Discrete-time port-Hamiltonian systems: A definition based on symplectic integration
Authors
Paul Kotyczka, Laurent Lefèvre
Abstract
We introduce a new definition of discrete-time port-Hamiltonian (PH) systems, which results from structure-preserving discretization of explicit PH systems in time. We discretize the underlying continuous-time Dirac structure with the collocation method and add discrete-time dynamics by the use of symplectic numerical integration schemes. The conservation of a structural discrete-time energy balance – expressed in terms of the discrete-time Dirac structure – extends the notion of symplecticity of geometric integration schemes to open systems. We discuss the energy approximation errors in the context of the presented definition and show that their order for linear PH systems is consistent with the order of the numerical integration scheme. Implicit Gauss–Legendre methods and Lobatto IIIA/IIIB pairs for partitioned systems are examples for integration schemes that are covered by our definition. The statements on the numerical energy errors are illustrated by elementary numerical experiments.
Keywords
Port-Hamiltonian systems; Dirac structures; Discrete-time systems; Geometric numerical integration; Symplectic methods
Citation
- Journal: Systems & Control Letters
- Year: 2019
- Volume: 133
- Issue:
- Pages: 104530
- Publisher: Elsevier BV
- DOI: 10.1016/j.sysconle.2019.104530
BibTeX
@article{Kotyczka_2019,
title={{Discrete-time port-Hamiltonian systems: A definition based on symplectic integration}},
volume={133},
ISSN={0167-6911},
DOI={10.1016/j.sysconle.2019.104530},
journal={Systems & Control Letters},
publisher={Elsevier BV},
author={Kotyczka, Paul and Lefèvre, Laurent},
year={2019},
pages={104530}
}
References
- Leimkuhler, (2004)
- Hairer, (2006)
- Lew, An overview of variational integrators. (2004)
- Duindam, (2009)
- Talasila, V., Clemente-Gallardo, J. & van der Schaft, A. J. Discrete port-Hamiltonian systems. Systems & Control Letters vol. 55 478–486 (2006) – 10.1016/j.sysconle.2005.10.001
- Gören-Sümer, L. & Yalçιn, Y. Gradient Based Discrete-Time Modeling and Control of Hamiltonian Systems. IFAC Proceedings Volumes vol. 41 212–217 (2008) – 10.3182/20080706-5-kr-1001.00036
- Falaize, A. & Hélie, T. Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach. Applied Sciences vol. 6 273 (2016) – 10.3390/app6100273
- Aoues, S., Di Loreto, M., Eberard, D. & Marquis-Favre, W. Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability. Systems & Control Letters vol. 110 9–14 (2017) – 10.1016/j.sysconle.2017.10.003
- Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems based on Gauss-Legendre collocation. IFAC-PapersOnLine vol. 51 125–130 (2018) – 10.1016/j.ifacol.2018.06.035
- 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, (2017)
- Aoues, Canonical interconnection of discrete linear port-Hamiltonian systems. (2013)
- Hairer, Long-time energy conservation of numerical integrators. (2006)
- Sun, Construction of high order symplectic Runge-Kutta methods. J. Comput. Math. (1993)
- Jay, L. Symplectic Partitioned Runge–Kutta Methods for Constrained Hamiltonian Systems. SIAM Journal on Numerical Analysis vol. 33 368–387 (1996) – 10.1137/0733019
- Sun, Symplectic partitioned Runge-Kutta methods. J. Comput. Math. (1993)
- van der Schaft, Port-Hamiltonian differential-algebraic systems. (2013)
- Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0223-3
- Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics vol. 361 442–476 (2018) – 10.1016/j.jcp.2018.02.006
- Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 53 2527–2542 (2008) – 10.1109/tac.2008.2006930