Reliable Computing

, Volume 6, Issue 3, pp 281–301 | Cite as

Analysis of the Robustness of Predictive Controllers via Modal Intervals

  • Josep Vehí
  • José Rodellar
  • Miguel Sainz
  • Joaquim Armengol
Article

Abstract

This paper aims to start exploring the application of interval techniques to deal with robustness issues in the context of predictive control. The robust stability problem is transformed into that of checking the positivity of a rational function. Modal intervals are presented as a useful tool to deal with this kind of function.

Modal interval analysis extends real numbers to intervals, identifying the intervals by the predicates that the real numbers fulfill, unlike classical interval analysis which identifies the intervals with the set of real numbers that they contain. Modal interval analysis not only simplifies the computation of interval functions but also allows semantic interpretations of the results. These interpretations are applied to the analysis and design of robust predictive controllers for parametric systems. Necessary, sufficient and, in some cases, necessary and sufficient conditions for robust performance are presented.

Specifically, an interval procedure is proposed to compute the stability margin of a predictive control law when applied to a class of plants described by discrete time transfer functions with coefficients that depend polynomially on uncertain parameters.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balakrishnan, V. and Boyd, S.: Global Optimization in Control Systems Analysis and Design, in: Leondes, C. (ed.), Control and Dynamic Systems: Advances in Theory and Applications, Vol. 53, 1992.Google Scholar
  2. 2.
    Bitmead, R. R., Gevers, M., and Wetz, V.: Adaptive Optimal Control, Prentice Hall, 1990.Google Scholar
  3. 3.
    Camacho, E. F. and Bordons, C.: Model Predictive Control in the Process Industry, Springer Verlag, 1994.Google Scholar
  4. 4.
    Gardeñes, E. and Mielgo, H.: Modal Intervals: Functions, in: Proc. Polish Symposium on Interval and Fuzzy Mathematics, 1986, pp. 39-58.Google Scholar
  5. 5.
    Gardeñes, E., Mielgo, H., and Sainz, M.: Presentation of the Research Group SIGLA/X, Research report IMA 95-10, Dept. IMA. University of Girona, Spain, 1995.Google Scholar
  6. 6.
    Gardeñes, E., Mielgo, H., and Trepat, A.: Modal Intervals: Reasons and Ground Semantics, in: Interval Mathematics 1985, Springer, Berlin, 1985, pp. 27-35.Google Scholar
  7. 7.
    Garloff, J. and Graf, B.: Solving Strict Polynomial Inequalities by Bernstein Expansion, in: Munro, N. (ed.), The Use of Symbolic Methods in Control System Analysis and Design, IEE London, Chapt. 14, 1999, pp. 339-351.Google Scholar
  8. 8.
    Hansen, E.: Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.Google Scholar
  9. 9.
    Isermann, R.: Digital Control Systems, Springer-Verlag, 1989.Google Scholar
  10. 10.
    Malan, S., Milanese, M., and Taragna, M.: Robust Analysis and Design of Control Systems Using Interval Arithmetic, Automatica 33(7) (1997), pp. 1363-1372.Google Scholar
  11. 11.
    Martín-Sánchez, J. M. and Rodellar, J.: Adaptive Predictive Control, Prentice Hall, 1995.Google Scholar
  12. 12.
    Martín-Sánchez, J. M. and Rodellar, J.: Adaptive Predictive Control: Limits of Stability, Int. Journal of Adaptive Control and Signal Processing 11 (1997), pp. 263-283.Google Scholar
  13. 13.
    Morari, M. and Zafiriou, E.: Robust Process Control, Prentice-Hall, 1989.Google Scholar
  14. 14.
    SIGLA/X: 1999, Modal Intervals, in: Prep. Workshop on Applications of Interval Analysis to Systems and Control. MISC99, Girona, Spain, pp. 139-210.Google Scholar
  15. 15.
    Vehí, J.: Analysis and Design of Robust Controllers by Means of Modal Intervals, PhD thesis, Universitat de Girona, Spain, 1998 (in Catalan).Google Scholar
  16. 16.
    Vehí, J., Armengol, J., Rodellar, J., and Sainz, M.: Using Interval Methods for Control Systems Design in the Parameter Space, in: 7th IFAC Symposium on Computer Aided Control Systems Design, Gent-Belgium, 1997, pp. 371-375.Google Scholar
  17. 17.
    Walter, E. and Jaulin, L.: Guaranteed Characterization of Stability Domains via Set Inversion, IEEE Trans. on Automatic Control 35 (1994), pp. 835-841.Google Scholar
  18. 18.
    Zettler, M. and Garloff, J.: Robustness Analysis of Polynomials with Polynomial Parameter Dependency Using Bernstein Expansion, IEEE Trans. on Aut. Control 43(3) (1998), pp. 425-431.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Josep Vehí
    • 1
  • José Rodellar
    • 2
  • Miguel Sainz
    • 3
  • Joaquim Armengol
    • 4
  1. 1.Institut d'Informàtica i AplicacionsUniversitat de Girona, Campus MontiliviSpain
  2. 2.Dept. of Applied Mathematics IIITechnical University of Catalonia, Campus NordBarcelonaSpain
  3. 3.Dept. of Applied Mathematics & Computer ScienceUniversitat de Girona, Campus MontiliviGironaSpain
  4. 4.Institut d'Informàtica i AplicacionsUniversitat de Girona, Campus MontiliviGironuSpain

Personalised recommendations