Bifurcation Control of Nonlinear Systems With Time-Periodic Coefficients

[+] Author and Article Information
Alexandra Dávid, S. C. Sinha

Nonlinear Systems Research Laboratory, Department of Mechanical Engineering, Auburn University, Auburn, AL 36849

J. Dyn. Sys., Meas., Control 125(4), 541-548 (Jan 29, 2004) (8 pages) doi:10.1115/1.1636194 History: Received October 18, 1999; Revised May 28, 2003; Online January 29, 2004
Copyright © 2003 by ASME
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Aeyels,  D., 1985, “Stabilization of a Class of Nonlinear Systems by a Smooth Feedback Control,” Systems and Control Letters,5, pp. 289–294.
Abed,  E. H., and Fu,  J.-H., 1986, “Local Feedback Stabilization and Bifurcation Control, I. Hopfbifurcation,” Systems and Control Letters,7, pp. 11–17.
Abed,  E. H., and Fu,  J.-H., 1987, “Local Feedback Stabilization and Bifurcation Control, II. Stationary bifurcation,” Systems and Control Letters,7, pp. 467–473.
Iooss, G., and Joseph, D. D., 1990, Elementary Stability and Bifurcation Theory, 2 ed., Springer-Verlag, New York, NY.
Abed, E. H., Wang, H. O., and Chen, R. C., 1992, “Stabilization of Period Doubling Bifurcation and Implications for Control of Chaos,” Proc. 31st IEEE Conference on Decision and Control, Tucson, AZ, Dec., 1992, pp. 2119–2124.
Liaw,  D.-Ch., and Abed,  E. H., 1996, “Active Control of Compressor Stall Interception: a Bifurcation-theoretic Approach,” Automatica, 32, No. 1, pp. 109–115.
Emad, P. F., and Abdelfatah, A. M., 1989, “Nonlinear Oscillations in Magnetic Bearing Systems,” Proc. 28th IEEE Conference on Decision and Control, Tampa, FL, Dec., 1989, pp. 548–553.
Kliemann, W., and Namachchivaya, N. S., 1995, Nonlinear Dynamics and Stochastic Mechanics, CRC Press, Boca Raton, FL.
Oueini, S. S., and Nayfeh, A. H., 1998, “Control of a System Under Principal Parametric Excitation,” Proc. 4th Intl. Conference on Motion and Vibration Control, Zurich, Switzerland, August 25–28, 2 , pp. 405–409.
Pandiyan,  R., and Sinha,  S. C., 1995, “Analysis of Time-Periodic Nonlinear Dynamical Systems Undergoing Bifurcations,” Nonlinear Dyn., 8, pp. 21–45.
Sinha,  S. C., Butcher,  E. A., and Dávid,  A., 1998, “Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems,” Nonlinear Dyn., 16, pp. 203–221.
Dávid,  A., and Sinha,  S. C., 2000, “Versal Deformation and Local Bifurcation Analysis of Time-Periodic Systems,” Nonlinear Dyn., 21, No. 4, pp. 317–336.
Sinha,  S. C. , 1996, “Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems,” J. Vibr. Acoust., 118, 209–219.
Malkin,  I. G., 1962, “Some Basic Theorems of the Theory of Stability of Motion in Critical Cases,” Stability and Dynamic Systems, Translations, American Mathematical Society, Series 1,5, pp. 242–290.
Perko, L., 1991, Differential Equations and Dynamical Systems, Springer-Verlag, New York, NY.
Chow, S.-N., and Hale, J. K., 1982, Methods of Bifurcation Theory, Springer-Verlag, New York, NY.
Arnold, V. I., 1988, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, NY.
Sinha,  S. C., and Butcher,  E. A., 1997, “Symbolic Computation of Fundamental Solution Matrices for Linear Time-Periodic Dynamical Systems,” STLE Tribol. Trans. Journal of Sound and Vibration, 206(1), pp. 61–85.
Dávid, A., and Sinha, S. C., 1999 “Some Ideas on the Local Control of Nonlinear Systems With Time-Periodic Coefficients,” Proc. 1999 Design Engineering Technical Conferences, September 12–16, Las Vegas, NE.
Khalil, H. K., 1992, Nonlinear Systems, Macmillan, New York, NY.


Grahic Jump Location
Bifurcation diagrams of uncontrolled and controlled flip bifurcations
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Parametrically excited simple inverted pendulum
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Bifurcation control of the simple pendulum, comparison of different control gains
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Comparison of the control effort for different control gains for the simple pendulum
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Comparison of different controllers for the simple pendulum
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Comparison of the torque required by the different controllers for the simple pendulum
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Double inverted pendulum with a periodic follower load
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Double pendulum, uncontrolled and controlled motions
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Double pendulum, the uncontrolled and controlled motions in a Poincaré map representation



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