no. 3
Flat outputs of two-input driftless control systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 774-798.

We study the problem of flatness of two-input driftless control systems. Although a characterization of flat systems of that class is known, the problems of describing all flat outputs and of calculating them is open and we solve it in the paper. We show that all x-flat outputs are parameterized by an arbitrary function of three canonically defined variables. We also construct a system of 1st order PDE's whose solutions give all x-flat outputs of two-input driftless systems. We illustrate our results by describing all x-flat outputs of models of a nonholonomic car and the n-trailer system.

DOI : 10.1051/cocv/2011181
Classification : 93B27, 93C10, 93C95
Mots clés : control system, flatness, flat output, feedback equivalence, characteristic distribution, n-trailer system 
@article{COCV_2012__18_3_774_0,
     author = {Li, Shun-Jie and Respondek, Witold},
     title = {Flat outputs of two-input driftless control systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {774--798},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {3},
     year = {2012},
     doi = {10.1051/cocv/2011181},
     mrnumber = {3041664},
     zbl = {1252.93041},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2011181/}
}
TY  - JOUR
AU  - Li, Shun-Jie
AU  - Respondek, Witold
TI  - Flat outputs of two-input driftless control systems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 774
EP  - 798
VL  - 18
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2011181/
DO  - 10.1051/cocv/2011181
LA  - en
ID  - COCV_2012__18_3_774_0
ER  - 
%0 Journal Article
%A Li, Shun-Jie
%A Respondek, Witold
%T Flat outputs of two-input driftless control systems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 774-798
%V 18
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2011181/
%R 10.1051/cocv/2011181
%G en
%F COCV_2012__18_3_774_0
Li, Shun-Jie; Respondek, Witold. Flat outputs of two-input driftless control systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 774-798. doi : 10.1051/cocv/2011181. http://archive.numdam.org/articles/10.1051/cocv/2011181/

[1] E. Aranda-Bricaire, C.H. Moog and J.-B. Pomet, Infinitesimal Brunovský form for nonlinear systems with applications to Dynamic Linearization, Geometry in nonlinear control and differential inclusions 32, edited by B. Jakuczyk, W. Respondek and T. Rzeżuchowski. Banach Center Publications, Warsaw (1995) 19-33. | MR | Zbl

[2] R. Bryant, S.-S. Chern, R. Gardner, H. Goldschmidt and P. Griffiths, Exterior Differential Systems. Mathematical Sciences Research Institute Publications, Springer-Verlag, New York (1991). | MR | Zbl

[3] E. Cartan, Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes, Bulletin de la Société Mathématique de France 42, Œuvres complètes 2. Part. II, Gauthiers-Villars, Paris (1914) 12-48. | JFM

[4] M. Cheaito and P. Mormul, Rank-2 distributions satisfying the Goursat condition :all their local models in dimension 7 and 8. ESAIM : COCV 4 (1999) 137-158. | Numdam | MR | Zbl

[5] M. Fliess, J. Lévine, P. Martin and P. Rouchon, Sur les systèmes non linéaires différentiellement plats. C. R. Acad. Sci. 315 (1992) 619-624. | MR | Zbl

[6] 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. | MR | Zbl

[7] M. Fliess, J. Lévine, P. Martin and P. Rouchon, A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 61 (1999) 1327-1361. | MR | Zbl

[8] A. Giaro, A. Kumpera and C. Ruiz, Sur la lecture correcte d'un resultat d'Élie Cartan. C. R. Acad. Sci. Paris 287 (1978) 241-244. | MR | Zbl

[9] E. Goursat, Leçons sur le problème de Pfaff. Hermann, Paris (1923). | JFM

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

[11] A. Isidori. Nonlinear Control Systems, 3rd edition. Springer-Verlag, London (1995). | Zbl

[12] 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 & Control. Athens (1986) 203-207.

[13] B. Jakubczyk, Invariants of dynamic feedback and free systems, in Proceedings of the European Control Conference. Groningen (1993) 1510-1513.

[14] F. Jean, The car with n trailers : Characterisation of the singular configurations. ESAIM : COCV 1 (1996) 241-266. | Numdam | MR | Zbl

[15] A. Kumpera and C. Ruiz, Sur l'équivalence locale des systèmes de Pfaff en drapeau, in Monge-Ampère equations and related topics, edited by F. Gherardelli. Instituto Nazionale di Alta Matematica Francesco Severi, Rome (1982) 201-247. | MR | Zbl

[16] J.P. Laumond, Controllability of a multibody robot. IEEE Trans. Robot. Autom. 9 (1991) 755-763.

[17] J.P. Laumond, Robot Motion Planning and Control, Lecture Notes on Control and Information Sciences 229. Springer-Verkag, New York (1997).

[18] Z. Li and J.F. Canny Eds., Nonholonomic Motion Plannging. Internqtional Series in Engineering and Computer Sciences, Kluwer, Dordrecht (1992). | Zbl

[19] P. Martin and P. Rouchon, Feedback linearization and driftless systems. CAS internal report No. 446, École des Mines (1993). | MR | Zbl

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

[21] P. Martin, R.M. Murray and P. Rouchon, Flat systems, in Mathematical Control Theory, Part 2, ICTP Lecture Notes 8, edited by A.A. Agrachev. ICTP Publications, Trieste (2002) 705-768. | MR | Zbl

[22] P. Mormul, Goursat flags : classification of codimension-one singularities. J. Dyn. Control Syst. 6 (2000) 311-330. | MR | Zbl

[23] R. Murray, Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Math. Control Signals Syst. 7 (1994) 58-75. | MR | Zbl

[24] R. Murray and S. Sastry, Nonholonomic motion planning : Steering using sinusoids. IEEE Trans. Autom. Control 38 (1993) 700-716. | MR | Zbl

[25] W. Pasillas-Lépine and W. Respondek, On the geometry of control systems equivalent to canonical contact systems : regular points, singular points and flatness, Proceedings of the 39th IEEE Conference of Decision and Control. Sydney, Australia (2000) 5151-5156.

[26] W. Pasillas-Lépine and W. Respondek, On the geometry of Goursat structures. ESAIM : COCV 6 (2001) 119-181. | Numdam | MR | Zbl

[27] P.S. Pereira Da Silva, and C. Corrêa Filho, Relative flatness and flatness of implicit systems. SIAM J. Control Optim. 39 (2001) 1929-1951. | MR | Zbl

[28] J.-B. Pomet, A differential geometric setting for dynamic equivalence and dynamic linearization, in Geometry in Nonlinear Control and Differential Inclusions 32, edited by B. Jakubczyk, W. Respondek and T. Rzeżuchowski. Banach Center Publications, Warsaw (1995) 319-339. | MR | Zbl

[29] W. Respondek, Symmetries and minimal flat outputs of nonlinear control systems, in New Trends in Nonlinear Dynamics and Control, and their Applications, Lecture Notes on Control and Information Sciences 295, edited by W. Kang, M. Xiao and C. Borges. Springer Verlag, Berlin, Heidelberg (2003) 65-86. | MR | Zbl

[30] O.J. Sørdalen, Conversion of the kinematics of a car with n trailers into a chained form, Proceeding of 1993 International Conference on Robotics and Automation, Atlanta, CA (1993) 382-387.

[31] M. Van Nieuwstadt, M. Rathinam and R.M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems. SIAM J. Control Optim. 36 (1998) 1225-1239. | MR | Zbl

[32] E. Von Weber, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellshaft der Wissenshaften Mathematisch-Physikalische Klasse, Leipzig 50 (1898) 207-229. | JFM

Cité par Sources :