Skip to main content

Effective algorithms for parametrizing linear control systems over Ore algebras


In this paper, we study linear control systems over Ore algebras. Within this mathematical framework, we can simultaneously deal with different classes of linear control systems such as time-varying systems of ordinary differential equations (ODEs), differential time-delay systems, underdetermined systems of partial differential equations (PDEs), multidimensional discrete systems, multidimensional convolutional codes, etc. We give effective algorithms which check whether or not a linear control system over some Ore algebra is controllable, parametrizable, flat or π-free.

This is a preview of subscription content, access via your institution.


  1. Adams, W.W., Loustaunau, P.: An Introduction to Gröbner Bases. American Mathematical Society, Providence 1994

  2. Becker, T., Weispfenning, V.: Gröbner Bases. A Computational Approach to Commutative Algebra. Springer, New York, 1993

  3. Bender, C.M., Dunne, G.V., Mead, L.R.: Underdetermined systems of partial differential equations. J. of Math. Phys. 41, 6388–6398 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bourbaki, N.: Algèbre, Chap. 10, Algèbre homologique. Masson, Paris, 1980

  5. Chyzak, F.: Fonctions holonomes en calcul formel. PhD thesis. Ecole Polytechnique (27/05/1998)

  6. Chyzak, F.: Mgfun project.

  7. Chyzak, F., Salvy, B.: Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Computation 26, 187–227 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chyzak, F., Quadrat, A., Robertz, D.: OreModules project.

  9. Chyzak, F., Quadrat, A., Robertz, D.: Linear control systems over Ore algebras: Effective algorithms for the computation of parametrizations. In: CDRom of the Workshop on Time-Delay Systems (TDS03), IFAC Workshop, INRIA Rocquencourt (France) (08-10/09/03)

  10. Chyzak, F., Quadrat, A., Robertz, D.: OreModules: A symbolic package for the study of multidimensional linear systems. In: Proceedings of MTNS04, Leuven (Belgium) (05-09/07/04)

  11. Cotroneo, T.: Algorithms in Behavioral Systems Theory. PhD thesis. University of Groningen (The Netherlands) (18/05/2001)

  12. Fliess, M., Mounier, H.: Controllability and observability of linear delay systems: an algebraic approach. ESAIM COCV 3, 301–314 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kredel, H.: Solvable Polynomial Rings. Shaker, Aachen, 1993

  14. La Scala, R., Stillman, M.: Strategies for computing minimal free resolutions. J. Symbolic Computation 26, 409–431 (1998)

    Article  MATH  Google Scholar 

  15. Landau, L., Lifchitz, E.: Physique théorique. Tome 2: Théorie des champs. MIR, Moscow, 1989

  16. Le Vey, G.: Some remarks on solvability and various indices for implicit differential equations. Numer. Algorithms 19, 127–145 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. McConnell, J. C., Robson, J. C.: Noncommutative Noetherian Rings. American Mathematical Society, Providence, 2000

  18. Oberst, U.: Multidimensional constant linear systems. Acta Appl. Math. 20, 1–175 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Manitius, A.: Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulations. IEEE Trans. Autom. Contr. 29, 1058–1068 (1984)

    Article  MathSciNet  Google Scholar 

  20. Mounier, H.: Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques. PhD Thesis. University of Orsay, France, 1995

  21. Mounier, H., Rudolph, J., Fliess, M., Rouchon, P.: Tracking control of a vibrating string with an interior mass viewed as delay system. ESAIM COCV 3, 315–321 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Pillai, H. K., Shankar, S.: A behavioral approach to control of distributed systems. SIAM J. Control & Optimization 37, 388–408 (1998)

    Article  MathSciNet  Google Scholar 

  23. Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory. A Behavioral Approach. TAM 26. Springer, New York, 1998

  24. Pommaret, J.-F.: Dualité différentielle et applications. C. R. Acad. Sci. Paris, Série I 320, 1225–1230 (1995)

    MATH  MathSciNet  Google Scholar 

  25. Pommaret, J.-F.: Partial Differential Control Theory. Kluwer, Dordrecht, 2001

  26. Pommaret, J.-F., Quadrat, A.: Localization and parametrization of linear multidimensional control systems. Systems & Control Letters 37, 247–260 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pommaret, J.-F., Quadrat, A.: Algebraic analysis of linear multidimensional control systems. IMA J. Control and Optimization 16, 275–297 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Pommaret, J.-F., Quadrat, A.: Equivalences of linear control systems. In: Proceedings of MTNS00, Perpignan (France), available at

  29. Pommaret, J.-F., Quadrat, A.: A functorial approach to the behaviour of multidimensional control systems. Appl. Math. and Computer Science 13, 7–13 (2003)

    MATH  MathSciNet  Google Scholar 

  30. Quadrat, A.: Analyse algébrique des systèmes de contrôle linéaires multidimensionnels. PhD thesis. Ecole Nationale des Ponts et Chaussées (France) (23/09/1999)

  31. Quadrat, A.: Extended Bézout identities. In: Proceedings of ECC01, Porto (Portugal), available at

  32. Quadrat, A., Robertz, D.: Parametrizing all solutions of uncontrollable multidimensional linear systems. In Proceedings of the 16th IFAC World Congress, Prague (Czech Republic) (04-08/07/05)

  33. Quadrat, A., Robertz, D.: On the blowing-up of stably free behaviours. To appear in the proceedings of CDC-ECC05, Seville (Spain) (12-15/12/05)

  34. Quadrat, A., Robertz, D.: Constructive computation of flat outputs of multidimensional linear systems. Submitted to MTNS06, Kyoto (Japan) (24-28/07/06)

  35. Rocha, P.: Structure and Representation of 2-D systems. PhD thesis, University of Groningen (The Netherlands), 1990

  36. Rotman, J.J.: An Introduction to Homological Algebra. Academic Press, New York, 1979

  37. Salamon, D.: Control and Observation of Neutral Systems. Pitman, London, 1984

  38. Seiler, W.M.: Involution analysis of the partial differential equations characterising Hamiltonian vector fields. J. Math. Phys. 44, 1173–1182 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wood, J.: Modules and behaviours in nD systems theory. Multidimensional Systems and Signal Processing 11, 11–48 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zerz, E.: Topics in Multidimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences 256. Springer, London, 2000

  41. Zerz, E.: An algebraic analysis approach to linear time-varying systems, to appear in IMA J. Mathematical Control & Information.

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. Quadrat.

Additional information

This paper is dedicated to the memory of our dear friend and colleague Manuel Bronstein.

The third author has been financially supported by the Control Training Site grant HPMT-CT-2001-00278 and the Deutsche Forschungsgemeinschaft during his stays at INRIA Sophia Antipolis.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chyzak, F., Quadrat, A. & Robertz, D. Effective algorithms for parametrizing linear control systems over Ore algebras. AAECC 16, 319–376 (2005).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • Linear systems over Ore algebras
  • Parametrization
  • Flatness
  • Constructive algorithms
  • Non-commutative Gröbner bases
  • Module theory

Mathematics Subject Classification (2000)

  • 93C05
  • 93B25
  • 16E30
  • 68W30
  • 13P10