Piezoelectric beam with distributed control ports: a power-preserving discretization using weak formulation.

Authors

Flávio Luiz Cardoso-Ribeiro, Denis Matignon, Valérie Pommier-Budinger

Abstract

A model reduction method for infinite-dimensional port-Hamiltonian systems with distributed ports is presented. The method is applied to the Euler-Bernoulli equation with piezoelectric patches. The voltage is considered as an external input of the system. This gives rise to an unbounded input operator. A weak formulation is used to overcome this difficulty. It also allows defining a discretization method which leads to a finite-dimensional port-Hamiltonian system; the energy flow of the original system is preserved. Numerical results are compared to experimental ones to validate the method. Further work should use this model to couple the approximated equations with a more complex system, and to design active control laws.

Keywords

Model Reduction for Control; Port-Hamiltonian systems; Piezoelectric materials

Citation

• Journal: IFAC-PapersOnLine
• Year: 2016
• Volume: 49
• Issue: 8
• Pages: 290–297
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2016.07.456
• Note: 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2016- Bertinoro, Italy, 13—15 June 2016

BibTeX

@article{Cardoso_Ribeiro_2016,
  title={{Piezoelectric beam with distributed control ports: a power-preserving discretization using weak formulation.}},
  volume={49},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2016.07.456},
  number={8},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Cardoso-Ribeiro, Flávio Luiz and Matignon, Denis and Pommier-Budinger, Valérie},
  year={2016},
  pages={290--297}
}

Download the bib file

References

• Aglietti, G. S., Gabriel, S. B., Langley, R. S. & Rogers, E. A modeling technique for active control design studies with application to spacecraft microvibrations. The Journal of the Acoustical Society of America vol. 102 2158–2166 (1997) – 10.1121/1.419631
• Banks, (1996)
• Boyd, (2001)
• Cardoso-Ribeiro, F. L., Matignon, D. & Pommier-Budinger, V. Modeling of a Fluid-structure coupled system using port-Hamiltonian formulation. IFAC-PapersOnLine vol. 48 217–222 (2015) – 10.1016/j.ifacol.2015.10.242
• Duindam, (2009)
• Geradin, (2015)
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Multi-variable port Hamiltonian model of piezoelectric material. 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566) vol. 1 897–902 – 10.1109/iros.2004.1389466
• Matignon, D. & Hélie, T. A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems. European Journal of Control vol. 19 486–494 (2013) – 10.1016/j.ejcon.2013.10.003
• Morris, K. & Ozer, A. O. Strong stabilization of piezoelectric beams with magnetic effects. 52nd IEEE Conference on Decision and Control 3014–3019 (2013) doi:10.1109/cdc.2013.6760341 – 10.1109/cdc.2013.6760341
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics vol. 231 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Preumont, A. Vibration Control of Active Structures. Solid Mechanics and Its Applications (Springer Netherlands, 2011). doi:10.1007/978-94-007-2033-6 – 10.1007/978-94-007-2033-6
• Trefethen, (2000)
• Voss, T. & Scherpen, J. M. A. Structure Preserving Spatial Discretization of a 1-D Piezoelectric Timoshenko Beam. Multiscale Modeling & Simulation vol. 9 129–154 (2011) – 10.1137/100789038
• Voß, T. & Scherpen, J. M. A. Port-Hamiltonian Modeling of a Nonlinear Timoshenko Beam with Piezo Actuation. SIAM Journal on Control and Optimization vol. 52 493–519 (2014) – 10.1137/090774598
• VU, N. M. T., LEFEVRE, L., NOUAILLETAS, R. & BREMOND, S. Geometric discretization for a plasma control model. IFAC Proceedings Volumes vol. 46 755–760 (2013) – 10.3182/20130204-3-fr-2033.00098