Shape optimization of piezoelectric sensors or actuators for the control of plates

Emmanuel Degryse; Stéphane Mottelet

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 4, page 673-690
  • ISSN: 1292-8119

Abstract

top
This paper deals with a new method to control flexible structures by designing non-collocated sensors and actuators satisfying a pseudo-collocation criterion in the low-frequency domain. This technique is applied to a simply supported plate with a point force actuator and a piezoelectric sensor, for which we give some theoretical and numerical results. We also compute low-order controllers which stabilize pseudo-collocated systems and the closed-loop behavior show that this approach is very promising.

How to cite

top

Degryse, Emmanuel, and Mottelet, Stéphane. "Shape optimization of piezoelectric sensors or actuators for the control of plates." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2010): 673-690. <http://eudml.org/doc/90782>.

@article{Degryse2010,
abstract = { This paper deals with a new method to control flexible structures by designing non-collocated sensors and actuators satisfying a pseudo-collocation criterion in the low-frequency domain. This technique is applied to a simply supported plate with a point force actuator and a piezoelectric sensor, for which we give some theoretical and numerical results. We also compute low-order controllers which stabilize pseudo-collocated systems and the closed-loop behavior show that this approach is very promising. },
author = {Degryse, Emmanuel, Mottelet, Stéphane},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Collocation; piezoelectric sensors/actuators; positive-real systems; topology optimization.; collocation; topology optimization},
language = {eng},
month = {3},
number = {4},
pages = {673-690},
publisher = {EDP Sciences},
title = {Shape optimization of piezoelectric sensors or actuators for the control of plates},
url = {http://eudml.org/doc/90782},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Degryse, Emmanuel
AU - Mottelet, Stéphane
TI - Shape optimization of piezoelectric sensors or actuators for the control of plates
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 4
SP - 673
EP - 690
AB - This paper deals with a new method to control flexible structures by designing non-collocated sensors and actuators satisfying a pseudo-collocation criterion in the low-frequency domain. This technique is applied to a simply supported plate with a point force actuator and a piezoelectric sensor, for which we give some theoretical and numerical results. We also compute low-order controllers which stabilize pseudo-collocated systems and the closed-loop behavior show that this approach is very promising.
LA - eng
KW - Collocation; piezoelectric sensors/actuators; positive-real systems; topology optimization.; collocation; topology optimization
UR - http://eudml.org/doc/90782
ER -

References

top
  1. G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002).  Zbl0990.35001
  2. H.T. Banks, R.C. Smith and Y. Wang, Smart material structures, modelling, estimation and control. Res. Appl. Math. Masson, Paris (1996).  Zbl0882.93001
  3. D. Chenais and E. Zuazua, Finite Element Approximation on Elliptic Optimal Design. C.R. Acad. Sci. Paris Ser. I338 729–734 (2004).  Zbl1052.65059
  4. M.J. Chen and C.A. Desoer, Necessary and sufficient conditions for robust stability of linear distributed feedback systems. Internat. J. Control35 (1982) 255–267.  Zbl0489.93041
  5. R.F. Curtain and B. Van Keulen, Robust control with respect to coprime factors of infinite-dimensional positive real systems. IEEE Trans. Autom. Control37 (1992) 868–871.  Zbl0760.93061
  6. R.F. Curtain and B. Van Keulen, Equivalence of input-output stability and exponential stability for infinite dimensional systems. J. Math. Syst. Theory21 (1988) 19–48.  Zbl0657.93050
  7. R.F. Curtain, A synthesis of Time and Frequency domain methods for the control of infinite dimensional systems: a system theoretic approach, in Control and Estimation in Distributed Parameter Systems, H.T. Banks Ed. SIAM (1988) 171–224.  
  8. R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim.26 (1988) 697–713.  Zbl0643.93050
  9. E. Degryse, Étude d'une nouvelle approche pour la conception de capteurs et d'actionneurs pour le contrôle des systèmes flexibles abstraits. Ph.D. Thesis, Université de Technologie de Compiègne, France (2002).  
  10. P.H. Destuynder, I. Legrain, L. Castel and N. Richard, Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction. Eur. J. Mech A/solids11 (1992) 181–213.  
  11. B.A. Francis, A Course in H∞ Control Theory. Lecture notes in control and information sciences. Springer-Verlag Berlin (1988).  
  12. P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping. J. Diff. Equations132 (1996) 338–352.  Zbl0878.35067
  13. J.S. Freudenberg and P.D. Looze, Right half plane poles and zeros and design tradeoffs in feedback systems. IEEE Trans. Autom. Control30 (1985) 555–565.  Zbl0562.93022
  14. J.S. Gibson and A. Adamian, Approximation theory for Linear-Quadratic-Gaussian control of flexible structures. SIAM J. Control Optim.29 (1991) 1–37.  Zbl0788.93027
  15. A. Haraux, Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990).  Zbl0726.58001
  16. P. Hébrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string. Syst. Control Lett.48 (2003) 199–209.  Zbl1134.93399
  17. P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim., to appear.  Zbl1083.49002
  18. C. Inniss and T. Williams, Sensitivity of the zeros of flexible structures to sensor and actuator location. IEEE Trans. Autom. Control45 (2000) 157–160.  Zbl0973.93006
  19. S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations145 (1998) 184–215.  Zbl0920.35029
  20. T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin (1980).  Zbl0435.47001
  21. B. van Keulen, H∞ control for distributed parameter systems: a state-space approach. Birkaüser, Boston (1993).  Zbl0788.93018
  22. I. Lasiecka and R. Triggiani, Non-dissipative boundary stabilization of the wave equation via boundary observation. J. Math. Pures Appl.63 (1984) 59–80.  
  23. D.G. Luenberger, Optimization by Vector Space Methods. John Wiley and Sons, New York (1969).  Zbl0176.12701
  24. F. Macia and E. Zuazua, On the lack of controllability of wave equations: a Gaussian beam approach. Asymptotic Analysis32 (2002) 1–26.  Zbl1024.35062
  25. M. Minoux, Programmation Mathématique: théorie et algorithmes, tome 2. Dunod, Paris (1983).  Zbl0546.90056
  26. O. Morgül, Dynamic boundary control of an Euler-Bernoulli beam. IEEE Trans. Autom. Control37 (1992) 639–642.  
  27. S. Mottelet, Controllability and stabilization of a canal with wave generators. SIAM J. Control Optim.38 (2000) 711–735.  Zbl0966.76015
  28. V.M. Popov, Hyperstability of Automatic Control Systems. Springer, New York (1973).  Zbl0276.93033
  29. F. Shimizu and S. Hara, A method of structure/control design Integration based on finite frequency conditions and its application to smart arm structure design,Proc. of SICE 2002, Osaka, (August 2002).  
  30. V.A. Spector and H. Flashner, Sensitivity of structural models for non collocated control systems. Trans. ASME111 (1989) 646–655.  Zbl0705.73154
  31. M. Tucsnak and S. Jaffard, Regularity of plate equations with control concentrated in interior curves. Proc. Roy. Soc. Edinburg A127 (1997) 1005–1025.  Zbl0889.35059
  32. Y. Zhang, Solving Large-Scale Linear Programs by Interior-Point Methods Under the MATLAB Environment. Technical Report TR96-01, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD (July 1995).  

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.