Shape optimization of piezoelectric sensors or actuators for the control of plates
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 4, pp. 673-690.

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.

DOI: 10.1051/cocv:2005025
Classification: 49N35, 49Q10, 74P15, 90C47, 93B40, 93C20
Keywords: collocation, piezoelectric sensors/actuators, positive-real systems, topology optimization
@article{COCV_2005__11_4_673_0,
     author = {Degryse, Emmanuel and Mottelet, St\'ephane},
     title = {Shape optimization of piezoelectric sensors or actuators for the control of plates},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {673--690},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {4},
     year = {2005},
     doi = {10.1051/cocv:2005025},
     mrnumber = {2167879},
     zbl = {1081.49029},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2005025/}
}
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
PY  - 2005
SP  - 673
EP  - 690
VL  - 11
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2005025/
DO  - 10.1051/cocv:2005025
LA  - en
ID  - COCV_2005__11_4_673_0
ER  - 
%0 Journal Article
%A Degryse, Emmanuel
%A Mottelet, Stéphane
%T Shape optimization of piezoelectric sensors or actuators for the control of plates
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2005
%P 673-690
%V 11
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2005025/
%R 10.1051/cocv:2005025
%G en
%F COCV_2005__11_4_673_0
Degryse, Emmanuel; Mottelet, Stéphane. Shape optimization of piezoelectric sensors or actuators for the control of plates. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 4, pp. 673-690. doi : 10.1051/cocv:2005025. http://archive.numdam.org/articles/10.1051/cocv:2005025/

[1] G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002). | MR | Zbl

[2] H.T. Banks, R.C. Smith and Y. Wang, Smart material structures, modelling, estimation and control. Res. Appl. Math. Masson, Paris (1996). | Zbl

[3] D. Chenais and E. Zuazua, Finite Element Approximation on Elliptic Optimal Design. C.R. Acad. Sci. Paris Ser. I 338 729-734 (2004). | Zbl

[4] M.J. Chen and C.A. Desoer, Necessary and sufficient conditions for robust stability of linear distributed feedback systems. Internat. J. Control 35 (1982) 255-267. | Zbl

[5] R.F. Curtain and B. Van Keulen, Robust control with respect to coprime factors of infinite-dimensional positive real systems. IEEE Trans. Autom. Control 37 (1992) 868-871. | Zbl

[6] R.F. Curtain and B. Van Keulen, Equivalence of input-output stability and exponential stability for infinite dimensional systems. J. Math. Syst. Theory 21 (1988) 19-48. | Zbl

[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. | Zbl

[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/solids 11 (1992) 181-213.

[11] B.A. Francis, A Course in H Control Theory. Lecture notes in control and information sciences. Springer-Verlag Berlin (1988). | Zbl

[12] P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping. J. Diff. Equations 132 (1996) 338-352. | Zbl

[13] J.S. Freudenberg and P.D. Looze, Right half plane poles and zeros and design tradeoffs in feedback systems. IEEE Trans. Autom. Control 30 (1985) 555-565. | Zbl

[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. | Zbl

[15] A. Haraux, Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990). | MR | Zbl

[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. | Zbl

[17] P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators. SIAM J. Control Optim., to appear. | MR | Zbl

[18] C. Inniss and T. Williams, Sensitivity of the zeros of flexible structures to sensor and actuator location. IEEE Trans. Autom. Control 45 (2000) 157-160. | Zbl

[19] S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215. | Zbl

[20] T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin (1980). | Zbl

[21] B. Van Keulen, H control for distributed parameter systems: a state-space approach. Birkaüser, Boston (1993). | MR | Zbl

[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). | MR | Zbl

[24] F. Macia and E. Zuazua, On the lack of controllability of wave equations: a Gaussian beam approach. Asymptotic Analysis 32 (2002) 1-26. | Zbl

[25] M. Minoux, Programmation Mathématique: théorie et algorithmes, tome 2. Dunod, Paris (1983). | MR | Zbl

[26] O. Morgül, Dynamic boundary control of an Euler-Bernoulli beam. IEEE Trans. Autom. Control 37 (1992) 639-642.

[27] S. Mottelet, Controllability and stabilization of a canal with wave generators. SIAM J. Control Optim. 38 (2000) 711-735. | Zbl

[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. ASME 111 (1989) 646-655. | Zbl

[31] M. Tucsnak and S. Jaffard, Regularity of plate equations with control concentrated in interior curves. Proc. Roy. Soc. Edinburg A 127 (1997) 1005-1025. | Zbl

[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). | Zbl

Cited by Sources: