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

Avanessoff, David; Pomet, Jean-Baptiste

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 237-264.

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.

Classification : 93B18, 93B29, 34C20
Mots clés : dynamic feedback linearization, flat control systems, Monge problem, Monge equations

[1] E. Aranda-Bricaire, C.H. Moog and J.-B. Pomet, An infinitesimal Brunovsky form for nonlinear systems with applications to dynamic linearization. Banach Center Publications 32 (1995) 19-33. | Zbl

[2] 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).

[3] 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). | MR | Zbl

[4] É. Cartan, Sur l'intégration de certains systèmes indéterminés d'équations différentielles. J. reine angew. Math. 145 (1915) 86-91. | JFM

[5] B. Charlet, J. Lévine and R. Marino, On dynamic feedback linearization. Syst. Control Lett. 13 (1989) 143-151. | Zbl

[6] B. Charlet, J. Lévine and R. Marino, Sufficient conditions for dynamic state feedback linearization. SIAM J. Control Optim. 29 (1991) 38-57. | Zbl

[7] 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. | Zbl

[8] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control 61 (1995) 1327-1361. | Zbl

[9] M. Fliess, J. Lévine, P. Martin and P. Rouchon, A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922-937. | Zbl

[10] 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.

[11] M. Golubitsky and V. Guillemin, Stable mappings and their singularities. Springer-Verlag, New York, GTM 14 (1973). | MR | Zbl

[12] D. Hilbert, Über den Begriff der Klasse von Differentialgleichungen. Math. Annalen 73 (1912) 95-108. | JFM

[13] 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). | Zbl

[14] 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.

[15] P. Martin, Contribution à l'étude des systèmes différentiellement plats. Ph.D. thesis, École des Mines, Paris (1992).

[16] 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. | Zbl

[17] P. Martin and P. Rouchon, Feedback linearization and driftless systems. Math. Control Signals Syst. 7 (1994) 235-254. | Zbl

[18] J.-B. Pomet, A differential geometric setting for dynamic equivalence and dynamic linearization. Banach Center Publications 32 (1995) 319-339. | Zbl

[19] J.-B. Pomet, 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] J.F. Ritt, Differential Algebra. AMS Coll. Publ. XXXIII. New York (1950). | MR | Zbl

[21] P. Rouchon, Flatness and oscillatory control: some theoretical results and case studies. Tech. report PR412, CAS, École des Mines, Paris (1992).

[22] P. Rouchon, Necessary condition and genericity of dynamic feedback linearization. J. Math. Syst. Estim. Contr. 4 (1994) 1-14. | Zbl

[23] W.M. Sluis, A necessary condition for dynamic feedback linearization. Syst. Control Lett. 21 (1993) 277-283. | Zbl

[24] M. Van Nieuwstadt, M. Rathinam and R. Murray, 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

[25] P. Zervos, Le problème de Monge. Mémorial des Sciences Mathématiques, LIII (1932). | JFM | Numdam | Zbl