Stabilization of a 1-D tank modeled by the shallow water equations

Christophe Prieur; Jonathan de Halleux

Journées équations aux dérivées partielles (2002)

  • Volume: 52, Issue: 3-4, page 1-13
  • ISSN: 0752-0360

Abstract

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We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by the shallow water equations. By means of a Lyapunov approach, we deduce control laws to stabilize the fluid's state and the tank's position. Although global asymptotic stability is yet to be proved, we numerically simulate the system and observe the stabilization for different control situations.

How to cite

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Prieur, Christophe, and de Halleux, Jonathan. "Stabilization of a 1-D tank modeled by the shallow water equations." Journées équations aux dérivées partielles 52.3-4 (2002): 1-13. <http://eudml.org/doc/93424>.

@article{Prieur2002,
abstract = {We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by the shallow water equations. By means of a Lyapunov approach, we deduce control laws to stabilize the fluid's state and the tank's position. Although global asymptotic stability is yet to be proved, we numerically simulate the system and observe the stabilization for different control situations.},
author = {Prieur, Christophe, de Halleux, Jonathan},
journal = {Journées équations aux dérivées partielles},
keywords = {Hyperbolic PDEs; Shallow water equations; Lyapunov approach; Boundary control; Numerical resolution},
language = {eng},
number = {3-4},
pages = {1-13},
publisher = {Université de Nantes},
title = {Stabilization of a 1-D tank modeled by the shallow water equations},
url = {http://eudml.org/doc/93424},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Prieur, Christophe
AU - de Halleux, Jonathan
TI - Stabilization of a 1-D tank modeled by the shallow water equations
JO - Journées équations aux dérivées partielles
PY - 2002
PB - Université de Nantes
VL - 52
IS - 3-4
SP - 1
EP - 13
AB - We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by the shallow water equations. By means of a Lyapunov approach, we deduce control laws to stabilize the fluid's state and the tank's position. Although global asymptotic stability is yet to be proved, we numerically simulate the system and observe the stabilization for different control situations.
LA - eng
KW - Hyperbolic PDEs; Shallow water equations; Lyapunov approach; Boundary control; Numerical resolution
UR - http://eudml.org/doc/93424
ER -

References

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  1. [1] Coron J.M. (2001). Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations. Preprint, Université Paris-Sud, number 2001-28. Zbl1071.76012MR1932962
  2. [2] Coron J.M., B. d'Andréa Novel and G. Bastin (1999). A lyapunov approach to control irrigation canals modeled by saint-venant equations. European Control Conference. Karlsruhe, Germany. 
  3. [3] Debnath L. (1994). Nonlinear water waves, Academic Press, Boston. Zbl0793.76001MR1266390
  4. [4] J.-F. Gerbeau et B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, rapport de recherche numéro RR- 4084, INRIA Rocquencourt, 2000. Zbl0997.76023MR1821555
  5. [5] Graf W.H. (1998). Fluvial Hydraulics. John Wiley & Sons. 
  6. [6] M. Grundelius, Iterative optimal control of liquid slosh in an industrial packaging machine, Conference on Decision and Control, Sydney, Australie, 2000. XIII-11 
  7. [7] Dubois F., N. Petit and P. Rouchon (1999).Motion planning and nonlinear simulations for a tank containing a fluid. European Control Conference. Karlsruhe, Germany. 
  8. [8] Li Ta-tsien and Wen-ci Yu (1995). Boundary value problems for quasilinear hyperbolic systems. Duke University, Durham. Zbl0627.35001MR823237
  9. [9] Malaterre P.O. (1994). Modelisation, Analysis and LQR Optimal Control of an Irrigation Canal. PhD thesis, LAAS-CNRS-ENGREF-Cemagref, France. 
  10. [10] Mazenc F. and L. Praly (1996). Adding integrations, saturated controls, and stabilization for feedforward systems. IEEE Trans. Automat. Control, 41 (11), 1559-1578. Zbl0865.93049MR1419682
  11. [11] Mottelet S. (2000). Controllability and stabilization of a canal with wave generators. SIAM J. Control Optimization, 38 (3), 711-735. Zbl0966.76015MR1741435
  12. [12] Mottelet S. (2000). Controllability and stabilizability of liquid vibration in a container during transportation. Conference on Decision and Control. Sydney, Australy. 
  13. [13] Petit N. and P. Rouchon (2000). Dynamics and solutions to some control problems for water-tank systems. CIT-CDC, (00-004). XIII-12 
  14. [14] Prieur C., Diverses méthodes pour des problèmes de stabilisation, thèse, université Paris-Sud, 2001. 
  15. [15] Saint-Venant B. de (1971). Théorie du mouvement non-permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit, Comptes-rendus de l'académie des Sciences, Paris, 73, 148-154, 237-240. JFM03.0482.04
  16. [16] Serre D. (1996 

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