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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 407))

Abstract

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

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References

  1. Anderson, R.L., Ibragimov, N.H.: Lie-Bäcklund Transformations in Applications. SIAM, Philadelphia (1979)

    MATH  Google Scholar 

  2. Antritter, F., Lévine, J.: Towards a computer algebraic algorithm for flat output determination. In: Proc. ISSAC 2008 (2008)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Charlet, B., Lévine, J., Marino, R.: Sufficient conditions for dynamic state feedback linearization. SIAM J. Control Optimiz. 29(1), 38–57 (1991)

    Article  MATH  Google Scholar 

  5. Chern, S., Chen, W., Lam, K.: Lectures on Differential Geometry. Series on University Mathematics, vol. 1. World Scientific, Singapore (2000)

    Google Scholar 

  6. Chetverikov, V.: New flatness conditions for control systems. In: Proceedings of NOLCOS 2001, St. Petersburg, pp. 168–173 (2001)

    Google Scholar 

  7. Chetverikov, V.: Flatness conditions for control systems. Preprint DIPS (2002), http://www.diffiety.ac.ru

  8. Cohn, P.: Free Rings and Their Relations. Academic Press, London (1985)

    MATH  Google Scholar 

  9. Fliess, M.: A remark on Willems’ trajectory characterization of linear controllability. Systems & Control Letters 19, 43–45 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  14. Franch, J.: Flatness, Tangent Systems and Flat Outputs. PhD thesis, Universitat Politècnica de Catalunya Jordi Girona (1999)

    Google Scholar 

  15. Jakubczyk, B.: Invariants of dynamic feedback and free systems. In: Proc. ECC 1993, Groningen, pp. 1510–1513 (1993)

    Google Scholar 

  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. Lévine, J.: On necessary and sufficient conditions for differential flatness. In: Proc. of IFAC NOLCOS 2004 Conference, Stuttgart (2004)

    Google Scholar 

  18. Lévine, J.: On necessary and sufficient conditions for differential flatness. arXiv:math.OC/0605405 (2006), http://www.arxiv.org

  19. Lévine, J.: Analysis and Control of Nonlinear Systems: A Flatness-based Approach. Mathematical Engineering Series. Springer, Heidelberg (2009)

    MATH  Google Scholar 

  20. Martin, P.: Contribution à l’Étude des Systèmes Différentiellement Plats. PhD thesis, École des Mines de Paris (1992)

    Google Scholar 

  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. Pereira da Silva, P., Filho, C.C.: Relative flatness and flatness of implicit systems. SIAM J. Control and Optimization 39(6), 1929–1951 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. Rathinam, M., Murray, R.: Configuration flatness of Lagrangian systems underactuated by one control. SIAM J. Control Optimiz. 36(1), 164–179 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rouchon, P.: Necessary condition and genericity of dynamic feedback linearization. J. Math. Systems Estim. & Control 4(2), 257–260 (1994)

    MATH  MathSciNet  Google Scholar 

  26. Rudolph, J.: Flatness Based Control of Distributed Parameter Systems. Shaker Verlag, Aachen (2003)

    Google Scholar 

  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. 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. 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. Shadwick, W.: Absolute equivalence and dynamic feedback linearization. Systems & Control Letters 15, 35–39 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sira-Ramirez, H., Agrawal, S.: Differentially Flat Systems. Marcel Dekker, New York (2004)

    MATH  Google Scholar 

  32. Sluis, W.: A necessary condition for dynamic feedback linearization. Systems & Control Letters 21, 277–283 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  34. Zharinov, V.: Geometrical Aspect of Partial Differential Equations. World Scientific, Singapore (1992)

    Google Scholar 

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Antritter, F., Lévine, J. (2010). Flatness Characterization: Two Approaches. In: Lévine, J., Müllhaupt, P. (eds) Advances in the Theory of Control, Signals and Systems with Physical Modeling. Lecture Notes in Control and Information Sciences, vol 407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16135-3_11

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  • DOI: https://doi.org/10.1007/978-3-642-16135-3_11

  • Publisher Name: Springer, Berlin, Heidelberg

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