On structural invariants in the energy-based in-domain control of infinite-dimensional port-Hamiltonian systems
Authors
Tobias Malzer, Hubert Rams, Markus Schöberl
Abstract
This contribution deals with energy-based in-domain control of systems governed by partial differential equations with spatial domain up to dimension two. We exploit a port-Hamiltonian system description based on an underlying jet-bundle formalism, where we restrict ourselves to systems with 2nd-order Hamiltonian. A certain power-conserving interconnection enables the application of a dynamic control law based on structural invariants. Furthermore, we use various examples such as beams and plates with in-domain actuation to demonstrate the capability of our approach.
Keywords
Infinite-dimensional systems; Partial-differential equations; Differential geometry; Port-Hamiltonian systems; In-domain actuation; Structural invariants
Citation
- Journal: Systems & Control Letters
- Year: 2020
- Volume: 145
- Issue:
- Pages: 104778
- Publisher: Elsevier BV
- DOI: 10.1016/j.sysconle.2020.104778
BibTeX
@article{Malzer_2020,
title={{On structural invariants in the energy-based in-domain control of infinite-dimensional port-Hamiltonian systems}},
volume={145},
ISSN={0167-6911},
DOI={10.1016/j.sysconle.2020.104778},
journal={Systems & Control Letters},
publisher={Elsevier BV},
author={Malzer, Tobias and Rams, Hubert and Schöberl, Markus},
year={2020},
pages={104778}
}
References
- van der Schaft, (2000)
- Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
- Schoberl, M. & Siuka, A. Analysis and comparison of port-Hamiltonian formulations for field theories - demonstrated by means of the Mindlin plate. 2013 European Control Conference (ECC) 548–553 (2013) doi:10.23919/ecc.2013.6669137 – 10.23919/ecc.2013.6669137
- 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
- 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
- Schöberl, M. & Siuka, A. Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica vol. 50 607–613 (2014) – 10.1016/j.automatica.2013.11.035
- Jacob, (2012)
- Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
- Macchelli, A., Le Gorrec, Y., Ramirez, H. & Zwart, H. On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 62 1700–1713 (2017) – 10.1109/tac.2016.2595263
- Schoberl, M. & Siuka, A. On Casimir Functionals for Infinite-Dimensional Port-Hamiltonian Control Systems. IEEE Transactions on Automatic Control vol. 58 1823–1828 (2013) – 10.1109/tac.2012.2235739
- Rams, H. & Schoberl, M. On structural invariants in the energy based control of port-Hamiltonian systems with second-order Hamiltonian. 2017 American Control Conference (ACC) 1139–1144 (2017) doi:10.23919/acc.2017.7963106 – 10.23919/acc.2017.7963106
- Trenchant, V., Vu, T., Ramirez, H., Lefevre, L. & Le Gorrec, Y. On the use of structural invariants for the distributed control of infinite dimensional port-Hamitonian systems. 2017 IEEE 56th Annual Conference on Decision and Control (CDC) 47–52 (2017) doi:10.1109/cdc.2017.8263641 – 10.1109/cdc.2017.8263641
- Malzer, Energy-based in-domain control of a piezo-actuated Euler–Bernoulli beam. (2019)
- Schöberl, M. & Schlacher, K. On the extraction of the boundary conditions and the boundary ports in second-order field theories. Journal of Mathematical Physics vol. 59 (2018) – 10.1063/1.5024847
- Saunders, (1989)
- Schöberl, M., Ennsbrunner, H. & Schlacher, K. Modelling of piezoelectric structures–a Hamiltonian approach. Mathematical and Computer Modelling of Dynamical Systems vol. 14 179–193 (2008) – 10.1080/13873950701844824
- Rams, (2018)
- Meirovitch, (1967)
- Meurer, T., Schröck, J. & Kugi, A. Trajektorienplanung für eine piezo-aktuierte elastische Kirchhoff-Platte. e & i Elektrotechnik und Informationstechnik vol. 129 11–17 (2012) – 10.1007/s00502-012-0068-2
- Schröck, (2011)
- Cardoso-Ribeiro, F. L., Matignon, D. & Pommier-Budinger, V. Piezoelectric beam with distributed control ports: a power-preserving discretization using weak formulation.. IFAC-PapersOnLine vol. 49 290–297 (2016) – 10.1016/j.ifacol.2016.07.456
- Siuka, A., Schöberl, M. & Schlacher, K. Port-Hamiltonian modelling and energy-based control of the Timoshenko beam. Acta Mechanica vol. 222 69–89 (2011) – 10.1007/s00707-011-0510-2