LMI-based stabilization of a class of fractional-order chaotic systems. (English) Zbl 1268.93121
Summary: Based on the theory of stabilization of fractional-order LTI interval systems, a simple controller for stabilization of a class of fractional-order chaotic systems is proposed in this paper. We consider the structure of the chaotic systems as fractional-order LTI interval systems due to the limited amplitude of chaotic trajectories. We introduce a simple feedback controller for the interval system and then, based on a recently established theorem for stabilization of interval systems, we reach to a linear matrix inequality (LMI) problem. Solving the LMI yields an appropriate decoupling feedback control law which suffices to bring the chaotic trajectories to the origin. Several illustrative examples are given which show the effectiveness of the method.
MSC:
| 93D15 | Stabilization of systems by feedback |
| 34H10 | Chaos control for problems involving ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
References:
| [1] | Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008 |
| [2] | Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I, Regul. Pap. 42, 485-490 (1995) · doi:10.1109/81.404062 |
| [3] | Matouk, A.E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol-Duffing circuit. Commun. Nonlinear Sc.i Numer. Simulations 16, 975-986 (2011) · Zbl 1221.93227 · doi:10.1016/j.cnsns.2010.04.027 |
| [4] | Bhalekar, S., Gejji, V.D.: Synchronization of different fractional order chaotic systems using active control. Commun. Nonlinear Sc.i Numer. Simulations 15, 3536-3546 (2010) · Zbl 1222.94031 · doi:10.1016/j.cnsns.2009.12.016 |
| [5] | Faieghi, M.R., Delavari, H.: Chaos in fractional-order Genesio-Tesi system and its synchronization. Commun. Nonlinear Sc.i Numer. Simulations 17, 731-741 (2012) · Zbl 1239.93020 · doi:10.1016/j.cnsns.2011.05.038 |
| [6] | Faieghi, M.R., Delavari, H., Baleanu, D.: Control of an uncertain fractional-order Liu system via fuzzy fractional-order. J. Vib. Control (2011). doi:10.1177/1077546311422243 · doi:10.1177/1077546311422243 |
| [7] | Zhou, P., Ding, R.: Chaotic synchronization between different fractional-order chaotic systems. J. Franklin Inst. 348, 2839-2848 (2011) · Zbl 1254.93078 · doi:10.1016/j.jfranklin.2011.09.004 |
| [8] | Chen, D., Liu, Y., Ma, X., Zhang, R.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 67(1), 893-901 (2011) · Zbl 1242.93027 · doi:10.1007/s11071-011-0002-x |
| [9] | Lin, T.C., Kou, C.H.: H∞ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach. ISA Trans. 50, 548-556 (2011) · doi:10.1016/j.isatra.2011.06.001 |
| [10] | Lu, J., Chen, Y.: Robust stability and stabilization of fractional-order interval systems with the fractional order α: the 0 < α<1 case. IEEE Trans. Autom. Control 55, 152-158 (2010) · Zbl 1368.93506 · doi:10.1109/TAC.2009.2033738 |
| [11] | Kuntanapreeda, S.: Robust synchronization of fractional-order unified chaotic systems via linear control. Comput. Math. Appl. 63, 183-190 (2012) · Zbl 1238.93045 · doi:10.1016/j.camwa.2011.11.007 |
| [12] | Tavazoei, M.S., Haeri, M.: Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Process. 1, 171-181 (2007) · doi:10.1049/iet-spr:20070053 |
| [13] | Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341 |
| [14] | Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be |
| [15] | Garrappa, R.: On linear stability of predictor-corrector algorithms for fractional differential equations. Int. J. Comput. Math. 87, 2281-2290 (2010) · Zbl 1206.65197 · doi:10.1080/00207160802624331 |
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