×

Chaos control using an adaptive fuzzy sliding mode controller with application to a nonlinear pendulum. (English) Zbl 1198.93115

Summary: Chaos control may be understood as the use of tiny perturbations for the stabilization of unstable periodic orbits embedded in a chaotic attractor. The idea that chaotic behavior may be controlled by small perturbations of physical parameters allows this kind of behavior to be desirable in different applications. In this work, chaos control is performed employing a variable structure controller. The approach is based on the sliding mode control strategy and enhanced by an adaptive fuzzy algorithm to cope with modeling inaccuracies. The convergence properties of the closed-loop system are analytically proven using Lyapunov’s direct method and Barbalat’s lemma. As an application of the control procedure, a nonlinear pendulum dynamics is investigated. Numerical results are presented in order to demonstrate the control system performance. A comparison between the stabilization of general orbits and unstable periodic orbits embedded in chaotic attractor is carried out showing that the chaos control can confer flexibility to the system by changing the response with low power consumption.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

93C42 Fuzzy control/observation systems
34H10 Chaos control for problems involving ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ott, E.; Grebogi, C.; Yorke, J.A., Controlling chaos, Phys rev lett, 64, 11, 1196-1199, (1990) · Zbl 0964.37501
[2] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys lett A, 170, 421-428, (1992)
[3] Pyragas, K., Delayed feedback control of chaos, Philos trans R soc A, 364, 2309-2334, (2006) · Zbl 1152.93477
[4] Dressler, U.; Nitsche, G., Controlling chaos using time delay coordinates, Phys rev lett, 68, 1, 1-4, (1992)
[5] Hübinger, B.; Doerner, R.; Martienssen, W.; Herdering, M.; Pitka, R.; Dressler, U., Controlling chaos experimentally in systems exhibiting large effective Lyapunov exponents, Phys rev E, 50, 2, 932-948, (1994)
[6] de Korte, R.J.; Schouten, J.C.; van den Bleek, C.M.V., Experimental control of a chaotic pendulum with unknown dynamics using delay coordinates, Phys rev E, 52, 4, 3358-3365, (1995)
[7] Otani M, Jones AJ. Guiding chaotic orbits, Tech. rep., Imperial College of Science Technology and Medicine, London, 1997.
[8] So, P.; Ott, E., Controlling chaos using time delay coordinates via stabilization of periodic orbits, Phys rev E, 51, 4, 2955-2962, (1995)
[9] De Paula AS, Savi MA. A multiparameter chaos control method based on OGY approach. Chaos, Solitons and Fractals. doi:10.1016/j.chaos.2007.09.056. · Zbl 1197.34110
[10] Savi, M.A.; Pereira-Pinto, F.H.I.; Ferreira, A.M., Chaos control in mechanical systems, Shock vibr, 13, 4/5, 301-314, (2006)
[11] Andrievskii, B.R.; Fradkov, A.L., Control of chaos: methods and applications, II - applications, Autom remote control, 65, 4, 505-533, (2004) · Zbl 1115.37315
[12] Moon, F.C.; Reddy, A.J.; Holmes, W.T., Experiments in control and anti-control of chaos in a dry friction oscillator, J vibr control, 9, 3/4, 387-397, (2003)
[13] Begley, C.J.; Virgin, L.N., On the OGY control of an impact-friction oscillator, J vibr control, 7, 6, 923-931, (2001) · Zbl 1006.70514
[14] Hu, H.Y., Controlling chaos of a periodically forced nonsmooth mechanical system, Acta mechanica sinica, 11, 3, 251-258, (1995) · Zbl 0855.70016
[15] Spano, M.L.; Ditto, W.L.; Rauseo, S.N., Exploitation of chaos for active control: an experiment, J intell mater syst struct, 2, 4, 482-493, (1990)
[16] Macau, E.E.N., Exploiting unstable periodic orbits of a chaotic invariant set for spacecraft control, Celestial mech dyn astronomy, 87, 3, 291-305, (2003) · Zbl 1106.70329
[17] Pereira-Pinto, F.H.I.; Ferreira, A.M.; Savi, M.A., Chaos control in a nonlinear pendulum using a semi-continuous method, Chaos, solitons and fractals, 22, 3, 653-668, (2004) · Zbl 1116.70335
[18] Pereira-Pinto, F.H.I.; Ferreira, A.M.; Savi, M.A., State space reconstruction using extended state observers to control chaos in a nonlinear pendulum, Int J bifurcat chaos, 15, 12, 4051-4063, (2005) · Zbl 1097.37029
[19] Wang, R.; Jing, Z., Chaos control of chaotic pendulum system, Chaos, solitons and fractals, 21, 1, 201-207, (2004) · Zbl 1045.37016
[20] Yagasaki, K.; Yamashita, S., Controlling chaos using nonlinear approximations for a pendulum with feedforward and feedback control, Int J bifurcat chaos, 9, 1, 233-241, (1999) · Zbl 0941.93532
[21] De Paula AS, Savi MA. Chaos control in a nonlinear pendulum using an extended time-delayed feedback method. submitted to Chaos, Solitons and Fractals, 2008.
[22] De Paula, A.S.; Savi, M.A.; Pereira-Pinto, F.H.I., Chaos and transient chaos in an experimental nonlinear pendulum, J sound vibr, 294, 585-595, (2006)
[23] Bessa WM, Barrêto RSS. Adaptive fuzzy sliding mode control of uncertain nonlinear systems. Revista Controle & Automačão, accepted for publication.
[24] Bessa, W.M.; Dutra, M.S.; Kreuzer, E., Depth control of remotely operated underwater vehicles using an adaptive fuzzy sliding mode controller, Robot autonom syst, 56, 670-677, (2008)
[25] Guan, P.; Liu, X.-J.; Liu, J.-Z., Adaptive fuzzy sliding mode control for flexible satellite, Eng appl artif intel, 18, 451-459, (2005)
[26] Slotine, J.-J.E., Sliding controller design for nonlinear systems, Int J control, 40, 2, 421-434, (1984) · Zbl 0541.93034
[27] Bessa, W.M., Some remarks on the boundedness and convergence properties of smooth sliding mode controllers, Int J autom comput, 6, 2, 154-158, (2009)
[28] Franca, L.F.P.; Savi, M.A., Distinguishing periodic and chaotic time series obtained from an experimental pendulum, Nonlinear dyn, 26, 253-271, (2001) · Zbl 1006.70500
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.