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Control of a class of fractional-order chaotic systems via sliding mode. (English) Zbl 1242.93027
Summary: This paper investigates the chaos control of a class of fractional-order chaotic systems via sliding mode. First, the sliding mode control law is derived to make the states of the fractional-order chaotic systems asymptotically stable. Second, the designed control scheme guarantees asymptotical stability of the uncertain fractional-order chaotic systems in the presence of an external disturbance. Finally, simulation results are given to demonstrate the effectiveness of the proposed sliding mode control method.

93B12Variable structure systems
93C15Control systems governed by ODE
34H10Chaos control (ODE)
34A08Fractional differential equations
Full Text: DOI
[1] Liu, Y.J.: Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system. Nonlinear Dyn. (2011). doi: 10.1007/s11071-011-9960-2 · doi:10.1007/s11071-010-9906-0
[2] Liu, Y.J., Yang, Q.G.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Anal., Real World Appl. 11, 2563--2572 (2010) · Zbl 1202.34083 · doi:10.1016/j.nonrwa.2009.09.001
[3] Harb, A.M., Abdel-Jabbar, N.: Controlling Hopf bifurcation and chaos in a small power system. Chaos Solitons Fractals 18, 1055--1063 (2003) · Zbl 1074.93522 · doi:10.1016/S0960-0779(03)00073-0
[4] Ditto, W.L.: Applications of chaos in biology and medicine. Chaos Chang. Nat. Sci. Med. 376, 175--202 (1996)
[5] Ma, J., Wang, C.N., Tang, J., Xia, Y.F.: Suppression of the spiral wave and turbulence in the excitability-modulated media. Int. J. Theor. Phys. 48, 150--157 (2009) · doi:10.1007/s10773-008-9790-2
[6] Lamba, P., Hudson, J.L.: Experiments on bifurcations to chaos in a forced chemical reactor. Chem. Eng. Sci. 42, 1--8 (1987) · doi:10.1016/0009-2509(87)80203-8
[7] Ross, B.: The development of fractional calculus 1695--1900. Hist. Math. 4, 75--89 (1977) · Zbl 0358.01008 · doi:10.1016/0315-0860(77)90039-8
[8] He, G.L., Zhou, S.P.: What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact box dimension. Chaos Solitons Fractals 26, 867--879 (2005) · Zbl 1072.37021 · doi:10.1016/j.chaos.2005.01.041
[9] Jumarie, G.: Fractional master equation: non-standard analysis and Liouville--Riemann derivative. Chaos Solitons Fractals 12, 2577--2587 (2001) · Zbl 0994.82062 · doi:10.1016/S0960-0779(00)00218-6
[10] Elwakil, S.A., Zahran, M.A.: Fractional integral representation of master equation. Chaos Solitons Fractals 10, 1545--1558 (1999) · Zbl 0988.82039 · doi:10.1016/S0960-0779(98)00176-3
[11] El-Misiery, A.E.M., Ahmed, E.: On a fractional model for earthquakes. Appl. Math. Comput. 178, 207--211 (2006) · Zbl 1130.86309 · doi:10.1016/j.amc.2005.10.011
[12] Bagley, R.L., Calico, R.A.: Fractional-order state equations for the control of viscoelastically damped structures. Guid. Control Dyn. 14, 304--311 (1991) · doi:10.2514/3.20641
[13] El-Sayed, A.M.A.: Fractional-order diffusion-wave equation. Int. J. Theor. Phys. 35, 311--322 (1996) · Zbl 0846.35001 · doi:10.1007/BF02083817
[14] Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44, 554--566 (1996) · Zbl 0944.78506 · doi:10.1109/8.489308
[15] Lazopoulos, K.A.: Non-local continuum mechanics and fractional calculus. Mech. Res. Commun. 33, 753--757 (2006) · Zbl 1192.74010 · doi:10.1016/j.mechrescom.2006.05.001
[16] Wu, X.J., Li, J., Chen, G.R.: Chaos in the fractional order unified system and its synchronization. J. Franklin Inst. 345, 392--401 (2008) · Zbl 1166.34030 · doi:10.1016/j.jfranklin.2007.11.003
[17] Ge, Z.M., Ou, C.Y.: Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals 35, 705--717 (2008) · doi:10.1016/j.chaos.2006.05.101
[18] Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Phys. A, Stat. Mech. Appl. 387, 57--70 (2008) · doi:10.1016/j.physa.2007.08.039
[19] Matouk, A.E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol--Duffing circuit. Commun. Nonlinear Sci. Numer. Simul. 16, 975--986 (2011) · Zbl 1221.93227 · doi:10.1016/j.cnsns.2010.04.027
[20] Delavari, H., Ghaderi, R., Ranjbar, A., Momani, S.: Fuzzy fractional order sliding mode controller for nonlinear systems. Commun. Nonlinear Sci. Numer. Simul. 15, 963--978 (2010) · Zbl 1221.93140 · doi:10.1016/j.cnsns.2009.05.025
[21] Wu, X.J., Lu, H.T., Shen, S.L.: Synchronization of a new fractional-order hyperchaotic system. Phys. Lett. A 373, 2329--2337 (2009) · Zbl 1231.34091 · doi:10.1016/j.physleta.2009.04.063
[22] Wang, J.R., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262--272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[23] Zhang, R.X., Yang, S.P.: Designing synchronization schemes for a fractional-order hyperchaotic system. Acta Phys. Sin. 57, 6837--6843 (2008) · Zbl 1199.37107
[24] Mophou, G.M.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61, 68--78 (2011) · Zbl 1207.49006 · doi:10.1016/j.camwa.2010.10.030
[25] Wang, X.Y., He, Y.J., Wang, M.J.: Chaos control of a fractional order modified coupled dynamos system. Nonlinear Anal. 71, 6126--6134 (2009) · Zbl 1187.34080 · doi:10.1016/j.na.2009.06.065
[26] 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
[27] Zheng, Y.G., Nian, Y.B., Wang, D.J.: Controlling fractional order chaotic systems based on Takagi--Sugeno fuzzy model and adaptive adjustment mechanism. Phys. Lett. A 375, 125--129 (2010) · Zbl 1241.34088 · doi:10.1016/j.physleta.2010.10.038
[28] Dadras, S., Momeni, H.R.: Control of a fractional-order economical system via sliding mode. Phys. A, Stat. Mech. Appl. 389, 2434--2442 (2010) · doi:10.1016/j.physa.2010.02.025
[29] Deng, W.H.: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal., Real World Appl. 72, 1768--1777 (2009) · Zbl 1182.26009
[30] Asheghan, M.M., Beheshti, M.T.H., Tavazoei, M.S.: Robust synchronization of perturbed Chen’s fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16, 1044--1051 (2011) · Zbl 1221.34007 · doi:10.1016/j.cnsns.2010.05.024
[31] Peng, G., Jiang, Y.: Two routes to chaos in the fractional Lorenz system with dimension continuously varying. Phys. A, Stat. Mech. Appl. 389, 4140--4148 (2010) · doi:10.1016/j.physa.2010.05.037
[32] Yang, Q.G., Zeng, C.B.: Chaos in fractional conjugate Lorenz system and its scaling attractors. Commun. Nonlinear Sci. Numer. Simul. 15, 4041--4051 (2010) · Zbl 1222.37037 · doi:10.1016/j.cnsns.2010.02.005