Modelling of piezoelectric structures–a Hamiltonian approach

Authors

M. Schöberl, H. Ennsbrunner, K. Schlacher

Abstract

This contribution is dedicated to the geometric description of infinite-dimensional port Hamiltonian systems with in- and output operators. Several approaches exist, which deal with the extension of the well-known lumped parameter case to the distributed one. In this article a description has been chosen, which preserves useful properties known from the class of port controlled Hamiltonian systems with dissipation in the lumped scenario. Furthermore, the introduced in- and output maps are defined by linear differential operators. The derived theory is applied to the piezoelectric field equations to obtain their port Hamiltonian representation. In this example, the electrical field strength is assumed to act as distributed input. Finally it is shown, that distributed inputs, that are in the kernel of the input map act similarly on the system as certain boundary inputs.

Citation

• Journal: Mathematical and Computer Modelling of Dynamical Systems
• Year: 2008
• Volume: 14
• Issue: 3
• Pages: 179–193
• Publisher: Informa UK Limited
• DOI: 10.1080/13873950701844824

BibTeX

@article{Sch_berl_2008,
  title={{Modelling of piezoelectric structures–a Hamiltonian approach}},
  volume={14},
  ISSN={1744-5051},
  DOI={10.1080/13873950701844824},
  number={3},
  journal={Mathematical and Computer Modelling of Dynamical Systems},
  publisher={Informa UK Limited},
  author={Schöberl, M. and Ennsbrunner, H. and Schlacher, K.},
  year={2008},
  pages={179--193}
}

Download the bib file

References

• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• Macchelli A.. Port Hamiltonian Systems – A unified approach for modeling and control (2002)
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Ennsbrunner, H. and Schlacher, K. 2005. On the geometrical representation and interconnection of infinite dimensional port controlled Hamiltonian systems. 44th IEEE, Conference on Decision and Control and European Control Conference. 2005, Sevilla, Spain.
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1986). doi:10.1007/978-1-4684-0274-2 – 10.1007/978-1-4684-0274-2
• Giachetta G.. New Lagrangian and Hamiltonian Methods in Field Theory (1994)
• Saunders, D. J. The Geometry of Jet Bundles. (1989) doi:10.1017/cbo9780511526411 – 10.1017/cbo9780511526411
• Boothby W. M.. An Introduction to Differentiable Manifolds and Riemanian Geometry (1986)
• Pommaret J. F.. Systems of Partial Differential Equations and Lie Pseudogroups (1978)
• Kugi A.. Non-linear Control Based on Physical Models (2001)
• Macchelli, A. & Melchiorri, C. Control by Interconnection and Energy Shaping of the Timoshenko Beam. Mathematical and Computer Modelling of Dynamical Systems vol. 10 231–251 (2004) – 10.1080/13873950412331335243
• Hebey, E. Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes (2000) doi:10.1090/cln/005 – 10.1090/cln/005
• Zeidler, E. Applied Functional Analysis. Applied Mathematical Sciences (Springer New York, 1995). doi:10.1007/978-1-4612-0815-0 – 10.1007/978-1-4612-0815-0