Advertisement

Continuous–Discrete-Time Observers for a Class of Uniformly Observable Systems

  • Mondher Farza
  • Mohammed M’Saad
  • Krishna BusawonEmail author
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 457)

Abstract

This chapter addresses the observer design problem for a class of continuous-time dynamical systems with nonuniformly sampled measurements. More specifically, an observer is proposed that runs in continuous-time with an output error correction term that is updated in a mixed continuous-discrete fashion. The proposed observer is actually an impulsive system as it is described by a set of differential equations with instantaneous state impulses corresponding to the measured samples and their estimates. In addition, it is shown that such an impulsive system can be put under the form of a hybrid system composed of a continuous-time high gain observer coupled with an inter-sample output predictor. The proposed observer present two design features that are worth noting: First, the observer calibration is achieved through the tuning of a scalar design parameter. Second, the exponential convergence to zero of the observation error is established under a well-defined condition on the maximum value of the sampling partition diameter. Simulations results dealing with a flexible joint robot arm are given in order to highlight the performance of the proposed observer.

Keywords

Continuous–Discrete-Time observers High gain observers Impulsive systems Nonlinear observers Sampled-data observers 

References

  1. 1.
    Gauthier, J., Kupka, I.: Deterministic Observation Theory and Applications. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kazantzis, N., Kravaris, C.: Nonlinear observer design using lyapunov’s auxiliary theorem. Syst. Contr. Lett. 34, 241–247 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Farza, M., M’Saad, M., Triki, M., Maatoug, T.: High gain observer for a class of non-triangular systems. Syst. Contr. Lett. 60(1), 27–35 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andrieu, V., Praly, L.: On the existence of Kazantzis-Kravaris/Luenberger observers. SIAM J. Contr. Optim. 45, 432–456 (2006)Google Scholar
  5. 5.
    Rajamani, R.: Observers for Lipschitz Nonlinear Systems. IEEE Trans. Autom. Contr. 43(3), 397–401 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Krener, A.J., Isidori, A.: Linearization by output injection and nonlinear observers. Syst. Contr. Lett. 3, 47–52 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deza, F., Busvelle, E., Gauthier, J., Rakotopara, D.: High gain estimation for nonlinear systems. Syst. Contr. Lett. 18, 295–299 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nadri, M., Hammouri, H., Zaragoza, C.A.: Observer design for continuous-discrete time state affine systems up to output injection. Eur. J. Contr. 10(3), 252–263 (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Andrieu, V., Nadri, M.: Observer design for lipschitz systems with discrete-time measurements. In: Proceedings of 49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA, 2010Google Scholar
  10. 10.
    Nadri, M., Hammouri, H., Grajales, R.: Observer design for uniformly observable systems with sampled measurements. IEEE Trans. Autom. Contr. 58, 757–762 (2013)CrossRefGoogle Scholar
  11. 11.
    Karafyllis, I., Kravaris, C.: From continuous-time design to sampled-data design of observers. IEEE Trans. Autom. Contr. 54(9), 2169–2174 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Raff, T., Kögel, M., Allgöwer, F.: Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements. In: Proceedings of the American Control Conference, Washington, USA, 2008Google Scholar
  13. 13.
    Naghshtabrizi, P., Hespanha, J., Teel, A.: Exponential stability of impulsive systems with application to uncertain sampled-data systems. Syst. Contr. Lett. 57, 378–385 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hammouri, H., Farza, M.: Nonlinear observers for locally uniformly observable systems. ESAIM J. Contr. Optim. Calc. Var. 9, 353–370 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Farza, M., M’Saad, M., Maatoug, T., Kamoun, M.: Adaptive observers for nonlinearly parameterized class of nonlinear systems. Automatica 45(10), 2292–2299 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shim, H., Son, Y.I., Seo, J.H.: Semi-global observer for multi-output nonlinear systems. Syst. Contr. Lett. 42, 233–244 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Farza, M., M’Saad, M., Rossignol, L.: Observer design for a class of MIMO nonlinear systems. Automatica 40(1), 135–143 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gauthier, J., Hammouri, H., Othman, S.: A simple observer for nonlinear systems - application to bioreactors. IEEE Trans. Auto. Contr. 37(6), 875–880 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Farza, M., Busawon, K., Hammouri, H.: Simple nonlinear observers for on-line estimation of kinetic rates in bioreactors. Automatica 34(3), 301–318 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Andrieu, V., Praly, L., Astolfi, A.: High gain observers with updated gain and homogeneous correction terms. Automatica 45, 422–428 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Boizot, N., Busvelle, E., Gauthier, J.: An adaptive high-gain observer for nonlinear systems. Automatica 46, 1483–1488 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Oueder, M., Farza, M., Abdennour, R.B., M’Saad, M.: A high gain observer with updated gain for a class of MIMO non-triangular systems. Syst. Contr. Lett. 61, 298–308 (2012)CrossRefzbMATHGoogle Scholar
  23. 23.
    Howell, A., Hedrick, J.K.: Nonlinear observer design via convex optimization. In: Proceedings of the American Control Conference, Washington, USA, 2008Google Scholar
  24. 24.
    Karafyllis, I., Krstic, M.: Nonlinear Stabilization Under Sampled and Delayed Measurements, and with Inputs Subject to Delay and Zero-Order Hold. Private Correspondance, 2012.Google Scholar
  25. 25.
    Germani, A., Manes, C., Pepe, P.: A new approach to state observation of nonlinear systems with delayed output. IEEE Trans. Autom. Contr. 47(1), 96–101 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kazantzis, N., Wright, R.A.: Nonlinear observer design in the presence of delayed output measurements. Syst. Contr. Lett. 54(9), 877–886 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Cacace, F., Germani, A., Manes, C.: An observer for a class of nonlinear systems with time varying observation delay. Syst. Contr. Lett. 59(5), 305–312 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mondher Farza
    • 1
  • Mohammed M’Saad
    • 1
  • Krishna Busawon
    • 2
    Email author
  1. 1.GREYC Laboratory (UMR 6072 CNRS)Université de Caen and ENSICAENCaen CedexFrance
  2. 2.Engineering and EnvironmentNorthumbria UniversityNewcastle upon TyneUK

Personalised recommendations