Abstract
A nonlinear system with a chaotic attractor produces random like motion on the attractor. As a consequence, there must exist a neighborhood about any given point on or near the attractor such that the system will visit the neighborhood in finite time. This observation leads to a very simple “Chaotic control” algorithm for bringing nonlinear systems to a fixed point. Suppose that the system to be controlled is either naturally chaotic or that chaotic motion can be produced by means of open loop control. Suppose also, a neighborhood of the desired fixed point can be found, such that, using standard feedback control techniques, the system is guaranteed to be driven to the fixed point. If this neighborhood also has points in common with the chaotic attractor it is then called a “controllable target” for the fixed point. The chaotic control algorithm consists of first using, if necessary, open loop control to generate chaotic motion and then wait for the system to move into the controllable target. At such a time the open loop control is turned off and the appropriate closed loop control applied. The basic requirement with this approach is, of coarse, being able to determine a large controllable target. The following method is used here: The system is first linearized about the desired fixed point solution. If necessary, a feedback controller is then designed so that this reference solution has suitable stability properties. A Lyapunov function is then obtained based on this stable linear system.
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References
G. Chen, G. and X. Dong, 1993. “From Chaos to Order — Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems”, International Journal of Bifurcation and Chaos 3(6), pp. 1363–1409.
Guckenheimer, J. and P. Holmes, 1993. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York.
Hénon, M., 1976, “A Two Dimensional Map with a Strange Attractor”, Commun. Math. Phys. 50,69.
Luenberger, D.G., 1979. Introduction to Dynamic Systems, Wiley, New York.
Ogata, K, 1987, Discrete-Time Control Systems, Prentice Hall, Englewood Cliffs, New Jersey.
Ott, E., C. Grebogi, and J.A. York, 1990. “Controlling Chaos”, Physical Review Letters 64, pp. 1196–1199.
Paskota, M., A.I. Mees, and K.L. Teo, 1994 “Stabilizing Higher Periodic Orbits”, International Journal of Bifurcation and Chaos 4(2), pp. 457–460.
Paskota, M., A.I. Mees, and K.L. Teo, 1995. “On Local Control of Chaos, the Neighborhood Size”, Manuscript, personal communication.
Tufillaro, N.B., T. Abbot, and Reilly, A., 1992. “An Experimental Approach to Nonlinear Dynamics and Chaos”, Ch 1, Addison Wesely.
Vincent, T.L. and J. Yu, 1991. “Control of a Chaotic System”, Dynamics and Control 1, pp. 35–52.
Vincent, T.L., T.J. Schmitt, and T.L. Vincent, 1994. “A Chaotic Controller for the Double Pendulum”, Mechanics and Control, edited by R.S. Guttalu, Plenum Press, New York, pp. 257–273.
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© 1997 Birkhäuser Boston
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Vincent, T.L. (1997). Controllable Targets Near a Chaotic Attractor. 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_17
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DOI: https://doi.org/10.1007/978-1-4612-2446-4_17
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