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Exponential stabilization of nonlinear driftless systemswith robustness to unmodeled dynamics

Published online by Cambridge University Press:  15 August 2002

Pascal Morin  and
Claude Samson
Pascal Morin
Affiliation:
INRIA, 2004 route des Lucioles, 06902 Sophia-Antipolis Cedex, France; [email protected].
Claude Samson
Affiliation:
INRIA, 2004 route des Lucioles, 06902 Sophia-Antipolis Cedex, France; [email protected].
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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.

Keywords

Nonlinear systemasymptotic stabilizationrobust controlChen-Fliess series.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 1 - 35
Copyright
© EDP Sciences, SMAI, 1999

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References

M.K. Bennani and P. Rouchon, Robust stabilization of flat and chained systems, in European Control Conference (ECC) (1995) 2642-2646.
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).
Canudas de, C. Wit and O. J. Sørdalen, Exponential stabilization of mobile robots with nonholonomic constraints. IEEE Trans. Automat. Control 37 (1992) 1791-1797.
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Flatness and defect of non-linear systems: introductory theory and examples. Internat. J. Control 61 (1995) 1327-1361. CrossRef
Hermes, H., Nilpotent and high-order approximations of vector field systems. SIAM Rev. 33 (1991) 238-264. CrossRef
A. Isidori, Nonlinear control systems. Springer Verlag, third edition (1995).
M. Kawski, Geometric homogeneity and stabilization, in IFAC Nonlinear Control Systems Design Symp. (NOLCOS) (1995) 164-169.
I. Kolmanovsky and N.H. McClamroch, Developments in nonholonomic control problems. IEEE Control Systems (1995) 20-36.
Kurzweil, J. and Jarnik, J., Iterated lie brackets in limit processes in ordinary differential equations. Results in Mathematics 14 (1988) 125-137. CrossRef
Z. Li and J.F. Canny, Nonholonomic motion planning. Kluwer Academic Press (1993).
Liu, W., An approximation algorithm for nonholonomic systems. SIAM J. Contr. Opt. 35 (1997) 1328-1365. CrossRef
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
M'Closkey, R.T. and Murray, R.M., Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. IEEE Trans. Automat. Contr. 42 (1997) 614-628. CrossRef
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.
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).
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.
P. Morin and C. Samson, Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics. Technical Report 3477, INRIA (1998).
Murray, R.M. and Sastry, S.S., Nonholonomic motion planning: Steering using sinusoids. IEEE Trans. Automat. Contr. 38 (1993) 700-716. CrossRef
L. Rosier, Étude de quelques problèmes de stabilisation. PhD thesis, École Normale de Cachan (1993).
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).
Sørdalen, O.J. and Egeland, O., Exponential stabilization of nonholonomic chained systems. IEEE Trans. Automat. Contr. 40 (1995) 35-49. CrossRef
G. Stefani, Polynomial approximations to control systems and local controllability, in IEEE Conf. on Decision and Control (CDC) (1985) 33-38.
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.
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.
Sussmann, H.J., Lie brackets and local controllability: a sufficient condition for scalar-input systems. SIAM J. Contr. Opt. 21 (1983) 686-713. CrossRef
Sussmann, H.J., A general theorem on local controllability. SIAM J. Contr. Opt. 25 (1987) 158-194. CrossRef