Clocks and Insensitivity to Small Measurement Errors

Eduardo D. Sontag

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 537-557
  • ISSN: 1292-8119

Abstract

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This paper deals with the problem of stabilizing a system in the presence of small measurement errors. It is known that, for general stabilizable systems, there may be no possible memoryless state feedback which is robust with respect to such errors. In contrast, a precise result is given here, showing that, if a (continuous-time, finite-dimensional) system is stabilizable in any way whatsoever (even by means of a dynamic, time varying, discontinuous, feedback) then it can also be semiglobally and practically stabilized in a way which is insensitive to small measurement errors, by means of a hybrid strategy based on the idea of sampling at a “slow enough” rate.

How to cite

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Sontag, Eduardo D.. "Clocks and Insensitivity to Small Measurement Errors." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 537-557. <http://eudml.org/doc/116549>.

@article{Sontag2010,
abstract = { This paper deals with the problem of stabilizing a system in the presence of small measurement errors. It is known that, for general stabilizable systems, there may be no possible memoryless state feedback which is robust with respect to such errors. In contrast, a precise result is given here, showing that, if a (continuous-time, finite-dimensional) system is stabilizable in any way whatsoever (even by means of a dynamic, time varying, discontinuous, feedback) then it can also be semiglobally and practically stabilized in a way which is insensitive to small measurement errors, by means of a hybrid strategy based on the idea of sampling at a “slow enough” rate. },
author = {Sontag, Eduardo D.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control systems; feedback stabilization; controllability.; nonlinear control system; discontinuous feedback; piecewise constant feedback; Lyapunov function; sampling schedule; stabilization},
language = {eng},
month = {3},
pages = {537-557},
publisher = {EDP Sciences},
title = {Clocks and Insensitivity to Small Measurement Errors},
url = {http://eudml.org/doc/116549},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Sontag, Eduardo D.
TI - Clocks and Insensitivity to Small Measurement Errors
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 537
EP - 557
AB - This paper deals with the problem of stabilizing a system in the presence of small measurement errors. It is known that, for general stabilizable systems, there may be no possible memoryless state feedback which is robust with respect to such errors. In contrast, a precise result is given here, showing that, if a (continuous-time, finite-dimensional) system is stabilizable in any way whatsoever (even by means of a dynamic, time varying, discontinuous, feedback) then it can also be semiglobally and practically stabilized in a way which is insensitive to small measurement errors, by means of a hybrid strategy based on the idea of sampling at a “slow enough” rate.
LA - eng
KW - Control systems; feedback stabilization; controllability.; nonlinear control system; discontinuous feedback; piecewise constant feedback; Lyapunov function; sampling schedule; stabilization
UR - http://eudml.org/doc/116549
ER -

References

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  1. F. Albertini and Sontag E.D., Continuous control-Lyapunov functions for asymptotically controllable time-varying systems, Internat. J. Control. to appear. (See also Control-Lyapunov functions for time-varying set stabilization, Proc. European Control Conf., Brussels, July 1997, Paper ECC515.)  Zbl0949.93035
  2. Z. Artstein, Stabilization with relaxed controls. Nonl. Anal. TMA7 (1983) 1163-1173.  Zbl0525.93053
  3. F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998).  Zbl1047.49500
  4. F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag and A.I. Subbotin, Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control42 (1997) 1394-1407.  Zbl0892.93053
  5. F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R. Stern, Feedback stabilization and Lyapunov functions. preprint, Univ. de Lyon (1999).  Zbl0961.93047
  6. J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems, Estimation, and Control4 (1994) 67-84.  Zbl0925.93827
  7. J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion. Amer. Math. Society Translations, Series 224 (1956) 19-77.  
  8. Yu.S. Ledyaev and E.D. Sontag, A remark on robust stabilization of general asymptotically controllable systems, in Proc. Conf. on Information Sciences and Systems (CISS 97), John Hopkins University, Baltimore (1997) 246-251.  
  9. Yu.S. Ledyaev and E.D. Sontag, A Lyapunov characterization of robust stabilization. J. Nonl. Anal.37 (1999) 813-840.  Zbl0947.34054
  10. Y. Lin, E.D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim.34 (1996) 124-160.  Zbl0856.93070
  11. E.P. Ryan, On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim.32 (1994) 1597-1604.  Zbl0806.93049
  12. E.D. Sontag A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim.21 (1983) 462-471.  
  13. E.D. Sontag, Mathematical Control Theory, Deterministic Finite Dimensional Systems, Second Edition. Springer-Verlag, New York (1998).  Zbl0945.93001
  14. E.D. Sontag, 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; F.H. Clarke and R.J. Stern, Eds., Kluwer, Dordrecht (1999) 551-598. See also Nonlinear Control Abstracts #NCA-8-2-981026, Oct 1998.  Zbl0937.93034
  15. E.D. Sontag and H.J. Sussmann, Nonsmooth Control Lyapunov Functions, in Proc. IEEE Conf. Decision and Control, New Orleans, IEEE Publications (1995) 2799-2805.  

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