Flatness Characterization: Two Approaches

  • Felix Antritter
  • Jean Lévine
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 407)


We survey two approaches to flatness necessary and sufficient conditions and compare them on examples.


Meromorphic Function Nonlinear Control System Distribute Parameter System Control Letter Principal Ideal Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, R.L., Ibragimov, N.H.: Lie-Bäcklund Transformations in Applications. SIAM, Philadelphia (1979)zbMATHGoogle Scholar
  2. 2.
    Antritter, F., Lévine, J.: Towards a computer algebraic algorithm for flat output determination. In: Proc. ISSAC 2008 (2008)Google Scholar
  3. 3.
    Aranda-Bricaire, E., Moog, C., Pomet, J.-B.: A linear algebraic framework for dynamic feedback linearization. IEEE Trans. Automat. Control 40(1), 127–132 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Charlet, B., Lévine, J., Marino, R.: Sufficient conditions for dynamic state feedback linearization. SIAM J. Control Optimiz. 29(1), 38–57 (1991)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chern, S., Chen, W., Lam, K.: Lectures on Differential Geometry. Series on University Mathematics, vol. 1. World Scientific, Singapore (2000)Google Scholar
  6. 6.
    Chetverikov, V.: New flatness conditions for control systems. In: Proceedings of NOLCOS 2001, St. Petersburg, pp. 168–173 (2001)Google Scholar
  7. 7.
    Chetverikov, V.: Flatness conditions for control systems. Preprint DIPS (2002),
  8. 8.
    Cohn, P.: Free Rings and Their Relations. Academic Press, London (1985)zbMATHGoogle Scholar
  9. 9.
    Fliess, M.: A remark on Willems’ trajectory characterization of linear controllability. Systems & Control Letters 19, 43–45 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fliess, M., Lévine, J., Martin, P., Ollivier, F., Rouchon, P.: Controlling nonlinear systems by flatness. In: Byrnes, C., Datta, B., Gilliam, D., Martin, C. (eds.) Systems and Control in the Twenty-First Century, pp. 137–154. Birkhäuser, Boston (1997)Google Scholar
  11. 11.
    Fliess, M., Lévine, J., Martin, P., Rouchon, P.: Sur les systèmes non linéaires différentiellement plats. C.R. Acad. Sci. Paris I(315), 619–624 (1992)Google Scholar
  12. 12.
    Fliess, M., Lévine, J., Martin, P., Rouchon, P.: Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control 61(6), 1327–1361 (1995)zbMATHCrossRefGoogle Scholar
  13. 13.
    Fliess, M., Lévine, J., Martin, P., Rouchon, P.: A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44(5), 922–937 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Franch, J.: Flatness, Tangent Systems and Flat Outputs. PhD thesis, Universitat Politècnica de Catalunya Jordi Girona (1999)Google Scholar
  15. 15.
    Jakubczyk, B.: Invariants of dynamic feedback and free systems. In: Proc. ECC 1993, Groningen, pp. 1510–1513 (1993)Google Scholar
  16. 16.
    Krasil’shchik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations. In: Gordon and Breach, New York (1986)Google Scholar
  17. 17.
    Lévine, J.: On necessary and sufficient conditions for differential flatness. In: Proc. of IFAC NOLCOS 2004 Conference, Stuttgart (2004)Google Scholar
  18. 18.
    Lévine, J.: On necessary and sufficient conditions for differential flatness. arXiv:math.OC/0605405 (2006),
  19. 19.
    Lévine, J.: Analysis and Control of Nonlinear Systems: A Flatness-based Approach. Mathematical Engineering Series. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  20. 20.
    Martin, P.: Contribution à l’Étude des Systèmes Différentiellement Plats. PhD thesis, École des Mines de Paris (1992)Google Scholar
  21. 21.
    Martin, P., Murray, R., Rouchon, P.: Flat systems. In: Bastin, G., Gevers, M. (eds.) Plenary Lectures and Minicourses, Proc. ECC 1997, Brussels, pp. 211–264 (1997)Google Scholar
  22. 22.
    Pereira da Silva, P., Filho, C.C.: Relative flatness and flatness of implicit systems. SIAM J. Control and Optimization 39(6), 1929–1951 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pomet, J.-B.: A differential geometric setting for dynamic equivalence and dynamic linearization. In: Jakubczyk, B., Respondek, W., Rzeżuchowski, T. (eds.) Geometry in Nonlinear Control and Differential Inclusions, pp. 319–339. Banach Center Publications, Warsaw (1993)Google Scholar
  24. 24.
    Rathinam, M., Murray, R.: Configuration flatness of Lagrangian systems underactuated by one control. SIAM J. Control Optimiz. 36(1), 164–179 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rouchon, P.: Necessary condition and genericity of dynamic feedback linearization. J. Math. Systems Estim. & Control 4(2), 257–260 (1994)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Rudolph, J.: Flatness Based Control of Distributed Parameter Systems. Shaker Verlag, Aachen (2003)Google Scholar
  27. 27.
    Rudolph, J., Winkler, J., Woittenek, F.: Flatness Based Control of Distributed Parameter Systems: Examples and Computer Exercises from Various Technological Domains. Shaker Verlag, Aachen (2003)Google Scholar
  28. 28.
    Schlacher, K., Schöberl, M.: Construction of flat outputs by reduction and elimination. In: Proc. 7th IFAC Symposium on Nonlinear Control Systems, Pretoria, South Africa, pp. 666–671 (August 2007)Google Scholar
  29. 29.
    Schlacher, K., Schöberl, M.: Construction of flat outputs by reduction and elimination. In: Lévine, J., Müllhaupt, P. (eds.) Advances in the Theory of Control, Signals and Systems, with Physical Modeling, Springer, Heidelberg (2010)Google Scholar
  30. 30.
    Shadwick, W.: Absolute equivalence and dynamic feedback linearization. Systems & Control Letters 15, 35–39 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Sira-Ramirez, H., Agrawal, S.: Differentially Flat Systems. Marcel Dekker, New York (2004)zbMATHGoogle Scholar
  32. 32.
    Sluis, W.: A necessary condition for dynamic feedback linearization. Systems & Control Letters 21, 277–283 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    van Nieuwstadt, M., Rathinam, M., Murray, R.: Differential flatness and absolute equivalence of nonlinear control systems. SIAM J. Control Optim. 36(4), 1225–1239 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Zharinov, V.: Geometrical Aspect of Partial Differential Equations. World Scientific, Singapore (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Felix Antritter
    • 1
  • Jean Lévine
    • 2
  1. 1.Automatisierungs- und RegelungstechnikUniversität der Bundeswehr MünchenNeubibergGermany
  2. 2.Mines ParisTechCAS- Centre Automatique et SystèmesFontainebleauFrance

Personalised recommendations