Dynamics and Control

, Volume 4, Issue 3, pp 277–298 | Cite as

Nonlinear regulation of a Lorenz system by feedback linearization techniques

  • Joaquín Alvarez-Gallegos


In this paper we analyze some dynamical properties of a chaotic Lorenz system driven by a control input. These properties are the input-state and the input-output feedback linearizability, the stability of the zero dynamics, and the phase minimality of the system. We show that the controlled Lorenz system is feedback equivalent to a controllable linear system. We also show that the zero dynamics are asymptotically stable when the output is an arbitrary state. These facts allow designing control laws such that the closed-loop system has asymptotically stable equilibrium points with dynamic behavior free from chaotic transients. The controllers are robust in the sense that the closed-loop system is stable and non chaotic around a nominal set of parameter values. The results also show that the proposed controllers give better responses compared to linear algorithms obtained from standard linearization techniques, and exhibit a good performance even when the control input is bounded.


Control Input Feedback Linearizability Lorenz System Arbitrary State Phase Minimality 
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  1. 1.
    Carroll, T. L., and Pecora, L. M., “Synchronizing chaotic circuits,”IEEE Trans. on Circuits and Systems, vol. 38, pp. 453–456, 1991.CrossRefGoogle Scholar
  2. 2.
    Fleming, W. H., et al., “Report of the panel on future directions in control theory: a mathematical perspective”, Society for Industrial and Applied Mathematics, Philadelphia, USA, 1988.Google Scholar
  3. 3.
    Hartley, T. T. and Mossayebi, F., “Classical control of a chaotic system”, 1st. IEEE Conference on Control Applications, Dayton, Ohio, USA, pp. 522–526, 1992.Google Scholar
  4. 4.
    Hunt, L. R., Su, R., and Meyer, G., “Global transformations of nonlinear systems”,IEEE Transactions on Automatic Control, vol. 28, pp. 24–31, 1983.CrossRefGoogle Scholar
  5. 5.
    Inaba, N. and Mori, S., “Chaos via torus breakdown in a piece wise-linear forced van der Pol oscillator with a diode,”IEEE Trans. on Circuits and Systems, vol. 38, pp. 398–409, 1991.CrossRefGoogle Scholar
  6. 6.
    Isidori, A.,Nonlinear control systems: an introduction. Springer Verlag, 1989.Google Scholar
  7. 7.
    Kirchgraber, U. and Stoffer, D., “Chaotic behavior in simple dynamical systems,”SIAM Review, vol. 32, pp. 424–452, 1990.CrossRefGoogle Scholar
  8. 8.
    Lorenz, E. N., “Deterministic non-periodic flow,”J. Atmos. Science, vol. 20, pp. 130–141, 1963.CrossRefGoogle Scholar
  9. 9.
    Khalil, H. K.,Nonlinear systems. MacMillan Publishing Company, 1992.Google Scholar
  10. 10.
    Mahfouz, I. A. and Badrakhan, F., “Chaotic behavior of some piece wise-linear systems, part II: systems with clearance,”Journal of Sound and Vibration, vol. 143, pp. 289–328, 1990.CrossRefGoogle Scholar
  11. 11.
    Moon, F. C.,Chaotic and fractal dynamics. John Wiley & Sons, Inc., 1992.Google Scholar
  12. 12.
    Ott, E., Grebogi, C., and Yorke, J. A., “Controlling chaotic dynamical systems,” inChaos/Xaoc: Soviet-American perspectives on nonlinear science, edited by D. K. Campbell. American Institute of Physics, pp. 153–172, 1990.Google Scholar
  13. 13.
    Spano, M. L. and Ditto, W. L., “Taming chaos experimentally: a primer,”Proc. of the 1st. Experimental Chaos Conference, Arlington, VA, October 1991.Google Scholar
  14. 14.
    Sparrow, C.,The Lorenz equations: bifurcations, chaos, and strange attractors. Springer Verlag, 1982.Google Scholar
  15. 15.
    Su, R., Hunt, L., and Meyer, G., “Theory of design using nonlinear transformations,”Proc. of the American Control Conference, USA, pp. 247–251, 1982.Google Scholar
  16. 16.
    Vincent, T. L. and Yu, J., “Control of a chaotic system,”Dynamics and Control, vol. 1, pp. 35–52, 1991.Google Scholar
  17. 17.
    Wu, X. and Schelly, Z. A., “Chaotic behavior of chemical systems,”React. Kinet. Catal. Letters, vol. 42, pp. 303–307, 1990.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Joaquín Alvarez-Gallegos
    • 1
  1. 1.Depto. de Electrónica y TelecomunicacionesCentro de Investigación Científica y de Educación Superior de Ensenada, B. C. (CICESE)EnsenadaMéxico

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