On the Observation Analysis and Observer Design for a Class of Hybrid Continuous-Discrete Dynamic System

  • Noureddine Manamanni
  • Mohamed Djemai
  • Jean Pierre Barbot
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 457)

Abstract

This chapter deals with observability conditions and state observer design for a class of hybrid systems whose the continuous part combines continuous and discrete dynamics. The main contribution of the work lies in the performed observability conditions for this class of systems and the design of a hybrid observer to reconstruct both continuous and discrete states starting only from the knowledge of a continuous output. Firstly, a high-order sliding mode based observer is used to estimate the continuous state and to generate a discrete output. Secondly, starting from this discrete output, a discrete state reconstructor is designed. An illustrative example is provided to show the efficiency of the proposed observer.

References

  1. 1.
    Antsaklis, P.J. (ed.): In: Proceedings of the IEEE Special Issue on Hybrid Systems: Theory and Applications, vol. 43, no. (2) (2000)Google Scholar
  2. 2.
    Babaali, M., Egerstedt, M., Kamen, E.W.: An observer for linear systems with randomly-switching measurement equations. In: Proceedings of the American Control Conference, Denver, Colorado, 4–6 June 2003Google Scholar
  3. 3.
    Barbot, J.-P., Saadaoui, H., Djemaï, M., Manamanni, N.: Nonlinear observer for autonomous switching systems with jumps. In: Proceedings of the Nonlinear Analysis: Hybrid Systems, vol. 1, pp. 537–547 (2007)Google Scholar
  4. 4.
    Barbot, J., Djemai, M., Manamanni, N.: State observer and observability conditions for a class of hybrid continuous-discrete dynamic system. In: Proceedings of the 46th IEEE Conference on Decision and Control (CDC 07), pp. 708–713. IEEE New Orleans, December 2007Google Scholar
  5. 5.
    Baglietto, M., Battistelli, G., Scardovi, L.: Active mode observability of switched linear systems. In: Proceedings of the 45th IEEE-CDC, San Diego, USA, pp. 145–150 (2006)Google Scholar
  6. 6.
    Balluchi, A., Benvenuti, L., Di Benedetto, M.D., Sangiovanni Vincentelliy, A.L.: A Hybrid Observer for the Driveline Dynamics. ECC, Porto (2001)Google Scholar
  7. 7.
    Balluchi, A., Benvenuti, L., Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L.: Observability for hybrid systems, In: Proceedings of the 42nd IEEE-CDC, Maui, Hawaii USA, Dec 2003Google Scholar
  8. 8.
    Barbot, J.P., Floquet, T.: State and unknown input estimation for linear discrete-time systems. In: Proceedings of the IFAC World Congress, Prague (2005)Google Scholar
  9. 9.
    Bemporad, A., Ferrari-Trecate, G., Morari, M.: Observability and controllability of piecewise affine and hybrid systems. IEEE Trans. Autom. Control 45(10), 1864–1876 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Baglietto, M., Battistelli, G., Scardovi, L.: Active mode observability of switching linear systems. Automatica 43, 1442–1449 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Di Benedetto, M.D., Di Gennaro, S., D’Innocenzo, A.: Discrete state observability of hybrid systems. Int. J. Robust Nonlinear Control 19(14), 1564–1580 (2009)CrossRefGoogle Scholar
  12. 12.
    De Santis, E., Di Benedetto, M.D., Pola, G.: Observability of internal variables in interconnected switching systems. In: Proceedings of the 45th IEEE-CDC, pp. 4121–4126 (2006)Google Scholar
  13. 13.
    DeCarlo, R., Branicky, M., Pettersson, S., Lennartson, B.: Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE 88(7), 1069–1082 (2000)CrossRefGoogle Scholar
  14. 14.
    De Santis, E., Di Benedetto, M.D.: Observability and observer-based control of hybrid systems. Int. J. Robust Nonlinear Control 19(14), 1519–1520 (2009)Google Scholar
  15. 15.
    Djemai, M., Barbot, J.P.: singularly perturbed method for the control design of a synchronous motor with its PWM inverter. In: Proceedings of the IEEE-Conference on Control Application, CCA’95, New York, USA (1995)Google Scholar
  16. 16.
    Djemai, M., Manamani, N., Barbot, J.P.: Sliding mode observer for triangular input hybrid system, IFAC World Congres Prague (2005)Google Scholar
  17. 17.
    Fliess, M.: Generalized controller canonical forms for linear and nonlinear dynamics. IEEE Trans. Autom. Control 35(9), 994–1000 (1990)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fliess, M., Join, C., Mboup, M., Sira-ramirez, H.: Analyse et représentation de signaux transitoires : application a la compression, au débruitage et á la détection de ruptures. In: Proceedings of the IEEE 20éme colloque sur le traitement du signal et de l’image GRETSI, Louvain-la-Neuve, Belgique (2005)Google Scholar
  19. 19.
    Gauthier, J.P., Bornanrd, G.: Observability for any u(t) of a class of nonlinear systems. IEEE Trans. Autom. Control 26(4), 922–926 (1981)CrossRefMATHGoogle Scholar
  20. 20.
    Hermann, R., Krener, A.J.: Nonlinear controllability and observability. IEEE Trans. Autom. Control 22(9), 728–740 (1977)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hirschorn, R.M.: Invertibilty of multivariable nonlinear control systems. IEEE Trans. Autom. Control 24, 855–865 (1979)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Levant, A.: Robust Exact Differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Menold, P.H., Findeisen, R., Allgöwer, F.: The peaking Finite time convergent observers for linear time-varying systems. In: Proceedings of the 42nd IEEE-CDC Conference. Maui, Hawaii USA, December 2003Google Scholar
  24. 24.
    Mansouri, B., Manamanni, N., Guelton, K., Djemai, M.: Robust pole placement controller design in LMI region for uncertain and disturbed switched systems. Nonlinear Anal.: Hybrid Syst. (Elsevier) 2(4), 1136–1143 (2008)MathSciNetMATHGoogle Scholar
  25. 25.
    Pettersson, S., Lennartson, B.: Hybrid system stability and robustness verification using linear matrix inequalities. Int. J. Control 75(16/17), 1335–1355 (2002)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Saadaoui, H., Manamanni, N., Djemaï, M., Barbot, J.P., Floquet, T.: Exact differentiation and sliding mode observer for switched Lagrangian systems. In: Proceedings of the Nonlinear Analysis: Theory, Methods and Appl., Elsevier, vol. 6, pp. 1050–1069 (2005)Google Scholar
  27. 27.
    Vidal, R., Chiuso, A., Soatto, S., Sastry, S.S.: Observability of linear hybrid systems. In: Proceedings of the Hybrid Systems: Comput. and Cont. Lecture Notes in Computer Science, Springer Verlag, vol. 2623, pp. 526–539 (2003)Google Scholar
  28. 28.
    Vas, P.: Sensorless Vector and Direct Torque Control (Monographs in Electrical and Electronic Engineering). Oxford Science Publications, Oxford (1998)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Noureddine Manamanni
    • 1
  • Mohamed Djemai
    • 2
  • Jean Pierre Barbot
    • 3
  1. 1.CReSTIC EA3804Université de Reims Champagne ArdenneReimsFrance
  2. 2.LAMIH, CNRS UMR 8201University of Valenciennes and Hainaut-CambresisValenciennesFrance
  3. 3.ENSEACergy-PontoiseFrance

Personalised recommendations