A subspace framework for L _∞ model reduction

References

[1] K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd ed. Philadelphia, PA, SIAM, 1996.Search in Google Scholar

[2] Y. Feng and M. Yagoubi, Robust Control of Linear Descriptor Systems, Singapore, Springer, 2017.Search in Google Scholar

[3] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, Zurich, Switzerland, EMS Press, 2006.Search in Google Scholar

[4] R. Riaza, Differential Algebraic Systems. Analytical Aspects and Circuit Applications. Notes on Mathematics and its Applications, Singapore, World Scientific, 2008.Search in Google Scholar

[5] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Volume 6 of Adv. Des. Control, Philadelphia, PA, SIAM Publications, 2005.Search in Google Scholar

[6] V. Mehrmann and T. Stykel, “Balanced truncation model reduction for large-scale systems in descriptor form,” in Dimension Reduction of Large-Scale Systems, P. Benner, D. C. Sorensen, and V. Mehrmann, Eds., Berlin, Springer, vol. 45, 2005, pp. 83–115.Search in Google Scholar

[7] B. Moore, “Principal component analysis in linear systems: controllability, observability, and model reduction,” IEEE Trans. Autom. Control, vol. 26, no. 1, pp. 17–32, 1981. https://doi.org/10.1109/tac.1981.1102568.Search in Google Scholar

[8] T. Stykel, “Gramian-based model reduction for descriptor systems,” Math. Contol Signals Syst., vol. 16, no. 4, pp. 297–319, 2004. https://doi.org/10.1007/s00498-004-0141-4.Search in Google Scholar

[9] S. Gugercin and A. C. Antoulas, “A survey of model reduction by balanced truncation and some new results,” Int. J. Control, vol. 77, no. 8, pp. 748–766, 2004. https://doi.org/10.1080/00207170410001713448.Search in Google Scholar

[10] S. Gugercin, D. C. Sorensen, and A. C. Antoulas, “A modified low-rank Smith method for large-scale Lyapunov equations,” Numer. Algorithm., vol. 32, no. 1, pp. 27–55, 2003. https://doi.org/10.1023/a:1022205420182.Search in Google Scholar

[11] T. Penzl, “A cyclic low-rank Smith method for large sparse Lyapunov equations,” SIAM J. Sci. Comput., vol. 21, no. 4, pp. 1401–1418, 1999. https://doi.org/10.1137/s1064827598347666.Search in Google Scholar

[12] K. Glover, “All optimal hankel-norm approximations of linear multivariable systems and their L∞${\mathcal{L}}_{\infty }$-error bounds,” Int. J. Control, vol. 39, no. 6, pp. 1115–1193, 1984. https://doi.org/10.1080/00207178408933239.Search in Google Scholar

[13] P. Benner and S. W. R. Werner, “Hankel-norm approximation of large-scale descriptor systems,” Adv. Comput. Math., vol. 46, Art.no. 40, 2020, https://doi.org/10.1007/s10444-020-09750-w.Search in Google Scholar

[14] P. Benner, E. S. Quintana-Orti, and G. Quintana-Orti, “Computing optimal hankel norm approximations of large-scale systems,” in Proceedings of 43rd IEEE Conference on Decision and Control, vol. 3, 2004, pp. 3078–3083.Search in Google Scholar

[15] S. Gugercin, A. C. Antoulas, and C. Beattie, “ H 2 model reduction for large-scale linear dynamical systems,” SIAM J. Matrix Anal. Appl., vol. 30, no. 2, pp. 609–638, 2008. https://doi.org/10.1137/060666123.Search in Google Scholar

[16] G. Flagg, C. A. Beattie, and S. Gugercin, “Interpolatory H∞ model reduction,” Syst. Control Lett., vol. 62, no. 7, pp. 567–574, 2013. https://doi.org/10.1016/j.sysconle.2013.03.006.Search in Google Scholar

[17] C. K. Krzysztof, Methods of Descent for Nondifferentiable Optimization, Berlin, New York, Springer-Verlag, 1985.Search in Google Scholar

[18] M. M. Mäkelä and P. Neittaanmäki, Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control, Singapore, World Scientific, 1992.Search in Google Scholar

[19] J. V. Burke, A. S. Lewis, and M. L. Overton, “Two numerical methods for optimizing matrix stability,” Linear Algebra Appl., vols. 351–352, pp. 117–145, 2002, https://doi.org/10.1016/s0024-3795(02)00260-4.Search in Google Scholar

[20] J. V. Burke, A. S. Lewis, and M. L. Overton, “A robust gradient sampling algorithm for nonsmooth, nonconvex optimization,” SIAM J. Opt., vol. 15, no. 3, pp. 751–779, 2005. https://doi.org/10.1137/030601296.Search in Google Scholar

[21] S. W. R. Werner, M. L. Overton, and B. Peherstorfer, “Multifidelity robust controller design with gradient sampling,” SIAM J. Sci. Comput., vol. 45, no. 2, pp. A933–A957, 2023. https://doi.org/10.1137/22m1500137.Search in Google Scholar

[22] A. Asl and M. L. Overton, “Analysis of the gradient method with an Armijo–Wolfe line search on a class of non-smooth convex functions,” Optim. Method Softw., vol. 35, no. 2, pp. 223–242, 2020. https://doi.org/10.1080/10556788.2019.1673388.Search in Google Scholar

[23] A. S. Lewis and M. L. Overton, “Nonsmooth optimization via quasi-Newton methods,” Math. Program., vol. 141, nos. 1–2, pp. 135–163, 2013, https://doi.org/10.1007/s10107-012-0514-2.Search in Google Scholar

[24] C. Lemaréchal, Numerical Experiments in Nonsmooth Optimization, Laxenburg, Austria, International Institute for Applied System Analysis (IIASA), 1982, pp. 61–84.Search in Google Scholar

[25] J. V. Burke, F. E. Curtis, A. S. Lewis, M. L. Overton, and L. E. A. Simões, “Gradient sampling methods for nonsmooth optimization,” in Numerical Nonsmooth Optimization: State of the Art Algorithms, A. M. Bagirov, M. Gaudioso, N. Karmitsa, M. M. Mäkelä, and S. Taheri, Eds., Cham, Switzerland, Springer International Publishing, 2020, pp. 201–225.Search in Google Scholar

[26] P. Schwerdtner and M. Voigt, “SOBMOR: structured optimization-based model order reduction,” SIAM J. Sci. Comput., vol. 45, no. 2, pp. A502–A529, 2023. https://doi.org/10.1137/20m1380235.Search in Google Scholar

[27] A. Megretski, “H-infinity model reduction with guaranteed suboptimality bound,” in 2006 American Control Conference, 2006, p. 6.Search in Google Scholar

[28] A. Aliyev, P. Benner, E. Mengi, and M. Voigt, “A subspace framework for H-infinity norm minimization,” SIAM J. Matrix Anal. Appl., vol. 41, no. 2, pp. 928–956, 2020. https://doi.org/10.1137/19m125892x.Search in Google Scholar

[29] S. Chellappa, L. Feng, V. de la Rubia, and P. Benner, “Adaptive interpolatory mor by learning the error estimator in the parameter domain,” in Model Reduction of Complex Dynamical Systems, P. Benner, T. Breiten, H. Faßbender, M. Hinze, T. Stykel, and R. Zimmermann, Eds., Cham, Switzerland, Birkhauser, 2021, pp. 97–117.Search in Google Scholar

[30] L. Feng and P. Benner, “A new error estimator for reduced-order modeling of linear parametric systems,” IEEE Trans. Microw. Theory, vol. 67, no. 12, pp. 4848–4859, 2019. https://doi.org/10.1109/tmtt.2019.2948858.Search in Google Scholar

[31] F. E. Curtis, T. Mitchell, and M. L. Overton, “A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles,” Optim. Method Softw., vol. 32, no. 1, pp. 148–181, 2017. https://doi.org/10.1080/10556788.2016.1208749.Search in Google Scholar

[32] F. E. Curtis and M. L. Overton, “A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization,” SIAM J. Opt., vol. 22, no. 2, pp. 474–500, 2012. https://doi.org/10.1137/090780201.Search in Google Scholar

[33] P. Lancaster, “On eigenvalues of matrices dependent on a parameter,” Numer. Math., vol. 6, no. 1, pp. 377–387, 1964. https://doi.org/10.1007/bf01386087.Search in Google Scholar

[34] E. Mengi, E. A. Yildirim, and M. Kiliç, “Numerical optimization of eigenvalues of Hermitian matrix functions,” SIAM J. Matrix Anal. Appl., vol. 35, no. 2, pp. 699–724, 2014. https://doi.org/10.1137/130933472.Search in Google Scholar

[35] C. Beattie and S. Gugercin, “Interpolatory projection methods for structure-preserving model reduction,” Syst. Control Lett., vol. 58, no. 3, pp. 225–232, 2009. https://doi.org/10.1016/j.sysconle.2008.10.016.Search in Google Scholar

[36] E. Mengi, “Large-scale estimation of dominant poles of a transfer function by an interpolatory framework,” SIAM J. Sci. Comput., vol. 44, no. 4, pp. A2412–A2438, 2022. https://doi.org/10.1137/21m1434349.Search in Google Scholar

[37] S. Boyd and V. Balakrishnan, “A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L∞-norm,” Syst. Control Lett., vol. 15, no. 1, pp. 1–7, 1990. https://doi.org/10.1016/0167-6911(90)90037-u.Search in Google Scholar

[38] N. A. Bruinsma and M. Steinbuch, “A fast algorithm to compute the H∞-norm of a transfer function matrix,” Syst. Control Lett., vol. 14, no. 4, pp. 287–293, 1990. https://doi.org/10.1016/0167-6911(90)90049-z.Search in Google Scholar

[39] E. Mengi, “Large-scale and global maximization of the distance to instability,” SIAM J. Matrix Anal. Appl., vol. 39, no. 4, pp. 1776–1809, 2018. https://doi.org/10.1137/18m1177019.Search in Google Scholar

[40] K. Carlberg, R. Tuminaro, and P. Boggs, “Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics,” SIAM J. Sci. Comput., vol. 37, no. 2, pp. B153–B184, 2015. https://doi.org/10.1137/140959602.Search in Google Scholar

[41] I. Kalashnikova, B. van Bloemen Waanders, S. Arunajatesan, and M. Barone, “Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment,” Comput. Methods Appl. Mech. Eng., vol. 272, pp. 251–270, 2014, https://doi.org/10.1016/j.cma.2014.01.011.Search in Google Scholar

[42] G. Serre, P. Lafon, X. Gloerfelt, and C. Bailly, “Reliable reduced-order models for time-dependent linearized Euler equations,” J. Comput. Phys., vol. 231, no. 15, pp. 5176–5194, 2012. https://doi.org/10.1016/j.jcp.2012.04.019.Search in Google Scholar

[43] F. R. Gantmacher, The Theory of Matrices, vol. II, New York, NY, USA, Chelsea Publishing Company, 1959.Search in Google Scholar

[44] A. Aliyev, P. Benner, E. Mengi, P. Schwerdtner, and M. Voigt, “Large-scale computation of L∞${\mathcal{L}}_{\infty }$-norms by a greedy subspace method,” SIAM J. Matrix Anal. Appl., vol. 38, no. 4, pp. 1496–1516, 2017. https://doi.org/10.1137/16m1086200.Search in Google Scholar

[45] P. Benner and T. Mitchell, “Faster and more accurate computation of the H∞${\mathcal{H}}_{\infty }$ norm via optimization,” SIAM J. Sci. Comput., vol. 40, no. 5, pp. A3609–A3635, 2018. https://doi.org/10.1137/17m1137966.Search in Google Scholar

[46] T. Mitchell, “ROSTAPACK: RObust STAbility PACKage v3.0,” 2024. http://www.timmitchell.com/software/ROSTAPACK/.Search in Google Scholar

[47] P. Benner and S. W. R. Werner, “MORLAB – model order reduction LABoratory (version 5.0),” 2019. Available at: http://www.mpi-magdeburg.mpg.de/projects/morlab.Search in Google Scholar

[48] Y. Chahlaoui and P. Van Dooren, “Benchmark examples for model reduction of linear time-invariant dynamical systems,” in Dimension Reduction of Large-Scale Systems, P. Benner, D. C. Sorrenson and V. Mehrmann, Eds., Berlin, Springer, vol. 45, 2005, pp. 379–392.Search in Google Scholar

[49] T. Penzl, “Algorithms for model reduction of large dynamical systems,” Linear Algebra Appl., vol. 415, nos. 2–3, pp. 322–343, 2006, https://doi.org/10.1016/j.laa.2006.01.007.Search in Google Scholar

[50] A. J. Mayo and A. C. Antoulas, “A framework for the solution of the generalized realization problem,” Linear Algebra Appl., vol. 425, nos. 2–3, pp. 634–662, 2007, https://doi.org/10.1016/j.laa.2007.03.008.Search in Google Scholar