Structure Preserving Truncation of Nonlinear Port Hamiltonian Systems

Authors

Yu Kawano, Jacquelien M.A. Scherpen

Abstract

In this paper, we present a novel balancing method for nonlinear port Hamiltonian systems based on the Hamiltonian and the controllability function. This corresponding balanced truncation method results in a reduced-order model that is still in port Hamiltonian form in contrast to the traditional balanced truncation method based on the controllability and observability functions.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2018
• Volume: 63
• Issue: 12
• Pages: 4286–4293
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2018.2811787

BibTeX

@article{Kawano_2018,
  title={{Structure Preserving Truncation of Nonlinear Port Hamiltonian Systems}},
  volume={63},
  ISSN={2334-3303},
  DOI={10.1109/tac.2018.2811787},
  number={12},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Kawano, Yu and Scherpen, Jacquelien M.A.},
  year={2018},
  pages={4286--4293}
}

Download the bib file

References

• 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
• Lopezlena, R., Scherpen, J. M. A. & Fujimoto, K. Energy-Storage Balanced Reduction of Port-Hamiltonian Systems. IFAC Proceedings Volumes vol. 36 69–74 (2003) – 10.1016/s1474-6670(17)38869-9
• Fujimoto, K. & Kajiura, H. Balanced realization and model reduction of port-Hamiltonian systems. 2007 American Control Conference 930–934 (2007) doi:10.1109/acc.2007.4282653 – 10.1109/acc.2007.4282653
• Scherpen, J. M. A. Balancing for nonlinear systems. Systems & Control Letters vol. 21 143–153 (1993) – 10.1016/0167-6911(93)90117-o
• Fujimoto, K. & Scherpen, J. M. A. Nonlinear input-normal realizations based on the differential eigenstructure of Hankel operators. IEEE Transactions on Automatic Control vol. 50 2–18 (2005) – 10.1109/tac.2004.840476
• Fujimoto, K. & Scherpen, J. M. A. Balanced Realization and Model Order Reduction for Nonlinear Systems Based on Singular Value Analysis. SIAM Journal on Control and Optimization vol. 48 4591–4623 (2010) – 10.1137/070695332
• Moore, B. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control vol. 26 17–32 (1981) – 10.1109/tac.1981.1102568
• meirovitch, Methods of Analytical Dynamics (1970)
• scherpen, Balanced model reduction of gradient systems. Proc 18th IFAC World Congr (2011)
• 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
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• van der schaft, -Gain and Passivity Techniques in Nonlinear Control (2012)
• Scherpen, J. M. A. & van der Schaft, A. J. A structure preserving minimal representation of a nonlinear port-Hamiltonian system. 2008 47th IEEE Conference on Decision and Control 4885–4890 (2008) doi:10.1109/cdc.2008.4739266 – 10.1109/cdc.2008.4739266
• polyuga, Structure preserving model reduction of port-Hamiltonian systems. Proc Int Symp Math Theory Netw Syst (2008)
• Ionescu, T. C. & Astolfi, A. Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems. Automatica vol. 49 2424–2434 (2013) – 10.1016/j.automatica.2013.05.006
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• maschke, Port-controlled Hamiltonian systems: Modeling origins and system-theoretic properties. Proc IFAC Symp Nonlinear Contr Syst Des (1991)
• ionescu, Moment matching for nonlinear port Hamiltonian and gradient systems. Proc 9th IFAC Symp Nonlinear Control Syst (2013)
• kawano, Structure preserving truncation for linear port Hamiltonian systems. Proc Int Symp Math Theory Netw Syst (2016)
• Milnor, J. Morse Theory. (AM-51). (1963) doi:10.1515/9781400881802 – 10.1515/9781400881802
• Peng, Z. K., Meng, G., Lang, Z. Q., Zhang, W. M. & Chu, F. L. Study of the effects of cubic nonlinear damping on vibration isolations using Harmonic Balance Method. International Journal of Non-Linear Mechanics vol. 47 1073–1080 (2012) – 10.1016/j.ijnonlinmec.2011.09.013
• Krener, A. J. Reduced Order Modeling of Nonlinear Control Systems. Analysis and Design of Nonlinear Control Systems 41–62 doi:10.1007/978-3-540-74358-3_4 – 10.1007/978-3-540-74358-3_4
• Antoulas, A. C. Approximation of Large-Scale Dynamical Systems. (2005) doi:10.1137/1.9780898718713 – 10.1137/1.9780898718713
• khalil, Nonlinear Systems (1996)
• van der schaft, On the relation between port-Hamiltonian and gradient systems. Proc 18th IFAC World Congr (2011)
• Pernebo, L. & Silverman, L. Model reduction via balanced state space representations. IEEE Transactions on Automatic Control vol. 27 382–387 (1982) – 10.1109/tac.1982.1102945