Mathematics in Computer Science

, Volume 8, Issue 3–4, pp 391–406 | Cite as

Determination of Set-Membership Identifiability Sets

  • Laleh Ravanbod
  • Nathalie Verdière
  • Carine Jauberthie
Article
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Abstract

This paper concerns the concept of set-membership identifiability introduced in Jauberthie et al. (Proceedings of the 18th IFAC World Congress. Milan, Italie, 12024–12029, 2011). Given a model, a set-membership identifiable set is a connected set in the parameter domain of the model such that its corresponding trajectories are distinct to trajectories arising from its complementary. For obtaining the so-called set-membership identifiable sets, we propose an algorithm based on interval analysis tools. The proposed algorithm is decomposed into three parts namely mincing, evaluating and regularization (Jaulin et al. in Applied interval analysis, with examples in parameter and state estimation, robust control and robotics. Springer, Londres, 2001). The latter step has been modified in order to obtain guaranteed set-membership identifiable sets. Our algorithm will be tested on two examples.

Keywords

Set-membership identifiability Interval analysis 

Mathematics Subject Classification

65G40 

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References

  1. 1.
    Bourbaki N.: Elements of Mathematics. Springer, Berlin (1989)Google Scholar
  2. 2.
    Braems, I., Jaulin, L., Kieffer, M., Walter, E.: Guaranteed numerical alternatives to structural identiability testing. In: Proceedings of the 40th IEEE CDC, pp. 3122–3127. Orlando, USA (2001)Google Scholar
  3. 3.
    Chabert G., Jaulin L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Jauberthie, C., Verdière, N., Travé-Massuyès, L.: Set-membership identifiability: definitions and analysis. In Proceedings of the 18th IFAC World Congress, pp. 12024–12029. Milan, Italie (2011)Google Scholar
  5. 5.
    Jauberthie C., Verdière N., Travé-Massuyès L.: Fault detection and identification relying on set-membership identifiability. Ann. Rev. Control 73(1), 129–136 (2013)CrossRefGoogle Scholar
  6. 6.
    Jaulin L., Kieffer M., Didrit O., Walter E.: Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, Londres (2001)MATHGoogle Scholar
  7. 7.
    Kolchin E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)MATHGoogle Scholar
  8. 8.
    Lagrange S., Delanoue N., Jaulin L.: On sufficient conditions of injectivity, development of a numerical test via interval analysis. J. Reliable Comput. 13, 409–421 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Lagrange S., Delanoue N., Jaulin L.: Injectivity analysis using interval analysis: application to structural identifiability. Automatica 44(11), 2959–2962 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Ljung L., Glad T.: On global identifiability for arbitrary model parametrizations. Automatica 30, 265–276 (1994)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Milanese M., Vicino A.: Estimation theory for nonlinear models and set membership uncertainty. Automatica 27(2), 403–408 (1991)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Munkres J.R.: Topology a first courses. Prentice Hall, New Jersey (1975)Google Scholar
  13. 13.
    Raıssi T., Ramdani N., Candau Y.: Set-membership state and parameter estimation for systems described by nonlinear differential equations. Automatica 40(10), 1771–1777 (2004)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Verdière N., Denis-Vidal L., Joly-Blanchard G., Domurado D.: Identifiability and estimation of pharmacokinetic parameters of ligands of macrophage mannose receptor. Int. J. Appl. Math. Comput. Sci 15, 101–110 (2005)Google Scholar
  15. 15.
    Verdière, N., Jauberthie, C., Travé-Massuyès, L.: Set-membership identifiability of nonlinear models. LAAS Report, Number 13001, p. 8 (2013)Google Scholar
  16. 16.
    Walter E., Norton J., Piet-Lahanier H.H., Milanese M.: Bounding Approaches to System Identification. Perseus Publishing, New York (1996)MATHGoogle Scholar
  17. 17.
    Walter E., Piet-Lahanier H.: Estimation of parameter bounds from bounded-error data: a survey. Math. Comput. Simul. 32(5), 449–468 (1990)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Laleh Ravanbod
    • 1
  • Nathalie Verdière
    • 2
  • Carine Jauberthie
    • 3
    • 4
  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  2. 2.Université du HavreLe Havre CedexFrance
  3. 3.CNRS, LAASToulouseFrance
  4. 4.Université de Toulouse, UPS, LAASToulouseFrance

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