Abstract
The investigation of this paper is related to the design of active real-time controls which provide the desired performance of flexible constructions subjected to varying disturbances. Physical controllers are piezoelectric elements that transform applied voltages into mechanical forces and moments. Control voltages are computed on the base of signals measured on piezoelectric sensors. The underlying structure is a thin plate or shell. The piezoelectric elements are either surface mounted or embedded within the structure. Conventional averaging procedures are used to eliminate the thickness in mathematical models. A homogenization procedure is used to reduce structures with large number of piezoelectric elements to the case of continuously distributed input or output signals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Preumont, Vibration Control of Active Structures. Solid Mechanics and its Applications. 50, Kluver Academic Publisher (Dordrecht-Boston-London) 1997.
H. T. Banks, R. C. Smith and Y. Wang, Smart material structures: modeling, estimation and control. Wiley, Chichester 1996.
J. Zelenka, Piezoelectric Resonators and their Applications. Studies in Electrical and Electronic Engineering. 24, Elsevier 1986.
G. A. Maugin, Continuum mechanics of electromagnetic solids. North-Holland series in Applied Mathematics and mechanics. 33, North-Holland 1987.
T. Basar and P. Bernhard, H∞ -Optimal Control and Related Minimax Design Problems. Birkhäuser, Boston 1987.
I. Lasiecka and R. Trigiani, Control theory for partial equations: continuous and approximation theories. II. Abstract hyperbolic-like systems over a finite time horizon. Encyclopedia of mathematics and its applications 75, Edited by G.-C. Rota, Cambridge University Press 2000.
L. D. Landau and E. M. Lifschitz, Elastizitätstheorie. Akademie-Verlag (Berlin) 1975.
M. Bernadou, Finite Element Methods for Thin Shell Problems. Wiley (Chichester) 1996.
K.-H. Hoffmann and N. D. Botkin, A Fully coupled model of a nonlinear thin plate excited by piezoelectric actuators. Smart materials. Proceedings of the first caesarium, Bonn, Nov. 17–19, 1999. Edited by K.-H. Hoffmann, Springer 2001, available on the preprint server1 .
K.-H. Hoffmann and N. D. Botkin, Homogenization of a fully coupled model of a non-linear thin plate excited by piezoelectric actuators. International conference: Optimal Control of Complex Structures, Mathematical Research Center Oberwolfach, June, 4— 10, 2000 (to appear in proceedings, available on the preprint server2.
K.-H. Hoffmann and N. D. Botkin, Oscillations of nonlinear thin plates excited by piezoelectric patches, ZAMM: Z. angew. Math. Mech., 78 (1998), 495–503.
N. D. Botkin, Estimation of parameters of a linear thin plate. Analysis 17 (1997) 367–378.
N. D. Botkin, W. G. Litvinov, Y. I. Rubezhansky, Models and problems for piezoelectric elastic bodies and thin shells. DFG-Project “Echtzeit-Optimierung großer Systeme”, Preprint 99–5. Preprintserver3
N. D. Botkin, Estimation of parameters of thin plates excited by piezoelectric patches. Preprint of the Technical University of Munich. M9704, 1997.
G. Allaire, Homogenization and two-scale Convergence, SIAM J. Math. Anal., 23 (6) (1992), 1482–1518.
H. Haller, Verbundwerkstoffe mit Formgedächtnislegierung — Mikromechanische Modellierung und Homogenisierung, Dissertation, TU-München 1997.
K.-H. Hoffmann and N. D. Botkin, Homogenization of von Kármán plates excited by piezoelectric patches, ZAMM: Z. angew. Math. Mech., 133 (1999), 191–200.
K.-H. Hoffmann and N. D. Botkin, Adaptive Materialien in der Echtzeitoptimierung. DFG-Projekt “Echtzeitoptimierung großer Systeme”, Arbeitsbericht über den ersten Förderungszeitraum von 1995–1997. München 1997 (available on the preprint server4).
H. T. Banks, R. C. H. Rosario and R. C. Smith, Reduced order model feedback control design: computational studies for thin cylindric shells. Report of the Center for Research in Scientific Computation. North Carolina State University. CRSC-TR98–24, June, 1998.
N. N. Krasovskii, Control for dynamic system (in Russian). Nauka (Moscow), 1985.
N. N. Krasovskii, Control problems with incomplete information (in Russian). Academy of Sciences of USSR, Ural Scientific Center, Institute of Mathematics and Mechanics (Sverdlovsk) 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hoffmann, KH., Botkin, N.D. (2001). Real-Time Optimization and Stabilization of Distributed Parameter Systems with Piezoelectric Elements. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds) Online Optimization of Large Scale Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04331-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-662-04331-8_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07633-6
Online ISBN: 978-3-662-04331-8
eBook Packages: Springer Book Archive