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Mathematical and Computer Modelling of Dynamical Systems
Methods, Tools and Applications in Engineering and Related Sciences
Volume 17, 2011 - Issue 2
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Articles

Efficient balancing-based MOR for large-scale second-order systems

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Pages 123-143 | Received 03 Mar 2010, Accepted 08 Nov 2010, Published online: 23 Feb 2011

References

  • Benner , P. and Saak , J. Efficient balancing based MOR for second order systems arising in control of machine tools . Proceedings of the MathMod 2009: No. 35, ARGESIM-Reports . Vienna, Austria. Edited by: Troch , I. and Breitenecker , F. pp. 1232 – 1243 . ARGE Simulation News .
  • Davison , E. 1966 . A method for simplifying linear dynamic systems . IEEE Trans. Automat. Control AC-11 , : 93 – 101 .
  • Marschall , S. 1966 . An approximate method for reducing the order of a linear system . Control Eng. , 10 : 642 – 648 .
  • Craig , R. and Bampton , M. 1968 . Coupling of substructures for dynamic analysis . AIAA J. , 6 : 1313 – 1319 .
  • Fehr , J. , Eberhard , P. and Lehner , M. Improving the Reduction Process in Flexible Multibody Dynamics by the Use of 2nd Order Position Gramian Matrices . Proceedings ENOC, St. Petersburg . Russia.
  • Wittig , T. , Schuhmann , R. and Weiland , T. 2006 . Model order reduction for large systems in computational electromagnetics . Linear Algebra Appl. , 415 : 499 – 530 .
  • Lehner , M. and Eberhard , P. 2006 . Modellreduktion in elastischen Mehrkörpersystemen (Model Reduction in Flexible Multibody Systems) . Automatisierungstechnik , 54 : 170 – 177 .
  • Lehner , M. and Eberhard , P. 2007 . A two-step approach for model reduction in flexible multibody dynamics . Multibody Syst. Dyn. , 17 : 157 – 176 .
  • Rudnyi , E.B. and Korvink , J.G. Model order reduction for large scale engineering models developed in ANSYS . Applied Parallel Computing. State of the Art in Scientific Computing: 7th International Conference, PARA 2004, Lecture Notes in Computer Science, 3732 . Verlag, Berlin, Heidelberg. pp. 349 – 356 . Springer .
  • Freund , R.W. 2003 . Model reduction methods based on Krylov subspaces . Acta Numerica , 12 : 267 – 319 .
  • Benner , P. 2006 . Numerical linear algebra for model reduction in control and simulation . GAMM Mitt. , 29 : 275 – 296 .
  • Faßbender , H. and Soppa , A. Machine tool simulation based on reduced order FE models . Proceedings of the MathMod 2009: No. 35, ARGESIM-Reports . Vienna, Austria. Edited by: Troch , I. and Breitenecker , F. pp. 1266 – 1277 . ARGE Simulation News .
  • Fehr , J. and Eberhard , P. 2010 . Error-controlled model reduction in flexible multibody dynamics . J. Comput. Nonlinear Dyn. , 5 : 031005
  • Chahlaoui , V. , Gallivan , K.A. , Vandendorpe , A. and Van Dooren , P. 2005 . “ Model reduction of second-order system, in Dimension Reduction of Large-Scale Systems ” . In Lecture Notes in Computer Science and Engineering , Edited by: Benner , P. , Mehrmann , V. and Sorensen , D. Vol. 45 , 149 – 172 . Berlin Heidelberg : Springer-Verlag .
  • Chahlaoui , Y. , Lemonnier , D. , Vandendorpe , A. and Van Dooren , P. 2006 . Second-order balanced truncation . Linear Algebra Appl. , 415 : 373 – 384 .
  • Reis , T. and Stykel , T. 2008 . Balanced truncation model reduction of second-order systems . Math. Comput. Model. Dyn. Syst. , 14 : 391 – 406 .
  • Meyer , D.G. and Srinivasan , S. 1996 . Balancing and model reduction for second-order form linear systems . IEEE Trans. Automat. Control , 41 : 1632 – 1644 .
  • Bai , Z. and Su , Y. 2005 . Dimension reduction of large-scale second-order dynamical systems via a secondorder Arnoldi method . SIAM J. Sci. Comput. , 26 : 1692 – 1709 .
  • Li , R.C. and Bai , Z. 2005 . Structure-preserving model reduction using a Krylov subspace projection formulation . Commun. Math. Sci. , 3 : 179 – 199 .
  • Bai , Z. , Meerbergen , K. and Su , Y. Arnoldi methods for structure-preserving dimension reduction of second-order dynamical systems Dimension Reduction of Large-Scale Systems . Lecture Notes in Computer Science and Engineering . Berlin Heidelberg. Edited by: Benner , P. , Mehrmann , V. and Sorensen , D. Vol. 45 , pp. 173 – 189 . Springer-Verlag .
  • Salimbahrami , B. and Lohmann , B. 2006 . Order reduction of large scale second-order systems using Krylov subspace methods . Linear Algebra Appl. , 415 : 385 – 405 .
  • Polyuga , R. and van der Schaft , A. Structure preserving port-Hamiltonian model reduction of electric circuits . Lecture Notes in Electrical Engineering . Berlin Heidelberg. Edited by: Benner , Vol. 74, P. , Hinze , M. and ter Maten , J. Springer-Verlag . andinandeds.2011.
  • Hartmann , C. , Vulcanov , V.M. and Schütte , C. 2010 . Balanced truncation of linear second-order systems: A Hamiltonian approach . Multiscale Model. Simul. , 8 : 1348 – 1367 .
  • Penzl , T. 2006 . Algorithms for model reduction of large dynamical systems . Linear Algebra Appl. , 415 : 322 – 343 .
  • Benner , P. 2004 . Solving large-scale control problems . IEEE Control Syst. Mag. , 14 : 44 – 59 . p.
  • Li , J.R. and White , J. 2002 . Low rank solution of Lyapunov equations . SIAM J. Matrix Anal. Appl. , 24 : 260 – 280 .
  • Benner , P. , Li , J.R. and Penzl , T. 2008 . Numerical solution of large Lyapunov equations, riccati equations, and linear-quadratic control problems . Numer. Linear Algebra. Appl. , 15 : 755 – 777 .
  • Moore , B.C. 1981 . Principal component analysis in linear systems: Controllability, observability, and model reduction . IEEE Trans. Automat. Control AC-26 , : 17 – 32 .
  • Tombs , M.S. and Postlethwaite , I. 1987 . Truncated balanced realization of a stable nonminimal state-space system . Int. J. Control , 46 : 1319 – 1330 .
  • Laub , A.J. , Heath , M.T. , Paige , C.C. and Ward , R.C. 1987 . Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms . IEEE Trans. Automat. Control , 32 : 115 – 122 .
  • Y. Saad Iterative Methods for Sparse Linear Systems, 1st ed., The PWS Series in Computer Science. PWS Publishing Company, Boston, MA, 1996. Available at(Accessed 6 December 2010). http://www-users.cs.umn.edu/~saad/books.html (http://www-users.cs.umn.edu/~saad/books.html)
  • Glover , K. 1984 . All optimal Hankel-norm approximations of linear multivariable systems and their L1 norms . Int. J. Control , 39 : 1115 – 1193 .
  • Benner , P. , Mena , H. and Saak , J. 2008 . On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations . Electron. Trans. Numer. Anal. , 29 : 136 – 149 .
  • P. Benner, E. Quintana-Ortí, and G. Quintana-Ortí, Parallel model reduction of large-scale linear descriptor systems via balanced truncation, in High Performance Computing for Computational Science. Proceedings of the 6th International Meeting VECPAR'04, June 28–30, Valencia, Spain, 2004, J. Dongarra, M. Daydé, J.M.L.M. Palma, and V. Hernández, eds. Lecture Notes in Computer Science 3402, Springer-Verlag, Berlin Heidelberg, 2005, pp. 65–78.
  • J. Saak Efficient numerical solution of large scale algebraic matrix equations in PDE control and model order reduction, Ph.D. thesis, Faculty of Mathematics, TU Chemnitz, 2009. Available at http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901642 (http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901642)
  • Higham , N.J. , Mackey , D.S. , Tisseur , F. and Garvey , S.D. 2008 . Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems . Int. J. Numer. Methods Eng. , 73 : 344 – 360 .
  • Fan , H.Y. , Lin , W.W. and Van Dooren , P. 2004 . Normwise scaling of second order polynomial matrices . SIAM J. Matrix Anal. Appl. , 26 : 252 – 256 .
  • Benner , P. and Saak , J. 2010 . A Galerkin-Newton-ADI Method for Solving Large-Scale Algebraic Riccati Equations andSPP1253-090, DFG-SPP1253submitted to SIMAX
  • Yan , B. , Tan , S.X.D. and McGaughy , B. 2008 . Second-order balanced truncation for passive-order reduction of RLCK circuits . IEEE Trans. Circuits Syst. II , 55 : 942 – 946 .
  • Billger , D. Dimension Reduction of Large-Scale Systems . Lecture Notes in Computer Science and Engineering . Berlin. The Butterfly Gyro , Edited by: Benner , P. , Mehrmann , V. and Sorensen , D. Vol. 45 , pp. 349 – 352 . Springer Verlag .
  • Haase , J. , Reitz , S. , ünsche , S. W , Schwarz , P. , Becker , U. , Lorenz , G. and Neul , R. 1997 . “ Netzwerk- und Verhaltensmodellierung eines mikromechanischen Beschleunigungssensors ” . In 6. Workshop ‘Methoden und Werkzeuge zum Entwurf von Microsystemen’ 23 – 30 . andinpp.
  • Benner , P. , Quintana-Ortí , E. and Quintana-Ortí , G. 2003 . State-space truncation methods for parallel model reduction of large-scale systems . Parallel Comput. , 29 : 1701 – 1722 .
  • Benner , P. , öhler , M. D , Pester , M. and Saak , J. 2008 . “ PLiCMR - Usage on CHiC ” . In Chemnitz Scientific Computing Preprint 08-01, TU Chemnitz
  • Penzl , T. 2000 . A cyclic low rank Smith method for large sparse Lyapunov equations . SIAM J. Sci. Comput. , 21 : 1401 – 1418 .

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