Representation of a general composition of Dirac structures

Authors

Carles Batlle, Imma Massana, Ester Simo

Abstract

We provide explicit representations for the Dirac structure obtained from an arbitrary number of component Dirac structures coupled by means of another interconnecting Dirac structure. Our work generalizes the results in [1] in two aspects. First, the interconnecting structure is not limited to the simple feedback case considered there, and this opens new possibilities for designing control systems. Second, the number of simultaneously interconnected systems is not limited to two, which allows for extra flexibility in modeling, particularly in the case of electrical networks. Several relevant particular cases are presented, and the application to the interconnection of port-Hamiltonian systems is discussed by means of an example.

Citation

• Journal: IEEE Conference on Decision and Control and European Control Conference
• Year: 2011
• Volume:
• Issue:
• Pages: 5199–5204
• Publisher: IEEE
• DOI: 10.1109/cdc.2011.6160588

BibTeX

@inproceedings{Batlle_2011,
  title={{Representation of a general composition of Dirac structures}},
  DOI={10.1109/cdc.2011.6160588},
  booktitle={{IEEE Conference on Decision and Control and European Control Conference}},
  publisher={IEEE},
  author={Batlle, Carles and Massana, Imma and Simo, Ester},
  year={2011},
  pages={5199--5204}
}

Download the bib file

References

• sjo?berg, Optimal control and model reduction of nonlinear DAE models (2008)
• Fujimoto, K. & Scherpen, J. M. A. Balanced Realization and Model Order Reduction for Nonlinear Systems Based on Singular Value Analysis. SIAM J. Control Optim. 48, 4591–4623 (2010) – 10.1137/070695332
• Sjoberg, J. & Glad, T. Computing the controllability function for nonlinear descriptor systems. 2006 American Control Conference 6 pp. (2006) doi:10.1109/acc.2006.1655493 – 10.1109/acc.2006.1655493
• sjo?berg, Model reduction of nonlinear differential-algebraic equations. Proceedings of the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS) (2007)
• Maschke, B. M., van der Schaft, A. J. & Breedveld, P. C. An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Trans. Circuits Syst. I 42, 73–82 (1995) – 10.1109/81.372847
• polyuga, Model Reduction of Port-Hamiltonian Systems (2010)
• 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
• maschke, Port-controlled hamiltonian systems: Modelling origins and system theoretic properties. Proceedings 2nd IFAC Symposium on Nonlinear Control Systems (NOLCOS 1992) (1992)
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica 43, 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• Batlle, C., Dòria-Cerezo, A., Espinosa-Pérez, G. & Ortega, R. Simultaneous interconnection and damping assignment passivity-based control: the induction machine case study. International Journal of Control 82, 241–255 (2009) – 10.1080/00207170802050817
• 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
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• 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
• Yoshimura, H. & Marsden, J. E. Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems. Journal of Geometry and Physics 57, 133–156 (2006) – 10.1016/j.geomphys.2006.02.009
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• Putting energy back in control. IEEE Control Syst. 21, 18–33 (2001) – 10.1109/37.915398