Abstract
The port-Hamiltonian (pH) modelling framework allows for models that preserve essential physical properties such as energy conservation or dissipative inequalities. If all subsystems are modelled as pH systems and the inputs are related to the output in a linear manner, the overall system can be modelled as a pH system, too, which preserves the properties of the underlying subsystems. If the coupling is given by a skew-symmetric matrix, as usual in many applications, the overall system can be easily derived from the subsystems without the need of introducing dummy variables and therefore artificially increasing the complexity of the system. Hence the framework of pH systems is especially suitable for modelling multiphysical systems.
In this paper, we show that pH systems are a natural generalization of Hamiltonian systems, define coupled pH systems as ordinary and differential-algebraic equations. To highlight the suitability for electrical engineering applications, we derive pH models for MNA network equations, electromagnetic devices and coupled systems thereof.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Arnold, M., Günther, M.: Preconditioned dynamic iteration for coupled differential-algebraic equations. BIT 41, 1–25 (2001)
Bartel, A., Günther, M.: Multirate schemes – an answer of numerical analysis to a demand from applications. In: Günther, M., Schilders, W. (eds.) Novel Mathematics Inspired by Industrial Challenges, pp. 5–27. Springer (2022). https://doi.org/10.1007/978-3-030-96173-2_1
Bartel, A., Günther, M., Jacob, B., et al.: Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems. Numer. Math. 155, 1–34 (2023). https://doi.org/10.1007/s00211-023-01369-5
Bartel, A., Brunk, M., Günther, M., Schöps, S.: Dynamic iteration for coupled problems of electric circuits and distributed devices. SIAM J. Sci. Comput. 35(2), B315–B335 (2013). https://doi.org/10.1137/120867111
Diab, M.: Splitting methods for partial differential-algebraic systems with application on coupled field-circuit DAEs. Ph.D. thesis, Humboldt Universität zu Berlin (2022)
Weiland, T.: Time domain electromagnetic field computation with finite difference methods. Int. J. Numer. Model. Electr. Netw. Devices Fields 9, 295–319 (1996)
Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 1432–1467 (1996)
Günther, M., Bartel, A., Jacob, B., Reis, T.: Dynamic iteration schemes and port-Hamiltonian formulation in coupled DAE circuit simulation. Int. J. Circuit Theory Appl. 49, 430–452 (2021)
Günther, M., Feldmann, U.: CAD based electric circuit modeling I: mathematical structure and index of network equations. Surv. Math. Ind. 8, 97–129 (1999)
Günther, M., Sandu, A.: Multirate generalized additive Runge Kutta methods. Numer. Math. 133, 497–524 (2016). https://doi.org/10.1007/s00211-015-0756-z
Jacob, B., Zwart, H.J.: Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. Birkhäuser Verlag, Basel (2012)
Jeltsema, D., van der Schaft, A.J.: Port-Hamiltonian systems theory: an introductory overview. Found. Trends Syst. Control 1(2–3), 173–387 (2014)
Mehrmann, V., Morandin, R.: Structure-preserving discretization for port-Hamiltonian descriptor systems. In: 2019 IEEE 58th Conference on Decision and Control (CDC), pp. 6863–6868 (2019)
van der Schaft, A.J.: Port-Hamiltonian systems: network modeling and control of nonlinear physical systems. In: Schlacher, K., Irschnik, H. (eds.) Advanced Dynamics and Control of Structures and Machines. CISM, vol. 444, pp. 127–167. Springer, Vienna (2004). https://doi.org/10.1007/978-3-7091-2774-2_9
van der Schaft, A.: Port-Hamiltonian systems: an introductory survey. In: Sanz-Sole, M., Soria, J., Varona, J.L., Verdera, J. (eds.) Proceedings of the International Congress of Mathematicians, vol. III, pp. 1339–1365. European Mathematical Society Publishing House (2006)
Acknowledgements
Michael Günther is indebted to the funding given by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 765374, ROMSOC.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bartel, A., Clemens, M., Günther, M., Jacob, B., Reis, T. (2024). Port-Hamiltonian Systems’ Modelling in Electrical Engineering. In: van Beurden, M., Budko, N.V., Ciuprina, G., Schilders, W., Bansal, H., Barbulescu, R. (eds) Scientific Computing in Electrical Engineering. SCEE 2022. Mathematics in Industry(), vol 43. Springer, Cham. https://doi.org/10.1007/978-3-031-54517-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-031-54517-7_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-54516-0
Online ISBN: 978-3-031-54517-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)