# Invariant tracking

Philippe Martin; Pierre Rouchon; Joachim Rudolph^{[1]}

- [1] Institut fur Regelungs- und Steuerungstheorie, Technische Universität Dresden, Mommsenstr. 13, 01062 Dresden, Germany

ESAIM: Control, Optimisation and Calculus of Variations (2004)

- Volume: 10, Issue: 1, page 1-13
- ISSN: 1292-8119

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topMartin, Philippe, Rouchon, Pierre, and Rudolph, Joachim. "Invariant tracking." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2004): 1-13. <http://eudml.org/doc/245522>.

@article{Martin2004,

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.},

affiliation = {Institut fur Regelungs- und Steuerungstheorie, Technische Universität Dresden, Mommsenstr. 13, 01062 Dresden, Germany},

author = {Martin, Philippe, Rouchon, Pierre, Rudolph, Joachim},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {symmetries; invariants; nonlinear control; output tracking; decoupling; Symmetries},

language = {eng},

number = {1},

pages = {1-13},

publisher = {EDP-Sciences},

title = {Invariant tracking},

url = {http://eudml.org/doc/245522},

volume = {10},

year = {2004},

}

TY - JOUR

AU - Martin, Philippe

AU - Rouchon, Pierre

AU - Rudolph, Joachim

TI - Invariant tracking

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2004

PB - EDP-Sciences

VL - 10

IS - 1

SP - 1

EP - 13

AB - 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.

LA - eng

KW - symmetries; invariants; nonlinear control; output tracking; decoupling; Symmetries

UR - http://eudml.org/doc/245522

ER -

## References

top- [1] A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden and R. Murray, Nonholonomic mechanical systems with symmetry. Arch. Rational Mech. Anal. 136 (1996) 21-99. Zbl0886.70014MR1423003
- [2] F. Bullo and R.M. Murray, Tracking for fully actuated mechanical systems: A geometric framework. Automatica 35 (1999) 17-34. Zbl0941.93014MR1827788
- [3] E. Delaleau and P.S. Pereira da Silva, Filtrations in feedback synthesis: Part I – Systems and feedbacks. Forum Math. 10 (1998) 147-174. Zbl0891.93022
- [4] J. Descusse and C.H. Moog, Dynamic decoupling for right invertible nonlinear systems. Systems Control Lett. 8 (1988) 345-349. Zbl0617.93024MR884884
- [5] F. Fagnani and J. Willems, Representations of symmetric linear dynamical systems. SIAM J. Control Optim. 31 (1993) 1267-1293. Zbl0785.93005MR1234003
- [6] J.W. Grizzle and S.I. Marcus, The structure of nonlinear systems possessing symmetries. IEEE Trans. Automat. Control 30 (1985) 248-258. Zbl0562.93041MR778428
- [7] A. Isidori, Nonlinear Control Systems, 2nd Edition. Springer, New York (1989). Zbl0693.93046MR1015932
- [8] B. Jakubczyk, Symmetries of nonlinear control systems and their symbols, in Canadian Math. Conf. Proceed., Vol. 25 (1998) 183-198. Zbl1126.93352MR1648715
- [9] W.S. Koon and J.E. Marsden, Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction. SIAM J. Control Optim. 35 (1997) 901-929. Zbl0880.70020MR1444343
- [10] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry. Springer-Verlag, New York (1994). Zbl0811.70002MR1304682
- [11] Ph. Martin, R. Murray and P. Rouchon, Flat systems, in Proc. of the $4$th European Control Conf.. Brussels (1997) 211-264. Plenary lectures and Mini-courses. Zbl1008.93019MR1972795
- [12] H. Nijmeijer, Right-invertibility for a class of nonlinear control systems: A geometric approach. Systems Control Lett. 7 (1986) 125-132. Zbl0622.93027MR836302
- [13] H. Nijmeijer and A.J. van der Schaft, Nonlinear Dynamical Control Systems. Springer-Verlag (1990). Zbl0701.93001MR1047663
- [14] P.J. Olver, Equivalence, Invariants and Symmetry. Cambridge University Press (1995). Zbl0837.58001MR1337276
- [15] P.J. Olver, Classical Invariant Theory. Cambridge University Press (1999). Zbl0971.13004MR1694364
- [16] W. Respondek and H. Nijmeijer, On local right-invertibility of nonlinear control system. Control Theory Adv. Tech. 4 (1988) 325-348. MR964370
- [17] W. Respondek and I.A. Tall, Nonlinearizable single-input control systems do not admit stationary symmetries. Systems Control Lett. 46 (2002) 1-16. Zbl1023.93017MR2011068
- [18] 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. Zbl0932.93066MR1714594
- [19] A.J. van der Schaft, Symmetries in optimal control. SIAM J. Control Optim. 25 (1987) 245-259. Zbl0616.49003MR877061
- [20] C. Woernle, Flatness-based control of a nonholonomic mobile platform. Z. Angew. Math. Mech. 78 (1998) 43-46. Zbl0915.70020

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