Invariant tracking
Martin, Philippe ; Rouchon, Pierre ; Rudolph, Joachim1
1 Institut fur Regelungs- und Steuerungstheorie, Technische Universität Dresden, Mommsenstr. 13, 01062 Dresden, Germany
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 1-13.

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.

DOI : 10.1051/cocv:2003037
Classification : 53A55, 93C10, 93D25, 70Q05
Mots-clés : symmetries, invariants, nonlinear control, output tracking, decoupling
Martin, Philippe  ; Rouchon, Pierre  ; Rudolph, Joachim 1

1 Institut fur Regelungs- und Steuerungstheorie, Technische Universität Dresden, Mommsenstr. 13, 01062 Dresden, Germany 
@article{COCV_2004__10_1_1_0,
     author = {Martin, Philippe and Rouchon, Pierre and Rudolph, Joachim},
     title = {Invariant tracking},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1--13},
     publisher = {EDP-Sciences},
     volume = {10},
     number = {1},
     year = {2004},
     doi = {10.1051/cocv:2003037},
     mrnumber = {2084252},
     zbl = {1088.93016},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2003037/}
}
TY  - JOUR
AU  - Martin, Philippe
AU  - Rouchon, Pierre
AU  - Rudolph, Joachim
TI  - Invariant tracking
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2004
SP  - 1
EP  - 13
VL  - 10
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2003037/
DO  - 10.1051/cocv:2003037
LA  - en
ID  - COCV_2004__10_1_1_0
ER  - 
%0 Journal Article
%A Martin, Philippe
%A Rouchon, Pierre
%A Rudolph, Joachim
%T Invariant tracking
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2004
%P 1-13
%V 10
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2003037/
%R 10.1051/cocv:2003037
%G en
%F COCV_2004__10_1_1_0
Martin, Philippe; Rouchon, Pierre; Rudolph, Joachim. Invariant tracking. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 1-13. doi : 10.1051/cocv:2003037. http://archive.numdam.org/articles/10.1051/cocv:2003037/

[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. | MR | Zbl

[2] F. Bullo and R.M. Murray, Tracking for fully actuated mechanical systems: A geometric framework. Automatica 35 (1999) 17-34. | MR | Zbl

[3] E. Delaleau and P.S. Pereira Da Silva, Filtrations in feedback synthesis: Part I - Systems and feedbacks. Forum Math. 10 (1998) 147-174. | Zbl

[4] J. Descusse and C.H. Moog, Dynamic decoupling for right invertible nonlinear systems. Systems Control Lett. 8 (1988) 345-349. | MR | Zbl

[5] F. Fagnani and J. Willems, Representations of symmetric linear dynamical systems. SIAM J. Control Optim. 31 (1993) 1267-1293. | MR | Zbl

[6] J.W. Grizzle and S.I. Marcus, The structure of nonlinear systems possessing symmetries. IEEE Trans. Automat. Control 30 (1985) 248-258. | MR | Zbl

[7] A. Isidori, Nonlinear Control Systems, 2nd Edition. Springer, New York (1989). | MR

[8] B. Jakubczyk, Symmetries of nonlinear control systems and their symbols, in Canadian Math. Conf. Proceed., Vol. 25 (1998) 183-198. | MR | Zbl

[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. | MR | Zbl

[10] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry. Springer-Verlag, New York (1994). | MR | Zbl

[11] Ph. Martin, R. Murray and P. Rouchon, Flat systems1997) 211-264. Plenary lectures and Mini-courses. | MR | Zbl

[12] H. Nijmeijer, Right-invertibility for a class of nonlinear control systems: A geometric approach. Systems Control Lett. 7 (1986) 125-132. | MR | Zbl

[13] H. Nijmeijer and A.J. Van Der Schaft, Nonlinear Dynamical Control Systems. Springer-Verlag (1990). | MR | Zbl

[14] P.J. Olver, Equivalence, Invariants and Symmetry. Cambridge University Press (1995). | MR | Zbl

[15] P.J. Olver, Classical Invariant Theory. Cambridge University Press (1999). | MR | Zbl

[16] W. Respondek and H. Nijmeijer, On local right-invertibility of nonlinear control system. Control Theory Adv. Tech. 4 (1988) 325-348. | MR

[17] W. Respondek and I.A. Tall, Nonlinearizable single-input control systems do not admit stationary symmetries. Systems Control Lett. 46 (2002) 1-16. | MR | Zbl

[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. | MR | Zbl

[19] A.J. Van Der Schaft, Symmetries in optimal control. SIAM J. Control Optim. 25 (1987) 245-259. | MR | Zbl

[20] C. Woernle, Flatness-based control of a nonholonomic mobile platform. Z. Angew. Math. Mech. 78 (1998) 43-46. | Zbl

Cité par Sources :