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.


93B12 Variable structure systems
34A08 Fractional ordinary differential equations
37N35 Dynamical systems in control
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI


[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) · Zbl 0948.93043
[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
[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
[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) · Zbl 1234.37055
[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
[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
[8] Salarieh, H.; Alasty, A., Control of stochastic chaos using sliding mode method, J Comput Appl Math, 225, 135-145 (2009) · Zbl 1162.65062
[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
[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
[11] Yin, C.; Zhong, S.; Chen, W., Robust \(H_\infty\) 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
[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
[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
[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
[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
[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
[18] Tavazoei, M. S.; Haeri, M., Chaos control via a simple fractional-order controller, Phys Lett A, 372, 798-807 (2008) · Zbl 1217.70022
[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
[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, 15, 665-674 (2010) · Zbl 1221.37222
[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
[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
[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
[26] Farges, C.; Moze, M.; Sabatier, J., Pseudo-state feedback stabilization of commensurate fractional order systems, Automatica, 46, 1730-1734 (2010) · Zbl 1204.93094
[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
[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
[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
[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
[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, 59, 1637-1643 (2010) · Zbl 1189.34011
[34] Monje, C. A.; Chen, Y.; Vinagre, B. M.; Xue, D.; Feliu, V., Fractional-order systems and controls (2010), Springer · Zbl 1211.93002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.