A differential geometric setting for dynamic equivalence and dynamic linearization
Banach Center Publications (1995)
- Volume: 32, Issue: 1, page 319-339
- ISSN: 0137-6934
Access Full Article
topAbstract
topHow to cite
topPomet, Jean-Baptiste. "A differential geometric setting for dynamic equivalence and dynamic linearization." Banach Center Publications 32.1 (1995): 319-339. <http://eudml.org/doc/262861>.
@article{Pomet1995,
abstract = {This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise: equivalence by endogenous dynamic feedback) is conjugation by diffeomorphisms. These diffeomorphisms are very much related to Lie-Bäcklund transformations. It is proved in this framework that dynamic equivalence of single-input systems is the same as static equivalence.},
author = {Pomet, Jean-Baptiste},
journal = {Banach Center Publications},
keywords = {flat systems; infinite jet bundles; Dynamic feedback equivalence; dynamic feedback linearization; Lie-Bäcklund transformations; contact transformations; dynamic linearization; dynamic equivalence},
language = {eng},
number = {1},
pages = {319-339},
title = {A differential geometric setting for dynamic equivalence and dynamic linearization},
url = {http://eudml.org/doc/262861},
volume = {32},
year = {1995},
}
TY - JOUR
AU - Pomet, Jean-Baptiste
TI - A differential geometric setting for dynamic equivalence and dynamic linearization
JO - Banach Center Publications
PY - 1995
VL - 32
IS - 1
SP - 319
EP - 339
AB - This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise: equivalence by endogenous dynamic feedback) is conjugation by diffeomorphisms. These diffeomorphisms are very much related to Lie-Bäcklund transformations. It is proved in this framework that dynamic equivalence of single-input systems is the same as static equivalence.
LA - eng
KW - flat systems; infinite jet bundles; Dynamic feedback equivalence; dynamic feedback linearization; Lie-Bäcklund transformations; contact transformations; dynamic linearization; dynamic equivalence
UR - http://eudml.org/doc/262861
ER -
References
top- [1] R. L. Anderson and N. H. Ibragimov, Lie-Bäcklund Transformations in Applications, SIAM Stud. Appl. Math., SIAM, Philadelphia, 1979.
- [2] E. Aranda-Bricaire, C. H. Moog and J.-B. Pomet, A linear algebraic framework for dynamic feedback linearization, IEEE Trans. Automat. Control 40 (1995), 127-132. Zbl0844.93025
- [3] E. Aranda-Bricaire, C. H. Moog and J.-B. Pomet, Infinitesimal Brunovský form for nonlinear systems with applications to dynamic linearization, this volume. Zbl0844.93024
- [4] N. Bourbaki, Eléments de Mathématique, Espaces Vectoriels Topologiques, chap. 1, Masson, Paris, 1981.
- [5] P. Brunovský, A classification of linear controllable systems, Kybernetika 6 (1970), 176-188. Zbl0199.48202
- [6] B. Charlet, J. Lévine and R. Marino, On dynamic feedback linearization, Systems Control Lett. 13 (1989), 143-151. Zbl0684.93043
- [7] B. Charlet, J. Lévine and R. Marino, Sufficient conditions for dynamic feedback linearization, SIAM J. Control Optim. 29 (1991), 38-57. Zbl0739.93021
- [8] G. Conte, A.-M. Perdon and C. Moog, The differential field associated to a general analytic nonlinear system, IEEE Trans. Automat. Control 38 (1993), 1120-1124. Zbl0800.93539
- [9] E. Delaleau, Sur les dérivées de l'entrée en représentation et commande des systèmes non-linéaires, Thèse de l'Unviversité Paris XI, Orsay, 1993.
- [10] M. Fliess, Automatique et corps différentiels, Forum Math. 1 (1989), 227-238. Zbl0701.93048
- [11] M. Fliess, Décomposition en cascade des systèmes automatiques et feuilletages invariants, Bull. Soc. Math. France 113 (1985), 285-293. Zbl0587.93031
- [12] M. Fliess, J. Lévine, P. Martin and P. Rouchon, On differentially flat nonlinear systems, in: 2nd IFAC NOLCOS Symposium, 1992, 408-412.
- [13] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Sur les systèmes non linéaires différentiellement plats, C. R. Acad. Sci. Paris Sér. I 315 (1992), 619-624. Zbl0776.93038
- [14] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Linéarisation par bouclage dynamique et transformations de Lie-Bäcklund, ibid., to appear.
- [15] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Towards a new differential geometric setting in nonlinear control, Presented at International Geometrical Colloquium, Moscow, May 1993, and to appear in the proceedings.
- [16] E. Goursat, Le problème de Bäcklund, Mém. Sci. Math. 6, Gauthier-Villars, Paris, 1925.
- [17] R. S. Hamilton, The Inverse Function Theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 65-222. Zbl0499.58003
- [18] B. Jakubczyk, Remarks on equivalence and linearization of nonlinear systems, in: 2nd IFAC NOLCOS Symposium, 1992, 393-397.
- [19] B. Jakubczyk, Dynamic feedback equivalence of nonlinear control systems, preprint, 1993.
- [20] I. S. Krasil'shchik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Adv. Stud. Contemp. Math. 1, Gordon & Breach, 1986.
- [21] P. Martin, Contribution à l'étude des systèmes non linéaires différentiellement plats, Thèse de Doctorat, Ecole des Mines de Paris, 1992.
- [22] P. Otterson and G. Svetlichny, On derivative-dependent deformations of differential maps, J. Differential Equations 36 (1980), 270-294. Zbl0406.58029
- [23] F. A. E. Pirani, D. C. Robinson and W. F. Shadwick, Local jet bundle formulation of Bäcklund transformations, Math. Phys. Stud., Reidel, Dordrecht, 1979. Zbl0427.58003
- [24] J.-B. Pomet, C. H. Moog and E. Aranda, A non-exact Brunovský form and dynamic feedback linearization, in: Proc. 31st. IEEE Conf. Dec. Cont., 1992, 2012-2017.
- [25] J.-F. Pommaret, Géométrie différentielle algébrique et théorie du contrôle, C. R. Acad. Sci. Paris Sér. I 302 (1986), 547-550. Zbl0594.93030
- [26] D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lecture Note Ser. 142, Cambridge University Press, Cambridge, 1989. Zbl0665.58002
- [27] W. F. Shadwick, Absolute equivalence and dynamic feedback linearization, Systems Control Lett. 15 (1990), 35-39. Zbl0704.93037
- [28] A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math. 2 (1984), 21-78. Zbl0534.58005
- [29] J. C. Willems, Paradigms and puzzles in the theory of dynamical systems, IEEE Trans. Automat. Control 36 (1991), 259-294. Zbl0737.93004
Citations in EuDML Documents
top- Jean-Baptiste Pomet, On dynamic feedback linearization of four-dimensional affine control systems with two inputs
- E. Aranda-Bricaire, C. Moog, J. Pomet, Infinitesimal Brunovský form for nonlinear systems with applications to Dynamic Linearization
- Michel Fliess, Jean Lévine, Philippe Martin, Pierre Rouchon, Differential flatness and defect: an overview
- David Avanessoff, Jean-Baptiste Pomet, Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states
- Shun-Jie Li, Witold Respondek, Flat outputs of two-input driftless control systems
- Shun-Jie Li, Witold Respondek, Flat outputs of two-input driftless control systems
- Shun-Jie Li, Witold Respondek, Flat outputs of two-input driftless control systems
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.