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

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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

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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

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  1. G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002).  
  2. H.T. Banks, R.C. Smith and Y. Wang, Smart material structures, modelling, estimation and control. Res. Appl. Math. Masson, Paris (1996).  
  3. D. Chenais and E. Zuazua, Finite Element Approximation on Elliptic Optimal Design. C.R. Acad. Sci. Paris Ser. I338 729–734 (2004).  
  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.  
  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.  
  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.  
  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.  
  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.  
  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.  
  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.  
  15. A. Haraux, Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990).  
  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.  
  17. P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim., to appear.  
  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.  
  19. S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations145 (1998) 184–215.  
  20. T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin (1980).  
  21. B. van Keulen, H∞ control for distributed parameter systems: a state-space approach. Birkaüser, Boston (1993).  
  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).  
  24. F. Macia and E. Zuazua, On the lack of controllability of wave equations: a Gaussian beam approach. Asymptotic Analysis32 (2002) 1–26.  
  25. M. Minoux, Programmation Mathématique: théorie et algorithmes, tome 2. Dunod, Paris (1983).  
  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.  
  28. V.M. Popov, Hyperstability of Automatic Control Systems. Springer, New York (1973).  
  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.  
  31. M. Tucsnak and S. Jaffard, Regularity of plate equations with control concentrated in interior curves. Proc. Roy. Soc. Edinburg A127 (1997) 1005–1025.  
  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).  

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