Control of a clamped-free beam by a piezoelectric actuator

Emmanuelle Crépeau; Christophe Prieur

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 3, page 545-563
  • ISSN: 1292-8119

Abstract

top
We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations.

How to cite

top

Crépeau, Emmanuelle, and Prieur, Christophe. "Control of a clamped-free beam by a piezoelectric actuator." ESAIM: Control, Optimisation and Calculus of Variations 12.3 (2006): 545-563. <http://eudml.org/doc/249668>.

@article{Crépeau2006,
abstract = { We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations. },
author = {Crépeau, Emmanuelle, Prieur, Christophe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {piezoelectric actuator; metallic beam; exact controlability.; exact controllability.},
language = {eng},
month = {6},
number = {3},
pages = {545-563},
publisher = {EDP Sciences},
title = {Control of a clamped-free beam by a piezoelectric actuator},
url = {http://eudml.org/doc/249668},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Crépeau, Emmanuelle
AU - Prieur, Christophe
TI - Control of a clamped-free beam by a piezoelectric actuator
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/6//
PB - EDP Sciences
VL - 12
IS - 3
SP - 545
EP - 563
AB - We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations.
LA - eng
KW - piezoelectric actuator; metallic beam; exact controlability.; exact controllability.
UR - http://eudml.org/doc/249668
ER -

References

top
  1. C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat.2 (1999) 33–63.  Zbl0924.42022
  2. V. Balamurugan and S. Narayanan, Shell finite element for smart piezoelectric composite plate/shell structures and its application to the study of active vibration control. Finite Elem. Anal. Des.37 (2001) 713–738.  Zbl1094.74683
  3. J.W.S. Cassels, An introduction to diophantine approximation. Moskau: Verlag 213 S. (1961).  Zbl0098.26301
  4. E. Crépeau, Exact boundary controllability of the Boussinesq equation on a bounded domain. Diff. Int. Equ.16 (2003) 303–326.  Zbl1161.93302
  5. E. Crépeau, Contrôlabilité exacte d'équations dispersives issues de la mécanique. Thèse de l'Université de Paris-Sud, avalaible at www.math.ursq.fr/~crepeau/PUBLI/these1.html.  
  6. P. Destuynder, I. Legrain, L. Castel and N. Richard, Theorical, numerical and experimental discussion of the use of piezoelectric devices for control-structure interaction. Eur. J. Mech., A/Solids11 (1992) 97–106.  
  7. P. Destuynder, A mathematical analysis of a smart-beam which is equipped with piezoelectric actuators. Control Cybern.28 (1999) 503–530.  Zbl0962.93049
  8. M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, Linear systems over Mikusinski operators and control of a flexible beam. ESAIM: Proc.2 (1997) 183–193.  Zbl0898.93018
  9. P. Germain, Mecanique, Tome II. Ellipses (1996).  
  10. D. Halim and S.O.R. Moheimani, Spatial Resonant Control of Flexible Structures – Applications to a Piezoelectric Laminate Beam. IEEE Trans. Control Syst. Tech.9 (2001) 37–53.  
  11. D. Halim and S.O.R. Moheimani, Spatial H2 control of a piezoelectric laminate beam: experimental implementation. IEEE Trans. Control Syst. Tech.10 (2002) 533–546.  
  12. W.S. Hwang and H.C. Park, Finite element modeling of piezoelectric sensors and actuator. AIAA Journal31 (1993) 930–937.  
  13. A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Zeitschr. 41 (1936) 367–379.  Zbl0014.21503
  14. S. Lang, Introduction to diophantine approximations. Springer-Verlag (1991).  Zbl0214.06203
  15. S. Leleu, Amortissement actif des vibrations d'une structure flexible de type plaque à l'aide de transducteurs piézoélectriques. Thèse, ENS de Cachan (2002).  
  16. J.L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Contrôlabilité exacte, Collection de recherche en mathématiques appliquées8 (Masson, Paris), 1988. distribués, Masson, Paris (1988).  Zbl0653.93002
  17. L. Meirovitch, Elements of vibration analysis, Düsseldorf, McGraw-Hill (1975).  Zbl0359.70039
  18. R. Rebarber, Spectral assignability for distribued parameter systems with unbounded scalar control. SIAM J. Control Optim.27 (1989) 148–169.  Zbl0681.93037
  19. L. Rosier, Exact boundary controllability for the linear KdV equation – a numerical study. ESAIM: Proc.4 (1998) 255–267.  Zbl0919.93039
  20. J. Rudolph and F. Woittennek, Flatness based boundary control of piezoelectric benders. Automatisierungstechnik50 (2002) 412–421.  
  21. R.S. Smith, C.C. Chu and J.L. Fanson, The design of H controllers for an experimental non-collocated flexible structure Problem. IEEE Trans. Control Syst. Tech.2 (1994) 101–109.  
  22. S. Tliba and H. Abou-Kandil, H controller design for active vibration damping of a smart flexible structure using piezoelectric transducers, in 4th Symp. IFAC on Robust Control Design (ROCOND 2003), Milan, Italy (2003).  
  23. M. Tucsnak, Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim.34 (1996) 922–930.  Zbl0853.73051

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.