On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks

Authors

H. Egger, T. Kugler, B. Liljegren-Sailer, N. Marheineke, V. Mehrmann

Abstract

We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semidiscretization in space via a mixed finite element method and also during the further dimension reduction by structure- preserving Galerkin projection, which is the main focus of this paper. Krylov subspace methods are employed for the construction of the reduced models, and we discuss certain modifications needed to satisfy some algebraic compatibility conditions which are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our arguments are based on a careful analysis of the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct inte…

Citation

• Journal: SIAM Journal on Scientific Computing
• Year: 2018
• Volume: 40
• Issue: 1
• Pages: A331–A365
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/17m1125303

BibTeX

@article{Egger_2018,
  title={{On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks}},
  volume={40},
  ISSN={1095-7197},
  DOI={10.1137/17m1125303},
  number={1},
  journal={SIAM Journal on Scientific Computing},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Egger, H. and Kugler, T. and Liljegren-Sailer, B. and Marheineke, N. and Mehrmann, V.},
  year={2018},
  pages={A331--A365}
}

Download the bib file

References

• Antoulas AC (2005) Approximation of Large-Scale Dynamical Systems. Society for Industrial and Applied Mathematics – 10.1137/1.9780898718713
• Bai Z (2002) Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Applied Numerical Mathematics 43(1-2):9–44. https://doi.org/10.1016/s0168-9274(02)00116-2 – 10.1016/s0168-9274(02)00116-2
• Bai Z, Freund RW (2001) A partial Padé-via-Lanczos method for reduced-order modeling. Linear Algebra and its Applications 332-334:139–164. https://doi.org/10.1016/s0024-3795(00)00291-3 – 10.1016/s0024-3795(00)00291-3
• Bai Z, Meerbergen K, Su Y (2005) Arnoldi Methods for Structure-Preserving Dimension Reduction of Second-Order Dynamical Systems. In: Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin/Heidelberg, pp 173–189 – 10.1007/3-540-27909-1_7
• Bai Z, Su Y (2005) Dimension Reduction of Large-Scale Second-Order Dynamical Systems via a Second-Order Arnoldi Method. SIAM J Sci Comput 26(5):1692–1709. https://doi.org/10.1137/040605552 – 10.1137/040605552
• Bai Z, Su Y (2005) SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem. SIAM J Matrix Anal Appl 26(3):640–659. https://doi.org/10.1137/s0895479803438523 – 10.1137/s0895479803438523
• Baker GA, Dougalis VA (1976) The Effect of Quadrature Errors on Finite Element Approximations for Second Order Hyperbolic Equations. SIAM J Numer Anal 13(4):577–598. https://doi.org/10.1137/0713049 – 10.1137/0713049
• 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
• Benner P., ScienceOpen Res. (2015)
• Benner P, Sorensen DC, Mehrmann V (eds) (2005) Dimension Reduction of Large-Scale Systems. Springer Berlin Heidelberg, Berlin, Heidelberg – 10.1007/3-540-27909-1
• Brenan K. E., Classics Appl. Math. 14 (1996)
• Brezzi F., Rev. Française Automat. Informat. Rech. Opér. Sér. Rouge (1974)
• Brouwer J, Gasser I, Herty M (2011) Gas Pipeline Models Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Networks. Multiscale Model Simul 9(2):601–623. https://doi.org/10.1137/100813580 – 10.1137/100813580
• Chahlaoui Y, Gallivan KA, Vandendorpe A, Van Dooren P (2005) Model Reduction of Second-Order Systems. In: Lecture Notes in Computational Science and Engineering. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 149–172 – 10.1007/3-540-27909-1_6
• Chapelle D, Gariah A, Sainte-Marie J (2012) Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples. ESAIM: M2AN 46(4):731–757. https://doi.org/10.1051/m2an/2011053 – 10.1051/m2an/2011053
• Dai L (ed) (1989) Singular Control Systems. Springer-Verlag, Berlin/Heidelberg – 10.1007/bfb0002475
• Egger H., Uniform Exponential Stability of Galerkin Approximations for Damped Wave Systems, preprint, https://arxiv.org/abs/1511.08341 (2015)
• Egger H., Numer. Math. (2017)
• Emmrich E, Mehrmann V (2013) Operator Differential-Algebraic Equations Arising in Fluid Dynamics. Computational Methods in Applied Mathematics 13(4):443–470. https://doi.org/10.1515/cmam-2013-0018 – 10.1515/cmam-2013-0018
• Freund RW (2000) Krylov-subspace methods for reduced-order modeling in circuit simulation. Journal of Computational and Applied Mathematics 123(1-2):395–421. https://doi.org/10.1016/s0377-0427(00)00396-4 – 10.1016/s0377-0427(00)00396-4
• Freund RW (2005) Padé-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems. In: Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin/Heidelberg, pp 191–223 – 10.1007/3-540-27909-1_8
• Golo G, Talasila V, van der Schaft A, Maschke B (2004) Hamiltonian discretization of boundary control systems. Automatica 40(5):757–771. https://doi.org/10.1016/j.automatica.2003.12.017 – 10.1016/j.automatica.2003.12.017
• Golub G. H., Matrix Computations (1996)
• Grimme E. J., Dissertation (1997)
• Grundel S, Jansen L, Hornung N, Clees T, Tischendorf C, Benner P (2014) Model Order Reduction of Differential Algebraic Equations Arising from the Simulation of Gas Transport Networks. In: Differential-Algebraic Equations Forum. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 183–205 – 10.1007/978-3-662-44926-4_9
• 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
• Harkort C, Deutscher J (2012) Stability and passivity preserving Petrov–Galerkin approximation of linear infinite-dimensional systems. Automatica 48(7):1347–1352. https://doi.org/10.1016/j.automatica.2012.04.010 – 10.1016/j.automatica.2012.04.010
• Hinze M, Volkwein S (2005) Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control. In: Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin/Heidelberg, pp 261–306 – 10.1007/3-540-27909-1_10
• Kunisch K, Volkwein S (2001) Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik 90(1):117–148. https://doi.org/10.1007/s002110100282 – 10.1007/s002110100282
• Kunisch K, Volkwein S (2002) Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics. SIAM J Numer Anal 40(2):492–515. https://doi.org/10.1137/s0036142900382612 – 10.1137/s0036142900382612
• Kunkel P, Mehrmann V (2006) Differential-Algebraic Equations. EMS Press – 10.4171/017
• Lagnese JE, Leugering G, Schmidt EJPG (1994) Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Birkhäuser Boston, Boston, MA – 10.1007/978-1-4612-0273-8
• 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
• Mehrmann V, Watkins D (2001) Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils. SIAM J Sci Comput 22(6):1905–1925. https://doi.org/10.1137/s1064827500366434 – 10.1137/s1064827500366434
• Mehrmann V. L., Lect. Notes Control Inform. Sci. 163 (1991)
• Meyer DG, Srinivasan S (1996) Balancing and model reduction for second-order form linear systems. IEEE Trans Automat Contr 41(11):1632–1644. https://doi.org/10.1109/9.544000 – 10.1109/9.544000
• Polyuga RV (2010) Discussion on: “Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces”. European Journal of Control 16(4):407–409. https://doi.org/10.1016/s0947-3580(10)70672-5 – 10.1016/s0947-3580(10)70672-5
• 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
• 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
• Salimbahrami B, Lohmann B (2006) Order reduction of large scale second-order systems using Krylov subspace methods. Linear Algebra and its Applications 415(2-3):385–405. https://doi.org/10.1016/j.laa.2004.12.013 – 10.1016/j.laa.2004.12.013
• Salimbahrami B, Lohmann B, Bunse-Gerstner A (2008) Passive reduced order modelling of second-order systems. Mathematical and Computer Modelling of Dynamical Systems 14(5):407–420. https://doi.org/10.1080/13873950701844279 – 10.1080/13873950701844279
• Schilders WHA, van der Vorst HA, Rommes J (eds) (2008) Model Order Reduction: Theory, Research Aspects and Applications. Springer Berlin Heidelberg, Berlin, Heidelberg – 10.1007/978-3-540-78841-6
• Sorensen DC, Antoulas AC (2005) On Model Reduction of Structured Systems. In: Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin/Heidelberg, pp 117–130 – 10.1007/3-540-27909-1_4
• van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2-3):173–378. https://doi.org/10.1561/2600000002 – 10.1561/2600000002
• van der Schaft AJ, Maschke BM (2013) Port-Hamiltonian Systems on Graphs. SIAM J Control Optim 51(2):906–937. https://doi.org/10.1137/110840091 – 10.1137/110840091
• Van Loan C (1985) Computing the CS and the generalized singular value decompositions. Numer Math 46(4):479–491. https://doi.org/10.1007/bf01389653 – 10.1007/bf01389653
• Wu Y, Hamroun B, Gorrec YL, Maschke B (2014) Port Hamiltonian System in Descriptor Form for Balanced Reduction: Application to a Nanotweezer. IFAC Proceedings Volumes 47(3):11404–11409. https://doi.org/10.3182/20140824-6-za-1003.01579 – 10.3182/20140824-6-za-1003.01579