Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics

Pascal Morin; Claude Samson

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 1-35
  • ISSN: 1292-8119

Abstract

top
Exponential stabilization of nonlinear driftless affine control systems is addressed with the concern of achieving robustness with respect to imperfect knowledge of the system's control vector fields. In order to satisfy this robustness requirement, and inspired by Bennani and Rouchon [1] where the same issue was first addressed, we consider a control strategy which consists in applying periodically updated open-loop controls that are continuous with respect to state initial conditions. These controllers are more precisely described as continuous time-periodic feedbacks associated with a specific dynamic extension of the original system. Sufficient conditions which, if they are satisfied by the control law, ensure that the control is a robust exponential stabilizer for the extended system are given. Explicit and simple control expressions which satisfy these conditions in the case of n-dimensional chained systems are proposed. A constructive algorithm for the design of such control laws, which applies to any (sufficiently regular) driftless control system, is described.

How to cite

top

Morin, Pascal, and Samson, Claude. "Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 1-35. <http://eudml.org/doc/197375>.

@article{Morin2010,
abstract = { Exponential stabilization of nonlinear driftless affine control systems is addressed with the concern of achieving robustness with respect to imperfect knowledge of the system's control vector fields. In order to satisfy this robustness requirement, and inspired by Bennani and Rouchon [1] where the same issue was first addressed, we consider a control strategy which consists in applying periodically updated open-loop controls that are continuous with respect to state initial conditions. These controllers are more precisely described as continuous time-periodic feedbacks associated with a specific dynamic extension of the original system. Sufficient conditions which, if they are satisfied by the control law, ensure that the control is a robust exponential stabilizer for the extended system are given. Explicit and simple control expressions which satisfy these conditions in the case of n-dimensional chained systems are proposed. A constructive algorithm for the design of such control laws, which applies to any (sufficiently regular) driftless control system, is described. },
author = {Morin, Pascal, Samson, Claude},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear system; asymptotic stabilization; robust control; Chen-Fliess series.; affine control system; robust exponential stability; feedback time-dependent control; periodic control; robust stabilization},
language = {eng},
month = {3},
pages = {1-35},
publisher = {EDP Sciences},
title = {Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics},
url = {http://eudml.org/doc/197375},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Morin, Pascal
AU - Samson, Claude
TI - Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 1
EP - 35
AB - Exponential stabilization of nonlinear driftless affine control systems is addressed with the concern of achieving robustness with respect to imperfect knowledge of the system's control vector fields. In order to satisfy this robustness requirement, and inspired by Bennani and Rouchon [1] where the same issue was first addressed, we consider a control strategy which consists in applying periodically updated open-loop controls that are continuous with respect to state initial conditions. These controllers are more precisely described as continuous time-periodic feedbacks associated with a specific dynamic extension of the original system. Sufficient conditions which, if they are satisfied by the control law, ensure that the control is a robust exponential stabilizer for the extended system are given. Explicit and simple control expressions which satisfy these conditions in the case of n-dimensional chained systems are proposed. A constructive algorithm for the design of such control laws, which applies to any (sufficiently regular) driftless control system, is described.
LA - eng
KW - Nonlinear system; asymptotic stabilization; robust control; Chen-Fliess series.; affine control system; robust exponential stability; feedback time-dependent control; periodic control; robust stabilization
UR - http://eudml.org/doc/197375
ER -

References

top
  1. M.K. Bennani and P. Rouchon, Robust stabilization of flat and chained systems, in European Control Conference (ECC) (1995) 2642-2646.  
  2. R.W. Brockett, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, R.S. Millman R.W. Brockett and H.H. Sussmann Eds., Birkauser (1983).  Zbl0528.93051
  3. C. Canudas de Wit and O. J. Sørdalen, Exponential stabilization of mobile robots with nonholonomic constraints. IEEE Trans. Automat. Control37 (1992) 1791-1797.  Zbl0778.93077
  4. M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples. Internat. J. Control61 (1995) 1327-1361.  Zbl0838.93022
  5. H. Hermes, Nilpotent and high-order approximations of vector field systems. SIAM Rev.33 (1991) 238-264.  Zbl0733.93062
  6. A. Isidori, Nonlinear control systems. Springer Verlag, third edition (1995).  Zbl0878.93001
  7. M. Kawski, Geometric homogeneity and stabilization, in IFAC Nonlinear Control Systems Design Symp. (NOLCOS) (1995) 164-169.  
  8. I. Kolmanovsky and N.H. McClamroch, Developments in nonholonomic control problems. IEEE Control Systems (1995) 20-36.  
  9. J. Kurzweil and J. Jarnik, Iterated lie brackets in limit processes in ordinary differential equations. Results in Mathematics14 (1988) 125-137.  Zbl0663.34043
  10. Z. Li and J.F. Canny, Nonholonomic motion planning. Kluwer Academic Press (1993).  Zbl0875.00053
  11. W. Liu, An approximation algorithm for nonholonomic systems. SIAM J. Contr. Opt.35 (1997) 1328-1365.  Zbl0887.34063
  12. D.A. Lizárraga, P. Morin and C. Samson, Non-robustness of continuous homogeneous stabilizers for affine systems. Technical Report 3508, INRIA (1998). Available at http://www.inria.fr/RRRT/RR-3508.html  
  13. R.T. M'Closkey and R.M. Murray, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. IEEE Trans. Automat. Contr.42 (1997) 614-628.  Zbl0882.93066
  14. S. Monaco and D. Normand-Cyrot, An introduction to motion planning using multirate digital control, in IEEE Conf. on Decision and Control (CDC) (1991) 1780-1785.  
  15. P. Morin, J.-B. Pomet and C. Samson, Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of lie brackets in closed-loop. SIAM J. Contr. Opt. (to appear).  Zbl0938.93055
  16. P. Morin, J.-B. Pomet and C. Samson, Developments in time-varying feedback stabilization of nonlinear systems, in IFAC Nonlinear Control Systems Design Symp. (NOLCOS) (1998) 587-594.  
  17. P. Morin and C. Samson, Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics. Technical Report 3477, INRIA (1998).  Zbl0919.93059
  18. R.M. Murray and S.S. Sastry, Nonholonomic motion planning: Steering using sinusoids. IEEE Trans. Automat. Contr.38 (1993) 700-716.  Zbl0800.93840
  19. L. Rosier, Étude de quelques problèmes de stabilisation. PhD thesis, École Normale de Cachan (1993).  
  20. C. Samson, Velocity and torque feedback control of a nonholonomic cart, in Int. Workshop in Adaptative and Nonlinear Control: Issues in Robotics. LNCIS, Vol. 162, Springer Verlag, 1991 (1990).  
  21. O.J. Sørdalen and O. Egeland, Exponential stabilization of nonholonomic chained systems. IEEE Trans. Automat. Contr.40 (1995) 35-49.  Zbl0828.93055
  22. G. Stefani, Polynomial approximations to control systems and local controllability, in IEEE Conf. on Decision and Control (CDC) (1985) 33-38.  
  23. G. Stefani, On the local controllability of scalar-input control systems, in Theory and Applications of Nonlinear Control Systems, Proc. of MTNS'84, C.I. Byrnes and A. Linsquist Eds., North-Holland (1986) 167-179.  
  24. H.J. Sussmann and W. Liu, Limits of highly oscillatory controls ans approximation of general paths by admissible trajectories, in IEEE Conf. on Decision and Control (CDC) (1991) 437-442.  
  25. H.J. Sussmann, Lie brackets and local controllability: a sufficient condition for scalar-input systems. SIAM J. Contr. Opt.21 (1983) 686-713.  Zbl0523.49026
  26. H.J. Sussmann, A general theorem on local controllability. SIAM J. Contr. Opt.25 (1987) 158-194.  Zbl0629.93012

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.