Abstract
In this paper, we discuss the model order reduction problem for descriptor systems, that is, systems with dynamics described by differential-algebraic equations. We focus on linear descriptor systems as a broad variety of methods for these exist, while model order reduction for nonlinear descriptor systems has not received sufficient attention up to now. Model order reduction for linear state-space systems has been a topic of research for about 50 years at the time of writing, and by now can be considered as a mature field. The extension to linear descriptor systems usually requires extra treatment of the constraints imposed by the algebraic part of the system. For almost all methods, this causes some technical difficulties, and these have only been thoroughly addressed in the last decade. We will focus on these developments in particular for the popular methods related to balanced truncation and rational interpolation. We will review efforts in extending these approaches to descriptor systems, and also add the extension of the so-called stochastic balanced truncation method to descriptor systems which so far cannot be found in the literature.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Usually, the term moments is used to denote the coefficients of the Taylor series at s 0 = 0.
- 2.
It should be noted that by abuse of notation, these factors are neither necessarily upper triangular nor square, but we assume them to be of full rank. In particular, for non-minimal systems, these factors will in general be rectangular as then the Gramians will be rank deficient.
- 3.
For partial convergence results, see [52].
References
Ahmad, M.I., Benner, P., Goyal, P.: Krylov subspace-based model reduction for a class of bilinear descriptor systems. J. Comput. Appl. Math. 315, 303–318 (2017)
Alì, G., Banagaaya, N., Schilders, W., Tischendorf, C.: Index-aware model order reduction for linear index-2 DAEs with constant coefficients. SIAM J. Sci. Comput. 35, A1487–A1510 (2013)
Alì, G., Banagaaya, N., Schilders, W., Tischendorf, C.: Index-aware model order reduction for differential-algebraic equations. Math. Comput. Model. Dyn. Syst. 20 (4), 345–373 (2014)
Amsallem, D., Farhat, C.: Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAAJ 46 (7), 1803–1813 (2008)
Amsallem, D., Farhat, C.: An online method for interpolating linear reduced-order models. SIAM J. Sci. Comput. 33 (5), 2169–2198 (2011)
Amsallem, D., Cortial, J., Carlberg, K., Farhat, C.: A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Methods Eng. 80 (9), 1241–1258 (2009)
Anderson, B., Vongpanitlerd, S.: Network Analysis and Synthesis. Prentice Hall, Englewood Cliffs, NJ (1973)
Antoulas, A.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia, PA (2005)
Bai, Z., Su, Y.: Dimension reduction of large-scale second-order dynamical systems via a second-order Arnoldi method. SIAM J. Sci. Comput. 26, 1692–1709 (2005)
Baur, U., Benner, P.: Modellreduktion für parametrisierte Systeme durch balanciertes Abschneiden und Interpolation. at-Automatisierungstechnik 57 (8), 411–419 (2009)
Baur, U., Beattie, C., Benner, P., Gugercin, S.: Interpolatory projection methods for parameterized model reduction. SIAM J. Sci. Comput. 33 (5), 2489–2518 (2011)
Baur, U., Benner, P., Feng, L.: Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Arch. Comput. Meth. Eng. 21 (4), 331–358 (2014)
Benner, P.: Solving large-scale control problems. IEEE Contr. Syst. Mag. 24 (1), 44–59 (2004)
Benner, P.: Numerical linear algebra for model reduction in control and simulation. GAMM Mitteilungen 29 (2), 275–296 (2006)
Benner, P.: System-theoretic methods for model reduction of large-scale systems: simulation, control, and inverse problems. In: Troch, I., Breitenecker, F. (eds.) Proceedings of MathMod 2009 (Vienna, 11–13 February 2009), ARGESIM-Reports, vol. 35, pp. 126–145. Argesim, Wien (2009)
Benner, P.: Advances in balancing-related model reduction for circuit simulation. In: Roos, J., Costa, L. (eds.) Scientific Computing in Electrical Engineering SCEE 2008. Mathematics in Industry, vol. 14, pp. 469–482. Springer, Berlin, Heidelberg (2010)
Benner, P.: Partial stabilization of descriptor systems using spectral projectors. In: Van Dooren, P., Bhattacharyya, S.P., Chan, R.H., Olshevsky, V., Routray, A. (eds.) Numerical Linear Algebra in Signals, Systems and Control. Lecture Notes in Electrical Engineering, vol. 80, pp. 55–76. Springer, Netherlands (2011)
Benner, P., Quintana-Ortí, E.: Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorim. 20 (1), 75–100 (1999)
Benner, P., Feng, L.: A robust algorithm for parametric model order reduction based on implicit moment matching. In: Quarteroni, A., Rozza, R. (eds.) Reduced Order Methods for Modeling and Computational Reduction, vol. 9, pp. 159–186. Springer, Berlin, Heidelberg (2014)
Benner, P., Goyal, P.: Multipoint interpolation of Volterra series and H 2-model reduction for a family of bilinear descriptor systems. Systems Control Lett. 96, 1–11 (2016)
Benner, P., Heiland, J.: LQG-balanced truncation low-order controller for stabilization of laminar flows. In: King, R. (ed.) Active Flow and Combustion Control 2014. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 127, pp. 365–379. Springer International Publishing, Cham (2015)
Benner, P., Saak, J.: Efficient balancing based MOR for large scale second order systems. Math. Comput. Model. Dyn. Syst. 17 (2), 123–143 (2011)
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 Mitteilungen 36 (1), 32–52 (2013)
Benner, P., Sokolov, V.: Partial realization of descriptor systems. Syst. Control Lett. 55 (11), 929–938 (2006)
Benner, P., Stykel, T.: Numerical solution of projected algebraic Riccati equations. SIAM J. Numer. Anal. 52 (2), 581–600 (2014)
Benner, P., Quintana-Ortí, E., Quintana-Ortí, G.: Efficient numerical algorithms for balanced stochastic truncation. Int. J. Appl. Math. Comput. Sci. 11 (5), 1123–1150 (2001)
Benner, P., Mehrmann, V., Sorensen, D. (eds.): Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45. Springer, Berlin, Heidelberg (2005)
Benner, P., Hinze, M., ter Maten, E.J.W. (eds.): Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74 Springer, Dodrecht (2011)
Benner, P., Hossain, M.S., Stykel, T.: Model reduction of periodic descriptor systems using balanced truncation. In: Benner, P., Hinze, M., ter Maten, E.J.W. (eds.) Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74, pp. 187–200. Springer, Dodrecht (2011)
Benner, P., Kürschner, P., Saak, J.: Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method. Numer. Algoritm. 62 (2), 225–251 (2013)
Benner, P., Kürschner, P., Saak, J.: An improved numerical method for balanced truncation for symmetric second-order systems. Math. Comput. Model. Dyn. Syst. 19 (6), 593–615 (2013)
Benner, P., Hossain, M.S., Stykel, T.: Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems. Numer. Algoritm. 67 (3), 669–690 (2014)
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)
Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric systems. SIAM Rev. 57 (4), 483–531 (2015)
Benner, P., Saak, J., Uddin, M.M.: Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numer. Alg. Cont. Opt. 6 (1), 1–20 (2016)
Berger, T., Reis, T.: Controllability of linear differential-algebraic systems - a survey. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations I, Differential-Algebraic Equations Forum, pp. 1–61. Springer, Berlin, Heidelberg (2013)
Berger, T., Reis, T., Trenn, S.: Observability of linear differential-algebraic systems: a survey. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations IV. Differential-Algebraic Equations Forum, pp. 161–220. Springer, Heidelberg/New York/Dordrecht/London (2017)
Bollhöfer, M., Eppler, A.: Low-rank Cholesky factor Krylov subspace methods for generalized projected Lyapunov equations. In: Benner, P. (ed.) System Reduction for Nanoscale IC Design. Mathematics in Industry, vol. 20. Springer, Berlin, Heidelberg (to appear)
Brenan, K., Campbell, S., Petzold, L.: The Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland Publishing Co., New York (1989)
Chu, E.W., Fan, H.Y., Lin, W.W.: Projected generalized discrete-time periodic Lyapunov equations and balanced realization of periodic descriptor systems. SIAM J. Matrix Anal. Appl. 29 (3), 982–1006 (2007)
Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, vol. 118. Springer, Berlin, Heidelberg (1989)
Daniel, L., Siong, O., Chay, L., Lee, K., White, J.: A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 23 (5), 678–693 (2004)
Degroote, J., Vierendeels, J., Willcox, K.: Interpolation among reduced-order matrices to obtain parameterized models for design, optimization and probabilistic analysis. Int. J. Numer. Methods Fluids 63, 207–230 (2010)
D’Elia, M., Dedé, L., Quarteroni, A.: Reduced basis method for parameterized differential algebraic equations. Bol. Soc. Esp. Math. Apl. 46, 45–73 (2009)
Desai, U., Pal, D.: A transformation approach to stochastic model reduction. IEEE Trans. Autom. Control AC-29 (12), 1097–1100 (1984)
Druskin, V., Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60, 546–560 (2011)
Druskin, V., Knizhnerman, L., Simoncini, V.: Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation. SIAM J. Numer. Anal. 49, 1875–1898 (2011)
Enns, D.: Model reduction with balanced realization: an error bound and a frequency weighted generalization. In: Proceedings of the 23rd IEEE Conference on Decision and Control (Las Vegas, 1984), pp. 127–132. IEEE, New York (1984)
Estévez Schwarz, D., Tischendorf, C.: Structural analysis for electric circuits and consequences for MNA. Int. J. Circ. Theor. Appl. 28, 131–162 (2000)
Farle, O., Hill, V., Ingelström, P., Dyczij-Edlinger, R.: Multi-parameter polynomial order reduction of linear finite element models. Math. Comput. Model. Dyn. Syst. 14, 421–434 (2008)
Ferranti, F., Antonini, G., Dhaene, T., Knockaert, L.: Passivity-preserving interpolation-based parameterized model order reduction of PEEC models based on scattered grids. Int. J. Numer. Model. 24 (5), 478–495 (2011)
Flagg, G.M., Beattie, C.A., Gugercin, S.: Convergence of the iterative rational krylov algorithm. Syst. Control Lett. 61 (6), 688–691 (2012)
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 (2008)
Freund, R.: Krylov-subspace methods for reduced-order modeling in circuit simulation. J. Comput. Appl. Math. 123 (1–2), 395–421 (2000)
Freund, R.: Model reduction methods based on Krylov subspaces. Acta Numerica 12, 267–319 (2003)
Freund, R.: The SPRIM algorithm for structure-preserving order reduction of general RCL circuits. Model reduction for circuit simulation. In: Benner, P., Hinze, M., ter Maten, E.J.W. (eds.) Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74, pp. 25–52. Springer, Dodrecht (2011)
Freund, R., Feldmann, P.: The SyMPVL algorithm and its applications in interconnect simulation. In: Proceedings of the 1997 International Conference on Simulation of Semiconductor Processes and Devices, pp. 113–116. Ney York (1997)
Freund, R., Jarre, F.: An extension of the positive real lemma to descriptor systems. Optim. Methods Softw. 19, 69–87 (2004)
Gantmacher, F.: Theory of Matrices. Chelsea Publishing Company, New York (1959)
Gear, C., Leimkuhler, B., Gupta, G.: Automatic integration of Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12–13, 77–90 (1985)
Georgiou, T., Smith, M.: Optimal robustness in the gap metric. IEEE Trans. Autom. Control 35 (6), 673–686 (1990)
Geuss, M., Panzer, H., Lohmann, B.: On parametric model order reduction by matrix interpolation. In: Proceedings of the European Control Conference (Zürich, Switzerland, 17–19 July 2013), pp. 3433–3438 (2013)
Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L ∞-error bounds. Int. J. Control 39 (6), 1115–1193 (1984)
Golub, G., Loan, C.V.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore, London (1996)
Green, M.: Balanced stochastic realizations. Linear Algebra Appl. 98, 211–247 (1988)
Green, M.: A relative error bound for balanced stochastic truncation. IEEE Trans. Autom. Control 33, 961–965 (1988)
Gugercin, S., Antoulas, A.: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77 (8), 748–766 (2004)
Gugercin, S., Antoulas, A., Beattie, C.: \(\mathcal{H}_{2}\) model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30 (2), 609–638 (2008)
Gugercin, S., Stykel, T., Wyatt, S.: Model reduction of descriptor systems by interpolatory projection methods. SIAM J. Sci. Comput. 35 (5), B1010–B1033 (2013)
Harshavardhana, P., Jonckheere, E., Silverman, L.: Stochastic balancing and approximation - stability and minimality. IEEE Trans. Autom. Control 29 (8), 744–746 (1984)
Heinkenschloss, M., Sorensen, D., 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)
Ho, C.W., Ruehli, A., Brennan, P.: The modified nodal approach to network analysis. IEEE Trans. Circuits Syst. 22 (6), 504–509 (1975)
Hossain, M.S., Benner, P.: Generalized inverses of periodic matrix pairs and model reduction for periodic control systems. In: Proceedings of the 1st International Conference on Electrical Engineering and Information and Communication Technology (ICEEICT), Dhaka, Bangladesh, pp. 1–6. IEEE Publications, Piscataway (2014)
Ishihara, J., Terra, M.: On the Lyapunov theorem for singular systems. IEEE Trans. Autom. Control 47 (11), 1926–1930 (2002)
Jaimoukha, I., Kasenally, E.: Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numer. Anal. 31 (1), 227–251 (1994)
Jbilou, K.: ADI preconditioned Krylov methods for large Lyapunov matrix equations. Linear Algebra Appl. 432 (10), 2473–2485 (2010)
Jbilou, K., Riquet, A.: Projection methods for large Lyapunov matrix equations. Linear Algebra Appl. 415 (2–3), 344–358 (2006)
Jonckheere, E., Silverman, L.: A new set of invariants for linear systems with application to reduced order compensator. IEEE Trans. Autom. Control 28 (10), 953–964 (1983)
Katayama, T., Minamino, K.: Linear quadratic regulator and spectral factorization for continuous-time descriptor system. In: Proceedings of the 31st IEEE Conference on Decision and Control (Tuscon, 1992), pp. 967–972. IEEE, New York (1992)
Kawamoto, A., Katayama, T.: The semi-stabilizing solution of generalized algebraic Riccati equation for descriptor systems. Automatica 38, 1651–1662 (2002)
Kawamoto, A., Takaba, K., Katayama, T.: On the generalized algebraic Riccati equation for continuous-time descriptor systems. Linear Algebra Appl. 296, 1–14 (1999)
Kerler-Back, J., Stykel, T.: Model reduction for linear and nonlinear magneto-quasistatic equations. Int. J. Numer. Methods Eng. (to appear). doi:10.1002/nme.5507
Knizhnerman, L., Simoncini, V.: Convergence analysis of the extended Krylov subspace method for the Lyapunov equation. Numer. Math. 118 (3), 567–586 (2011)
Knockaert, L., De Zutter, D.: Laguerre-SVD reduced-order modeling. IEEE Trans. Microw. Theory Tech. 48 (9), 1469–1475 (2000)
Kowal, P.: Null space of a sparse matrix. MATLAB Central (2006). http://www.mathworks.fr/matlabcentral/fileexchange/11120
Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich, Switzerland (2006)
Larin, V., Aliev, F.: Construction of square root factor for solution of the Lyapunov matrix equation. Syst. Control Lett. 20 (2), 109–112 (1993)
Laub, A., Heath, M., Paige, C., Ward, R.: Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Automat. Control AC-32 (2), 115–122 (1987)
Li, J.R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24 (1), 260–280 (2002)
Li, Y., Bai, Z., Su, Y.: A two-directional Arnoldi process and its application to parametric model order reduction. J. Comput. Appl. Math. 226, 10–21 (2009)
Liebermeister, W., Baur, U., Klipp, E.: Biochemical network models simplified by balanced truncation. FEBS J. 272, 4034–4043 (2005)
Liu, W., Sreeram, V.: Model reduction of singular systems. Internat. J. Syst. Sci. 32 (10), 1205–1215 (2001)
Mehrmann, V., Stykel, T.: Descriptor systems: a general mathematical framework for modelling, simulation and control. at-Automatisierungstechnik 54 (8), 405–415 (2006)
Meier, III., L., Luenberger, D.: Approximation of linear constant systems. IEEE Trans. Autom. Control AC-12 (10), 585–588 (1967)
Möckel, J., Reis, T., Stykel, T.: Linear-quadratic gaussian balancing for model reduction of differential-algebraic systems. Int. J. Control 84 (10), 1621–1643 (2011)
Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control AC-26 (1), 17–32 (1981)
Model Order Reduction Wiki. http://www.modelreduction.org (visited 2015-11-02)
Ober, R.: Balanced parametrization of classes of linear systems. SIAM J. Control Optim. 29 (6), 1251–1287 (1991)
Odabasioglu, A., Celik, M., Pileggi, L.: PRIMA: Passive reduced-order interconnect macromodeling algorithm. IEEE Trans. Circuits Syst. 17 (8), 645–654 (1998)
Opdenacker, P., Jonckheere, E.: A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds. IEEE Trans. Circuits Syst. I 35 (2), 184–189 (1988)
Panzer, H., Mohring, J., Eid, R., Lohmann, B.: Parametric model order reduction by matrix interpolation. at-Automatisierungtechnik 58 (8), 475–484 (2010)
Penzl, T.: A cyclic low-rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21 (4), 1401–1418 (1999/2000)
Perev, K., Shafai, B.: Balanced realization and model reduction of singular systems. Int. J. Syst. Sci. 25 (6), 1039–1052 (1994)
Pernebo, L., Silverman, L.: Model reduction via balanced state space representation. IEEE Trans. Autom. Control AC-27, 382–387 (1982)
Phillips, J., Daniel, L., Miguel Silveira, L.: Guaranteed passive balancing transformations for model order reduction. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 22, 1027–1041 (2003)
Phillips, J., Miguel Silveira, L.: Poor Man’s TBR: a simple model reduction scheme. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24 (1), 43–55 (2005)
Poloni, F., Reis, T.: A structured doubling algorithm for the numerical solution of Lur’e equations. Preprint 748–2011, DFG Research Center MATHEON, Technische Universität Berlin (2011)
Poloni, F., Reis, T.: A deflation approach for large-scale Lur’e equations. SIAM J. Matrix Anal. Appl. 33 (4), 1339–1368 (2012)
Reis, T.: Circuit synthesis of passive descriptor systems - a modified nodal approach. Int. J. Circuit Theory Appl. 38 (1), 44–68 (2010)
Reis, T.: Lur’e equations and even matrix pencils. Linear Algebra Appl. 434 (1), 152–173 (2011)
Reis, T.: Mathematical modeling and analysis of nonlinear time-invariant RLC circuits. In: Benner, P., Findeisen, R., Flockerzi, D., Reichl, U., Sundmacher, K. (eds.) Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology, pp. 125–198. Birkhäuser, Basel (2014). Chapter 2
Reis, T., Rendel, O.: Projection-free balanced truncation for differential-algebraic systems. In: Workshop on Model Reduction of Complex Dynamical Systems MODRED 2013, Magdeburg, 13 December 2013
Reis, T., Stykel, T.: Balanced truncation model reduction of second-order systems. Math. Comput. Model. Dyn. Syst. 14 (5), 391–406 (2008)
Reis, T., Stykel, T.: PABTEC: Passivity-preserving balanced truncation for electrical circuits. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 29 (9), 1354–1367 (2010)
Reis, T., Stykel, T.: Positive real and bounded real balancing for model reduction of descriptor systems. Int. J. Control 83 (1), 74–88 (2010)
Reis, T., Stykel, T.: Lyapunov balancing for passivity-preserving model reduction of RC circuits. SIAM J. Appl. Dyn. Syst. 10 (1), 1–34 (2011)
Reis, T., Voigt, M.: Linear-quadratic infinite time horizon optimal control for differential-algebraic equations - a new algebraic criterion. In: Proceedings of the International Symposium on Mathematical Theory of Networks and Systems (MTNS 2012), Melbourne, Australia, 9–13 July 2012 (2012)
Riaza, R.: Differential-Algebraic Systems: Analytical Aspects and Circuit Applications. World Scientific Publishing Co. Pte. Ltd., Hackensack (2008)
Roberts, J.: Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Int. J. Control 32 (4), 677–687 (1980). Reprint of Technical Report TR-13, CUED/B-Control, Engineering Department, Cambridge University, 1971
Rommes, J., Martins, N.: Exploiting structure in large-scale electrical circuit and power system problems. Linear Algebra Appl. 431 (3–4), 318–333 (2009)
Rosenbrock, H.: The zeros of a system. Int. J. Control 18 (2), 297–299 (1973)
Saad, Y.: Numerical solution of large Lyapunov equations. In: Kaashoek, M., Schuppen, J.V., Ran, A. (eds.) Signal Processing, Scattering, Operator Theory, and Numerical Methods, pp. 503–511. Birkhäuser, Boston (1990)
Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996)
Sabino, J.: Solution of large-scale Lyapunov equations via the block modified Smith method. Ph.D. thesis, Rice University, Houston (2006)
Salimbahrami, B., Lohmann, B.: Order reduction of large scale second-order systems using Krylov subspace methods. Linear Algebra Appl. 415, 385–405 (2006)
Schöps, S.: Multiscale modeling and multirate time-integration of field/circuit coupled problems. Ph.D. thesis, Bergische Universität Wuppertal (2011)
Schöps, S., De Gersem, H., Weiland, T.: Winding functions in transient magnetoquasistatic field-circuit coupled simulations. COMPEL 32 (6), 2063–2083 (2013)
Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29 (3), 1268–1288 (2007)
Son, N.: Interpolation based parametric model order reduction. Ph.D. thesis, Universität Bremen, Germany (2012)
Son, N.: A real time procedure for affinely dependent parametric model order reduction using interpolation on Grassmann manifolds. Int. J. Numer. Methods Eng. 93 (8), 818–833 (2013)
Son, N., Stykel, T.: Model order reduction of parameterized circuit equations based on interpolation. Adv. Comput. Math. 41 (5), 1321–1342 (2015)
Steedhar, J., Van Dooren, P., Misra, P.: Minimal order time invariant representation of periodic descriptor systems. In: Proceedings of the American Control Conference, San Diego, California, June 1999, vol. 2, pp. 1309–1313 (1999)
Stykel, T.: Numerical solution and perturbation theory for generalized Lyapunov equations. Linear Algebra Appl. 349, 155–185 (2002)
Stykel, T.: Gramian-based model reduction for descriptor systems. Math. Control Signals Syst. 16, 297–319 (2004)
Stykel, T.: Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra Appl. 415, 262–289 (2006)
Stykel, T.: A modified matrix sign function method for projected Lyapunov equations. Syst. Control Lett. 56, 695–701 (2007)
Stykel, T.: Low-rank iterative methods for projected generalized Lyapunov equations. Electron. Trans. Numer. Anal. 30, 187–202 (2008)
Stykel, T.: Balancing-related model reduction of circuit equations using topological structure. In: Benner, P., Hinze, M., ter Maten, E.J.W. (eds.) Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74, pp. 53–80. Springer, Dodrecht (2011)
Stykel, T., Reis, T.: The PABTEC algorithm for passivity-preserving model reduction of circuit equations. In: Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010, Budapest, Hungary, 5–9 July 2010), paper 363. ELTE, Budapest, Hungary (2010)
Stykel, T., Simoncini, V.: Krylov subspace methods for projected Lyapunov equations. Appl. Numer. Math. 62, 35–50 (2012)
Takaba, K., Morihira, N., Katayama, T.: A generalized Lyapunov theorem for descriptor system. Syst. Control Lett. 24, 49–51 (1995)
Tan, S., He, L.: Advanced Model Order Reduction Techniques in VLSI Design. Cambridge University Press, New York (2007)
Tombs, M., Postlethweite, I.: Truncated balanced realization of a stable non-minimal state-space system. Int. J. Control 46 (4), 1319–1330 (1987)
Uddin, M., Saak, J., Kranz, B., Benner, P.: Computation of a compact state space model for an adaptive spindle head configuration with piezo actuators using balanced truncation. Prod. Eng. Res. Dev. 6 (6), 577–586 (2012)
Unneland, K., Van Dooren, P., Egeland, O.: New schemes for positive real truncation. Model. Identif. Control 28, 53–65 (2007)
Varga, A., Fasol, K.: A new square-root balancing-free stochastic truncation model reduction algorithm. In: Proceedings of 12th IFAC World Congress, Sydney, Australia, vol. 7, pp. 153–156 (1993)
Verghese, G., Lévy, B., Kailath, T.: A generalized state-space for singular systems. IEEE Trans. Autom. Control AC-26 (4), 811–831 (1981)
Wachspress, E.: The ADI minimax problem for complex spectra. In: Kincaid, D., Hayes, L. (eds.) Iterative Methods for Large Linear Systems, pp. 251–271. Academic Press, Boston (1990)
Wang, H.S., Chang, F.R.: The generalized state-space description of positive realness and bounded realness. In: Proceedings of the 39th IEEE Midwest Symposium on Circuits and Systems, vol. 2, pp. 893–896. IEEE, New York (1996)
Wang, H.S., Yung, C.F., Chang, F.R.: Bounded real lemma and H ∞ control for descriptor systems. In: IEE Proceedings on Control Theory and Applications, vol. 145, pp. 316–322. IEE, Stevenage (1998)
Wang, H.S., Yung, C.F., Chang, F.R.: The positive real control problem and the generalized algebraic Riccati equation for descriptor systems. J. Chin. Inst. Eng. 24 (2), 203–220 (2001)
Willcox, K., Lassaux, G.: Model reduction of an actively controlled supersonic diffuser. In: Benner, P., Mehrmann, V., Sorensen, D. (eds.) Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45, pp. 357–361. Springer, Berlin, Heidelberg (2005). Chapter 20
Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (11), 2323–2330 (2002)
Xin, X.: Strong solutions and maximal solutions of generalized algebraic Riccati equations. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 528–533. IEEE, Piscataway (2008)
Yan, B., Tan, S.D., McGaughy, B.: Second-order balanced truncation for passive-order reduction of RLCK circuits. IEEE Trans. Circuits Syst. II 55 (9), 942–946 (2008)
Zhang, L., Lam, J., Xu, S.: On positive realness of descriptor systems. IEEE Trans. Circuits Syst. 49 (3), 401–407 (2002)
Zhang, Z., Wong, N.: An efficient projector-based passivity test for descriptor systems. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 29 (8), 1203–1214 (2010)
Zhou, K.: Error bounds for frequency weighted balanced truncation and relative error model reduction. In: Proceedings of the IEEE Conference on Decision and Control (San Antonio, Texas, December 1993), pp. 3347–3352 (1993)
Zhou, K.: Frequency-weighted \(\mathcal{L}_{\infty }\) norm and optimal Hankel norm model reduction. IEEE Trans. Autom. Control 40, 1687–1699 (1995)
Acknowledgements
The first author acknowledges support by the collaborative project nanoCOPS: “Nanoelectronic COupled Problems Solutions” funded by the European Union in the FP7-ICT-2013-11 Program under Grant Agreement Number 619166.
The second author was supported by the Research Network KoSMos: Model reduction based simulation of coupled PDAE systems funded by the German Federal Ministry of Education and Science (BMBF), grant 05M13WAA, and by the project Model reduction for elastic multibody systems with moving interactions funded by the German Research Foundation (DFG), grant STY 58/1–2.
The authors gratefully acknowledge the careful proofreading by a reviewer and the editors which greatly enhanced the presentation of this survey.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Benner, P., Stykel, T. (2017). 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. Springer, Cham. https://doi.org/10.1007/978-3-319-46618-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-46618-7_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46617-0
Online ISBN: 978-3-319-46618-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)