Authors

K. Morris, A. Ozkan Ozer

Abstract

It is widely accepted in the literature that magnetic effects in the piezoelectric beams is relatively small, and does not change the overall dynamics. Therefore, most models for piezoelectric beams completely ignore the magnetic energy. These models are known to be exponentially stabilizable by a mechanical feedback controller in the energy space. In this paper, we use a variational approach to derive the differential equations and boundary conditions that model a single piezoelectric beam with magnetic effects. Next, we show that the resulting control system can be formulated as a port-Hamiltonian system and is hence well-posed. Finally, by using only an electrical feedback controller (the current flowing through the electrodes), we show that the closed-loop system is strongly stable in the energy space for a dense set of system parameters. The difference between this result and that for models that neglect magnetic effects is discussed.

Citation

  • Journal: 52nd IEEE Conference on Decision and Control
  • Year: 2013
  • Volume:
  • Issue:
  • Pages: 3014–3019
  • Publisher: IEEE
  • DOI: 10.1109/cdc.2013.6760341

BibTeX

@inproceedings{Morris_2013,
  title={{Strong stabilization of piezoelectric beams with magnetic effects}},
  DOI={10.1109/cdc.2013.6760341},
  booktitle={{52nd IEEE Conference on Decision and Control}},
  publisher={IEEE},
  author={Morris, K. and Ozer, A. Ozkan},
  year={2013},
  pages={3014--3019}
}

Download the bib file

References

  • Peng Yang & Kin Choong Yow. Passive cached clustering routing for MANETs. Proceedings. 2004 International Conference on Information and Communication Technologies: From Theory to Applications, 2004. 233–234 doi:10.1109/ictta.2004.1307710 – 10.1109/ictta.2004.1307710
  • Tucsnak, M. & Weiss, G. Simultaneous Exact Controllability and Some Applications. SIAM J. Control Optim. 38, 1408–1427 (2000) – 10.1137/s0363012999352716
  • Tzou, H. S. Piezoelectric Shells. Solid Mechanics and its Applications (Springer Netherlands, 1993). doi:10.1007/978-94-011-1783-8 – 10.1007/978-94-011-1783-8
  • Smith, R. C. Smart Material Systems. (2005) doi:10.1137/1.9780898717471 – 10.1137/1.9780898717471
  • Tiersten, H. F. Linear Piezoelectric Plate Vibrations. (Springer US, 1969). doi:10.1007/978-1-4899-6453-3 – 10.1007/978-1-4899-6453-3
  • rogacheva, Theory of Piezoelectric Shells and Plates (1994)
  • Russell, D. L. The Dirichlet–Neumann Boundary Control Problem Associated with Maxwell’s Equations in a Cylindrical Region. SIAM J. Control Optim. 24, 199–229 (1986) – 10.1137/0324012
  • morris, Modeling and Stabilizability of Voltageactuated Piezoelectric Beams with Magnetic Effects (0)
  • Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences (Springer New York, 1983). doi:10.1007/978-1-4612-5561-1 – 10.1007/978-1-4612-5561-1
  • Dáger, R. & Zuazua, E. Wave Progagation, Observation and Control in 1-d Flexible Multi-Structures. Mathématiques et Applications (Springer-Verlag, 2006). doi:10.1007/3-540-37726-3 – 10.1007/3-540-37726-3
  • Yang, J. A Review of a Few Topics in Piezoelectricity. Applied Mechanics Reviews 59, 335–345 (2006) – 10.1115/1.2345378
  • banks, Smart Material Structures Modelling (1996)
  • Arendt, W. & Batty, C. J. K. Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306, 837–852 (1988) – 10.1090/s0002-9947-1988-0933321-3
  • lions, Controlabilite exacte perturbations et stabilisation de syst’emes distribues 1. Rech Math Appl (1988)
  • komornik, Exact Controllability and Stabilization the Multiplier Method (1994)
  • Kapitonov, B., Miara, B. & Menzala, G. P. Stabilization of a layered piezoelectric 3-D body by boundary dissipation. ESAIM: COCV 12, 198–215 (2006) – 10.1051/cocv:2005028
  • Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
  • hansen, Analysis of a plate with a localized piezoelectric patch. Conference on Decision & Control (1998)
  • lasiecka, Exact Controllability of a 3D Piezoelectric Body (2009)
  • Komornik, V. & Loreti, P. Fourier Series in Control Theory. Springer Monographs in Mathematics (Springer New York, 2005). doi:10.1007/b139040 – 10.1007/b139040