Structure-Preserving Interpolatory Model Reduction for Port-Hamiltonian Differential-Algebraic Systems

Authors

Christopher Beattie, Serkan Gugercin, Volker Mehrmann

Abstract

We examine interpolatory model reduction methods that are particularly well-suited for treating large-scale port-Hamiltonian differential-algebraic systems. We are able to take advantage of underlying structural features of the system in a way that preserves them in the reduced model, using approaches that incorporate regularization and a prudent selection of interpolation data. We focus on linear time-invariant systems and present a systematic treatment of a variety of model classes that include combinations of index-1 and index-2 systems, describing in particular how constraints may be represented in the transfer function so that the polynomial part can be preserved with interpolatory methods. We propose an algorithm to generate effective interpolatory models and illustrate its effectiveness on a numerical example.

Keywords

Port-Hamiltonian descriptor system; Model reduction; Tangential interpolation; Regularization of descriptor system; Staircase form

Citation

• ISBN: 9783030951566
• Publisher: Springer International Publishing
• DOI: 10.1007/978-3-030-95157-3_13

BibTeX

@inbook{Beattie_2022,
  title={{Structure-Preserving Interpolatory Model Reduction for Port-Hamiltonian Differential-Algebraic Systems}},
  ISBN={9783030951573},
  DOI={10.1007/978-3-030-95157-3_13},
  booktitle={{Realization and Model Reduction of Dynamical Systems}},
  publisher={Springer International Publishing},
  author={Beattie, Christopher and Gugercin, Serkan and Mehrmann, Volker},
  year={2022},
  pages={235--254}
}

Download the bib file

References

• Antoulas AC, Beattie CA, Güğercin S (2020) Interpolatory Methods for Model Reduction. Society for Industrial and Applied Mathematics, Philadelphia, PA – 10.1137/1.9781611976083
• Beattie C, Gugercin S (2009) Interpolatory projection methods for structure-preserving model reduction. Systems & Control Letters 58(3):225–232. https://doi.org/10.1016/j.sysconle.2008.10.016 – 10.1016/j.sysconle.2008.10.016
• Beattie C, Gugercin S (2011) Structure-preserving model reduction for nonlinear port-Hamiltonian systems. In: IEEE Conference on Decision and Control and European Control Conference. IEEE, pp 6564–6569 – 10.1109/cdc.2011.6161504
• Beattie C, Gugercin S (2017) Chapter 7: Model Reduction by Rational Interpolation. In: Model Reduction and Approximation. Society for Industrial and Applied Mathematics, Philadelphia, PA, pp 297–334 – 10.1137/1.9781611974829.ch7
• Beattie C, Mehrmann V, Xu H, Zwart H (2018) Linear port-Hamiltonian descriptor systems. Math Control Signals Syst 30(4):17. https://doi.org/10.1007/s00498-018-0223-3 – 10.1007/s00498-018-0223-3
• Binder, A., Mehrmann, V., Miedlar, A., Schulze, P.: A Matlab toolbox for the regularization of descriptor systems arising from generalized realization procedures. Preprint 24–2015, Institut für Mathematik, TU Berlin (2015)
• Bunse-Gerstner A, Byers R, Mehrmann V, Nichols NK (1999) Feedback design for regularizing descriptor systems. Linear Algebra and its Applications 299(1-3):119–151. https://doi.org/10.1016/s0024-3795(99)00167-6 – 10.1016/s0024-3795(99)00167-6
• Campbell SL, Kunkel P, Mehrmann V (2012) Chapter 2: Regularization of Linear and Nonlinear Descriptor Systems. In: Control and Optimization with Differential-Algebraic Constraints. Society for Industrial and Applied Mathematics, Philadelphia, PA, pp 17–36 – 10.1137/9781611972252.ch2
• Chaturantabut S, Beattie C, Gugercin S (2016) Structure-Preserving Model Reduction for Nonlinear Port-Hamiltonian Systems. SIAM J Sci Comput 38(5):B837–B865. https://doi.org/10.1137/15m1055085 – 10.1137/15m1055085
• Dai L (ed) (1989) Singular Control Systems. Springer-Verlag, Berlin/Heidelberg – 10.1007/bfb0002475
• Egger H, Kugler T, Liljegren-Sailer B, Marheineke N, Mehrmann V (2018) On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks. SIAM J Sci Comput 40(1):A331–A365. https://doi.org/10.1137/17m1125303 – 10.1137/17m1125303
• Gallivan K, Vandendorpe A, Van Dooren P (2004) Model Reduction of MIMO Systems via Tangential Interpolation. SIAM J Matrix Anal Appl 26(2):328–349. https://doi.org/10.1137/s0895479803423925 – 10.1137/s0895479803423925
• Gantmacher, F.R.: The Theory of Matrices, vol. II. Chelsea Publishing Company, New York, N.Y. (1959)
• Gugercin S, Antoulas AC, Beattie C (2008) $\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems. SIAM J Matrix Anal Appl 30(2):609–638. https://doi.org/10.1137/060666123 – 10.1137/060666123
• Gugercin S, Polyuga RV, Beattie CA, van der Schaft AJ (2009) Interpolation-based \(H_{2}\) model reduction for port-Hamiltonian systems. In: Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference. IEEE, pp 5362–5369 – 10.1109/cdc.2009.5400626
• Gugercin S, Polyuga RV, Beattie C, van der Schaft A (2012) Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica 48(9):1963–1974. https://doi.org/10.1016/j.automatica.2012.05.052 – 10.1016/j.automatica.2012.05.052
• Gugercin S, Stykel T, Wyatt S (2013) Model Reduction of Descriptor Systems by Interpolatory Projection Methods. SIAM J Sci Comput 35(5):B1010–B1033. https://doi.org/10.1137/130906635 – 10.1137/130906635
• Hauschild S-A, Marheineke N, Mehrmann V (2019) Model reduction techniques for port‐Hamiltonian differential‐algebraic systems. Proc Appl Math and Mech 19(1):e201900040. https://doi.org/10.1002/pamm.201900040 – 10.1002/pamm.201900040
• Heinkenschloss M, Sorensen DC, Sun K (2008) Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations. SIAM J Sci Comput 30(2):1038–1063. https://doi.org/10.1137/070681910 – 10.1137/070681910
• Kunkel P, Mehrmann V (2006) Differential-Algebraic Equations. EMS Press – 10.4171/017
• Mayo AJ, Antoulas AC (2007) A framework for the solution of the generalized realization problem. Linear Algebra and its Applications 425(2-3):634–662. https://doi.org/10.1016/j.laa.2007.03.008 – 10.1016/j.laa.2007.03.008
• Mehl C, Mehrmann V, Wojtylak M (2018) Linear Algebra Properties of Dissipative Hamiltonian Descriptor Systems. SIAM J Matrix Anal Appl 39(3):1489–1519. https://doi.org/10.1137/18m1164275 – 10.1137/18m1164275
• Mehl, C., Mehrmann, V., Wojtylak, M.: Distance problems for dissipative Hamiltonian systems and related matrix polynomials. Technical Report 01-2020, Institut für Mathematik, TU Berlin (2020)
• Mehrmann V, Morandin R (2019) Structure-preserving discretization for port-Hamiltonian descriptor systems. In: 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, pp 6863–6868 – 10.1109/cdc40024.2019.9030180
• Mehrmann V, Stykel T (2005) Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form. In: Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin/Heidelberg, pp 83–115 – 10.1007/3-540-27909-1_3
• Polyuga RV, van der Schaft A (2011) Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos. IEEE Trans Automat Contr 56(6):1458–1462. https://doi.org/10.1109/tac.2011.2128650 – 10.1109/tac.2011.2128650
• Polyuga RV, van der Schaft AJ (2012) Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems. Systems & Control Letters 61(3):412–421. https://doi.org/10.1016/j.sysconle.2011.12.008 – 10.1016/j.sysconle.2011.12.008
• Polyuga RV, van der Schaft A (2010) Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica 46(4):665–672. https://doi.org/10.1016/j.automatica.2010.01.018 – 10.1016/j.automatica.2010.01.018
• Stykel T (2006) Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra and its Applications 415(2-3):262–289. https://doi.org/10.1016/j.laa.2004.01.015 – 10.1016/j.laa.2004.01.015
• van der Schaft AJ (2013) Port-Hamiltonian Differential-Algebraic Systems. In: Surveys in Differential-Algebraic Equations I. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 173–226 – 10.1007/978-3-642-34928-7_5
• Wolf T, Lohmann B, Eid R, Kotyczka P (2010) Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces. European Journal of Control 16(4):401–406. https://doi.org/10.3166/ejc.16.401-406 – 10.3166/ejc.16.401-406