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.
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References
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.
Bitmead, R. R., Gevers, M., and Wetz, V.: Adaptive Optimal Control, Prentice Hall, 1990.
Camacho, E. F. and Bordons, C.: Model Predictive Control in the Process Industry, Springer Verlag, 1994.
Gardeñes, E. and Mielgo, H.: Modal Intervals: Functions, in: Proc. Polish Symposium on Interval and Fuzzy Mathematics, 1986, pp. 39-58.
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.
Gardeñes, E., Mielgo, H., and Trepat, A.: Modal Intervals: Reasons and Ground Semantics, in: Interval Mathematics 1985, Springer, Berlin, 1985, pp. 27-35.
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.
Hansen, E.: Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.
Isermann, R.: Digital Control Systems, Springer-Verlag, 1989.
Malan, S., Milanese, M., and Taragna, M.: Robust Analysis and Design of Control Systems Using Interval Arithmetic, Automatica 33(7) (1997), pp. 1363-1372.
Martín-Sánchez, J. M. and Rodellar, J.: Adaptive Predictive Control, Prentice Hall, 1995.
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.
Morari, M. and Zafiriou, E.: Robust Process Control, Prentice-Hall, 1989.
SIGLA/X: 1999, Modal Intervals, in: Prep. Workshop on Applications of Interval Analysis to Systems and Control. MISC99, Girona, Spain, pp. 139-210.
Vehí, J.: Analysis and Design of Robust Controllers by Means of Modal Intervals, PhD thesis, Universitat de Girona, Spain, 1998 (in Catalan).
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.
Walter, E. and Jaulin, L.: Guaranteed Characterization of Stability Domains via Set Inversion, IEEE Trans. on Automatic Control 35 (1994), pp. 835-841.
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.
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Vehí, J., Rodellar, J., Sainz, M. et al. Analysis of the Robustness of Predictive Controllers via Modal Intervals. Reliable Computing 6, 281–301 (2000). https://doi.org/10.1023/A:1009982530323
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DOI: https://doi.org/10.1023/A:1009982530323