This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and two controls. Some of them are known to be flat, and this implies admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about “-flatness” of these systems, with much more elementary techniques.
Keywords: dynamic feedback linearization, flat control systems, Monge problem, Monge equations
@article{COCV_2007__13_2_237_0, author = {Avanessoff, David and Pomet, Jean-Baptiste}, title = {Flatness and {Monge} parameterization of two-input systems, control-affine with 4 states or general with 3 states}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {237--264}, publisher = {EDP-Sciences}, volume = {13}, number = {2}, year = {2007}, doi = {10.1051/cocv:2007011}, mrnumber = {2306635}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007011/} }
TY - JOUR AU - Avanessoff, David AU - Pomet, Jean-Baptiste TI - Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 237 EP - 264 VL - 13 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007011/ DO - 10.1051/cocv:2007011 LA - en ID - COCV_2007__13_2_237_0 ER -
%0 Journal Article %A Avanessoff, David %A Pomet, Jean-Baptiste %T Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 237-264 %V 13 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007011/ %R 10.1051/cocv:2007011 %G en %F COCV_2007__13_2_237_0
Avanessoff, David; Pomet, Jean-Baptiste. Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 2, pp. 237-264. doi : 10.1051/cocv:2007011. http://www.numdam.org/articles/10.1051/cocv:2007011/
[1] An infinitesimal Brunovsky form for nonlinear systems with applications to dynamic linearization. Banach Center Publications 32 (1995) 19-33. | Zbl
, and ,[2] Linéarisation dynamique des systèmes non linéaires et paramétrage de l'ensemble des solutions. Ph.D. thesis, University of Nice-Sophia Antipolis (June 2005).
,[3] Exterior Differential Systems, Springer-Verlag, M.S.R.I. Publications 18 (1991). | MR | Zbl
, , , and ,[4] Sur l'intégration de certains systèmes indéterminés d'équations différentielles. J. reine angew. Math. 145 (1915) 86-91. | JFM
,[5] On dynamic feedback linearization. Syst. Control Lett. 13 (1989) 143-151. | Zbl
, and ,[6] Sufficient conditions for dynamic state feedback linearization. SIAM J. Control Optim. 29 (1991) 38-57. | Zbl
, and ,[7] Sur les systèmes non linéaires différentiellement plats. C. R. Acad. Sci. Paris Sér. I 315 (1992) 619-624. | Zbl
, , and ,[8] Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control 61 (1995) 1327-1361. | Zbl
, , and ,[9] A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922-937. | Zbl
, , and ,[10] Some open questions related to flat nonlinear systems, in Open problems in mathematical systems and control theory, Springer, London (1999) 99-103.
, , and ,[11] Stable mappings and their singularities. Springer-Verlag, New York, GTM 14 (1973). | MR | Zbl
and ,[12] Über den Begriff der Klasse von Differentialgleichungen. Math. Annalen 73 (1912) 95-108. | JFM
,[13] Notes on triangular sets and triangulation-decomposition algorithms. I: Polynomial systems. II: Differential systems. In F. Winkler et al. eds., Symbolic and Numerical Scientific Computing 2630, 1-87. Lect. Notes Comput. Sci. (2003). | Zbl
,[14] A sufficient condition for full linearization via dynamic state feedback, in Proc. 25th IEEE Conf. on Decision and Control, Athens (1986) 203-207.
, and ,[15] Contribution à l'étude des systèmes différentiellement plats. Ph.D. thesis, École des Mines, Paris (1992).
,[16] Flat systems, in Mathematical control theory, Part 1, 2 (Trieste, 2001), ICTP Lect. Notes VIII, (electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002) 705-768. | Zbl
, and ,[17] Feedback linearization and driftless systems. Math. Control Signals Syst. 7 (1994) 235-254. | Zbl
and ,[18] A differential geometric setting for dynamic equivalence and dynamic linearization. Banach Center Publications 32 (1995) 319-339. | Zbl
,[19] On dynamic feedback linearization of four-dimensional affine control systems with two inputs. ESAIM Control Optim. Calc. Var. 2 (1997) 151-230. http://www.edpsciences.org/cocv/. | Numdam | Zbl
,[20] Differential Algebra. AMS Coll. Publ. XXXIII. New York (1950). | MR | Zbl
,[21] Flatness and oscillatory control: some theoretical results and case studies. Tech. report PR412, CAS, École des Mines, Paris (1992).
,[22] Necessary condition and genericity of dynamic feedback linearization. J. Math. Syst. Estim. Contr. 4 (1994) 1-14. | Zbl
,[23] A necessary condition for dynamic feedback linearization. Syst. Control Lett. 21 (1993) 277-283. | Zbl
,[24] Differential flatness and absolute equivalence of nonlinear control systems. SIAM J. Control Optim. 36 (1998) 1225-1239. http://epubs.siam.org:80/sam-bin/dbq/article/27402. | Zbl
, and ,[25] Le problème de Monge. Mémorial des Sciences Mathématiques, LIII (1932). | JFM | Numdam | Zbl
,Cited by Sources: