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Flat outputs of two-input driftless controlsystems

Published online by Cambridge University Press:  14 October 2011

Shun-Jie Li  and
Witold Respondek
Shun-Jie Li
Affiliation:
INSA-Rouen, Laboratoire de Mathématiques, Avenue de l’Université, 76801 Saint-Etienne-du-Rouvray, France. [email protected]; [email protected]
Witold Respondek
Affiliation:
INSA-Rouen, Laboratoire de Mathématiques, Avenue de l’Université, 76801 Saint-Etienne-du-Rouvray, France. [email protected]; [email protected]
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Abstract

We study the problem of flatness of two-input driftless control systems. Although acharacterization of flat systems of that class is known, the problems of describing allflat outputs and of calculating them is open and we solve it in the paper. We show thatall x-flat outputs are parameterized by an arbitrary function of threecanonically defined variables. We also construct a system of 1st order PDE’s whosesolutions give all x-flat outputs of two-input driftless systems. Weillustrate our results by describing all x-flat outputs of models of anonholonomic car and the n-trailer system.

Keywords

Control systemflatnessflat outputfeedback equivalencecharacteristic distributionn-trailer system
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 774 - 798
Copyright
© EDP Sciences, SMAI, 2011

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References

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.
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).
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.
Cheaito, M. and Mormul, P., Rank-2 distributions satisfying the Goursat condition :all their local models in dimension 7 and 8. ESAIM : COCV 4 (1999) 137–158. Google Scholar
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Sur les systèmes non linéaires différentiellement plats. C. R. Acad. Sci. 315 (1992) 619–624. Google Scholar
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. Google Scholar
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 61 (1999) 1327–1361. Google Scholar
Giaro, A., Kumpera, A. and Ruiz, C., Sur la lecture correcte d’un resultat d’Élie Cartan. C. R. Acad. Sci. Paris 287 (1978) 241–244. Google Scholar
E. Goursat, Leçons sur le problème de Pfaff. Hermann, Paris (1923).
Hilbert, D., Über den Begriff der Klasse von Differentialgleichungen. Math. Ann. 73 (1912) 95–108. Google Scholar
A. Isidori. Nonlinear Control Systems, 3rd edition. Springer-Verlag, London (1995).
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.
B. Jakubczyk, Invariants of dynamic feedback and free systems, in Proceedings of the European Control Conference. Groningen (1993) 1510–1513.
Jean, F., The car with n trailers : Characterisation of the singular configurations. ESAIM : COCV 1 (1996) 241–266. Google Scholar
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.
Laumond, J.P., Controllability of a multibody robot. IEEE Trans. Robot. Autom. 9 (1991) 755–763. Google Scholar
J.P. Laumond, Robot Motion Planning and Control, Lecture Notes on Control and Information Sciences 229. Springer-Verkag, New York (1997).
Z. Li and J.F. Canny Eds., Nonholonomic Motion Plannging. Internqtional Series in Engineering and Computer Sciences, Kluwer, Dordrecht (1992).
P. Martin and P. Rouchon, Feedback linearization and driftless systems. CAS internal report No. 446, École des Mines (1993).
Martin, P. and Rouchon, P., Feedback linearization and driftless systems. Math. Contr. Signals Syst. 7 (1994) 235–254. Google Scholar
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.
Mormul, P., Goursat flags : classification of codimension-one singularities. J. Dyn. Control Syst. 6 (2000) 311–330. Google Scholar
Murray, R., Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Math. Control Signals Syst. 7 (1994) 58–75. Google Scholar
Murray, R. and Sastry, S., Nonholonomic motion planning : Steering using sinusoids. IEEE Trans. Autom. Control 38 (1993) 700–716. Google Scholar
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.
Pasillas-Lépine, W. and Respondek, W., On the geometry of Goursat structures. ESAIM : COCV 6 (2001) 119–181. Google Scholar
Pereira da Silva, P.S., and Corrêa Filho, C., Relative flatness and flatness of implicit systems. SIAM J. Control Optim. 39 (2001) 1929–1951. Google Scholar
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.
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.
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.
van Nieuwstadt, M., Rathinam, M. and Murray, R.M., Differential Flatness and Absolute Equivalence of Nonlinear Control Systems. SIAM J. Control Optim. 36 (1998) 1225–1239. Google Scholar
von Weber, E., Zur Invariantentheorie der Systeme Pfaff’scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellshaft der Wissenshaften Mathematisch-Physikalische Klasse, Leipzig 50 (1898) 207–229. Google Scholar