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Stabilization of generalized fractional order chaotic systems using state feedback control. (English) Zbl 1060.93515

Summary: We address the problem of chaos control of three types of fractional order systems using simple state feedback gains. Electronic chaotic oscillators, mechanical ”jerk” systems, and the Chen system are investigated when they assume generalized fractional orders. We design the static gains to place the eigenvalues of the system Jacobian matrices in a stable region whose boundaries are determined by the orders of the fractional derivatives. We numerically demonstrate the effectiveness of the controller in eliminating the chaotic behavior from the state trajectories, and driving the states to the nearest equilibrium point in the basin of attraction. For the recently introduced Chen system, in particular, we demonstrate that with a proper choice of model parameters, chaotic behavior is preserved when the system order becomes fractional. Both state and output feedback controllers are then designed to stabilize a generalized fractional order Chen system.

MSC:

93C10 Nonlinear systems in control theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93D15 Stabilization of systems by feedback
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[1] Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D., Nonlinear noninteger order circuits and systems-an introduction (2000), World Scientific · Zbl 0966.93006
[2] Oldham, K.; Spainer, J., Fractional calculus (1974), Academic Press: Academic Press New York
[3] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[4] (Hilfer, R., Applications of fractional calculus in physics (2001), World Scientific: World Scientific New Jersey) · Zbl 0998.26002
[5] Ahmad, W.; El-Khazali, R.; El-Wakil, A., Fractional-order Wien-bridge oscillators, Electron. Lett., 37, 18, 1110-1112 (2001)
[6] Ahmad, W.; Sprott, J. C., Chaos in fractional order system autonomous nonlinear systems, Chaos, Solitons & Fractals, 16, 339-351 (2003) · Zbl 1033.37019
[7] (Chen, G.; Yu, X., Chaos control: theory and applications (2003), Springer Verlag: Springer Verlag New York) · Zbl 1029.00015
[8] Nayfeh, A. H.; Balachandran, B., Applied nonlinear dynamics (1994), John Wiley: John Wiley New York
[9] Harb A, Ahmad W. Control of chaotic oscillators using a nonlinear recursive backstepping controller. In: IASTED Conf on Applied Simulations and Modeling, Crete, Greece, June 2002. p. 451-3; Harb A, Ahmad W. Control of chaotic oscillators using a nonlinear recursive backstepping controller. In: IASTED Conf on Applied Simulations and Modeling, Crete, Greece, June 2002. p. 451-3
[10] Ahmad, W.; Harb, A., On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, Solitons & Fractals, 18, 693-701 (2003) · Zbl 1073.93027
[11] Khazali R. Fractional systems variable structure control. In: IASTED International Conference on Control and Applications, Cancun, Mexico, May 2002. p. 7-11; Khazali R. Fractional systems variable structure control. In: IASTED International Conference on Control and Applications, Cancun, Mexico, May 2002. p. 7-11
[12] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B., Fractal system as represented by singularity function, IEEE Trans. Automat. Control, 37, 9 (1992) · Zbl 0825.58027
[13] Elwakil, A. S.; Kennedy, M. P., Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices, IEEE Trans. Circuits Syst., 48, 3 (2001) · Zbl 0998.94048
[14] Sprott, J. C., Simple chaotic systems and circuits, Am. J. Phys., 68, 758-763 (2000)
[15] Chen, G.; Ueta, T., Yet another chaotic attractor, Int. J. Bifurcat. Chaos, 9, 7, 1465-1466 (1999) · Zbl 0962.37013
[16] Matignon D. Stability results of fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC, Lille, France, 1996. p. 963-8; Matignon D. Stability results of fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC, Lille, France, 1996. p. 963-8
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