Reduced order controller design for Timoshenko beam: A port Hamiltonian approach
Authors
Yongxin Wu, Boussad Hamroun, Yann Le Gorrec, Bernhard Maschke
Abstract
This paper deals with the structure and passivity preserving model reduction and the reduced order controller design for a class of distributed controlled port Hamiltonian systems - Timoshenko beam. The boundary conditions of the beam lead to physical constraints which are hardly considered in the reduction procedure. In this work we propose to use the descriptor system realization of port Hamiltonian system to conserve the physical constraints. A passive LQG control design method is proposed for this type of system. This LQG method defines a balanced coordinate which allows us to reduce the system. Using the obtained reduced model, a reduced order passive controller which stabilizes the full order system is designed using the LQG method. At last we give the numerical simulations to show the effectiveness of the proposed reduced passive controller.
Keywords
Port Hamiltonian systems; distributed control; Hyperbolic PDEs; LQG method; passivity preserving reduction; reduced order control design
Citation
- Journal: IFAC-PapersOnLine
- Year: 2017
- Volume: 50
- Issue: 1
- Pages: 7121–7126
- Publisher: Elsevier BV
- DOI: 10.1016/j.ifacol.2017.08.547
- Note: 20th IFAC World Congress
BibTeX
@article{Wu_2017,
title={{Reduced order controller design for Timoshenko beam: A port Hamiltonian approach}},
volume={50},
ISSN={2405-8963},
DOI={10.1016/j.ifacol.2017.08.547},
number={1},
journal={IFAC-PapersOnLine},
publisher={Elsevier BV},
author={Wu, Yongxin and Hamroun, Boussad and Le Gorrec, Yann and Maschke, Bernhard},
year={2017},
pages={7121--7126}
}
References
- Baaiu, A., Couenne, F., Lefevre, L., Le Gorrec, Y. & Tayakout, M. Structure-preserving infinite dimensional model reduction: Application to adsorption processes. Journal of Process Control vol. 19 394–404 (2009) – 10.1016/j.jprocont.2008.07.002
- Singular Control Systems. Lecture Notes in Control and Information Sciences (Springer-Verlag, 1989). doi:10.1007/bfb0002475 – 10.1007/bfb0002475
- Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
- 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
- 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
- Hamroun, (2009)
- Jacob, (2012)
- 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
- Macchelli, A., Melchiorri, C. & Stramigioli, S. Port-Based Modeling and Simulation of Mechanical Systems With Rigid and Flexible Links. IEEE Transactions on Robotics vol. 25 1016–1029 (2009) – 10.1109/tro.2009.2026504
- 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
- Nishida, G., Takagi, K., Maschke, B. & Osada, T. Multi-scale distributed parameter modeling of ionic polymer-metal composite soft actuator. Control Engineering Practice vol. 19 321–334 (2011) – 10.1016/j.conengprac.2010.10.005
- Polyuga, R. V. & van der Schaft, A. J. Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems. Systems & Control Letters vol. 61 412–421 (2012) – 10.1016/j.sysconle.2011.12.008
- Polyuga, R. V. & van der Schaft, A. Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos. IEEE Transactions on Automatic Control vol. 56 1458–1462 (2011) – 10.1109/tac.2011.2128650
- van der Schaft, The Hamil-tonian formulation of energy conserving physical systems with external ports. Archiv für Elektronik und Übertragungstechnik (1995)
- 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
- Wu, (2014)