Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T23:21:28.579Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant tracking

Published online by Cambridge University Press:  15 February 2004

Philippe Martin ,
Pierre Rouchon  and
Joachim Rudolph
Philippe Martin
Affiliation:
Centre Automatique et Systèmes, École des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; philippe.martin@ensmp.fr., pierre.rouchon@ensmp.fr.
Pierre Rouchon
Affiliation:
Centre Automatique et Systèmes, École des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; philippe.martin@ensmp.fr., pierre.rouchon@ensmp.fr.
Joachim Rudolph
Affiliation:
Institut fur Regelungs- und Steuerungstheorie, Technische Universität Dresden, Mommsenstr. 13, 01062 Dresden, Germany; rudolph@erss11.et.tu-dresden.de.
Get access

Abstract

The problem of invariant output tracking is considered: given a control system admitting a symmetry group G, design a feedback such that the closed-loop system tracks a desired output reference and is invariant under the action of G. Invariant output errors are defined as a set of scalar invariants of G; they are calculated with the Cartan moving frame method. It is shown that standard tracking methods based on input-output linearization can be applied to these invariant errors to yield the required “symmetry-preserving” feedback.

Keywords

Symmetriesinvariantsnonlinear controloutputtrackingdecoupling.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 1 - 13
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E. and Murray, R., Nonholonomic mechanical systems with symmetry. Arch. Rational Mech. Anal. 136 (1996) 21-99. CrossRef
Bullo, F. and Murray, R.M., Tracking for fully actuated mechanical systems: A geometric framework. Automatica 35 (1999) 17-34. CrossRef
Delaleau, E. and Pereira da, P.S. Silva, Filtrations in feedback synthesis: Part I – Systems and feedbacks. Forum Math. 10 (1998) 147-174.
Descusse, J. and Moog, C.H., Dynamic decoupling for right invertible nonlinear systems. Systems Control Lett. 8 (1988) 345-349. CrossRef
Fagnani, F. and Willems, J., Representations of symmetric linear dynamical systems. SIAM J. Control Optim. 31 (1993) 1267-1293. CrossRef
Grizzle, J.W. and Marcus, S.I., The structure of nonlinear systems possessing symmetries. IEEE Trans. Automat. Control 30 (1985) 248-258. CrossRef
A. Isidori, Nonlinear Control Systems, 2nd Edition. Springer, New York (1989).
B. Jakubczyk, Symmetries of nonlinear control systems and their symbols, in Canadian Math. Conf. Proceed., Vol. 25 (1998) 183-198.
Koon, W.S. and Marsden, J.E., Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction. SIAM J. Control Optim. 35 (1997) 901-929. CrossRef
J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry. Springer-Verlag, New York (1994).
Ph. Martin, R. Murray and P. Rouchon, Flat systems, in Proc. of the 4th European Control Conf.. Brussels (1997) 211-264. Plenary lectures and Mini-courses.
Nijmeijer, H., Right-invertibility for a class of nonlinear control systems: A geometric approach. Systems Control Lett. 7 (1986) 125-132. CrossRef
H. Nijmeijer and A.J. van der Schaft, Nonlinear Dynamical Control Systems. Springer-Verlag (1990).
P.J. Olver, Equivalence, Invariants and Symmetry. Cambridge University Press (1995).
P.J. Olver, Classical Invariant Theory. Cambridge University Press (1999).
Respondek, W. and Nijmeijer, H., On local right-invertibility of nonlinear control system. Control Theory Adv. Tech. 4 (1988) 325-348.
Respondek, W. and Tall, I.A., Nonlinearizable single-input control systems do not admit stationary symmetries. Systems Control Lett. 46 (2002) 1-16. CrossRef
P. Rouchon and J. Rudolph, Invariant tracking and stabilization: problem formulation and examples. Springer, Lecture Notes in Control and Inform. Sci. 246 (1999) 261-273. CrossRef
van der Schaft, A.J., Symmetries in optimal control. SIAM J. Control Optim. 25 (1987) 245-259. CrossRef
Woernle, C., Flatness-based control of a nonholonomic mobile platform. Z. Angew. Math. Mech. 78 (1998) 43-46.