Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-27T15:19:17.115Z Has data issue: false hasContentIssue false
  • English
  • Français

Shape optimization of piezoelectric sensors or actuators for the controlof plates

Published online by Cambridge University Press:  15 September 2005

Emmanuel Degryse  and
Stéphane Mottelet
Emmanuel Degryse
Affiliation:
Laboratoire de Mathématiques Appliquées de Compiègne, Département de Génie Informatique, Université de Technologie de Compiègne, BP 20529, 60205 Compiegne Cedex, France; degryse@iutc3.unicaen.fr stephane.mottelet@utc.fr
Stéphane Mottelet
Affiliation:
Laboratoire de Mathématiques Appliquées de Compiègne, Département de Génie Informatique, Université de Technologie de Compiègne, BP 20529, 60205 Compiegne Cedex, France; degryse@iutc3.unicaen.fr stephane.mottelet@utc.fr
Get access

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.

Keywords

Collocationpiezoelectric sensors/actuatorspositive-real systemstopology optimization.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 4 , October 2005 , pp. 673 - 690
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002).
H.T. Banks, R.C. Smith and Y. Wang, Smart material structures, modelling, estimation and control. Res. Appl. Math. Masson, Paris (1996).
D. Chenais and E. Zuazua, Finite Element Approximation on Elliptic Optimal Design. C.R. Acad. Sci. Paris Ser. I 338 729–734 (2004).
Chen, M.J. and Desoer, C.A., Necessary and sufficient conditions for robust stability of linear distributed feedback systems. Internat. J. Control 35 (1982) 255–267. CrossRef
Curtain, R.F. and Van Keulen, B., Robust control with respect to coprime factors of infinite-dimensional positive real systems. IEEE Trans. Autom. Control 37 (1992) 868–871. CrossRef
Curtain, R.F. and Van Keulen, B., Equivalence of input-output stability and exponential stability for infinite dimensional systems. J. Math. Syst. Theory 21 (1988) 19–48. CrossRef
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.
Datko, R., 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. CrossRef
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).
Destuynder, P.H., Legrain, I., Castel, L. and Richard, N., Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction. Eur. J. Mech A/solids 11 (1992) 181–213.
B.A. Francis, A Course in H∞ Control Theory. Lecture notes in control and information sciences. Springer-Verlag Berlin (1988).
Freitas, P. and Zuazua, E., Stability results for the wave equation with indefinite damping. J. Diff. Equations 132 (1996) 338–352. CrossRef
Freudenberg, J.S. and Looze, P.D., Right half plane poles and zeros and design tradeoffs in feedback systems. IEEE Trans. Autom. Control 30 (1985) 555–565. CrossRef
Gibson, J.S. and Adamian, A., Approximation theory for Linear-Quadratic-Gaussian control of flexible structures. SIAM J. Control Optim. 29 (1991) 1–37. CrossRef
A. Haraux, Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990).
Hébrard, P. and Henrot, A., Optimal shape and position of the actuators for the stabilization of a string. Syst. Control Lett. 48 (2003) 199–209. CrossRef
P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim., to appear.
Inniss, C. and Williams, T., Sensitivity of the zeros of flexible structures to sensor and actuator location. IEEE Trans. Autom. Control 45 (2000) 157–160. CrossRef
Jaffard, S., Tucsnak, M. and Zuazua, E., Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184–215. CrossRef
T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin (1980).
B. van Keulen, H∞ control for distributed parameter systems: a state-space approach. Birkaüser, Boston (1993).
Lasiecka, I. and Triggiani, R., Non-dissipative boundary stabilization of the wave equation via boundary observation. J. Math. Pures Appl. 63 (1984) 59–80.
D.G. Luenberger, Optimization by Vector Space Methods. John Wiley and Sons, New York (1969).
Macia, F. and Zuazua, E., On the lack of controllability of wave equations: a Gaussian beam approach. Asymptotic Analysis 32 (2002) 1–26.
M. Minoux, Programmation Mathématique: théorie et algorithmes, tome 2. Dunod, Paris (1983).
Morgül, O., Dynamic boundary control of an Euler-Bernoulli beam. IEEE Trans. Autom. Control 37 (1992) 639–642. CrossRef
Mottelet, S., Controllability and stabilization of a canal with wave generators. SIAM J. Control Optim. 38 (2000) 711–735. CrossRef
V.M. Popov, Hyperstability of Automatic Control Systems. Springer, New York (1973).
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).
Spector, V.A. and Flashner, H., Sensitivity of structural models for non collocated control systems. Trans. ASME 111 (1989) 646–655.
Tucsnak, M. and Jaffard, S., Regularity of plate equations with control concentrated in interior curves. Proc. Roy. Soc. Edinburg A 127 (1997) 1005–1025.
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).