Port Hamiltonian System in Descriptor Form for Balanced Reduction: Application to a Nanotweezer

Authors

Yongxin Wu, Boussad Hamroun, Yann Le Gorrec, Bernhard Maschke

Abstract

This paper proposes a method of balanced model reduction for constrained linear port Hamiltonian systems. Constrained linear port Hamiltonian systems are first written in a canonical descriptor form such that the Hamiltonian structure is preserved. The computations of the controllability and observability Gramians are then used to derive the balanced port Hamiltonian representation of the system. The method of flow constraint is applied to reduce the system. Finally, numerical simulations for the reduction of a micro mechanical actuator model is given to illustrate the effectiveness of the proposed method.

Keywords

Port Hamiltonian system; Descriptor system; Balanced reduction; Gramian; flow constraint method

Citation

• Journal: IFAC Proceedings Volumes
• Year: 2014
• Volume: 47
• Issue: 3
• Pages: 11404–11409
• Publisher: Elsevier BV
• DOI: 10.3182/20140824-6-za-1003.01579
• Note: 19th IFAC World Congress

BibTeX

@article{Wu_2014,
  title={{Port Hamiltonian System in Descriptor Form for Balanced Reduction: Application to a Nanotweezer}},
  volume={47},
  ISSN={1474-6670},
  DOI={10.3182/20140824-6-za-1003.01579},
  number={3},
  journal={IFAC Proceedings Volumes},
  publisher={Elsevier BV},
  author={Wu, Yongxin and Hamroun, Boussad and Gorrec, Yann Le and Maschke, Bernhard},
  year={2014},
  pages={11404--11409}
}

Download the bib file

References

• Antoulas, A. C. Approximation of Large-Scale Dynamical Systems. (2005) doi:10.1137/1.9780898718713 – 10.1137/1.9780898718713
• Boudaoud, M., Haddab, Y. & Le Gorrec, Y. Modeling and Optimal Force Control of a Nonlinear Electrostatic Microgripper. IEEE/ASME Trans. Mechatron. 18, 1130–1139 (2013) – 10.1109/tmech.2012.2197216
• Dai, (1989)
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J. Control Optim. 37, 54–91 (1998) – 10.1137/s0363012996312039
• Duindam, (2009)
• Gentili, L., Macchelli, A., Melchiorri, C. & Mameli, A. Mastering the complexity of an Ultrasonic Sealing System: The port-Hamiltonian approach. Mechatronics 21, 594–603 (2011) – 10.1016/j.mechatronics.2011.02.009
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica 40, 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Gugercin, S. & Antoulas, A. C. A Survey of Model Reduction by Balanced Truncation and Some New Results. International Journal of Control 77, 748–766 (2004) – 10.1080/00207170410001713448
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica 48, 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• 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
• Macchelli, A. Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation. Systems & Control Letters 60, 579–589 (2011) – 10.1016/j.sysconle.2011.04.016
• Polyuga, R. V. & van der Schaft, A. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica 46, 665–672 (2010) – 10.1016/j.automatica.2010.01.018
• Polyuga, R. V. & van der Schaft, A. Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos. IEEE Trans. Automat. Contr. 56, 1458–1462 (2011) – 10.1109/tac.2011.2128650
• 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 61, 412–421 (2012) – 10.1016/j.sysconle.2011.12.008
• Ramirez, H. & Le Gorrec, Y. Exponential stability of a class of PDE’s with dynamic boundary control. 2013 American Control Conference 3290–3295 (2013) doi:10.1109/acc.2013.6580339 – 10.1109/acc.2013.6580339
• Reis, T. & Stykel, T. Positive real and bounded real balancing for model reduction of descriptor systems. International Journal of Control 83, 74–88 (2009) – 10.1080/00207170903100214
• Stykel, T. Gramian-Based Model Reduction for Descriptor Systems. Mathematics of Control, Signals, and Systems (MCSS) 16, 297–319 (2004) – 10.1007/s00498-004-0141-4
• van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM J. Control Optim. 51, 906–937 (2013) – 10.1137/110840091
• Van Der Schaft, A. J. & Maschke, B. M. On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics 34, 225–233 (1994) – 10.1016/0034-4877(94)90038-8
• van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv für Elektronik und Übertragungstechnik (1995)