A new framework for H2-optimal model reduction

References

• T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput. 21 (4) (2000), pp. 1401–1418. doi:10.1137/S1064827598347666
   Web of Science ®Google Scholar
• V. Druskin, V. Simoncini, and M. Zaslavsky, Adaptive tangential interpolation in rational Krylov subspaces for MIMO dynamical systems, SIAM J. Matrix Anal. Appl. 35 (2) (2014), pp. 476–498. doi:10.1137/120898784
   Web of Science ®Google Scholar
• P. Kürschner, Efficient low-rank solution of large-scale matrix equations, Ph.D. thesis, Max Planck Institute for Dynamics of Complex Technical Systems, 2016.
   Google Scholar
• E.J. Grimme, Krylov projection methods for model reduction, Ph.D. Thesis, University of Illinois at Urbana-Champaign, USA, 1997.
   Google Scholar
• C.A. Beattie and S. Gugercin, Model reduction by rational interpolation, in Model Reduction and Approximation: Theory and Algorithms, P. Benner, A.M. Cohen, M. Ohlberger, eds., et al., math.NA. SIAM, Sep. 2017, Available at http://arxiv.org/abs/1409.2140v1.
   Google Scholar
• K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error norms, Int. J. Control 39 (6) (1984), pp. 1115–1193. doi:10.1080/00207178408933239
   Web of Science ®Google Scholar
• A. Castagnotto, C.A. Beattie, and S. Gugercin, Interpolatory methods for H2∞ model reduction of multi-input/multi-output systems, in Model Reduction of Parametrized Systems, P. Benner, M. Ohlberger, A. Patera, eds., et al., Vol. 17, Modeling, Simulation & Applications, Springer, Cham, 2017, Chap. 22, pp. 349–365. doi:10.1007/978-3-319-58786-8_22
   Google Scholar
• S. Gugercin, A.C. Antoulas, and C.A. Beattie, H2 model reduction for largescale dynamical systems, SIAM J. Matrix Anal. Appl. 30 (2) (2008), pp. 609–638. doi:10.1137/060666123
   Web of Science ®Google Scholar
• P. Van Dooren, K. Gallivan, and P.-A. Absil, H2-optimal model reduction of MIMO systems, Appl. Math. Lett. 21 (2008), pp. 1267–1273. doi:10.1016/j.aml.2007.09.015
   Google Scholar
• C.A. Beattie and S. Gugercin, A trust region method for optimal H2 model reduction, in IEEE Conference on Decision and Control, IEEE, 2009, pp. 5370–5375. doi:10.1109/CDC.2009.5400605
   Google Scholar
• H.K.F. Panzer, S. Jaensch, T. Wolf, et al. A Greedy Rational Krylov Method for H2-pseudooptimal model order reduction with preservation of stability, in Proceedings of the American Control Conference, 2013, pp. 5512–5517.
   Google Scholar
• H.K.F. Panzer, Model order reduction by Krylov subspace methods with global error bounds and automatic choice of parameters, Ph.D. thesis, Technische Universität München, 2014.
   Google Scholar
• A. Castagnotto, H.K.F. Panzer, and B. Lohmann, Fast H2-optimal model order reduction exploiting the local nature of Krylov-subspace methods, in European Control Conference 2016, Aalborg, Denmark, 2016, pp. 1958–1963. doi:10.1109/ECC.2016.7810578
   Google Scholar
• K. Gallivan, A. Vandendorpe, and P. Van Dooren, Model reduction of MIMO systems via tangential interpolation, SIAM J. Matrix Anal. Appl. 26 (2) (2004), pp. 328–349. doi:10.1137/S0895479803423925
   Web of Science ®Google Scholar
• K. Gallivan, A. Vandendorpe, and P. Van Dooren, Sylvester equations and projection-based model reduction, J. Comput. Appl. Math. 162 (1) (2004), pp. 213–229. doi:10.1016/j.cam.2003.08.026
   Web of Science ®Google Scholar
• A.C. Antoulas, Approximation of Large-Scale Dynamical Systems, Vol. 6, Advances in Design and Control, SIAM Publications, Philadelphia, PA, 2005, ISBN 9780898715293. doi:10.1137/1.9780898718713
   Google Scholar
• A.C. Antoulas, C.A. Beattie, and S. Gugercin, Interpolatory model reduction of large-scale dynamical systems, in Efficient Modeling and Control of Large-Scale Systems, Springer US, 2010, pp. 3–58. doi:10.1007/978-1-4419-57573_1
   Google Scholar
• L. Meier and D.G. Luenberger, Approximation of linear constant systems, IEEE Trans. Autom. Control 12 (5) (Oct., 1967), pp. 585–588. doi:10.1109/TAC.1967.1098680
   Google Scholar
• C.A. Beattie and S. Gugercin, Realization-independent H22-approximation, in 51st IEEE Conference on Decision and Control, IEEE, 2012, pp. 4953–4958. doi:10.1109/CDC.2012.6426344
   Google Scholar
• Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003.
   Google Scholar
• Y. Chahlaoui and P. Van Dooren, A collection of benchmark examples for model reduction of linear time invariant dynamical systems, Tech. rep. 2002-2, SLICOT Working Note, 2002. Available at www.slicot.org.
   Google Scholar
• J.G. Korvink and E.B. Rudnyi, Oberwolfach benchmark collection, in Dimension Reduction of Large-Scale Systems, P. Benner, D.C. Sorensen, and V. Mehrmann, eds., Vol. 45, Lecture Notes in Computational Science and Engineering, Springer Berlin Heidelberg, 2005, pp. 311–315. doi:10.1007/3-54027909-1_11
   Google Scholar
• G.H. Golub and C.F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1996.
   Google Scholar
• A. Castagnotto, M. Cruz Varona, and L. Jeschek, sss & sssMOR: analysis and reduction of large-scale dynamic systems in MATLAB, atAutomatisierungstechnik, 65 (2) (2017), pp. 134–150. doi:10.1515/auto-20160137
   Web of Science ®Google Scholar
• N.V. Queipo, R.T. Haftka, W. Shyy, T. Goel, R. Vaidyanathan, and P. Kevin Tucker, Surrogate-based analysis and optimization, Prog. Aerosp. Sci. 41 (1) (2005), pp. 1–28. doi:10.1016/j.paerosci.2005.02.001
   Web of Science ®Google Scholar
• P. Benner, Z. Tomljanović, and N. Truhar, Optimal damping of selected eigenfrequencies using dimension reduction, Numer. Lin. Alg. Appl. 20 (1) (2013), pp. 1–17, ISSN . doi:10.1002/nla.833
   Web of Science ®Google Scholar
• J. Rommes and N. Martins, Effcient computation of transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst. 21 (4) (2006), pp. 1218–1226. doi:10.1109/TPWRS.2006.876671
   Web of Science ®Google Scholar
• J.D. Pintér, Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications, Vol. 6, Springer Science & Business Media, 2013.
   Google Scholar
• A. Castagnotto, S. Hu, and B. Lohmann, An approach for globalized H2-optimal model reduction, in Proceedings of the 9th Vienna International Conference on Mathematical Modelling, 2018.
   Google Scholar
• J. Lienemann, D. Billger, E.B. Rudnyi, et al., MEMS compact modeling meets model order reduction: Examples of the application of Arnoldi methods to microsystem devices, in The Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show, Nanotech, Vol. 4, 2004.
   Google Scholar
• S. Gugercin, Projection methods for model reduction of large-scale dynamical systems, Ph.D. thesis, Rice University, 2003. Available at http://hdl.%20handle.%20net/1911/18536.
   Google Scholar
• S. Gugercin and A.C. Antoulas, An H2 error expression for the Lanczos procedure, in 42nd IEEE Conference on Decision and Control, Vol. 2, IEEE, 2003, pp. 1869–1872.
   Google Scholar
• S. Gugercin, T. Stykel, and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods, SIAM J. Sci. Comput. 35 (5) (2013), pp. B1010–B1033. doi:10.1137/130906635
   Web of Science ®Google Scholar
• I. Pontes Duff, S. Gugercin, C. Beattie, C. Poussot-Vassal, and C. Seren, H2-optimality conditions for reduced time-delay systems of dimension one, IFAC Pap. OnLine 49 (10) (2016), 13th IFAC Workshop on Time Delay Systems TDS 2016, pp. 7–12. doi:10.1016/j.ifacol.2016.07.464
   Google Scholar
• S. Gugercin, R.V. Polyuga, C. Beattie, and A. van der Schaft, Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems, Autom. 48 (9) (2012), pp. 1963–1974. doi:10.1016/j.automatica.2012.05.052.
   Web of Science ®Google Scholar
• P. Benner and T. Breiten, Interpolation-based H2-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl. 33 (3) (2012), pp. 859–885. doi:10.1137/110836742
   Web of Science ®Google Scholar
• G.M. Flagg and S. Gugercin, Multipoint volterra series interpolation and H2Optimal model reduction of bilinear systems, SIAM J. Numer. Anal. 36 (2) (2015), pp. 549–579. doi:10.1137/130947830
   Google Scholar
• P. Benner, P. Goyal, and S. Gugercin, H2-quasi-optimal model order reduction for quadratic-bilinear control systems, arXiv Preprint arXiv:1610.03279 (2016).
   Google Scholar
• P. Benner and T. Stykel, Model Order Reduction for Differential-Algebraic Equations: A Survey, Springer, 2017, pp. 107–160.
   Google Scholar
• A. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, 3rd ed., Springer, 2017.
   Google Scholar
• T. Wolf, B. Lohmann, R. Eid, and P. Kotyczka, Passivity and structure preserving order reduction of linear port-Hamiltonian systems using Krylov subspaces, Eur. J. Control 16 (4) (2010), pp. 401–406. doi:10.3166/ejc.16.401-406
   Web of Science ®Google Scholar
• A.J. Mayo and A.C. Antoulas, A framework for the solution of the generalized realization problem, Linear Algebra Appl. 425 (2–3) (2007), Special Issue in honor of P. A. Fuhrmann, Edited by A. C. Antoulas, U. Helmke, J. Rosenthal, V. Vinnikov, and E. Zerz, pp. 634–662, ISSN: . doi:10.1016/j.laa.2007.03.008
   Web of Science ®Google Scholar
• I. Pontes Duff, C. Poussot-Vassal, and C. Seren, Realization independent single time-delay dynamical model interpolation and H2-optimal approximation, in 2015 54th IEEE Conference on Decision and Control (CDC), Dec. 2015, pp. 4662–4667. doi:10.1109/CDC.2015.7402946
   Google Scholar
• C. Beattie, Z. Drmač, and S. Gugercin, Quadrature-based IRKA for optimal H2 model reduction, IFAC Pap. OnLine 48 (1) (2015), 8th Vienna International Conferenceon Mathematical Modelling, pp. 5–6. doi:10.1016/j.ifacol.2015.05.196.
   Google Scholar
• Z. Drmač, S. Gugercin, and C. Beattie, Quadrature-based vector fitting for discretized H2 approximation, SIAM J. Sci. Comput. 37 (2) (2015), pp. A625–A652. doi:10.1137/140961511
   Google Scholar
• W.J. Rugh, Nonlinear System Theory, Johns Hopkins University Press Baltimore, 1981.
   Google Scholar
• G.M. Flagg, Interpolation methods for the model reduction of bilinear systems, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, Apr. 2012.
   Google Scholar
• P. Benner and P. Goyal, Multipoint interpolation of volterra series and H2Model reduction for a family of bilinear descriptor systems, Syst. Control Lett. 97 (2016), pp. 1–11. doi:10.1016/j.sysconle.2016.08.008
   Web of Science ®Google Scholar
• T. Wolf, Pseudo-optimal model order reduction, Dissertation, Technische Universität München, Munich, Germany, 2015.
   Google Scholar
• J. Peters, Parametrische modellordnungsreduktion mit dem H₂-optimalen Modellfunktions-Framework, Bachelor Thesis, Technical University of Munich, Oct. 2017.
   Google Scholar