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Design of sliding mode controller for a class of fractional-order chaotic systems. (English) Zbl 1248.93041
Summary: In this paper, a sliding mode control law is designed to control chaos in a class of fractional-order chaotic systems. A class of unknown fractional-order systems is introduced. Based on the sliding mode control method, the states of the fractional-order system have been established, even if the system with uncertainty is subjected to external disturbances. In addition, chaos control is implemented in the fractional-order Chen system, the fractional-order Lorenz system, and the same to the fractional-order financial system by utilizing this method. Effectiveness of the proposed control scheme is illustrated through numerical simulations.

MSC:
93B12Variable structure systems
34A08Fractional differential equations
37N35Dynamical systems in control
93C15Control systems governed by ODE
References:
[1]Konishi, K.; Hirai, M.; Kokame, H.: Sliding mode control for a class of chaotic systems, Phys lett A 245, 511-517 (1998)
[2]Yang, G.; Wei, S.; Wei, J.; Zeng, J.; Wu, Z.; Sun, G.: Stabilization of unstable periodic orbits for a chaotic system, Systems control lett 38, 21-26 (1999)
[3]Gouaisbaut, F.; Dambrine, M.; Richard, J. P.: Robust control of delay systems: a sliding mode control design via LMI, Syst control lett 46, 219-230 (2002) · Zbl 0994.93004 · doi:10.1016/S0167-6911(01)00199-2
[4]Yau, H. T.; Yan, J. J.: Design of sliding mode controller for Lorenz chaotic system with nonlinear input, Chaos soliton fract 19, 891-898 (2004) · Zbl 1064.93010 · doi:10.1016/S0960-0779(03)00255-8
[5]Guo, H.; Lin, S.; Liu, J.: A radial basis function sliding mode controller for chaotic Lorenz system, Phys lett A 351, 257-261 (2006)
[6]Laghrouche, S.; Plestan, F.; Glumineau, A.: Higher order sliding mode control based on integral sliding mode, Automatica 43, 531-537 (2007) · Zbl 1137.93338 · doi:10.1016/j.automatica.2006.09.017
[7]Layeghi, H.; Arjmand, M. T.; Salarieh, H.; Alasty, A.: Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control, Chaos soliton fract 37, 1125-1135 (2008) · Zbl 1153.37352 · doi:10.1016/j.chaos.2006.10.021
[8]Salarieh, H.; Alasty, A.: Control of stochastic chaos using sliding mode method, J comput appl math 225, 135-145 (2009) · Zbl 1162.65062 · doi:10.1016/j.cam.2008.07.032
[9]Roopaei, M.; Sahraei, B. R.; Lin, T. C.: Adaptive sliding mode control in a novel class of chaotic systems, Commun nonlinear sci numer simulat 15, 4158-4170 (2010) · Zbl 1222.93124 · doi:10.1016/j.cnsns.2010.02.017
[10]Yin, C.; Zhong, S.; Chen, W.: Design PD controller for master-slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities, Commun nonlinear sci numer simulat 16, 1632-1639 (2011) · Zbl 1221.93112 · doi:10.1016/j.cnsns.2010.05.031
[11]Yin, C.; Zhong, S.; Chen, W.: Robust H control for uncertain Lur’e systems with sector and slope restricted nonlinearities by PD state feedback, Nonlinear anal real world appl 12, 501-512 (2011) · Zbl 1203.93166 · doi:10.1016/j.nonrwa.2010.06.035
[12]Wang, J.; Xiong, X.; Zhang, Y.: Extending synchronization scheme to chaotic fractional-order Chen systems, Phys A 370, 279-285 (2006)
[13]Lu, J. G.; Chen, G.: A note on the fractional-order Chen system, Chaos soliton fract 27, 685-688 (2006) · Zbl 1101.37307 · doi:10.1016/j.chaos.2005.04.037
[14]Asheghan, M. M.; Beheshti, M. T. H.; Tavazoei, M. S.: Robust synchronization of perturbed Chen’s fractional-order chaotic systems, Commun nonlinear sci numer simulat 16, 1044-1051 (2011) · Zbl 1221.34007 · doi:10.1016/j.cnsns.2010.05.024
[15]Alomari, A. K.; Noorani, M. S. M.; Nazar, R.; Li, C. P.: Homotopy analysis method for solving fractional Lorenz system, Commun nonlinear sci numer simulat 15, 1864-1872 (2010) · Zbl 1222.65082 · doi:10.1016/j.cnsns.2009.08.005
[16]Ahmad, W. M.; El-Khazali, R.; Al-Assaf, Y.: Stabilization of generalized fractional order chaotic systems using state feedback control, Chaos soliton fract 22, 141-150 (2004) · Zbl 1060.93515 · doi:10.1016/j.chaos.2004.01.018
[17]Chen, Y. Q.; Ahn, H. S.; Xue, D.: Robust controllability of interval fractional order linear time invariant systems, Signal process 86, 2794-2802 (2006) · Zbl 1172.94386 · doi:10.1016/j.sigpro.2006.02.021
[18]Tavazoei, M. S.; Haeri, M.: Chaos control via a simple fractional-order controller, Phys lett A 372, 798-807 (2008) · Zbl 1217.70022 · doi:10.1016/j.physleta.2007.08.040
[19]Peng, G.; Jiang, Y. Q.: Two routes to chaos in the fractional Lorenz system with dimension continuously varying, Phys A 389, 4140-4148 (2010)
[20]Luo, Y.; Chen, Y. Q.; Ahn, H. S.; Pi, Youguo: Fractional order robust control for cogging effect compensation in PMSM position servo systems: stability analysis and experiments, Control eng practice 18, 1022-1036 (2010)
[21]Tricaud, C.; Chen, Y.: An approximate method for numerically solving fractional order optimal control problems of general form, Comput math appl 59, 1644-1655 (2010) · Zbl 1189.49045 · doi:10.1016/j.camwa.2009.08.006
[22]Shahiri, M.; Ghaderi, R.; Ranjbar, N. A.; Hosseinnia, S. H.; Momani, S.: Chaotic fractional-order coullet system: synchronization and control approach, Commun nonlinear sci numer simulat, No. 15, 665-674 (2010) · Zbl 1221.37222 · doi:10.1016/j.cnsns.2009.05.054
[23]Pan, L.; Zhou, W.; Fang, Jian’an; Li, Dequan: Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control, Commun nonlinear sci numer simulat 15, 3754-3762 (2010) · Zbl 1222.34063 · doi:10.1016/j.cnsns.2010.01.025
[24]Hu, J.; Han, Y.; Zhao, L.: Synchronizing chaotic systems using control based on a special matrix structure and extending to fractional chaotic systems, Commun nonlinear sci numer simulat 15, 115-123 (2010) · Zbl 1221.37212 · doi:10.1016/j.cnsns.2009.03.017
[25]Jesus, I. S.; Machado, J. A. T.; Barbosa, R. S.: Control of a heat diffusion system through a fractional order nonlinear algorithm, Comput math appl 59, 1687-1694 (2010) · Zbl 1189.93047 · doi:10.1016/j.camwa.2009.08.010
[26]Farges, C.; Moze, M.; Sabatier, J.: Pseudo-state feedback stabilization of commensurate fractional order systems, Automatica 46, 1730-1734 (2010) · Zbl 1204.93094 · doi:10.1016/j.automatica.2010.06.038
[27]Zhou, P.; Zhu, W.: Function projective synchronization for fractional-order chaotic systems, Nonlinear anal real world appl 12, 811-816 (2011) · Zbl 1209.34065 · doi:10.1016/j.nonrwa.2010.08.008
[28]Matouk, A. E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol-Duffing circuit, Commun nonlinear sci numer simulat 16, 975-986 (2011) · Zbl 1221.93227 · doi:10.1016/j.cnsns.2010.04.027
[29]Balochian, S.; Sedigh, A. K.; Zare, A.: Variable structure control of linear time invariant fractional order systems using a finite number of state feedback law, Commun nonlinear sci numer simulat 16, 1433-1442 (2011) · Zbl 1221.93041 · doi:10.1016/j.cnsns.2010.06.030
[30]Tavazoei, Mo.S.; Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller, Phys A 387, 57-70 (2008)
[31]Si-Ammour, A.; Djennoune, S.; Bettayeb, M.: A sliding mode control for linear fractional systems with input and state delays, Commun nonlinear sci numer simulat 14, 2310-2318 (2009) · Zbl 1221.93048 · doi:10.1016/j.cnsns.2008.05.011
[32]Dadras, S.; Momeni, H. R.: Control of a fractional-order economical system via sliding mode, Phys A 389, 2434-2442 (2010)
[33]Hosseinnia, S. H.; Ghaderi, R.; Ranjbar, N. A.; Mahmoudian, M.; Momani, S.: Sliding mode synchronization of an uncertain fractional order chaotic system, Comput math appl, No. 59, 1637-1643 (2010) · Zbl 1189.34011 · doi:10.1016/j.camwa.2009.08.021
[34]Monje, C. A.; Chen, Y.; Vinagre, B. M.; Xue, D.; Feliu, V.: Fractional-order systems and controls, (2010)