Dynamic iteration schemes and port‐Hamiltonian formulation in coupled differential‐algebraic equation circuit simulation

Authors

Michael Günther, Andreas Bartel, Birgit Jacob, Timo Reis

Abstract

Electric circuits are usually described by charge/flux‐oriented modified nodal analysis. Here, we derive models as port‐Hamiltonian systems on several levels: overall systems, multiply coupled systems, and systems within dynamic iteration procedures. To this end, we introduce new classes of port‐Hamiltonian differential‐algebraic equations. Thereby, we additionally allow for nonlinear dissipation on a subspace of the state space. Both, each subsystem and the overall system possess a port‐Hamiltonian structure. A structural analysis is performed for the new setups. Dynamic iteration schemes are investigated, and we show that the Jacobi approach as well as an adapted Gauss‐Seidel approach lead to port‐Hamiltonian differential‐algebraic equations.

Citation

• Journal: International Journal of Circuit Theory and Applications
• Year: 2021
• Volume: 49
• Issue: 2
• Pages: 430–452
• Publisher: Wiley
• DOI: 10.1002/cta.2870

BibTeX

@article{G_nther_2020,
  title={{Dynamic iteration schemes and port‐Hamiltonian formulation in coupled differential‐algebraic equation circuit simulation}},
  volume={49},
  ISSN={1097-007X},
  DOI={10.1002/cta.2870},
  number={2},
  journal={International Journal of Circuit Theory and Applications},
  publisher={Wiley},
  author={Günther, Michael and Bartel, Andreas and Jacob, Birgit and Reis, Timo},
  year={2020},
  pages={430--452}
}

Download the bib file

References

• Günther M, CAD based electric circuit modeling in industry I: mathematical structure and index of network equations. Surv Math Ind (1999)
• Schaft, A. J. Port-Hamiltonian Systems: Network Modeling and Control of Nonlinear Physical Systems. Advanced Dynamics and Control of Structures and Machines 127–167 (2004) doi:10.1007/978-3-7091-2774-2_9 – 10.1007/978-3-7091-2774-2_9
• 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
• Gernandt, H., Haller, F. E., Reis, T. & Schaft, A. J. van der. Port-Hamiltonian formulation of nonlinear electrical circuits. Journal of Geometry and Physics vol. 159 103959 (2021) – 10.1016/j.geomphys.2020.103959
• van der Schaft, A. J. Port-Hamiltonian Differential-Algebraic Systems. Surveys in Differential-Algebraic Equations I 173–226 (2013) doi:10.1007/978-3-642-34928-7_5 – 10.1007/978-3-642-34928-7_5
• van der Schaft, A. & Maschke, B. Generalized port-Hamiltonian DAE systems. Systems & Control Letters vol. 121 31–37 (2018) – 10.1016/j.sysconle.2018.09.008
• Beattie C, Linear port‐Hamiltonian descriptor systems. Math Contr Sig Syst (2018)
• Mehrmann, V. & Morandin, R. Structure-preserving discretization for port-Hamiltonian descriptor systems. 2019 IEEE 58th Conference on Decision and Control (CDC) 6863–6868 (2019) doi:10.1109/cdc40024.2019.9030180 – 10.1109/cdc40024.2019.9030180
• Burrage, K. Parallel and Sequential Methods for Ordinary Differential Equations. (1995) doi:10.1093/oso/9780198534327.001.0001 – 10.1093/oso/9780198534327.001.0001
• Lelarasmee, E., Ruehli, A. E. & Sangiovanni-Vincentelli, A. L. The Waveform Relaxation Method for Time-Domain Analysis of Large Scale Integrated Circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol. 1 131–145 (1982) – 10.1109/tcad.1982.1270004
• Jackiewicz, Z. & Kwapisz, M. Convergence of Waveform Relaxation Methods for Differential-Algebraic Systems. SIAM Journal on Numerical Analysis vol. 33 2303–2317 (1996) – 10.1137/s0036142992233098
• Arnold, M. & Günther, M. Bit Numerical Mathematics vol. 41 1–25 (2001) – 10.1023/a:1021909032551
• Bartel, A., Baumanns, S. & Schöps, S. Structural analysis of electrical circuits including magnetoquasistatic devices. Applied Numerical Mathematics vol. 61 1257–1270 (2011) – 10.1016/j.apnum.2011.08.004
• Bartel, A. & Günther, M. PDAEs in Refined Electrical Network Modeling. SIAM Review vol. 60 56–91 (2018) – 10.1137/17m1113643
• Andrásfai B, Graph Theory: Flows, Matrices (1991)
• Tischendorf C, Topological index calculation of differential‐algebraic equations in circuit simulation. Surv Math Ind (1999)
• Est�vez Schwarz, D. & Tischendorf, C. Structural analysis of electric circuits and consequences for MNA. International Journal of Circuit Theory and Applications vol. 28 131–162 (2000) – 10.1002/(sici)1097-007x(200003/04)28:2<131::aid-cta100>3.0.co;2-w
• Plastock, R. Homeomorphisms between Banach spaces. Transactions of the American Mathematical Society vol. 200 169–169 (1974) – 10.2307/1997252
• Hairer E, Solving Ordinary Differential Equation II: Stiff and Differential‐Algebraic Problems (2002)
• Griepentrog E, Differential‐Algebraic Equations and Their Numerical Treatment (1986)
• Lamour, R., März, R. & Tischendorf, C. Differential-Algebraic Equations: A Projector Based Analysis. (Springer Berlin Heidelberg, 2013). doi:10.1007/978-3-642-27555-5 – 10.1007/978-3-642-27555-5
• Reis, T. Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits. Modeling and Simulation in Science, Engineering and Technology 125–198 (2014) doi:10.1007/978-3-319-08437-4_2 – 10.1007/978-3-319-08437-4_2