Hostname: page-component-5c6d5d7d68-thh2z Total loading time: 0 Render date: 2024-08-16T00:28:03.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states

Published online by Cambridge University Press:  12 May 2007

David Avanessoff  and
Jean-Baptiste Pomet
David Avanessoff
Affiliation:
INRIA Sophia Antipolis, B.P. 93, 06902 Sophia Antipolis cedex, France; David.Avanessoff@sophia.inria.fr; Jean-Baptiste.Pomet@sophia.inria.fr
Jean-Baptiste Pomet
Affiliation:
INRIA Sophia Antipolis, B.P. 93, 06902 Sophia Antipolis cedex, France; David.Avanessoff@sophia.inria.fr; Jean-Baptiste.Pomet@sophia.inria.fr
Get access

Abstract

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 “(x,u)-flatness” of these systems, with much more elementary techniques.

Keywords

Dynamic feedback linearizationflat control systemsMonge problemMonge equations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 2 , April 2007 , pp. 237 - 264
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aranda-Bricaire, E., Moog, C.H. and Pomet, J.-B., An infinitesimal Brunovsky form for nonlinear systems with applications to dynamic linearization. Banach Center Publications 32 (1995) 19–33.
D. Avanessoff, Linéarisation dynamique des systèmes non linéaires et paramétrage de l'ensemble des solutions. Ph.D. thesis, University of Nice-Sophia Antipolis (June 2005).
R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmitt and P.A. Griffiths, Exterior Differential Systems, Springer-Verlag, M.S.R.I. Publications 18 (1991).
Cartan, É., Sur l'intégration de certains systèmes indéterminés d'équations différentielles. J. reine angew. Math. 145 (1915) 86–91.
Charlet, B., Lévine, J. and Marino, R., On dynamic feedback linearization. Syst. Control Lett. 13 (1989) 143–151. CrossRef
Charlet, B., Lévine, J. and Marino, R., Sufficient conditions for dynamic state feedback linearization. SIAM J. Control Optim. 29 (1991) 38–57. CrossRef
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.
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control 61 (1995) 1327–1361. CrossRef
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Lie-Bäcklund, A approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922–937. CrossRef
M. Fliess, J. Lévine, P. Martin and P. Rouchon, Some open questions related to flat nonlinear systems, in Open problems in mathematical systems and control theory, Springer, London (1999) 99–103.
M. Golubitsky and V. Guillemin, Stable mappings and their singularities. Springer-Verlag, New York, GTM 14 (1973).
Hilbert, D., Über den Begriff der Klasse von Differentialgleichungen. Math. Annalen 73 (1912) 95–108. CrossRef
E. Hubert, 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).
A. Isidori, C.H. Moog and A. de Luca, A sufficient condition for full linearization via dynamic state feedback, in Proc. 25th IEEE Conf. on Decision and Control, Athens (1986) 203–207.
P. Martin, Contribution à l'étude des systèmes différentiellement plats. Ph.D. thesis, École des Mines, Paris (1992).
P. Martin, R.M. Murray and P. Rouchon, 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.
Martin, P. and Rouchon, P., Feedback linearization and driftless systems. Math. Control Signals Syst. 7 (1994) 235–254. CrossRef
Pomet, J.-B., A differential geometric setting for dynamic equivalence and dynamic linearization. Banach Center Publications 32 (1995) 319–339.
Pomet, J.-B., 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/. CrossRef
J.F. Ritt, Differential Algebra. AMS Coll. Publ. XXXIII. New York (1950).
P. Rouchon, Flatness and oscillatory control: some theoretical results and case studies. Tech. report PR412, CAS, École des Mines, Paris (1992).
Rouchon, P., Necessary condition and genericity of dynamic feedback linearization. J. Math. Syst. Estim. Contr. 4 (1994) 1–14.
Sluis, W.M., A necessary condition for dynamic feedback linearization. Syst. Control Lett. 21 (1993) 277–283. CrossRef
van Nieuwstadt, M., Rathinam, M. and Murray, R., 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. CrossRef
P. Zervos, Le problème de Monge. Mémorial des Sciences Mathématiques, LIII (1932).