Abstract
Chaotic motion can sometimes be desirable or undesirable, and hence control over such a phenomenon has become a topic of considerable interest. Currently available methods involve making systematic time-varying small perturbations in the system parameters. A new method is presented here to achieve control over chaotic motion using notch filter output feedback control. The notch filter controller uses an active negative feedback with fixed controller parameters without affecting the original system parameters. The motivation for using a notch filter in the feedback is to disturb the balance of power at the lower end of the participating frequencies in the power spectrum. This results in a truncation of the period-doubling route to chaos. For low-period motions the harmonic balance method is used to show that a single participating frequency can indeed be eliminated. To deal with relatively complex nonlinear plants, and higher-period motions, a numerical optimal parameter selection scheme is presented to choose the notch filter parameters. The procedures are tested on Duffing’s oscillator with a notch filter feedback to achieve desired k-period motion.
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References
Ott, E., Grebogi, C., and Yorke, J. A., “Controlling Chaos”, Phys. Rev. Let., Vol. 64, No. 11, pp. 1196–1199, 1990.
Ott, E., Grebogi, C, and Yorke, J. A., “Controlling Chaotic Dynamical Systems”, CHAOS/XAOS, Soviet-American Perspectives on Nonlinear Science, American Institute of Physics, pp. 227–258, 1990.
Romeiras, F.J., Grebogi, C, Ott, E., and DayawansaW. P., “Controlling Chaotic Dynamical Systems”, Physica D, Vol. 58, No. 1–4, pp. 165–192, 1992.
Wang, Y., Singer, J., and Bau H. H., “Controlling Chaos in a Thermal Convection Loop”, J. of Fluid Mechanics, Vol. 237, pp. 479–498, 1992.
Roy, R. T., Murphy Jr., T. W., Maier, T. D., and Gills Z., “Dynamical Control of a Chaotic Laser: Experimental Stabilization of a Globally Coupled System”, Phys. Rev. Let., Vol. 68, No. 9, pp. 1259–1262, 1992.
Azevedo, A. and Rezende, S. M., “Controlling Chaos in Spin-Wave Instabilities”, Phys. Rev. Let., Vol. 66, No. 10, pp. 1342–1345, 1990.
Hunt, E. R., “Stabilizing High-Period Orbits in a Chaotic System: The Diode Resonator”, Phys. Rev. Let., Vol. 67, No. 15, pp. 1953–1955, 1991.
Athalye, A. M., Notch Filter Control of a Chaotic System, Ph.D. dissertation, Department of Mechanical and Materials Engineering, Washington State University, Pullman, WA, 1993.
Pezeshki, C, Elgar, S., and Krishna, R. C., “Bispectral Analysis of Systems Possessing Chaotic Motion”, J. of Sound and Vibr., Vol. 173, No. 3, pp. 357–368, 1990.
Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
Genesio, R. and Tesi, A., “Harmonic Balance Methods for Analysis of Chaotic Dynamics in Nonlinear Systems”, Automatica, Vol. 28, No. 3, pp. 531–548, 1992.
V. I. Arnold. Mathematical Methods of Classical Mechanics, volume 60. Springer, New York, 1978.
E. Ott. Chaos in Dynamical Systems. Cambridge University Press, Cambridge, 1993.
A. Hubler, R. Mettin, A. Scheeline, and W. Lauterborn Parametric Entrainment Control. Phys. Rev. E. (1995), pp. 4065–4075.
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© 1997 Birkhäuser Boston
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Grantham, W.J., Athalye, A.M. (1997). Notch Filter Feedback Control for k-Period Motion in a Chaotic System. In: Judd, K., Mees, A., Teo, K.L., Vincent, T.L. (eds) Control and Chaos. Mathematical Modelling, vol 8. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2446-4_9
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DOI: https://doi.org/10.1007/978-1-4612-2446-4_9
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