Flatness Characterization: Two Approaches

  • Felix Antritter
  • Jean Lévine


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


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  1. 1.
    Anderson, R.L., Ibragimov, N.H.: Lie-Bäcklund Transformations in Applications. SIAM, Philadelphia (1979)MATHGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefGoogle 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), http://www.diffiety.ac.ru
  8. 8.
    Cohn, P.: Free Rings and Their Relations. Academic Press, London (1985)MATHGoogle Scholar
  9. 9.
    Fliess, M.: A remark on Willems’ trajectory characterization of linear controllability. Systems & Control Letters 19, 43–45 (1992)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefGoogle 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)MATHCrossRefMathSciNetGoogle 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), http://www.arxiv.org
  19. 19.
    Lévine, J.: Analysis and Control of Nonlinear Systems: A Flatness-based Approach. Mathematical Engineering Series. Springer, Heidelberg (2009)MATHGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rouchon, P.: Necessary condition and genericity of dynamic feedback linearization. J. Math. Systems Estim. & Control 4(2), 257–260 (1994)MATHMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Sira-Ramirez, H., Agrawal, S.: Differentially Flat Systems. Marcel Dekker, New York (2004)MATHGoogle Scholar
  32. 32.
    Sluis, W.: A necessary condition for dynamic feedback linearization. Systems & Control Letters 21, 277–283 (1993)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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

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