Skip to main content
SpringerLink
Log in

Application of Modal Intervals to the Generation of Error-Bounded Envelopes

  • Published:
Reliable Computing

Abstract

An interval model can express the imprecision and the uncertainty associated to the modeling of a system. The result of the simulation of one of these models can be represented in the form of envelope trajectories. These envelopes can be characterized by several properties such as completeness or soundness, that lead to the concepts of overbounded and underbounded envelopes. The simulation of such interval models can be performed by several means including qualitative, semiqualitative and quantitative methods. A brief description of the different types of simulators is presented and their respective properties are outlined and compared in relation to model-based fault detection. Whereas the existing simulators do not provide any information about the "error" with respect to the exact envelope, a method to obtain error-bounded envelopes is proposed. It is based on the simultaneous computation of an underbounded and an overbounded envelope by means of Modal Interval Analysis. A way of controlling the error of the envelopes and adjusting it to a desired specified value is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Armengol, J., Travé-Massuyès, L., Vehí, J., and de La Rosa, J. Ll.: A Survey on Interval Model Simulators and Their Properties Related to Fault Detection, in: 14th IFAC World Congress 1999, Beijing, China, 1999, Vol. O, pp. 511-519.

    Google Scholar 

  2. Armengol, J., Travé-Massuyès, L., Vehí, J., and de La Rosa, J. Ll.: On Modal Interval Analysis for Envelope Determination within the Ca∼En Qualitative Simulator, in: 7th Information Processing and Management of Uncertainty in Knowledge-Based Systems Conference IPMU 98, Paris, France, 1998, pp. 110-117.

  3. Berleant, D. and Kuipers, B. J.: Recent Advances in Qualitative Physics, chapter Qualitative and Numerical Simulation with Q3, The MIT Press, 1992, pp. 3-16.

  4. Bonarini, A. and Bontempi, G.: AQualitative Simulation Approach for Fuzzy Dynamical Models, ACM Transactions on Modeling and Computer Simulation 4 (4) (1994), pp. 258-313.

    Google Scholar 

  5. Bontempi, G.: Qua.Si. III: A Software Tool for Simulation of Fuzzy Dynamical Systems, in: Proceedings of the European Simulation Multiconference ESM 96, Ghent, Belgium, 1996, pp. 615-619.

  6. Bousson, K. and Travé-Massuyès, L.: Putting More Numbers in the Qualitative Simulator Ca∼En, in: 2nd International Conference on Intelligent Systems Engineering ISE 94, Hamburg-Harburg, Germany, 1994, pp. 62-69.

  7. Chen, J. and Patton, R. J.: Robust Model-Based Fault Diagnosis for Dynamic Systems, Kluwer Academic Publishers, 1998.

  8. Coghill, G.: Mycroft: A Framework for Constraint Based Fuzzy Qualitative Reasoning, PhD thesis, Heriot-Watt University, Edinburgh, 1996.

    Google Scholar 

  9. Gardeñes, E. and Mielgo, H.: Modal Intervals: Functions, in: Polish Symposium on Interval and Fuzzy Mathematics, Poznan, Poland, 1986.

  10. Gardeñes, E., Mielgo, H., and Trepat, A.: Modal Intervals: Reasons and Ground Semantics, Lecture Notes in Computer Science 212 (1986), pp. 27-35.

    Google Scholar 

  11. Gardeñes, E., Sainz, M. Á., Jorba, L., Calm, R., Estela, R., Mielgo, H., and Trepat, A.: Modal Intervals, this volume, pp. 77-111.

  12. Gasca, R. M., Ortega, J. A., and Toro, M.: SimulaciÓn semicualitativa, in: Jornades Hispano-Franceses de Sistemes Intel·ligents i Control Avanc¸at, Barcelona, Catalonia, Spain, 1996.

  13. Hansen, E.: Global Optimization Using Interval Analysis, Marcel Dekker, 1992.

  14. Kay, H.: Refining Imprecise Models and Their Behaviours, PhD thesis, University of Texas at Austin, 1996.

  15. Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, 1996.

  16. Keller, U., Wyatt, T., and Leitch, R. R.: Frensi—a Fuzzy Qualitative Simulation Method, in: Workshop on Applications of Interval Analysis to Systems and Control with Special Emphasis on Recent Advances in Modal Interval Analysis MISC 99, Girona, Catalonia, Spain, 1999, pp. 305-313.

  17. Kuipers, B. J.: Qualitative Reasoning. Modeling and Simulation with Incomplete Knowledge, The MIT Press, 1994.

  18. Kuipers, B. J.: Readings in Qualitative Reasoning about Physical Systems, chapter Qualitative Simulation, Morgan Kaufmann Publishers, 1990, pp. 236-260.

  19. Kuipers, B. J. and Berleant, D.: Using Incomplete Quantitative Knowledge in Qualitative Reasoning, in: AAAI 88, 1988, pp. 324-329.

  20. Lohner, R.: Einschliessung der Lösung gewöhnlicher Anfangs-und Randwertaufgaben, PhD thesis, Universität Karlsruhe, Germany, 1988.

  21. Maple V: mailto:info@maplesoft.com, http://www.maplesoft.com, Waterloo Maple, Inc.

  22. Markov, S. and Angelov, R.: An Interval Method for Systems of ODE, Lecture Notes in Computer Science 212 (1985), pp. 103-108.

    Google Scholar 

  23. Matlab-Simulink: mailto:info@mathworks.com, http://www.mathworks.com, The Math Works, Inc.

  24. Missier, A.: Structures mathematiques pour le calcul qualitatif—Contribution a la simulation qualitative, PhD thesis, INSA, Toulouse, France, 1991.

  25. Montmain, J.: From Diapason Research Program to Its Industrial Application in Nuclear Fuel Reprocessing, in: SAFEPROCESS 97, Kingston upon Hull, UK, 1997, pp. 209-216.

  26. Moore, R. E.: Interval Analysis, Prentice-Hall, 1966.

  27. Saludes, J., Puig, V., and Quevedo, J.: Determination of Window Length for a New Algorithm in Adaptive Threshold Generation, in: Symposium on Qualitative System Modeling, Qualitative Fault Diagnosis and Fuzzy Logic and Control, Budapest, Hungary, 1997.

  28. Shen, Q. and Leitch, R. R.: Fuzzy Qualitative Simulation, IEEE Transactions on Systems, Man and Cybernetics 23 (4) (1993), pp. 1038-1061.

    Google Scholar 

  29. SIGLA/X: Applications of Interval Analysis to Systems and Control, chapter Modal Intervals, Universitat de Girona, Catalonia, 1999, pp. 157-227.

  30. Struss, P.: Readings in Qualitative Reasoning about Physical Systems, chapter Problems of Interval-Based Qualitative Reasoning, Morgan Kaufmann Publishers, 1990, pp. 288-305.

  31. Travé-Massuyès, L., Dague, Ph., and Guerrin, F.: Le raisonnement qualitatif pour les sciences de l'ingénieur, Editions Hermes, 1997.

  32. Vehí, J.: Anàlisi i disseny de controladors robustos mitjançant intervals modals, PhD thesis, Universitat de Girona, Catalonia, 1998.

  33. Vehí, J., Ferrer, I., and Armengol, J.: Using Interval and Symbolic Computations to Deal with Uncertain Dynamic Systems, in: 1st NICONET Workshop on Numerical Software in Control Engineering, València, Spain, 1998, p. 13.

  34. Vescovi, M. R., Farquhar, A., and Iwasaki, Y.: Numerical Interval Simulation, in: International Joint Conference on Artificial Intelligence IJCAI 95, Montreal, Quebec, Canada, 1995, pp. 1806-1813.

  35. Wiegand, M. E.: Constructive Qualitative Simulation of Continuous Dynamic Systems, PhD thesis, Heriot-Watt University, Edinburgh, 1991.

Download references

Author information

Authors and Affiliations

  1. Institut d'Informàtica i Aplicacions, Universitat de Girona & LEA-SICA, Campus de Montilivi, E-17071, Girona, Catalonia, Spain

    Joaquim Armengol & Josep Vehí

  2. LAAS-CNRS & LEA-SICA, 7, Avenue du Colonel Roche, F-31077, Toulouse, France

    Louise Travé-Massuyès

  3. Departament d'Informàtica i Matemàtica Aplicada, Universitat de Girona, Campus de Montilivi, E-17071, Girona, Catalonia, Spain

    Miguel Ángel Sainz

Authors
  1. Joaquim Armengol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josep Vehí
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Louise Travé-Massuyès
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Miguel Ángel Sainz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Armengol, J., Vehí, J., Travé-Massuyès, L. et al. Application of Modal Intervals to the Generation of Error-Bounded Envelopes. Reliable Computing 7, 171–185 (2001). https://doi.org/10.1023/A:1011426300135

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011426300135

Keywords

Search

Navigation