Trajectory control of an elastic beam based on port-Hamiltonian numerical models

Authors

Mei Wang, Paul Kotyczka

Abstract

We present a systematic approach to realize the precise observer-based trajectory tracking of the tip of a flexible beam using the energy-based port-Hamiltonian (PH) system representation. The first design steps are the structure-preserving spatial discretization by means of a pseudo-spectral method and the structure-preserving order reduction. The model structure is exploited for inversion-based feedforward control, and the control loop is closed via an observer for the friction torque and the state difference. Experimental results for the reference input and the disturbance response illustrate the quality of the design.

Citation

• Journal: at - Automatisierungstechnik
• Year: 2021
• Volume: 69
• Issue: 6
• Pages: 457–471
• Publisher: Walter de Gruyter GmbH
• DOI: 10.1515/auto-2020-0159

BibTeX

@article{Wang_2021,
  title={{Trajectory control of an elastic beam based on port-Hamiltonian numerical models}},
  volume={69},
  ISSN={0178-2312},
  DOI={10.1515/auto-2020-0159},
  number={6},
  journal={at - Automatisierungstechnik},
  publisher={Walter de Gruyter GmbH},
  author={Wang, Mei and Kotyczka, Paul},
  year={2021},
  pages={457--471}
}

Download the bib file

References

• Aldraihem, O. J., Wetherhold, R. C. & Singh, T. Distributed Control of Laminated Beams: Timoshenko Theory vs. Euler-Bernoulli Theory. Journal of Intelligent Material Systems and Structures vol. 8 149–157 (1997) – 10.1177/1045389x9700800205
• Aoues, S., Cardoso-Ribeiro, F. L., Matignon, D. & Alazard, D. Modeling and Control of a Rotating Flexible Spacecraft: A Port-Hamiltonian Approach. IEEE Transactions on Control Systems Technology vol. 27 355–362 (2019) – 10.1109/tcst.2017.2771244
• Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0223-3
• Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A structure-preserving Partitioned Finite Element Method for the 2D wave equation. IFAC-PapersOnLine vol. 51 119–124 (2018) – 10.1016/j.ifacol.2018.06.033
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• O. Farle, R.-B. Baltes and R. Dyczij-Edlinger. A Port-Hamiltonian finite-element formulation for the transmission line. In Proceedings of 21st International Symposium on Mathematical Theory of Networks and Systems, pages 724–728, 2014.
• O. Föllinger, U. Konigorski, B. Lohmann, G. Roppenecker and A. Trächtler. Regelungstechnik. VDE-Verlag, 11 edition, 2013.
• Gao, Y., Wang, F.-Y. & Xiao, Z.-Q. Modeling of Flexible Manipulators. Flexible Manipulators 15–58 (2012) doi:10.1016/b978-0-12-397036-7.00003-9 – 10.1016/b978-0-12-397036-7.00003-9
• Gattringer, H. Starr-elastische Robotersysteme. (Springer Berlin Heidelberg, 2011). doi:10.1007/978-3-642-22828-5 – 10.1007/978-3-642-22828-5
• 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
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• 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
• Kotyczka, P. Finite Volume Structure-Preserving Discretization of 1D Distributed-Parameter Port-Hamiltonian Systems. IFAC-PapersOnLine vol. 49 298–303 (2016) – 10.1016/j.ifacol.2016.07.457
• P. Kotyczka. Numerical Methods for Distributed Parameter Port-Hamiltonian Systems. TUM University Press, 2019. ISBN 978-3-95884-028-7.
• P. Kotyczka and S. Brandstäter. Inversion-based feedforward control for discretized port-Hamiltonian systems. In Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), Groningen, The Netherlands, pages 729–735, 2014.
• Kotyczka, P. & Lefèvre, L. Discrete-Time Control Design Based on Symplectic Integration: Linear Systems. IFAC-PapersOnLine vol. 53 7563–7568 (2020) – 10.1016/j.ifacol.2020.12.1352
• 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. & 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
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• 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
• Ortega, R. & García-Canseco, E. Interconnection and Damping Assignment Passivity-Based Control: A Survey. European Journal of Control vol. 10 432–450 (2004) – 10.3166/ejc.10.432-450
• Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 53 2527–2542 (2008) – 10.1109/tac.2008.2006930
• J. N. Reddy. Energy Principles and Variational Methods in Applied Mechanics. John Wiley & Sons, 2002.
• Rudolph, J. & Woittennek, F. Flachheitsbasierte Steuerung eines Timoshenko‐Balkens. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik vol. 83 119–127 (2003) – 10.1002/zamm.200310011
• Simeon, B. Computational Flexible Multibody Dynamics. (Springer Berlin Heidelberg, 2013). doi:10.1007/978-3-642-35158-7 – 10.1007/978-3-642-35158-7
• 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
• S. P. Timoshenko and J. M. Gere. Theory of Elastic Stability. McGraw-Hill, 2nd edition, 1961.
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• 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
• M. Wang and P. Kotyczka. Port-Hamiltonian model of a flexible manipulator test rig. In G. Roppenecker and B. Lohmann, editors, Methoden und Anwendungen der Regelungstechnik – Erlangen-Münchener Workshops 2015 und 2016, pages 37–52. Shaker Verlag, 2017.
• Wang, M., Bestler, A. & Kotyczka, P. Modeling, discretization and motion control of a flexible beam in the port-Hamiltonian framework. IFAC-PapersOnLine vol. 50 6799–6806 (2017) – 10.1016/j.ifacol.2017.08.2511
• Wu, Y., Hamroun, B., Gorrec, Y. L. & Maschke, B. Port Hamiltonian System in Descriptor Form for Balanced Reduction: Application to a Nanotweezer. IFAC Proceedings Volumes vol. 47 11404–11409 (2014) – 10.3182/20140824-6-za-1003.01579
• Q. Zou and S. Devasia. Preview-based stable-inversion for output tracking. In American Control Conference, 1999. Proceedings of the 1999, volume 5, pages 3544–3548. IEEE, 1999.