Numerical Algorithms

, Volume 37, Issue 1–4, pp 187–198 | Cite as

Guaranteed Nonlinear State Estimator for Cooperative Systems

  • Michel Kieffer
  • Eric Walter


This paper is about state estimation for continuous-time nonlinear models, in a context where all uncertain variables can be bounded. More precisely, cooperative models are considered, i.e., models that satisfy some constraints on the signs of the entries of the Jacobian of their dynamic equation. In this context, interval observers and a guaranteed recursive state estimation algorithm are combined to enclose the state at any given instant of time in a subpaving. The approach is illustrated on the state estimation of a waste-water treatment process.

bounded errors cooperative models interval observer nonlinear models set estimator state estimator 


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  1. [1]
    V. Alcaraz-González, A. Genovesi, J. Harmand, A. González, A. Rapaport and J. Steyer, Robust exponential nonlinear interval observer for a class of lumped models useful in chemical and biochemical engineering. Application to a wastewater treatment process, in: Proc. of MISC'99 Workshop on Applications of Interval Analysis to Systems and Control, Girona, 24–26 February 1999, pp. 225–235.Google Scholar
  2. [2]
    F.L. Chernousko, State Estimation for Dynamic Systems (CRC Press, Boca Raton, 1994).Google Scholar
  3. [3]
    A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, 1974).Google Scholar
  4. [4]
    J.L. Gouzé, A. Rapaport and Z.M. Hadj-Sadok, Interval observers for uncertain biological systems, J. Ecological Modelling 133 (2000) 45–56.Google Scholar
  5. [5]
    Z.M. Hadj-Sadok, Modélisation et estimation dans les bioréacteurs; prise en compte des incertitudes: application au traitement de l'eau, Ph.D. thesis, Université de Nice, Sophia Antipolis (1999).Google Scholar
  6. [6]
    Z.M. Hadj-Sadok, J.L. Gouzé and A. Rapaport, State observers for uncertain models of activated sludge processes, in: CD-ROM of the IFAC-EurAgEng Internat. Workshop on Decision and Control in Waste Bio-Processing, Narbonne, 25–27 February 1998.Google Scholar
  7. [7]
    J. Hoefkens, M. Berz and K. Makino, Efficient high-order methods for ODEs and DAEs, in: Automatic Differentiation: From Simulation to Optimization, eds. G. Corliss, C. Faure and A. Griewank (Springer, NewYork, 2001) pp. 341–351.Google Scholar
  8. [8]
    J. Hoefkens, M. Berz and K. Makino, Verified high-order integration of DAEs and ODEs, in: Scientific Computing, Validated Numerics, Interval Methods, eds. W. Kraemer and J.W. von Gudenberg(Kluwer, Boston, 2001) pp. 281–292.Google Scholar
  9. [9]
    L. Jaulin, Nonlinear bounded-error state estimation of continuous-time systems, Automatica 38 (2002) 1079–1082.Google Scholar
  10. [10]
    L. Jaulin, M. Kieffer, I. Braems and E. Walter, Guaranteed nonlinear estimation using constraint propagation on sets, Internat. J. Control 74(18) (2001) 1772–1782.Google Scholar
  11. [11]
    L. Jaulin, M. Kieffer, O. Didrit and E. Walter, Applied Interval Analysis (Springer, London, 2001).Google Scholar
  12. [12]
    L. Jaulin and E. Walter, Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis, Math. Comput. Simulation 35(2) (1993) 123–137.Google Scholar
  13. [13]
    L. Jaulin and E. Walter, Set inversion via interval analysis for nonlinear bounded-error estimation, Automatica 29(4) (1993) 1053–1064.Google Scholar
  14. [14]
    M. Kieffer, Estimation ensembliste par analyse par intervalles, application à la localisation d'un véhicule, Ph.D. thesis, Université Paris-Sud, Orsay, France (1999).Google Scholar
  15. [15]
    M. Kieffer, L. Jaulin, I. Braems and E. Walter, Guaranteed set computation with subpavings, in: Scientific Computing, Validated Numerics, Interval Methods, eds. W. Kraemer and J.W. von Gudenberg (Kluwer, Boston, 2001) pp. 167–178.Google Scholar
  16. [16]
    M. Kieffer, L. Jaulin and E. Walter, Guaranteed recursive nonlinear state bounding using interval analysis, Internat. J. Adaptative Control Signal Process. 6(3) (2002) 193–218.Google Scholar
  17. [17]
    A. Kurzhanski and I. Valyi, Ellipsoidal Calculus for Estimation and Control (Birkhäuser, Boston, MA, 1997).Google Scholar
  18. [18]
    R. Lohner, Enclosing the solutions of ordinary initial and boundary value problems, in: Computer Arithmetic: Scientific Computation and Programming Languages, eds. E. Kaucher, U. Kulisch and C. Ullrich (BG Teubner, Stuttgart, 1987) pp. 255–286.Google Scholar
  19. [19]
    R. Lohner, Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value-problem, in: Computational Ordinary Differential Equations, eds. J.R. Cash and I. Gladwell (Clarendon Press, Oxford, 1992) pp. 425–435.Google Scholar
  20. [20]
    M. Milanese, J. Norton, H. Piet-Lahanier and E. Walter, eds., Bounding Approaches to System Identification (Plenum Press, New York, 1996).Google Scholar
  21. [21]
    N.S. Nedialkov and K.R. Jackson, Methods for initial value problems for ordinary differential equations, in: Perspectives on Enclosure Methods (Springer, Vienna, 2001) pp. 219–264.Google Scholar
  22. [22]
    H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41 (Amer. Math. Soc., Providence, 1995).Google Scholar
  23. [23]
    G. Stephanopoulos, Chemical Process Control. An Introduction to Theory and Practice (Prentice-Hall, Englewood Cliffs, 1984).Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Michel Kieffer
    • 1
  • Eric Walter
    • 1
  1. 1.LSS, CNRS-Supélec-Université Paris-Sud Plateau de MoulonGif-sur-YvetteFrance

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