Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-11T22:10:32.797Z Has data issue: false hasContentIssue false
  • English
  • Français

Clocks and Insensitivity to Small Measurement Errors

Published online by Cambridge University Press:  15 August 2002

Eduardo D. Sontag
Eduardo D. Sontag*
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A.; sontag@hilbert.rutgers.edu.
Get access

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.

Keywords

Control systemsfeedback stabilizationcontrollability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 537 - 557
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.)
Artstein, Z., Stabilization with relaxed controls. Nonl. Anal. TMA 7 (1983) 1163-1173. CrossRef
F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998).
Clarke, F.H., Ledyaev, Yu.S., Sontag, E.D. and Subbotin, A.I., Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42 (1997) 1394-1407. CrossRef
F.H. Clarke, Yu.S. Ledyaev, L. Rifford and R. Stern, Feedback stabilization and Lyapunov functions. preprint, Univ. de Lyon (1999).
J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems, Estimation, and Control 4 (1994) 67-84.
J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion. Amer. Math. Society Translations, Series 2 24 (1956) 19-77.
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.
Yu.S. Ledyaev, E.D. Sontag, A Lyapunov characterization of robust stabilization. J. Nonl. Anal. 37 (1999) 813-840. CrossRef
Lin, Y., Sontag, E.D. and Wang, Y., A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim. 34 (1996) 124-160. CrossRef
Ryan, E.P., Brockett's, On condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim. 32 (1994) 1597-1604. CrossRef
E.D. Sontag A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim. 21 (1983) 462-471.
E.D. Sontag, Mathematical Control Theory, Deterministic Finite Dimensional Systems, Second Edition. Springer-Verlag, New York (1998).
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.
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.