Hankel-norm approximation of large-scale descriptor systems. (English) Zbl 1441.93044

Summary: Hankel-norm approximation is a model reduction method for linear time-invariant systems, which provides the best approximation in the Hankel semi-norm. In this paper, the computation of the optimal Hankel-norm approximation is generalized to the case of linear time-invariant continuous-time descriptor systems. A new algebraic characterization of all-pass descriptor systems is developed and used to construct an efficient algorithm by refining the generalized balanced truncation square root method. For a wide practical usage, adaptations of the introduced algorithm towards stable computations and sparse systems are suggested, as well as an approach for a projection-free algorithm. To show the approximation behavior of the introduced method, numerical examples are presented.


93B11 System structure simplification
93A15 Large-scale systems
93B25 Algebraic methods
93C05 Linear systems in control theory


Full Text: DOI arXiv


[1] Adamjan, VM; Arov, DZ; Kreı̆n, MG, Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Mathematics of the USSR-Sbornik, 15, 1, 31-73 (1971) · Zbl 0248.47019
[2] Antoulas, AC, Approximation of large-scale dynamical systems, Adv. Des Control, vol. 6 (2005), Philadelphia: SIAM Publications, Philadelphia
[3] Bai, Z.; Demmel, J.; Gu, M., An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems, J Numer Math., 76, 3, 279-308 (1997) · Zbl 0876.65021
[4] Benner, P., Quintana-Ortí, E.S.: Model reduction based on spectral projection methods. In: Benner, P., Mehrmann, V., Sorensen, D. (eds.) Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng., vol. 45, pp 5-45. Springer, Berlin (2005), 10.1007/3-540-27909-1_1 · Zbl 1106.93015
[5] Benner, P., Quintana-Ortí, E.S., Quintana-Ortí, G.: Parallel model reduction of large-scale linear descriptor systems via balanced truncation. In: Daydé, M., Dongarra, J.J., Hernández, V., Palma, J.M.L.M. (eds.) High Performance Computing for Computational Science - VECPAR 2004, Lecture Notes in Comput. Sci., vol. 3402, pp 340-353. Springer, Berlin (2005), 10.1007/11403937_27 · Zbl 1118.65399
[6] Benner, P., Stykel, T.: Model order reduction for 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, Cham (2017), 10.1007/978-3-319-46618-7_3 · Zbl 1402.93069
[7] Benner, P., Werner, S.W.R.: MORLAB-3.0 - model order reduction laboratory. 10.5281/zenodo.842659. See also: https://www.mpi-magdeburg.mpg.de/projects/morlab (2017)
[8] Benner, P.; Werner, SWR, On the transformation formulas of the Hankel-norm approximation, Proc Appl Math Mech., 17, 1, 823-824 (2017)
[9] Benner, P.; Werner, SWR, Model reduction of descriptor systems with the MORLAB toolbox, IFAC-PapersOnLine 9th Vienna International Conference on Mathematical Modelling MATHMOD 2018, Vienna Austria, 21-23 February 2018, 51, 2, 547-552 (2018)
[10] Cao, X., Saltik, M.B., Weiland, S.: Hankel model reduction for descriptor systems. In: 2015 54th IEEE Conference on Decision and Control (CDC), pp 4668-4673 (2015), 10.1109/CDC.2015.7402947
[11] Freitas, F.; Rommes, J.; Martins, N., Gramian-based reduction method applied to large sparse power system descriptor models, IEEE Trans Power Del., 23, 3, 1258-1270 (2008)
[12] Glover, K., All optimal Hankel-norm approximations of linear multivariable systems and their \(L^{\infty }\)∞-error norms, Internat J. Control, 39, 6, 1115-1193 (1984) · Zbl 0543.93036
[13] Golub, GH; Van Loan, CF, Matrix Computations (2013), Baltimore: Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore
[14] Gugercin, S.; Stykel, T.; Wyatt, S., Model reduction of descriptor systems by interpolatory projection methods, SIAM J. Sci. Comput., 35, 5, B1010-B1033 (2013) · Zbl 1290.41001
[15] Kågström, B.; Van Dooren, P., A generalized state-space approach for the additive decomposition of a transfer matrix, Numer. Lin. Alg. Appl., 1, 2, 165-181 (1992)
[16] Mehrmann, V., Stykel, T.: Balanced truncation model reduction for large-scale systems in descriptor form. In: Benner, P., Mehrmann, V., Sorensen, D.C. (eds.) Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng., vol. 45, pp 83-115. Springer, Berlin (2005), 10.1007/3-540-27909-1_3 · Zbl 1107.93013
[17] Saak, J., Köhler, M., Benner, P.: M-M.E.S.S.-1.0.1 - the matrix equations sparse solvers library. 10.5281/zenodo.50575. See also: https://www.mpi-magdeburg.mpg.de/projects/mess (2016)
[18] Schmidt, M., Systematic discretization of input/output maps and other contributions to the control of distributed parameter systems. Ph.D. Thesis (2007), Berlin: Technische Universität, Berlin
[19] Sokolov, V.: Contributions to the minimal realization problem for descriptor systems. Dissertation, Fakultät für Mathematik, TU Chemnitz, Chemnitz (2006). http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600965
[20] Stewart, GW; Sun, JG, Matrix Perturbation Theory (1990), New York: Academic Press, New York
[21] Stykel, T.: Analysis and Numerical Solution of generalized Lyapunov equations. Dissertation, TU Berlin (2002). http://webdoc.sub.gwdg.de/ebook/e/2003/tu-berlin/stykel_tatjana.pdf · Zbl 1097.65074
[22] Stykel, T., Gramian-based model reduction for descriptor systems, Math Control Signals Syst., 16, 4, 297-319 (2004) · Zbl 1067.93011
[23] Stykel, T., Balanced truncation model reduction for semidiscretized Stokes equation, Linear Algebra Appl., 415, 2-3, 262-289 (2006) · Zbl 1102.65075
[24] Stykel, T., Low-rank iterative methods for projected generalized Lyapunov equations, Electron Trans Numer Anal., 30, 187-202 (2008) · Zbl 1171.65385
[25] Werner, S.: Hankel-norm approximation of descriptor systems. Master’s thesis, Otto-von-Guericke-Universität, Magdeburg, Germany (2016). http://nbn-resolving.de/urn:nbn:de:gbv:ma9:1-8845
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.