An Algorithm to Discretize One-Dimensional Distributed Port Hamiltonian Systems

Authors

Luca Bassi, Alessandro Macchelli, Claudio Melchiorri

Abstract

A key issue when dealing with distributed parameter systems is the solution of the set of partial differential equations that compose the system model. Even if we restrict our analysis to the linear case, it is often impossible to find a closedform solution for this kind of equations, especially in control applications, where the forcing action on the system generates time-varying boundary conditions. Therefore, numerical solvers play a key role as support tools for the analysis of this kind of systems.

Citation

• ISBN: 9783540738893
• Publisher: Springer Berlin Heidelberg
• DOI: 10.1007/978-3-540-73890-9_4

BibTeX

@inbook{Bassi,
  title={{An Algorithm to Discretize One-Dimensional Distributed Port Hamiltonian Systems}},
  ISBN={9783540738893},
  DOI={10.1007/978-3-540-73890-9_4},
  booktitle={{Lagrangian and Hamiltonian Methods for Nonlinear Control 2006}},
  publisher={Springer Berlin Heidelberg},
  author={Bassi, Luca and Macchelli, Alessandro and Melchiorri, Claudio},
  pages={61--73}
}

Download the bib file

References

• A. Bossavit. A. Bossavit. Differential forms and the computation of fields and forces in electromagnetism. European Journal of Mechanics, B/Fluids., 10(5):474–488, 1991. (1991)
• Curtain, R. F. & Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics (Springer New York, 1995). doi:10.1007/978-1-4612-4224-6 – 10.1007/978-1-4612-4224-6
• 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
• Frankel, T. The Geometry of Physics. (2003) doi:10.1017/cbo9780511817977 – 10.1017/cbo9780511817977
• G. Golo, V. Talasila, and A. J. van der Schaft. A Hamiltonian formulation of the Timoshenko beam model. In Proc. of Mechatronics 2002. University of Twente, June 2002.
• 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
• Y. Le Gorrec, H. Zwart, and B. J. Maschke. A semigroup approach to port Hamiltonian systems associated with linear skew symmetric operator. In Proc. Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2004), Leuven, 2004.
• A. Macchelli. Port Hamiltonian systems. A unified approach for modeling and control finite and infinite dimensional physical systems. PhD thesis, University of Bologna—DEIS, 2003. Available at http://www-lar.deis.unibo.it/woda/spider/e499.htm .
• 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. & Bassi, L. Port-based Modelling and Control of the Mindlin Plate. Proceedings of the 44th IEEE Conference on Decision and Control 5989–5994 doi:10.1109/cdc.2005.1583120 – 10.1109/cdc.2005.1583120
• A. Macchelli, S. Stramigioli, and C. Melchiorri. Network modelling and simulation of robots with flexible links A port-based approach. In Proc. 17th International Symposium on Mathematical Theory of Networks, and Systems (MTNS2006) 2006.
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Multi-variable port Hamiltonian model of piezoelectric material. 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566) vol. 1 897–902 – 10.1109/iros.2004.1389466
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
• 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
• 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