A triangular canonical form for a class of 0-flat nonlinear systems

International Journal of Control

Volumn 84, Issue 2, 2011, Pages 261-269

  Bououden, S ^a   Boutat, D ^a   Zheng, G ^b   Barbot, J P ^b,c   Kratz, F ^a  

^a ENSI de Bourges   (France)

^b INRIA   (France)

^c ENSEA   (France)

Author keywords

canonical form; flatness; geometrical condition

Indexed keywords

CANONICAL FORM; DIFFEOMORPHISMS; FLAT OUTPUT; FLATNESS; GEOMETRICAL CONDITION; GEOMETRICAL CONDITIONS;

NONLINEAR SYSTEMS;

EID: 79952907535     PISSN: 00207179     EISSN: 13665820     Source Type: Journal    
DOI: 10.1080/00207179.2010.549844     Document Type: Article

References (39)

1. Anderson, R., and Ibragimov, N (1979), Lie-Bäcklund Transformations in Applications, Philadelphia: SIAM.
2. Bououden, S., Boutat, D., Barbot, J. P., and Kratz, F. (2009), 'A Geometrical Characterisation of a Class of 0-flat Affine Dynamical Systems', in Proceedings of IEEE ACC, St. Louis, MO, pp. 796-801.
3. Cartan, E. (1914), 'Sur l'Equivalence Absolue de Certains Systemes d'Equations Differentielles et sur Certaines Familles de Courbes', Bulletin de Societe Mathematique de France, 42, 12-48.
4. Charlet, B., Levine, J., and Marino, R. (1989), 'On Dynamic Feedback Linearisation', Systems & Control Letters, 13, 143-151.
5. da Silva, P. S. P. (2000a), 'Flatness of Nonlinear Control Systems: A Cartan-Kahler Approach', in Proceedings of Mathematical Theory of Networks and Systems, pp. 1-11.
6. da Silva, P. S. P. (2000b), 'Flatness of Nonlinear Control Systems and Exterior Differential Systems', in Nonlinear Control in the Year 2000, eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek, Berlin: Springer-Verlag, pp. 205-227.
7. Delaleau, E., and da Silva, P. S. P. (1998), 'Filtrations in Feedback Synthesis: Part I - Systems and Feedbacks', Forum Mathematicum, 10, 147-174.
8. Descusse, J., and Moog, C. (1985), 'Decoupling with Dynamic Compensation for Strong Invertible Affine Nonlinear Systems', International Journal of Control, 42, 1387-1398. (Pubitemid 15604197)
9. Fliess, M., Lévine, J., Martin, P., and Rouchon, P. (1992), 'Sur les Systèmes Non-linénaires Différentiellement Plats', Comptes Rendus Mathématique, Académie des Sciences, Paris, Séries I, Mathématiques, 315, 619-624.
10. Fliess, M., Lévine, J., Martin, P., and Rouchon, P. (1993), 'Linéarisation par Bouclage Dynamique et Transformations de Lie-Bäcklund', Comptes Rendus Mathématique, Académie des Sciences, Paris, Series I, Mathématiques, 317, 981-986.
11. Fliess, M., Lévine, J., Martin, P., and Rouchon, P. (1995), 'Flatness and Defect of Non-linear Systems: Introductory Theory and Examples', International Journal of Control, 61, 1327-1361.
12. Fliess, M., Lévine, J., Martin, P., and Rouchon, P. (1997), 'Deux Applications de la geometrie Locale des Diffietes', Annales de l'Institut Henri Poincare, Physics Théorique, 66, 275-292.
13. Fliess, M., Lévine, J., Martin, P., and Rouchon, P. (1998), 'Nonlinear Control and Diffieties, with an Application to Physics', in Secondary Calculus and Cohomological Physics (Vol. 219), eds. J. Krasilshchik, M. Henneaux and A. Vinogradov, Providence, RI: American Mathematical Society, pp. 81-92.
14. Fliess, M., Levine, J., Martin, P., and Rouchon, P. (1999a), 'A Lie-Backlund Approach to Equivalence and Flatness of Nonlinear Systems', IEEE Transactions on Automatic Control, 44, 922-937.
15. Fliess, M., Levine, J., Martin, P., and Rouchon, P. (1999b), 'Some Open Question Related to Flat Nonlinear Systems', in Open Problems in Mathematical Systems and Control Theory, eds. V. D. Blondel, E. Sontag, M. Vidyasagar and J. C. Willems, London: Springer Verlag, pp. 99-103.
16. Hagenmeyer, V., and Delaleau, E. (2003), 'Robustness Analysis of Exact Feedforward Linearisation Based on Differential Fatness', Automatica, 39, 1941-1946.
17. Hunt, L. R., Su, R., and Meyer, G. (1983), 'Design for Multiinput Nonlinear Systems', in Differential Geometric Methods in Nonlinear Control Theory, eds. R. Brocket, R. Millmann and H. J. Sussmann, Boston, MA: Birkhauser, pp. 268-298.
18. Jakubczyk, B., and Respondek, W. (1980), 'On Linearisation of Control Systems', Bulletin de l'Academie Polonaise des Sciences, Series des Sciences Mathematiques, 28, 517-522.
19. Levine, J. (2004), 'On Flatness Necessary and Sufficient Conditions', in Proceedings of NOLCOS, pp. 123-128.
20. Levine, J. (2009), Analysis and Control of Nonlinear Systems: A Flatness-based Approach, Heidelberg: Springer-Verlag.
21. Martin, P., Murray, R., and Rouchon, P. (1997), 'Flat Systems', in European Control Conference, pp. 211-264.
22. Martin, P., and Rouchon, P. (1995a), 'Any (Controllable) Driftless System with 3 Inputs and 5 States is Flat', Systems & Control Letters, 25, 167-173.
23. Martin, P., and Rouchon, P. (1995b), 'Any (controllable) Driftless System with m Inputs and m+2 States is Flat', in Proceedings of the IEEE Conference on Decision and Control, pp. 2886-2891.
24. Pomet, J. B. (1995a), 'A Differential Geometric Setting for Dynamic Equivalence and Dynamic Linearization', in Geometry in Nonlinear Control and Differential inclusions, eds. B. Jackubczyk, W. Respondek and T. Rzezuchowski, Warsaw: Banach Center Publications, pp. 319-339.
25. Pomet, J. B. (1995b), 'On Dynamic Feedback Linearization of Four-dimensional Affine Control Systems with Two Inputs', Technical, Memorandum RR-2751, I. N. R. I. A., Books and Monographs, http://www.emath.fr/
26. Rotella, F. and Zambettakis, I. (2007), 'Commande des Systemes par Platitude', Techniques de l'Ingenieurs, http://www.techniques-ingenieur.fr, S7450.
27. Rouchon, P. (1995), 'Necessary Condition and Genericity of Dynamic Feedback Linearisation', Journal of Mathematical Systems, Estimation, and Control, 5, 345-358.
28. Rudolph, J. (1995), 'Well-formed Dynamics Under Quasistatic State Feedback', in Geometry in Nonlinear Control and Differential Inclusions, eds. B. Jackubczyk, W. Respondek and T. Rzezuchowski, Warsaw: Banach Center Publications, pp. 349-360.
29. Schlacher, K., and Schoberl, M. (2007), 'Construction of Flat Outputs by Reduction and Elimination', in Proceedings of NOLCOS, pp. 666-671.
30. Shadwick, W. (1990), 'Absolute Equivalence and Dynamic Feedback Linearisation', Systems and Control Letters, 15, 35-39. (Pubitemid 20723318)
31. Shadwick, W., and Sluis, W. (1994), 'Dynamic Feedback for Classical Geometries', in Differential Geometry and Mathematical Physics, eds. J. K. Beem and K. L. Duggal, Providence, RI: American Mathematical Society, pp. 207-213.
32. Silveira, H. (2009), 'Formas Triangulares para Sistemas nao Lineares com duas Entradas e Controle de Sistemas sem Arrasto em su(n) com Aplicações em Mecânica Quântica', Ph. D. Thesis, Universidade de São Paulo, Brazil.
33. Silveira, H., and da Silva, P. S. P. (2007), 'A Triangular Form for Flat Nonlinear Systems with Two Controls and Five or Less States', in Proceedings of the 3rd IFAC Symposium on System, Structure and Control, pp. 127-132.
34. Singh, S. (1981), 'A Modified Algorithm for Invertibility in Nonlinear Systems', IEEE Transactions on Automatic Control, 26, 595-598. (Pubitemid 11505969)
35. Sira-Ramirez, H., and Agrawal, S. (2004), Differentially Flat Systems, New York: Marcel Dekker.
36. Sluis, W. (1993), 'A Necessary Condition for Dynamic Feedback Linearisation', Systems and Control Letters, 21, 277-283.
37. Sluis, W., and Tilbury, D. (1996), 'A Bound on the Number of Integrators Needed to Linearise a Control System', Systems and Control Letters, 29, 43-50. (Pubitemid 126411626)
38. Tilbury, D., Murray, R., and Sastry, S. (1995), 'Trajectory Generation for the n-trailer Problem using Goursat Normal Form', IEEE Transactions on Automatic Control, 40, 802-819.
39. van Nieuwstadt, M., Rathinam, M., and Murray, R. (1998), 'Differential Flatness and Absolute Equivalence of Nonlinear Control Systems', SIAM Journal of Control Optimization, 36, 1225-1239. (Pubitemid 128490845)