Mathematics of Control, Signals and Systems

, Volume 18, Issue 1, pp 32–65 | Cite as

Global complete observability and output-to-state stability imply the existence of a globally convergent observer

  • Alessandro Astolfi
  • Laurent Praly


We consider systems which are globally completely observable and output-to-state stable. The former property guarantees the existence of coordinates such that the dynamics can be expressed in observability form. The latter property guarantees the existence of a state norm observer and therefore the possibility of bounding any continuous state functions. Both properties allow to conceptually build an observer from an approximation of an exponentially attractive invariant manifold in the space of the system state and an output driven dynamic extension. The proposed observer provides convergence to zero of the estimation error within the domain of definition of the solutions.


Non linear observers Output-to-state stability Invariant manifolds 


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  1. 1.
    Angeli D, Sontag E (1999) Forward completeness, unboundedness observability, and their Lyapunov characterizations. Syst Control Lett 38:209–217Google Scholar
  2. 2.
    Besançon G (1998) State affine systems and observer-based control. In: Proceedings of IFAC nonlinear control systems designs symposium, Enschede, The Netherlands, July 1998, pp 399–404Google Scholar
  3. 3.
    Chen C-T (1984) Linear systems theory and design. Holt (Rinehart and Winston), New YorkGoogle Scholar
  4. 4.
    Ciccarella G, Dalla Mora M, Germani A (1993) A Luenberger-like observer for nonlinear systems. Int J Control 57(3):537–556Google Scholar
  5. 5.
    Deheuvels P (1980) L'intégrale. Presses universitaires de France, ParisGoogle Scholar
  6. 6.
    Engel R, Kreisselmeier G (2002) A continuous-time observer which converges in finite time. IEEE Trans Automat Control 47(7):1202–1204Google Scholar
  7. 7.
    Filippov A (1988) Differential equations with discontinuous right hand sides. Mathematics and its applications, Kluwer, DordrechtGoogle Scholar
  8. 8.
    Gauthier J-P, Kupka I (1994) Observability and observers for nonlinear systems. SIAM J Control Optim 32(4) 975–994Google Scholar
  9. 9.
    Gauthier J-P, Kupka I (2001) Deterministic observation theory and applications. Cambridge University Press, CambridgeGoogle Scholar
  10. 10.
    Gauthier J-P, Hammouri H, Kupka I (1992) A simple observer for nonlinear systems, application to bioreactor. IEEE Trans Automat Control 37:875–880Google Scholar
  11. 11.
    Hammouri H, de Leon Morales J (1991) On systems equivalence and observer synthesis. In: Proceedings of joint conference, Genoa, Italy, pp 341–347Google Scholar
  12. 12.
    Isidori A (1999) Nonlinear control systems II. Springer, Berlin Heidelberg New YorkGoogle Scholar
  13. 13.
    Jiang Z-P, Praly L (1998) Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties. Automatica 34(7):835–840Google Scholar
  14. 14.
    Kazantzis N, Kravaris C (1998) Nonlinear observer design using Lyapunov's auxiliary theorem. Systems & Control Lett 34:241–247Google Scholar
  15. 15.
    Khalil HK, Esfandiari F (1993) Semiglobal stabilization of a class of nonlinear systems using output feedback. IEEE Trans Automat Control 38:1412–1415Google Scholar
  16. 16.
    Kreisselmeier G, Engel R (2003) Nonlinear observers for autonomous Lipschitz continuous systems. IEEE Trans Automat Control 48(3):451–464Google Scholar
  17. 17.
    Krener A, Isidori A (1983) Linearization by output injection and nonlinear observers. Syst Control Lett 3:47–52Google Scholar
  18. 18.
    Krener A, Respondek W (1985) Nonlinear observers with linearizable error dynamics. SIAM J Control Optim 23(2):197–216Google Scholar
  19. 19.
    Luenberger D (1964) Observing the state of a linear system. IEEE Trans Mil Electron MIL-8:74–80Google Scholar
  20. 20.
    Mazenc F, Praly L, Dayawansa WP (1994) Global stabilization by output feedback: Examples and Counter-Examples. Syst Control Lett 23:119–125Google Scholar
  21. 21.
    Nicosia S, Tornambè A (1989) High-gain observers in the state and parameter estimation of robots having elastic joints. Syst Control Lett 13:331–337Google Scholar
  22. 22.
    Sontag E, Wang Y (1996) New characterizations of input-to-state stability. IEEE Trans Automat Control 41(9)1283–1294Google Scholar
  23. 23.
    Sontag E, Wang Y (1997) Output-to-state stability and detectability of nonlinear systems. Syst Control Lett 29:279–290Google Scholar
  24. 24.
    Teel A, Praly L (1994) Global stabilizability and observability imply semi-global stabilizability by output feedback. Syst Control Lett 22:313–325Google Scholar
  25. 25.
    Teel A, Praly L (2000) A smooth Lyapunov function from a class-KL estimate involving two positive semi-definite functions. ESAIM: Control Optim Calc Var 5:313–367Google Scholar
  26. 26.
    Xiao M-Q, Krener AJ (2002) Design of reduced-order observers of nonlinear systems through change of coordinates. In: Proceedings of the 41st IEEE conference on decision and controlGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentImperial College LondonLondonUK
  2. 2.Dipartimento di Informatica, Sistemi e ProduzioneUniversità di Roma Tor VergataRomaItaly
  3. 3.Centre Automatique et SystèmesÉcole des Mines de ParisFontainebleauFrance

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