On structural invariants in the energy based control of port-Hamiltonian systems with second-order Hamiltonian

Authors

Hubert Rams, Markus Schoberl

Abstract

This paper deals with the energy based control of infinite-dimensional port-Hamiltonian systems, by exploiting structural invariants—so-called Casimir functionals—of the closed loop system. We confine ourselves to port-Hamiltonian systems, modeled within a jet-space approach, with one dimensional spatial domain and 2nd-order Hamiltonian. Therefore, the concept of structural invariants is extended from systems with 1st-order Hamiltonian to systems with 2nd-order Hamiltonian. To demonstrate the capability of this closed loop control methodology, a boundary controlled Euler-Bernoulli beam is used. Finally, simulation results conclude the paper.

Citation

• Journal: 2017 American Control Conference (ACC)
• Year: 2017
• Volume:
• Issue:
• Pages: 1139–1144
• Publisher: IEEE
• DOI: 10.23919/acc.2017.7963106

BibTeX

@inproceedings{Rams_2017,
  title={{On structural invariants in the energy based control of port-Hamiltonian systems with second-order Hamiltonian}},
  DOI={10.23919/acc.2017.7963106},
  booktitle={{2017 American Control Conference (ACC)}},
  publisher={IEEE},
  author={Rams, Hubert and Schoberl, Markus},
  year={2017},
  pages={1139--1144}
}

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References

• schöberl, Analysis and Comparison of Port-Hamiltonian Formulations for Field Theories-demonstrated by means of the Mindlin plate. Proceedings of the European Control Conference (2013)
• Schoberl, M. & Siuka, A. On Casimir Functionals for Infinite-Dimensional Port-Hamiltonian Control Systems. IEEE Trans. Automat. Contr. 58, 1823–1828 (2013) – 10.1109/tac.2012.2235739
• Schöberl, M. & Siuka, A. Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica 50, 607–613 (2014) – 10.1016/j.automatica.2013.11.035
• Schöberl, M., Ennsbrunner, H. & Schlacher, K. Modelling of piezoelectric structures–a Hamiltonian approach. Mathematical and Computer Modelling of Dynamical Systems 14, 179–193 (2008) – 10.1080/13873950701844824
• Schöberl, M. & Schlacher, K. First-order Hamiltonian field theory and mechanics. Mathematical and Computer Modelling of Dynamical Systems 17, 105–121 (2011) – 10.1080/13873954.2010.537526
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 10.1007/978-1-4612-4350-2
• Putting energy back in control. IEEE Control Syst. 21, 18–33 (2001) – 10.1109/37.915398
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J. Control Optim. 44, 1864–1892 (2005) – 10.1137/040611677
• guo, Stability and Stabilization of Infinite Dimensional Systems with Applications (1998)
• Macchelli, A., Le Gorrec, Y., Ramirez, H. & Zwart, H. On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems. IEEE Trans. Automat. Contr. 62, 1700–1713 (2017) – 10.1109/tac.2016.2595263
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM J. Control Optim. 43, 743–767 (2004) – 10.1137/s0363012903429530
• 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
• Siuka, A., Schöberl, M. & Schlacher, K. Port-Hamiltonian modelling and energy-based control of the Timoshenko beam. Acta Mech 222, 69–89 (2011) – 10.1007/s00707-011-0510-2
• Ennsbrunner, H. & Schlacher, K. On the geometrical representation and interconnection of infinite dimensional port controlled Hamiltonian systems. Proceedings of the 44th IEEE Conference on Decision and Control 5263–5268 doi:10.1109/cdc.2005.1582998 – 10.1109/cdc.2005.1582998
• Giachetta, G., Mangiarotti, L. & Sardanashvily, G. New Lagrangian and Hamiltonian Methods in Field Theory. (1997) doi:10.1142/2199 – 10.1142/2199
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• 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
• meirovitch, Principles and Techniques of Vibrations (1997)