# zbMATH — the first resource for mathematics

Frequency- and time-limited balanced truncation for large-scale second-order systems. (English) Zbl 07355232
Summary: Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefore, it usually pays off to compute local approximations of the used dynamical systems in the frequency and time domain. In this paper, we consider a structure-preserving extension of the frequency- and time-limited balanced truncation methods to second-order dynamical systems. We give a full overview about the first-order limited balanced truncation methods and extend those to second-order systems by using the different second-order balanced truncation formulas from the literature. Also, we present numerical methods for solving the arising large-scale sparse matrix equations and give numerical modifications to deal with the problematic case of second-order systems. The results are then illustrated on three numerical examples.
##### MSC:
 93B11 System structure simplification 93C70 Time-scale analysis and singular perturbations in control/observation systems 93C80 Frequency-response methods in control theory 93C05 Linear systems in control theory 93A15 Large-scale systems
##### Software:
DPA_TDEFL; MESS ; MORLAB; QDPA; mftoolbox
Full Text:
##### References:
 [1] Baker, J.; Embree, M.; Sabino, J., Fast singular value decay for Lyapunov solutions with nonnormal coefficients, SIAM J. Matrix Anal. Appl., 36, 2, 656-668 (2015) · Zbl 1320.15011 [2] Beattie, C. A.; Gugercin, S., Krylov-based model reduction of second-order systems with proportional damping, (Proceedings of the 44th IEEE Conference on Decision and Control (December 2005)), 2278-2283 [3] Benner, P.; Claver, J. M.; Quintana-Ortí, E. S., Efficient solution of coupled Lyapunov equations via matrix sign function iteration, (Proc. 3rd Portuguese Conf. on Automatic Control CONTROLO’98. Proc. 3rd Portuguese Conf. on Automatic Control CONTROLO’98, Coimbra (1998)), 205-210 [4] Benner, P.; Kürschner, P.; Saak, J., Improved second-order balanced truncation for symmetric systems, IFAC Proc. Vol. (7th Vienna International Conference on Mathematical Modelling), 45, 2, 758-762 (2012) [5] Benner, P.; Kürschner, P.; Saak, J., A reformulated low-rank ADI iteration with explicit residual factors, Proc. Appl. Math. Mech., 13, 1, 585-586 (2013) [6] Benner, P.; Kürschner, P.; Saak, J., Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations, Electron. Trans. Numer. Anal., 43, 142-162 (2014) · Zbl 1312.65068 [7] Benner, P.; Kürschner, P.; Saak, J., Frequency-limited balanced truncation with low-rank approximations, SIAM J. Sci. Comput., 38, 1, A471-A499 (February 2016) [8] Benner, P.; Kürschner, P.; Tomljanović, Z.; Truhar, N., Semi-active damping optimization of vibrational systems using the parametric dominant pole algorithm, Z. Angew. Math. Mech., 96, 5, 604-619 (2016) [9] Benner, P.; Li, J.-R.; Penzl, T., Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems, Numer. Linear Algebra Appl., 15, 9, 755-777 (2008) · Zbl 1212.65245 [10] Benner, P.; Quintana-Ortí, E. S.; Quintana-Ortí, G., Balanced truncation model reduction of large-scale dense systems on parallel computers, Math. Comput. Model. Dyn. Syst., 6, 4, 383-405 (2000) · Zbl 0978.93013 [11] Benner, P.; Saak, J., Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM-Mitt., 36, 1, 32-52 (August 2013) [12] Benner, P.; Stykel, T., Model order reduction for differential-algebraic equations: a survey, (Ilchmann, Achim; Reis, Timo, Surveys in Differential-Algebraic Equations IV, Differential-Algebraic Equations Forum (March 2017), Springer International Publishing: Springer International Publishing Cham), 107-160 · Zbl 1402.93069 [13] Benner, P.; Werner, S. W.R., MORLAB - Model Order Reduction LABoratory (version 5.0) (2019), see also: [14] Benner, P.; Werner, S. W.R., Limited balanced truncation for large-scale sparse second-order systems (version 2.0) (2020) [15] Breiten, T., Structure-preserving model reduction for integro-differential equations, SIAM J. Control Optim., 54, 6, 2992-3015 (2016) · Zbl 1350.93025 [16] Chahlaoui, V.; Gallivan, K. A.; Vandendorpe, A.; Van Dooren, P., Model reduction of second-order systems, (Benner, P.; Mehrmann, V.; Sorensen, D. C., Dimension Reduction of Large-Scale Systems. Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng., vol. 45 (2005), Springer-Verlag: Springer-Verlag Berlin/Heidelberg, Germany), 149-172 · Zbl 1079.65531 [17] Chahlaoui, Y.; Lemonnier, D.; Vandendorpe, A.; Van Dooren, P., Second-order balanced truncation, Linear Algebra Appl., 415, 2-3, 373-384 (2006) · Zbl 1102.93008 [18] Druskin, V.; Simoncini, V., Adaptive rational Krylov subspaces for large-scale dynamical systems, Syst. Control Lett., 60, 8, 546-560 (2011) · Zbl 1236.93035 [19] Fehr, J.; Eberhard, P., Error-controlled model reduction in flexible multibody dynamics, J. Comput. Nonlinear Dyn., 5, 3, Article 031005 pp. (2010) [20] Freitas, F.; Rommes, J.; Martins, N., Gramian-based reduction method applied to large sparse power system descriptor models, IEEE Trans. Power Syst., 23, 3, 1258-1270 (August 2008) [21] Gawronski, W.; Juang, J.-N., Model reduction in limited time and frequency intervals, Int. J. Syst. Sci., 21, 2, 349-376 (1990) · Zbl 0692.93007 [22] Gugercin, S.; Antoulas, A. C., A survey of model reduction by balanced truncation and some new results, Int. J. Control, 77, 8, 748-766 (2004) · Zbl 1061.93022 [23] Haider, K.; Ghafoor, A.; Imran, M.; Malik, F. M., Model reduction of large scale descriptor systems using time limited Gramians, Asian J. Control, 19, 3, 1217-1227 (2017) · Zbl 1366.93090 [24] Haider, K.; Ghafoor, A.; Imran, M.; Malik, F. M., Frequency interval Gramians based structure preserving model reduction for second-order systems, Asian J. Control, 20, 2, 790-801 (2018) · Zbl 1390.93551 [25] Haider, K.; Ghafoor, A.; Imran, M.; Malik, F. M., Time-limited Gramian-based model order reduction for second-order form systems, Trans. Inst. Meas. Control, 41, 8, 2310-2318 (2019) [26] Higham, N. J., Functions of Matrices: Theory and Computation, Applied Mathematics (2008), SIAM Publications: SIAM Publications Philadelphia, PA · Zbl 1167.15001 [27] Imran, M.; Ghafoor, A., Model reduction of descriptor systems using frequency limited Gramians, J. Franklin Inst., 352, 1, 33-51 (2015) · Zbl 1307.93091 [28] Jaimoukha, I. M.; Kasenally, E. M., Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31, 1, 227-251 (1994) · Zbl 0798.65060 [29] Kürschner, P., Balanced truncation model order reduction in limited time intervals for large systems, Adv. Comput. Math., 44, 6, 1821-1844 (2018) · Zbl 1453.65093 [30] Lang, N.; Mena, H.; Saak, J., On the benefits of the $$L D L^T$$ factorization for large-scale differential matrix equation solvers, Linear Algebra Appl., 480, 44-71 (September 2015) [31] Lehner, M.; Eberhard, P., A two-step approach for model reduction in flexible multibody dynamics, Multibody Syst. Dyn., 17, 2-3, 157-176 (2007) · Zbl 1113.70004 [32] Li, J.-R.; White, J., Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., 24, 1, 260-280 (2002) · Zbl 1016.65024 [33] Mehrmann, V.; Stykel, T., Balanced truncation model reduction for large-scale systems in descriptor form, (Benner, P.; Mehrmann, V.; Sorensen, D. C., Dimension Reduction of Large-Scale Systems. Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng., vol. 45 (2005), Springer-Verlag: Springer-Verlag Berlin/Heidelberg, Germany), 83-115 · Zbl 1107.93013 [34] Meyer, D. G.; Srinivasan, S., Balancing and model reduction for second-order form linear systems, IEEE Trans. Autom. Control, 41, 11, 1632-1644 (1996) · Zbl 0859.93015 [35] Moore, B. C., Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Autom. Control, AC-26, 1, 17-32 (1981) · Zbl 0464.93022 [36] Nowakowski, C.; Kürschner, P.; Eberhard, P.; Benner, P., Model reduction of an elastic crankshaft for elastic multibody simulations, Z. Angew. Math. Mech., 93, 198-216 (2013) · Zbl 1275.70005 [37] Panzer, H.; Wolf, T.; Lohamnn, B., A strictly dissipative state space representation of second order systems, Automatisierungstechnik, 60, 7, 392-397 (2012) [38] Petersson, D., A Nonlinear Optimization Approach to $$\mathcal{H}_2$$-Optimal Modeling and Control (2013), Linköping University, Dissertation [39] Reis, T.; Stykel, T., Balanced truncation model reduction of second-order systems, Math. Comput. Model. Dyn. Syst., 14, 5, 391-406 (2008) · Zbl 1151.93010 [40] Rommes, J.; Martins, N., Computing transfer function dominant poles of large-scale second-order dynamical systems, IEEE Trans. Power Syst., 21, 4, 1471-1483 (November 2006) [41] Saak, J.; Köhler, M.; Benner, P., M-M.E.S.S.-2.0 - the matrix equations sparse solvers library (August 2019), see also: [42] Saak, J.; Siebelts, D.; Werner, S. W.R., A comparison of second-order model order reduction methods for an artificial fishtail, Automatisierungstechnik, 67, 8, 648-667 (2019) [43] Salimbahrami, B., Structure Preserving Order Reduction of Large Scale Second Order Models (2005), Technische Universität München: Technische Universität München Munich, Germany, Dissertation [44] Salimbahrami, B.; Lohmann, B., Order reduction of large scale second-order systems using Krylov subspace methods, Linear Algebra Appl., 415, 2-3, 385-405 (2006) · Zbl 1103.93017 [45] Siebelts, D.; Kater, A.; Meurer, T.; Andrej, J., Matrices for an artificial fishtail (2019), Hosted at MORwiki - Model Order Reduction Wiki [46] Simoncini, V., A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput., 29, 3, 1268-1288 (2007) · Zbl 1146.65038 [47] Stykel, T., Analysis and Numerical Solution of Generalized Lyapunov Equations (2002), Dissertation, TU Berlin · Zbl 1014.34037 [48] Wolf, T.; Panzer, H. K.F.; Lohmann, B., Model order reduction by approximate balanced truncation: a unifying framework, Automatisierungstechnik, 61, 8, 545-556 (2013) [49] Wyatt, S., Issues in Interpolatory Model Reduction: Inexact Solves, Second Order Systems and DAEs (May 2012), Virginia Polytechnic Institute and State University: Virginia Polytechnic Institute and State University Blacksburg, Virginia, USA, PhD thesis
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.