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A new method based on least square support vector machines (LS-SVM) is proposed for chaos control of fractional order system. Based on the stability theory of fractional order linear system, the system is decomposed into stable linear parts and the corresponding nonlinear parts. The active controller is designed to compensate the nonlinear parts by using the excellent nonlinearity approximation ability and better generalization capacity of LS-SVM. Thus fractional order chaotic system is suppressed to the equilibrium point. Fractional order Liu system and fractional order Chen system are illustrated respectively. The simulation results verify the effectiveness and feasibility of the proposed method.
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Keywords:
- fractional order /
- chaos control /
- support vector machines /
- least square support vector machines
[1] ［1］Bagley R L, Calico R A 1991 J. Guid. Control Dyn. 14 304
[2] ［2］Sun H H, Abdelwahad A A, Onaral B 1984 IEEE Trans.Automat. Control. 29 441
[3] ［3］Ichise M, Nagayanagi Y, Kojima T 1971 J. Electro-Anal.Chem. 33 253
[4] ［4］Li C P, Peng G J 2004 Chaos Soliton. Fract. 22 443
[5] ［5］Grigorenko I, Grigorenko E 2003 Phys. Rev. Lett. 91 034101
[6] ［6］Ge Z M, Ou C Y 2007 Chaos Soliton. Fract. 34 262
[7] ［7］Lu J G 2006 Phys. Lett. A 354 305
[8] ［8］Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) ［刘崇新 2007 物理学报 56 6865］
[9] ［9］Li C G, Chen G R 2004 Chaos Soliton. Fract. 22 549
[10] ］Gao X, Yu J B 2005 Chin. Phys. 14 908
[11] ］Zhang C F, Gao J F, Xu L 2007 Acta Phys. Sin. 56 5124 (in Chinese) ［张成芬、高金峰、徐磊 2007 物理学报 56 5124］
[12] ］Zhong Q S, Bao J F, Yu Y B, Liao X F 2009 Chin. Phys. Lett. 25 2812
[13] ］Vapnik V N 2000 The nature of statistical learning theory (2nd ed) (New York: Springer-Vedag) p125
[14] ］Suykens J A K, Vandewalle J 1999 Int. J. Circ. Theor. Appl. 27 605
[15] ］Ye M Y 2005 Acta Phys. Sin. 54 30 (in Chinese) ［叶美盈 2005 物理学报 54 30 ］
[16] ］Liu H, Liu D, Ren H P 2005 Acta Phys. Sin. 54 4019 (in Chinese) ［刘涵、刘丁、任海鹏 2005 物理学报 54 4019］
[17] ］Podlubny I 1999 Fractional differential equations (1st ed) (New York: Academic Press) p41
[18] ］Matignon D 1996 IMACS- SMC Proceedings, Lille, France, July, 1996 p963
[19] ］Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542
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[1] ［1］Bagley R L, Calico R A 1991 J. Guid. Control Dyn. 14 304
[2] ［2］Sun H H, Abdelwahad A A, Onaral B 1984 IEEE Trans.Automat. Control. 29 441
[3] ［3］Ichise M, Nagayanagi Y, Kojima T 1971 J. Electro-Anal.Chem. 33 253
[4] ［4］Li C P, Peng G J 2004 Chaos Soliton. Fract. 22 443
[5] ［5］Grigorenko I, Grigorenko E 2003 Phys. Rev. Lett. 91 034101
[6] ［6］Ge Z M, Ou C Y 2007 Chaos Soliton. Fract. 34 262
[7] ［7］Lu J G 2006 Phys. Lett. A 354 305
[8] ［8］Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) ［刘崇新 2007 物理学报 56 6865］
[9] ［9］Li C G, Chen G R 2004 Chaos Soliton. Fract. 22 549
[10] ］Gao X, Yu J B 2005 Chin. Phys. 14 908
[11] ］Zhang C F, Gao J F, Xu L 2007 Acta Phys. Sin. 56 5124 (in Chinese) ［张成芬、高金峰、徐磊 2007 物理学报 56 5124］
[12] ］Zhong Q S, Bao J F, Yu Y B, Liao X F 2009 Chin. Phys. Lett. 25 2812
[13] ］Vapnik V N 2000 The nature of statistical learning theory (2nd ed) (New York: Springer-Vedag) p125
[14] ］Suykens J A K, Vandewalle J 1999 Int. J. Circ. Theor. Appl. 27 605
[15] ］Ye M Y 2005 Acta Phys. Sin. 54 30 (in Chinese) ［叶美盈 2005 物理学报 54 30 ］
[16] ］Liu H, Liu D, Ren H P 2005 Acta Phys. Sin. 54 4019 (in Chinese) ［刘涵、刘丁、任海鹏 2005 物理学报 54 4019］
[17] ］Podlubny I 1999 Fractional differential equations (1st ed) (New York: Academic Press) p41
[18] ］Matignon D 1996 IMACS- SMC Proceedings, Lille, France, July, 1996 p963
[19] ］Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542
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