A Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems

Authors

Tim Moser, Boris Lohmann

Abstract

We present a new structure-preserving model order reduction (MOR) framework for large-scale port-Hamiltonian descriptor systems (pH-DAEs). Our method exploits the structural properties of the Rosenbrock system matrix for this system class and utilizes condensed forms which often arise in applications and reveal the solution behaviour of a system. Provided that the original system has such a form, our method produces reduced-order models (ROMs) of minimal dimension, which tangentially interpolate the original model’s transfer function and are guaranteed to be again in pH-DAE form. This allows the ROM to be safely coupled with other dynamical systems when modelling large system networks, which is useful, for instance, in electric circuit simulation.

Citation

• Journal: Mathematical and Computer Modelling of Dynamical Systems
• Year: 2023
• Volume: 29
• Issue: 1
• Pages: 210–235
• Publisher: Informa UK Limited
• DOI: 10.1080/13873954.2023.2209798

BibTeX

@article{Moser_2023,
  title={{A Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems}},
  volume={29},
  ISSN={1744-5051},
  DOI={10.1080/13873954.2023.2209798},
  number={1},
  journal={Mathematical and Computer Modelling of Dynamical Systems},
  publisher={Informa UK Limited},
  author={Moser, Tim and Lohmann, Boris},
  year={2023},
  pages={210--235}
}

Download the bib file

References

• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• 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
• V. Mehrmann and B. Unger Control of port-Hamiltonian differential-algebraic systems and applications (2022). arXiv Preprint arXiv:2201.06590. http://arxiv.org/abs/2201.06590.
• Gugercin, S., Stykel, T. & Wyatt, S. Model Reduction of Descriptor Systems by Interpolatory Projection Methods. SIAM Journal on Scientific Computing vol. 35 B1010–B1033 (2013) – 10.1137/130906635
• Moser, T., Durmann, J., Bonauer, M. & Lohmann, B. MORpH: Model reduction of linear port-Hamiltonian systems in MATLAB. at - Automatisierungstechnik vol. 71 476–489 (2023) – 10.1515/auto-2022-0119
• Antoulas, A. C., Beattie, C. A. & Güğercin, S. Interpolatory Methods for Model Reduction. (Society for Industrial and Applied Mathematics, 2020). doi:10.1137/1.9781611976083 – 10.1137/1.9781611976083
• Gugercin, S., Antoulas, A. C. & Beattie, C. $\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems. SIAM Journal on Matrix Analysis and Applications vol. 30 609–638 (2008) – 10.1137/060666123
• Polyuga, R. V. & van der Schaft, A. Moment matching for linear port-Hamiltonian systems. 2009 European Control Conference (ECC) 4715–4720 (2009) doi:10.23919/ecc.2009.7075145 – 10.23919/ecc.2009.7075145
• 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
• Polyuga, R. V. & van der Schaft, A. Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos. IEEE Transactions on Automatic Control vol. 56 1458–1462 (2011) – 10.1109/tac.2011.2128650
• Gugercin, S., Polyuga, R. V., Beattie, C. A. & van der Schaft, A. J. Interpolation-based &#x210C;<inf>2</inf> model reduction for port-Hamiltonian systems. Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference 5362–5369 (2009) doi:10.1109/cdc.2009.5400626 – 10.1109/cdc.2009.5400626
• Hauschild S.A.. S.A. Hauschild, N. Marheineke, and V. Mehrmann, Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems, Contrl. Cybernet. 19 (2019), pp. 125–152. (2019)
• Beattie, C., Gugercin, S. & Mehrmann, V. Structure-Preserving Interpolatory Model Reduction for Port-Hamiltonian Differential-Algebraic Systems. Realization and Model Reduction of Dynamical Systems 235–254 (2022) doi:10.1007/978-3-030-95157-3_13 – 10.1007/978-3-030-95157-3_13
• Achleitner F.. F. Achleitner, A. Arnold, and V. Mehrmann, Hypocoercivity and controllability in linear semi-dissipative Hamiltonian ordinary differential equations and differential-algebraic equations, ZAMM. Z. Angew. Math. Mech. (2021). (2021)
• Byers M.V.X.H.. M.V.X.H. Byers, A structured staircase algorithm for skew-symmetric/symmetric pencils, ETNA. Electron Trans. Numer. Anal. [Electronic Only]. 26 (2007), pp. 1–33. http://eudml.org/doc/127545 (2007)
• Rosenbrock H.H.. H.H. Rosenbrock, State-Space and Multivariable Theory, London, Thomas Nelson and Sons Ltd, 1970. (1970)
• ROSENBROCK, H. H. Structural properties of linear dynamical systems. International Journal of Control vol. 20 191–202 (1974) – 10.1080/00207177408932729
• Zhou K.. K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, 1996. (1996)
• 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
• M. Mamunuzzaman and H. Zwart Structure preserving model order reduction of port-Hamiltonian systems (2022). arXiv Preprint arXiv:2203.07751. https://arxiv.org/abs/2203.07751.
• Schwerdtner, P., Moser, T., Mehrmann, V. & Voigt, M. Optimization-based model order reduction of port-Hamiltonian descriptor systems. Systems & Control Letters vol. 182 105655 (2023) – 10.1016/j.sysconle.2023.105655
• 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
• Mehrmann, V. & Stykel, T. Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form. Lecture Notes in Computational Science and Engineering 83–115 doi:10.1007/3-540-27909-1_3 – 10.1007/3-540-27909-1_3
• Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N. & Mehrmann, V. On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks. SIAM Journal on Scientific Computing vol. 40 A331–A365 (2018) – 10.1137/17m1125303
• Cherifi, K., Gernandt, H. & Hinsen, D. The difference between port-Hamiltonian, passive and positive real descriptor systems. Mathematics of Control, Signals, and Systems vol. 36 451–482 (2023) – 10.1007/s00498-023-00373-2
• C. Güdücü J. Liesen V. Mehrmann and D.B. Szyld On non-hermitian positive (semi)definite linear algebraic systems arising from dissipative Hamiltonian DAEs (2021). arXiv Preprint arXiv:2111.05616. https://arxiv.org/abs/2111.05616.
• Moser, T., Durmann, J. & Lohmann, B. Surrogate-Based ℋ2 Model Reduction of Port-Hamiltonian Systems. 2021 European Control Conference (ECC) 2058–2065 (2021) doi:10.23919/ecc54610.2021.9655109 – 10.23919/ecc54610.2021.9655109
• Druskin, V. & Simoncini, V. Adaptive rational Krylov subspaces for large-scale dynamical systems. Systems & Control Letters vol. 60 546–560 (2011) – 10.1016/j.sysconle.2011.04.013
• Druskin, V., Simoncini, V. & Zaslavsky, M. Adaptive Tangential Interpolation in Rational Krylov Subspaces for MIMO Dynamical Systems. SIAM Journal on Matrix Analysis and Applications vol. 35 476–498 (2014) – 10.1137/120898784
• Beattie, C. & Gugercin, S. Interpolatory projection methods for structure-preserving model reduction. Systems & Control Letters vol. 58 225–232 (2009) – 10.1016/j.sysconle.2008.10.016
• Castagnotto A.. A. Castagnotto, C. Beattie, and S. Gugercin, Interpolatory methods for H∞ model reduction of multi-input/multi-output systems, Model Simulat Appl. 2017, pp. 349–365. (2017)
• Flagg, G., Beattie, C. A. & Gugercin, S. Interpolatory model reduction. Systems & Control Letters vol. 62 567–574 (2013) – 10.1016/j.sysconle.2013.03.006
• P. Schwerdtner and M. Voigt Structure preserving model order reduction by parameter optimization (2020). arXiv Preprint arXiv:2011.07567. https://arxiv.org/abs/2011.07567.
• P. Schwerdtner Port-Hamiltonian system identification from noisy frequency response data (2021). arXiv Preprint arXiv:2106.11355. https://arxiv.org/abs/2106.11355.
• Breiten, T. & Unger, B. Passivity preserving model reduction via spectral factorization. Automatica vol. 142 110368 (2022) – 10.1016/j.automatica.2022.110368
• Beattie, C. A., Mehrmann, V. & Van Dooren, P. Robust port-Hamiltonian representations of passive systems. Automatica vol. 100 182–186 (2019) – 10.1016/j.automatica.2018.11.013
• Freund, R. W. The SPRIM Algorithm for Structure-Preserving Order Reduction of General RCL Circuits. Lecture Notes in Electrical Engineering 25–52 (2011) doi:10.1007/978-94-007-0089-5_2 – 10.1007/978-94-007-0089-5_2
• Odabasioglu, A., Celik, M. & Pileggi, L. T. PRIMA: passive reduced-order interconnect macromodeling algorithm. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol. 17 645–654 (1998) – 10.1109/43.712097
• Schwerdtner, P., Moser, T., Mehrmann, V. & Voigt, M. Optimization-based model order reduction of port-Hamiltonian descriptor systems. Systems & Control Letters vol. 182 105655 (2023) – 10.1016/j.sysconle.2023.105655
• Reis, T. & Stykel, T. Positive real and bounded real balancing for model reduction of descriptor systems. International Journal of Control vol. 83 74–88 (2009) – 10.1080/00207170903100214