# Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics

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

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

## Access Full Article

top## Abstract

top## How to cite

topMorin, 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- 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). Zbl0528.93051
- 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
- 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
- H. Hermes, Nilpotent and high-order approximations of vector field systems. SIAM Rev.33 (1991) 238-264. Zbl0733.93062
- A. Isidori, Nonlinear control systems. Springer Verlag, third edition (1995). Zbl0878.93001
- 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.
- J. Kurzweil and J. Jarnik, Iterated lie brackets in limit processes in ordinary differential equations. Results in Mathematics14 (1988) 125-137. Zbl0663.34043
- Z. Li and J.F. Canny, Nonholonomic motion planning. Kluwer Academic Press (1993). Zbl0875.00053
- W. Liu, An approximation algorithm for nonholonomic systems. SIAM J. Contr. Opt.35 (1997) 1328-1365. Zbl0887.34063
- 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
- 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
- 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). Zbl0938.93055
- 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). Zbl0919.93059
- R.M. Murray and S.S. Sastry, Nonholonomic motion planning: Steering using sinusoids. IEEE Trans. Automat. Contr.38 (1993) 700-716. Zbl0800.93840
- 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).
- O.J. Sørdalen and O. Egeland, Exponential stabilization of nonholonomic chained systems. IEEE Trans. Automat. Contr.40 (1995) 35-49. Zbl0828.93055
- 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.
- 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
- H.J. Sussmann, A general theorem on local controllability. SIAM J. Contr. Opt.25 (1987) 158-194. Zbl0629.93012

## NotesEmbed ?

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