Flat outputs of two-input driftless control systems

Shun-Jie Li; Witold Respondek

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 774-798
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Li, Shun-Jie, and Respondek, Witold. "Flat outputs of two-input driftless control systems." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 774-798. <http://eudml.org/doc/277812>.

@article{Li2012,
abstract = {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. },
author = {Li, Shun-Jie, Respondek, Witold},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control system; flatness; flat output; feedback equivalence; characteristic distribution; n-trailer system; control system; -trailer system},
language = {eng},
month = {11},
number = {3},
pages = {774-798},
publisher = {EDP Sciences},
title = {Flat outputs of two-input driftless control systems},
url = {http://eudml.org/doc/277812},
volume = {18},
year = {2012},
}

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
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 774
EP - 798
AB - 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.
LA - eng
KW - Control system; flatness; flat output; feedback equivalence; characteristic distribution; n-trailer system; control system; -trailer system
UR - http://eudml.org/doc/277812
ER -

References

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  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 inclusions32, edited by B. Jakuczyk, W. Respondek and T. Rzeżuchowski. Banach Center Publications, Warsaw (1995) 19–33.  Zbl0844.93024
  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).  
  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 France42, Œuvres complètes2. Part. II, Gauthiers-Villars, Paris (1914) 12–48.  Zbl45.0472.04
  4. M. Cheaito and P. Mormul, Rank-2 distributions satisfying the Goursat condition :all their local models in dimension 7 and 8. ESAIM : COCV4 (1999) 137–158.  Zbl0957.58002
  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.  Zbl0776.93038
  6. M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems : Introductory theory and examples. Int. J. Control61 (1995) 1327–1361.  Zbl0838.93022
  7. M. Fliess, J. Lévine, P. Martin and P. Rouchon, A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control61 (1999) 1327–1361.  Zbl0838.93022
  8. A. Giaro, A. Kumpera and C. Ruiz, Sur la lecture correcte d’un resultat d’Élie Cartan. C. R. Acad. Sci. Paris287 (1978) 241–244.  Zbl0398.58003
  9. E. Goursat, Leçons sur le problème de Pfaff. Hermann, Paris (1923).  
  10. D. Hilbert, Über den Begriff der Klasse von Differentialgleichungen. Math. Ann.73 (1912) 95–108.  Zbl43.0378.01
  11. A. Isidori. Nonlinear Control Systems, 3rd edition. Springer-Verlag, London (1995).  Zbl0878.93001
  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 : COCV1 (1996) 241–266.  Zbl0874.93033
  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.  Zbl0516.58004
  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 Sciences229. 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).  
  19. P. Martin and P. Rouchon, Feedback linearization and driftless systems. CAS internal report No. 446, École des Mines (1993).  Zbl0842.93015
  20. P. Martin and P. Rouchon, Feedback linearization and driftless systems. Math. Contr. Signals Syst.7 (1994) 235–254.  Zbl0842.93015
  21. P. Martin, R.M. Murray and P. Rouchon, Flat systems, in Mathematical Control Theory, Part 2, ICTP Lecture Notes8, edited by A.A. Agrachev. ICTP Publications, Trieste (2002) 705–768.  Zbl1013.93007
  22. P. Mormul, Goursat flags : classification of codimension-one singularities. J. Dyn. Control Syst.6 (2000) 311–330.  Zbl1040.58019
  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.  Zbl0825.93319
  24. R. Murray and S. Sastry, Nonholonomic motion planning : Steering using sinusoids. IEEE Trans. Autom. Control38 (1993) 700–716.  Zbl0800.93840
  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 : COCV6 (2001) 119–181.  Zbl0966.58002
  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.  Zbl1097.93019
  28. J.-B. Pomet, A differential geometric setting for dynamic equivalence and dynamic linearization, in Geometry in Nonlinear Control and Differential Inclusions32, edited by B. Jakubczyk, W. Respondek and T. Rzeżuchowski. Banach Center Publications, Warsaw (1995) 319–339.  Zbl0838.93019
  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 Sciences295, edited by W. Kang, M. Xiao and C. Borges. Springer Verlag, Berlin, Heidelberg (2003) 65–86.  Zbl1203.93093
  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.  Zbl0910.93023
  32. E. von Weber, Zur Invariantentheorie der Systeme Pfaff’scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellshaft der Wissenshaften Mathematisch-Physikalische Klasse, Leipzig50 (1898) 207–229.  Zbl29.0302.01

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