A structure preserving minimal representation of a nonlinear port-Hamiltonian system

Authors

Jacquelien M.A. Scherpen, Arjan J. van der Schaft

Abstract

In this paper an approach to reduce nonlinear non-observable and non-strongly accessible port-Hamiltonian systems to an observable and strongly accessible port-Hamiltonian system, respectively, is treated. A local state decomposition (the nonlinear version of the Kalman decomposition) is instrumental for the approach that preserves the port-Hamiltonian structure. The strongly accessible reduction scheme goes along similar lines as the linear scheme. However, the observable reduction scheme is somewhat more involved. Under some additional assumptions, the reduction can be performed along the lines of the linear scheme. If these assumptions are not fulfilled, a reduction scheme for a zero-observable representation using duality in the co-energy coordinates is developed. Finally, the possibilities to apply the approaches of this paper to approximate order reduction by e.g., use of balancing procedures, is discussed.

Citation

• Journal: 2008 47th IEEE Conference on Decision and Control
• Year: 2008
• Volume:
• Issue:
• Pages: 4885–4890
• Publisher: IEEE
• DOI: 10.1109/cdc.2008.4739266

BibTeX

@inproceedings{Scherpen_2008,
  title={{A structure preserving minimal representation of a nonlinear port-Hamiltonian system}},
  DOI={10.1109/cdc.2008.4739266},
  booktitle={{2008 47th IEEE Conference on Decision and Control}},
  publisher={IEEE},
  author={Scherpen, Jacquelien M.A. and van der Schaft, Arjan J.},
  year={2008},
  pages={4885--4890}
}

Download the bib file

References

• Hill, D. & Moylan, P. The stability of nonlinear dissipative systems. IEEE Transactions on Automatic Control vol. 21 708–711 (1976) – 10.1109/tac.1976.1101352
• Fujimoto, K., Scherpen, J. M. A. & Gray, W. S. Hamiltonian realizations of nonlinear adjoint operators. Automatica vol. 38 1769–1775 (2002) – 10.1016/s0005-1098(02)00079-1
• Sorensen, D. C. Passivity preserving model reduction via interpolation of spectral zeros. Systems & Control Letters vol. 54 347–360 (2005) – 10.1016/j.sysconle.2004.07.006
• freund, structure-preserving model order reduction of rcl circuit equations. (2007)
• Polyuga, R. V. & van der Schaft, A. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica vol. 46 665–672 (2010) – 10.1016/j.automatica.2010.01.018
• Nijmeijer, H. & van der Schaft, A. Nonlinear Dynamical Control Systems. (Springer New York, 1990). doi:10.1007/978-1-4757-2101-0 – 10.1007/978-1-4757-2101-0
• Meyer, D. G. & Srinivasan, S. Balancing and model reduction for second-order form linear systems. IEEE Transactions on Automatic Control vol. 41 1632–1644 (1996) – 10.1109/9.544000
• Isidori, A. Nonlinear Control Systems. Communications and Control Engineering (Springer London, 1995). doi:10.1007/978-1-84628-615-5 – 10.1007/978-1-84628-615-5
• Gray, W. S. & Scherpen, J. M. A. Minimality and local state decompositions of a nonlinear state space realization using energy functions. IEEE Transactions on Automatic Control vol. 45 2079–2086 (2000) – 10.1109/9.887630
• Scherpen, J. M. A. Balancing for nonlinear systems. Systems & Control Letters vol. 21 143–153 (1993) – 10.1016/0167-6911(93)90117-o
• van der Schaft, A. J. L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control. IEEE Transactions on Automatic Control vol. 37 770–784 (1992) – 10.1109/9.256331