Abstract
We propose an extension of the Loewner framework to descriptor linear systems that preserves the DAE (differential algebraic equation) structure of the underlying system. More precisely, by means of post-processing the data, the behavior at infinity is matched. As it turns out, the conventional procedure constructs a reduced model by directly compressing the data and hence losing information at infinity. By transforming the matrix pencil composed of the E and A matrices into a generalized block diagonal form, we can separate the descriptor system into two subsystems; one corresponding to the polynomial part and the other to the strictly proper part of the transfer function. Different algorithms are implemented to transform the matrix pencil into block diagonal form. Furthermore, a data-driven splitting of the descriptor system can be achieved in the Loewner framework. Hence, the coefficients of the polynomial part can be estimated directly from data. Several numerical examples are presented to illustrate the theoretical discussion.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)
Antoulas, A.C.: The Loewner framework and transfer functions of singular/rectangular systems. Appl. Math. Lett. 54, 36–47 (2016)
Antoulas, A.C., Beattie, C., Gugercin, S.: Interpolatory Model Reduction of Large-Scale Dynamical Systems, pp. 3–58. Efficient Modeling and Control of Large-Scale Systems. Springer, Boston (2010)
Antoulas, A.C., Beattie, C.A., Gugercin, S.: Interpolatory methods for model reduction. Computational Science and Engineering Series. SIAM (2019)
Antoulas, A.C., Gosea, I.V., Ionita, A.C.: Model reduction of bilinear systems in the Loewner framework. SIAM J. Sci. Comput. 38(5), B889–B916 (2016)
Antoulas, A.C., Lefteriu, S., Ionita, A.C.: A tutorial introduction to the Loewner framework for model reduction. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.) Model Reduction and Approximation for Complex Systems. Series: Computational Science & Engineering, pp. 335–376. SIAM (2017)
Baur, U., Benner, P., Feng, L.: Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Archives of Computational Methods in Engineering 21, 331–358 (2014)
Beelen, T., Van Dooren, P.: An improved algorithm for the computation of Kronecker’s canonical form of a singular pencil. Linear Algebra Appl. 105, 9–65 (1988)
Benner, P.: Partial stabilization of descriptor systems using spectral projectors. In: Van Dooren, P., Bhattacharyya, P.S., Chan, H.R., Olshevsky, V., Routray, A. (eds.) Numerical Linear Algebra in Signals, Systems and Control, pp. 55–76. Springer, Netherlands, Dordrecht (2011)
Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)
Benner, P., Losse, P., Mehrmann, V., Voigt, M.: Numerical linear algebra methods for linear differential-algebraic equations. In: Surveys in Differential-Algebraic Equations III, pp. 117–175. Springer (2015)
Benner, P., Mehrmann, V., Sorensen, D.C.: Dimension Reduction of Large-Scale Systems. Springer, Berlin (2005)
Benner, P., Sokolov, V.I.: Partial realization of descriptor systems. Systems & Control Letters 55(11), 929–938 (2006)
Benner, P., Stykel, T.: Model order reduction of differential-algebraic equations: a survey. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations IV. Differential-Algebraic Equations Forum, pp. 107–160. Springer International Publishing (2017)
Berger, T., Ilchmann, A., Trenn, S.: The quasi-Weierstrass form for regular matrix pencils. Linear Algebra Appl. 436(10), 4052–4069 (2012)
Berger, T., Reis, T.: Controllability of linear differential-algebraic systems - a survey. In: Surveys in Differential-Algebraic Equations I, pp. 1–61. Springer, Berlin (2013)
Berger, T., Trenn, S.: The quasi-Kronecker form for matrix pencils. SIAM Journal on Matrix Analysis and Applications 33(2), 336–368 (2012)
Chahlaoui, Y., Van Dooren, P.: A collection of benchmark examples for model reduction of linear time invariant dynamical systems. SLICOT Working Note 2002-2, http://slicot.org/20-site/126-benchmark-examples-for-model-reduction
Demmel, J., Kågström, B.: The generalized Schur decomposition of an arbitrary pencil A - zB: robust software with error bounds and applications. part i: theory and algorithms. ACM Trans. Math. Softw. 19(2), 160–174 (1993)
Demmel, J., Kågström, B.: The generalized Schur decomposition of an arbitrary pencil A - zB,: robust software with error bounds and applications. part ii: software and applications. ACM Trans. Math. Softw. 19(2), 175–201 (1993)
Demmel, J., Kågström, B.: Guptri Software for singular pencils. http://www8.cs.umu.se/research/nla/singular_pairs/guptri/ (1993)
Duan, G.R.: Analysis and Design of Descriptor Linear Systems. Springer, Berlin (2010)
Gosea, I.V., Antoulas, A.C.: Data-driven model order reduction of quadratic-bilinear systems. Special Issue of METT VII, Numercial Linear Algebra with Applications (NLAA), e2200 https://doi.org/10.1002/nla.2200 (2018)
Gosea, I.V., Petreczky, M., Antoulas, A.C.: Data-driven model order reduction of linear switched systems in the loewner framework. SIAM J. Sci. Comput. 40(2), B572–B610 (2018)
Gugercin, S., Stykel, T., Wyatt, S.: Model reduction of descriptor systems by interpolatory projection methods. SIAM J. Sci. Comput. 35(5), B1010–B1033 (2013)
Heinkenschloss, M., Sorensen, D.C., Sun, K.: Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations. SIAM J. Sci. Comput. 30(2), 1038–1063 (2008)
Jonsson, I., Kågström, B.: Recursive blocked algorithms for solving triangular systems – part i: One- sided and coupled Sylvester-type matrix equations. ACM Trans. Math. Software 28(4), 392–415 (2002)
Kågström, B.: RGSVD-An algorithm for computing the Kronecker structure and reducing subspaces of singular A - λ B pencils. SIAM J. Sci. Statist. Comput. 7, 185–211 (1986)
Kågström, B., Poromaa, P.: Lapack-style algorithms and software for solving the generalized sylvester equation and estimating the separation between regular matrix pairs. ACM Trans. Math. Software 22(1), 78–103 (1996)
Kågström, B., Van Dooren, P.: A generalized state-space approach for the additive decomposition of a transfer matrix. J Numer Linear Algebra Appl 1, 165–181 (1992)
Kågström, B., Westin, L.: Generalized schur methods with condition estimators for solving the generalized sylvester equation. IEEE Trans. Automat. Control 34, 745–751 (1989)
Kublanovskaya, V.N.: An approach to solving the spectral problem of A − λB, Matrix Pencils, Lecture Notes in Mathematics. In: Kågström, B., Ruhe, A. (eds.) , vol. 973, pp. 17–29. Springer (1983)
Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)
Mayo, A.J., Antoulas, A.C.: A framework for the generalized realization problem. Linear Algebra and Its Applications 426, 634–662 (2007)
Mehrmann, V., Stykel, T.: Balanced truncation model reduction for large-scale systems in descriptor form. Dimension Reduction of Large-Scale Systems 45, 83–115 (2005)
Mehrmann, V., Stykel, T.: Descriptor systems: a general mathematical framework for modelling, simulation and control. Automatisierungstechnik 54(8), 405–415 (2006)
Seiwald, P., Castagnotto, A., Stykel, T., Lohmann, B.: \({\mathscr{H}}_{2}\) pseudo-optimal reduction of structured DAEs by rational interpolation. Tech. rep. https://arxiv.org/pdf/1804.08755.pdf (2018)
Stykel, T.: Gramian based model reduction for descriptor systems. Mathematics of Control, Signals, and Systems 16, 297–319 (2004)
Trenn, S.: Solution concepts for linear DAEs: a survey. In: Surveys in Differential-Algebraic Equations I, pp. 137–172. Springer, Berlin (2013)
Van Dooren, P.: The computation of Kronecker’s canonical form of a singular pencil. Linear Algebra Appl. 27, 103–140 (1979)
Werner, S.: Hankel-norm approximation of descriptor systems. Ph.D. thesis, Department of Mathematics, Otto-von-Guerricke-University Magdeburg (2016)
Funding
Open access funding provided by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Anthony Nouy
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article belongs to the Topical Collection: Model reduction of parametrized Systems
Guest Editors: Anthony Nouy, Peter Benner, Mario Ohlberger, Gianluigi Rozza, Karsten Urban and Karen Willcox
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gosea, I.V., Zhang, Q. & Antoulas, A.C. Preserving the DAE structure in the Loewner model reduction and identification framework. Adv Comput Math 46, 3 (2020). https://doi.org/10.1007/s10444-020-09752-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-020-09752-8