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On the Effective Computation of Stabilizing Controllers of 2D Systems

  • Yacine Bouzidi
  • Thomas Cluzeau
  • Alban QuadratEmail author
  • Fabrice Rouillier
Conference paper
  • 43 Downloads
Part of the Communications in Computer and Information Science book series (CCIS, volume 1125)

Abstract

In this paper, we show how stabilizing controllers for 2D systems can effectively be computed based on computer algebra methods dedicated to polynomial systems, module theory and homological algebra. The complete chain of algorithms for the computation of stabilizing controllers, implemented in Maple, is illustrated with an explicit example.

Keywords

Multidimensional systems theory 2D systems Stability analysis Stabilization Computation of stabilizing controllers Polynomial systems Module theory Homological algebra 

References

  1. 1.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2006).  https://doi.org/10.1007/3-540-33099-2CrossRefzbMATHGoogle Scholar
  2. 2.
    Bouzidi, Y., Quadrat, A., Rouillier, F.: Computer algebra methods for testing the structural stability of multidimensional systems. In: Proceedings of the IEEE 9th International Workshop on Multidimensional (nD) Systems (2015)Google Scholar
  3. 3.
    Bouzidi, Y., Rouillier, F.: Certified Algorithms for proving the structural stability of two-dimensional systems possibly with parameters. In: Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS 2016) (2016)Google Scholar
  4. 4.
    Bouzidi, Y., Quadrat, A., Rouillier, F.: Certified non-conservative tests for the structural stability of discrete multidimensional systems. Multidimens. Syst. Sig. Process. 30(3), 1205–1235 (2019)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bouzidi, Y., Cluzeau, T., Moroz, G., Quadrat, A.: Computing effectively stabilizing controllers for a class of \(n\)D systems. In: Proceedings of IFAC 2017 Workshop Congress (2017)Google Scholar
  6. 6.
    Bouzidi, Y., Cluzeau, T., Quadrat, A.: On the computation of stabilizing controllers of multidimensional systems. In: Proceedings of Joint IFAC Conference 7th SSSC 2019 and 15th TDS 2019 (2019)CrossRefGoogle Scholar
  7. 7.
    Bridges, D., Mines, R., Richman, F., Schuster, P.: The polydisk Nullstellensatz. Proc. Am. Math. Soc. 132(7), 2133–2140 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chyzak, F., Quadrat, A., Robertz, D.: Effective algorithms for parametrizing linear control systems over Ore algebras. Appl. Algebra Engrg. Comm. Comput. 16, 319–376 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chyzak, F., Quadrat, A., Robertz, D.: OreModules: a symbolic package for the study of multidimensional linear systems. In: Chiasson, J., Loiseau, J.J. (eds.) Applications of Time Delay Systems. LNCIS, vol. 352, pp. 233–264. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-49556-7_15CrossRefzbMATHGoogle Scholar
  10. 10.
    Curtain, R.F., Zwart, H.J.: An Introduction to Infinite-Dimensional Linear Systems Theory. TAM, vol. 21. Springer, New York (1995).  https://doi.org/10.1007/978-1-4612-4224-6CrossRefzbMATHGoogle Scholar
  11. 11.
    Decarlo, R.A., Murray, J., Saeks, R.: Multivariable Nyquist theory. Int. J. Control 25(5), 657–675 (1977)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Desoer, C.A., Liu, R.W., Murray, J., Saeks, R.: Feedback system design: the fractional representation approach to analysis and synthesis. IEEE Trans. Automat. Control 25, 399–412 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Eisenbud, D.: Commutative Algebra: with a View Toward Algebraic Geometry. GTM, vol. 150. Springer, New York (1995).  https://doi.org/10.1007/978-1-4612-5350-1CrossRefzbMATHGoogle Scholar
  14. 14.
    Li, X., Saito, O., Abe, K.: Output feedback stabilizability and stabilization algorithms for 2D systems. Multidimension. Syst. Sig. Process. 5, 41–60 (1994)CrossRefGoogle Scholar
  15. 15.
    Li, L., Lin, Z.: Stability and stabilisation of linear multidimensional discrete systems in the frequency domain. Int. J. Control 86(11), 1969–1989 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lin, Z.: Output feedback stabilizability and stabilization of linear \(n\)D systems. In: Galkowski, K., Wood, J. (eds.) Multidimensional Signals, Circuits and Systems, pp. 59–76. Taylor & FrancisGoogle Scholar
  17. 17.
    Lin, Z., Lam, J., Galkowski, K., Xu, S.: A constructive approach to stabilizability and stabilization of a class of \(nD\) systems. Multidimension. Syst. Sig. Process. 12, 329–343 (2001)CrossRefGoogle Scholar
  18. 18.
    Lin, Z.: Feedback stabilizability of MIMO \(n\)D linear systems. Multidimension. Syst. Sig. Process. 9, 149–172 (1998)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lin, Z.: Feedback stabilization of MIMO \(n\)D linear systems. IEEE Trans. Autom. Control 45, 2419–2424 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Quadrat, A.: The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. Part I: (weakly) doubly coprime factorizations. SIAM J. Control Optim. 42, 266–299 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Quadrat, A.: The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. Part II: internal stabilization. SIAM J. Control. Optim. 42, 300–320 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Quadrat, A.: An introduction to constructive algebraic analysis and its applications. Les cours du CIRM 1(2), 281–471 (2010). Journées Nationales de Calcul Formel. INRIA Research Report n. 7354MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rotman, J.J.: An Introduction to Homological Algebra, 2nd edn. Springer, New York (2009).  https://doi.org/10.1007/b98977CrossRefzbMATHGoogle Scholar
  24. 24.
    Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. Appl. Algebra Eng. Commun. Comput. 9, 433–461 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rouillier, F.: Algorithmes pour l’étude des solutions réelles des systèmes polynomiaux. Habilitation, University of Paris 6 (2007)Google Scholar
  26. 26.
    Strintzis, M.: Tests of stability of multidimensional filters. IEEE Trans. Circ. Syst, 24, 432–437 (1977)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Vidyasagar, M.: Control System Synthesis: A Factorization Approach. MIT Press, Cambridge (1985)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Yacine Bouzidi
    • 1
  • Thomas Cluzeau
    • 2
  • Alban Quadrat
    • 3
    Email author
  • Fabrice Rouillier
    • 3
  1. 1.Inria Lille - Nord EuropeVilleneuve d’AscqFrance
  2. 2.CNRS, XLIM UMR 7252Limoges CedexFrance
  3. 3.Inria Paris, Institut de Mathématiques de Jussieu Paris-Rive Gauche, Sorbonne Université, Paris UniversitéParisFrance

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