chaos and hyperchaos in the fractional-order Rossler equations.Physica A 341 (2004) 55-61
[78] Jun Guo Lu.Chaotic dynamics and synchronization of fractional-order arneodo’s systems.Chaos,Solitons and Fractals, 26:1125–1133,2005.
[79] Hao Zhu,Shangbo Zhou,Jun Zhang.Chaos and synchronization of the fractional-order Chua’s system.Chaos,Solitons and Fractals 39 (2009) 1595-1603
[80] Lorenzo, C. F. and Hartley, T. T., Dynamics and Control of Initialized Fractional-Order Systems,2002(09):201-233
[81] Lorenzo, C. F. and Hartley, T. T.,‘Generalized functions for the fractional calculus’, NASA TP-1999-209424, 1999.
[82] Lorenzo, C. F. and Hartley, T. T.,‘R-function relationships for application in the fractional calculus’, NASA TM-2000-210361, 2000.
[83] T. T. Hartley, and C. F. Lorenzo,“Insights into the Initialization of Fractional Order Operators via emi-Infinite Lines”, NASA TM–1998-208407, Dec. 1998
[84] C. F. Lorenzo, and T. T. Hartley,“Initialization, Conceptualization, and Application in the Generalized Fractional Calculus”NASA TP-1998-208415, Dec. 1998
[85]刘宗华.混沌动力学基础及应用.高等教育出版社,2006
[86]尹万建.汽车空气弹簧悬架系统的非线性动力学行为研究[D].北京交通大学博士学位论文.2007
[87]翁建生.基于磁流变阻尼器的车辆悬架系统半主动控制[D].南京航空航天大学博士学位论文,2002
[88]于志生主编,汽车理论,机械工业出版社,1981
[89]台永鹏.基于分数阶微积分的车辆半主动悬架研究[M].南京林业大学硕士学位论文.2009
[90]郑玲,邓兆祥,李以农.汽车半主动悬架的滑模变结构控制.振动工程学报.Vol 16 No.4 Dec 2003
[91]段虎明,石峰,谢飞,张开斌.路面不平度研究综述.振动与冲击.Vol 28 No. 9 2009
[92]刘京津.基于滑模观测器的故障诊断技术及其在飞控系统中的应用研究.南京航空航天大学硕士论文.2008
[93]申忠宇,顾幸生,赵瑾.基于奇异值分解的鲁棒滑模观测器设计与仿真.系统仿真学报.Vol.21 No.3 Feb,2009
[94]李会艳,王江,胡龙根.基于滑模观测器的故障诊断及在主动悬架系统中的应用.上海海运学院学报.Vol.22 No.3Sep.2001
[95] Edwards C. Spurgeon S K. On the development of discontinuous observers [J]. International Journal of Control.1994,59(5):1211-1229
[96] Tan P C, Edwards C. An LMI approach for designing sliding mode observers[A]. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney[C]. Australia: IEEE,2000.2587-2592
[97] D.Karnopp, M.J.Crosby, R.A.Harwood.“Vibration control Using Semi-active force generators”. Transactions of ASME(1974):619-626
[98] Ning Chen, Yongpeng Tai, Application of Fractional Order Skyhook Damping Control on Semi-Active Suspension. 2008 International Pre-Olympic Workshop on Modelling and Simulation,Nanjin,China,2008.