Perturbed hybrid systems, applications in control theory

  • Christophe Prieur
Conference paper
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 281)


We study a class of perturbed hybrid systems, i.e. dynamical systems with a mixed continuous/discrete state in presence of disturbances. We introduce a natural notion of trajectories, but it is very sensitive to noise. Therefore we define a new notion of trajectories and, to investigate the sensitivity, we enlarge this class of trajectories. Finally we consider two problems in control theory (the uniting problem and the problem of the robust stabilization of asymptotically controllable systems) which have no solution in terms of (dis)continuous controller in presence of disturbances and we give a solution with a robust hybrid controller.


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  1. 1.
    Artstein Z. (1983) Stabilization with relaxed controls. Nonlin. Th., Meth., App. 7, 1163–1173zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Branicky M.S. (1998) Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Clarke F.H., Ledyaev Yu.S., Sontag E.D., Subbotin A.I. (1997) Asymptotic controllability implies feedback stabilization. IEEE Trans. Autom. Control 42, 1394–1407zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hermes H. (1967) Discontinuous vector fields and feedback control. In Differential Equations and Dynamic Systems, (Hale J.K. and La Salle J.P., eds.), Academic Press, New York London, 155–165Google Scholar
  5. 5.
    Prieur C. (2000) A Robust globally asymptotically stabilizing feedback: the example of the Artstein’s circles. In Nonlinear Control in the Year 2000 (Isidori A. et all, eds.), Springer Verlag, London, 279–3000Google Scholar
  6. 6.
    Prieur C. (2001) Uniting local and global controllers with robustness to vanishing noise. Math. Control Signals Systems 14, 143–172zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Prieur C. (2001) Asymptotic controllability and robust asymptotic stabilizability. Preprint 2001-03 Université Paris-Sud, FranceGoogle Scholar
  8. 8.
    Sontag E.D. (1999) Stability and stabilization: discontinuities and the effect of disturbances. In Nonlinear Analysis, Differential Equations, and Control, (Proc. NATO Advanced Study Institute, Montreal, Jul/Aug 1998; Clarke F.H., Stern R.J., eds.), Kluwer, 551–598Google Scholar
  9. 9.
    Sontag E.D. (1999) Clocks and insensitivity to small measurement errors. ESAIM: COCV, 4, 537–557zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Christophe Prieur
    • 1
  1. 1.Laboratoire d’analyse numérique et EDPUniversité Paris-SudOrsayFrance

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