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A sampled normal form for feedback linearization

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Abstract

This paper discusses the problem of preserving approximated feedback linearization under digital control. Starting from a partially feedback linearizable affine continuous-time dynamics, a digital control procedure which maintains the dimension of the maximally feedback linearizable part up to any order of approximation with respect to the sampling period is proposed. The result is based on the introduction of a sampled normal form, a canonical structure which naturally appears when studying feedback linearization.

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References

  1. A. Arapostathis, B. Jakubczyk, H. G. Lee, S. I. Marcus, and E. D. Sontag. The effect of sampling on linear equivalence and feedback linearization.Systems Control Lett.,13, 373–381, 1989.

    Google Scholar 

  2. K. G. Aström and B. Wittenmark.Computer Controlled Systems Theory and Design. Information and System Sciences Series. Prentice-Hall, Englewood Cliffs, NJ, 1984.

    Google Scholar 

  3. J. P. Barbot. Application of linearizing laws to helicopter flight.Proc. 12th IMACS World Congress on Scientific Computation, pages 87–89, 1988.

  4. J. P. Barbot, N. Pantalos, S. Monaco, and D. Normand-Cyrot. On the control of regularlyɛ-perturbed nonlinear systems.Internat. J. Control,59(5), 1255–1276, 1994.

    Google Scholar 

  5. J. W. Grizzle and P. V. Kokotovic. Feedback linearization of sampled-data systems.IEEE Tran. Automat. Control,33(9), 1988.

  6. J. W. Grizzle. Feedback linearization of discrete-time systems. InLecture Notes in Control and Information Sciences (A. Bensoussan and J. L. Lions, eds.), Vol. 83. Springer-Verlag, Berlin, pages 273–281, 1986.

    Google Scholar 

  7. A. O. Guelfond.Calcul des Différences Finies. Collection Universitaire de Mathématiques. Dunod, Paris, 1963.

    Google Scholar 

  8. A. Isidori.Nonlinear Control Systems: An Introduction, 2nd edn. Communications and Control Engineering Series. Springer-Verlag, New York, 1989.

    Google Scholar 

  9. B. Jakubczyk. Feedback linearization of discrete-time systems.Systems Control Lett.,11 411–416, 1987.

    Google Scholar 

  10. P, V. Kokotovic, H. K. Khalil, and J. O'Reilly.Singular Perturbation Methods in Control: Analysis and Design. Academic Press, New York, 1986.

    Google Scholar 

  11. H. G. Lee, A. Arapostathis, and S. I. Marcus. Linearization of discrete-time systems.Internat. J. Control,45(5), 1803–1822, 1987.

    Google Scholar 

  12. H. G. Lee, A. Arapostathis, and S. I. Marcus. On the digital control of nonlinear systems.Proc. 27th IEEE Conf. on Decision and Control, pages 480–481, 1988.

  13. R. H. Middelton and G. C. Goodwin.Digital Control and Estimation: a Unified Approach. Prentice-Hall, Englewood Cliffs, NJ, 1990.

    Google Scholar 

  14. S. Monaco and D. Normand-Cyrot. The immersion under feedback of a multidimensional discrete-time non-linear system into a linear system.Internat. J. Control,38(1), 245–261, 1983.

    Google Scholar 

  15. S. Monaco and D. Normand-Cyrot. On the sampling of a linear analytic control system.Proc. 24th IEEE Conf. on Decision and Control, pages 1457–1462, 1985.

  16. S. Monaco and D. Normand-Cyrot. Invariant distributions under sampling. InTheory and Applications of Nonlinear Control Systems (C. I. Byrnes and A. Linquist, eds.). North Holland, Amsterdam, pages 215–221, 1986.

    Google Scholar 

  17. S. Monaco and D. Normand-Cyrot. Multirate digital control. Internal Report, D.I.S, University of Rome “La Sapienza,” 1988.

  18. S. Monaco and D. Normand-Cyrot. Sur la commande digitale d'un système non linéaire à déphasage minimal. InLecture Notes in Control and Information Sciences (A. Bensoussan and J. L. Lions, eds.), Vol. 111, Springer-Verlag, Berlin, pages 193–204, 1988.

    Google Scholar 

  19. S. Monaco and D. Normand-Cyrot. Zero dynamics of sampled nonlinear systems.System Control Lett.,11, 229–234, 1988.

    Google Scholar 

  20. S. Monaco and D. Normand-Cyrot. Functional expansions for nonlinear discrete-time systems.Math. Systems Theory,21, 235–254, 1989.

    Google Scholar 

  21. S. Monaco and D. Normand-Cyrot. Multirate sampling and zero dynamics, from linear to nonlinear. InNonlinear Synthesis (C. I. Byrnes and A. Kurzhansky, eds.). Progress in Systems and Control Theory. Birkhäuser, Boston, pages 200–214, 1991.

    Google Scholar 

  22. S. Monaco and D. Normand-Cyrot. A unified representation for nonlinear discrete-time and sampled dynamics.J. Math. Systems Estimation Control,5(1), 103–106, 1995.

    Google Scholar 

  23. D. A. Recker and P. V. Kokotovic. Observers for nonlinear systems.Proc. American Control Conf., 1991.

  24. E. D. Sontag. An eigenvalue condition for sampled weak controllability of bilinear systems.Systems Control Lett. 7, 313–316, 1986.

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire des Signaux et Systémes, CNRS/ESE, Plateau de Moulon, 91190, Gif sur Yvette, France

    J. P. Barbot & D. Normand-Cyrot

  2. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, 18 Via Eudossiana, 0018, Roma, Italy

    S. Monaco

Authors
  1. J. P. Barbot
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  2. S. Monaco
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  3. D. Normand-Cyrot
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Additional information

This work was supported by an Italian 40% M.U.R.S.T. grant and a French M.E.N.-D.R.E.D. grant.

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Barbot, J.P., Monaco, S. & Normand-Cyrot, D. A sampled normal form for feedback linearization. Math. Control Signal Systems 9, 162–188 (1996). https://doi.org/10.1007/BF01211752

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  • DOI: https://doi.org/10.1007/BF01211752

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