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Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control
1. | Max Planck Institute for Dynamics of Complex Technical Systems, Computational Methods in Systems and Control Theory, Magdeburg, Germany, Germany, Germany |
References:
[1] |
M. I. Ahmad and P. Benner, Interpolatory model reduction techniques for linear second-order descriptor systems,, in Proc. European Control Conf. ECC 2014, (2014), 1075. Google Scholar |
[2] |
L. Amodei and J.-M. Buchot, A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation,, Numer. Lin. Alg. Appl., 19 (2012), 700.
doi: 10.1002/nla.799. |
[3] |
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems,, SIAM Publications, (2005).
doi: 10.1137/1.9780898718713. |
[4] |
A. C. Antoulas, D. C. Sorensen and Y. Zhou, On the decay rate of Hankel singular values and related issues,, Systems Control Lett., 46 (2002), 323.
doi: 10.1016/S0167-6911(02)00147-0. |
[5] |
U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations,, SIAM, (1998).
doi: 10.1137/1.9781611971392. |
[6] |
E. Bänsch, P. Benner, J. Saak and H. K. Weichelt, Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flows,, SIAM J. Sci. Comput., 37 (2015).
doi: 10.1137/140980016. |
[7] |
S. Barrachina, P. Benner and E. S. Quintana-Ortí, Efficient algorithms for generalized algebraic Bernoulli equations based on the matrix sign function,, Numer. Algorithms, 46 (2007), 351.
doi: 10.1007/s11075-007-9143-x. |
[8] |
P. Benner and T. Breiten, On optimality of approximate low rank solutions of large-scale matrix equations,, Systems Control Lett., 67 (2014), 55.
doi: 10.1016/j.sysconle.2014.02.005. |
[9] |
P. Benner, P. Kürschner and J. Saak, Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method,, Numer. Algorithms, 62 (2013), 225.
doi: 10.1007/s11075-012-9569-7. |
[10] |
P. Benner, P. Kürschner and J. Saak, An improved numerical method for balanced truncation for symmetric second order systems,, Math. Comput. Model. Dyn. Syst., 19 (2013), 593.
doi: 10.1080/13873954.2013.794363. |
[11] |
P. Benner, P. Kürschner and J. Saak, Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations,, Electron. Trans. Numer. Anal., 43 (2014), 142.
|
[12] |
P. Benner, J. R. Li and T. Penzl, Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems,, Numer. Lin. Alg. Appl., 15 (2008), 755.
doi: 10.1002/nla.622. |
[13] |
P. Benner, V. Mehrmann and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, vol. 45 of Lect. Notes Comput. Sci. Eng.,, Springer-Verlag, (2005).
doi: 10.1007/3-540-27909-1. |
[14] |
P. Benner and T. Stykel, Numerical solution of projected algebraic Riccati equations,, SIAM J. Numer. Anal, 52 (2014), 581.
doi: 10.1137/130923993. |
[15] |
K. A. Cliffe, T. J. Garratt and A. Spence, Eigenvalues of block matrices arising from problems in fluid mechanics,, SIAM J. Matrix Anal. Appl., 15 (1994), 1310.
doi: 10.1137/S0895479892233230. |
[16] |
B. N. Datta, Numerical Methods for Linear Control Systems,, Elsevier Academic Press, (2004).
|
[17] |
E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics,, European Consortium for Mathematics in Industry, (1998).
doi: 10.1007/978-3-663-09828-7. |
[18] |
K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L∞ norms,, Internat. J. Control, 39 (1984), 1115.
doi: 10.1080/00207178408933239. |
[19] |
G. H. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University Press, (1983).
|
[20] |
L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations,, SIAM J. Matrix Anal. Appl., 29 (2007), 870.
doi: 10.1137/040618102. |
[21] |
S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods,, SIAM J. Sci. Comput., 35 (2013).
doi: 10.1137/130906635. |
[22] |
M. Heinkenschloss, D. C. Sorensen and K. Sun, Balanced truncation model reduction for a class of descriptor systems with applications to the Oseen equations,, SIAM J. Sci. Comput., 30 (2008), 1038.
doi: 10.1137/070681910. |
[23] |
P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation,, in Finite Element Methods in Flow Problems (eds. J. T. Oden, (1974), 121. Google Scholar |
[24] |
P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution,, Textbooks in Mathematics, (2006).
doi: 10.4171/017. |
[25] |
J. R. Li and J. White, Low rank solution of Lyapunov equations,, SIAM J. Matrix Anal. Appl., 24 (2002), 260.
doi: 10.1137/S0895479801384937. |
[26] |
T. Penzl, A cyclic low rank Smith method for large sparse Lyapunov equations,, SIAM J. Sci. Comput., 21 (2000), 1401.
doi: 10.1137/S1064827598347666. |
[27] |
Y. Saad, Numerical Methods for Large Eigenvalue Problems,, Manchester University Press, (1992).
|
[28] |
W. H. A. Schilders, H. A. van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications,, Springer-Verlag, (2008).
doi: 10.1007/978-3-540-78841-6. |
[29] |
V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations,, SIAM J. Sci. Comput., 29 (2007), 1268.
doi: 10.1137/06066120X. |
[30] |
E. D. Sontag, Mathematical Control Theory,, 2nd edition, (1998).
doi: 10.1007/978-1-4612-0577-7. |
[31] |
T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation,, Linear Algebra Appl., 415 (2006), 262.
doi: 10.1016/j.laa.2004.01.015. |
[32] |
T. Stykel, Gramian-based model reduction for descriptor systems,, Math. Control Signals Systems, 16 (2004), 297.
doi: 10.1007/s00498-004-0141-4. |
[33] |
M. S. Tombs and I. Postlethwaite, Truncated balanced realization of a stable nonminimal state-space system,, Internat. J. Control, 46 (1987), 1319.
doi: 10.1080/00207178708933971. |
[34] |
B. Vandereycken and S. Vandewalle, A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations,, SIAM J. Matrix Anal. Appl., 31 (2010), 2553.
doi: 10.1137/090764566. |
[35] |
T. Wolf, H. K. F. Panzer and B. Lohmann, ADI iteration for Lyapunov equations: a tangential approach and adaptive shift selection,, arXiv e-prints 1312.1142v1, (1312). Google Scholar |
[36] |
K. Zhou, G. Salomon and E. Wu, Balanced realization and model reduction for unstable systems,, Internat. J. Robust and Nonlinear Cont., 9 (1999), 183.
doi: 10.1002/(SICI)1099-1239(199903)9:3<121::AID-RNC395>3.0.CO;2-1. |
show all references
References:
[1] |
M. I. Ahmad and P. Benner, Interpolatory model reduction techniques for linear second-order descriptor systems,, in Proc. European Control Conf. ECC 2014, (2014), 1075. Google Scholar |
[2] |
L. Amodei and J.-M. Buchot, A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation,, Numer. Lin. Alg. Appl., 19 (2012), 700.
doi: 10.1002/nla.799. |
[3] |
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems,, SIAM Publications, (2005).
doi: 10.1137/1.9780898718713. |
[4] |
A. C. Antoulas, D. C. Sorensen and Y. Zhou, On the decay rate of Hankel singular values and related issues,, Systems Control Lett., 46 (2002), 323.
doi: 10.1016/S0167-6911(02)00147-0. |
[5] |
U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations,, SIAM, (1998).
doi: 10.1137/1.9781611971392. |
[6] |
E. Bänsch, P. Benner, J. Saak and H. K. Weichelt, Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flows,, SIAM J. Sci. Comput., 37 (2015).
doi: 10.1137/140980016. |
[7] |
S. Barrachina, P. Benner and E. S. Quintana-Ortí, Efficient algorithms for generalized algebraic Bernoulli equations based on the matrix sign function,, Numer. Algorithms, 46 (2007), 351.
doi: 10.1007/s11075-007-9143-x. |
[8] |
P. Benner and T. Breiten, On optimality of approximate low rank solutions of large-scale matrix equations,, Systems Control Lett., 67 (2014), 55.
doi: 10.1016/j.sysconle.2014.02.005. |
[9] |
P. Benner, P. Kürschner and J. Saak, Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method,, Numer. Algorithms, 62 (2013), 225.
doi: 10.1007/s11075-012-9569-7. |
[10] |
P. Benner, P. Kürschner and J. Saak, An improved numerical method for balanced truncation for symmetric second order systems,, Math. Comput. Model. Dyn. Syst., 19 (2013), 593.
doi: 10.1080/13873954.2013.794363. |
[11] |
P. Benner, P. Kürschner and J. Saak, Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations,, Electron. Trans. Numer. Anal., 43 (2014), 142.
|
[12] |
P. Benner, J. R. Li and T. Penzl, Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems,, Numer. Lin. Alg. Appl., 15 (2008), 755.
doi: 10.1002/nla.622. |
[13] |
P. Benner, V. Mehrmann and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, vol. 45 of Lect. Notes Comput. Sci. Eng.,, Springer-Verlag, (2005).
doi: 10.1007/3-540-27909-1. |
[14] |
P. Benner and T. Stykel, Numerical solution of projected algebraic Riccati equations,, SIAM J. Numer. Anal, 52 (2014), 581.
doi: 10.1137/130923993. |
[15] |
K. A. Cliffe, T. J. Garratt and A. Spence, Eigenvalues of block matrices arising from problems in fluid mechanics,, SIAM J. Matrix Anal. Appl., 15 (1994), 1310.
doi: 10.1137/S0895479892233230. |
[16] |
B. N. Datta, Numerical Methods for Linear Control Systems,, Elsevier Academic Press, (2004).
|
[17] |
E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics,, European Consortium for Mathematics in Industry, (1998).
doi: 10.1007/978-3-663-09828-7. |
[18] |
K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L∞ norms,, Internat. J. Control, 39 (1984), 1115.
doi: 10.1080/00207178408933239. |
[19] |
G. H. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University Press, (1983).
|
[20] |
L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations,, SIAM J. Matrix Anal. Appl., 29 (2007), 870.
doi: 10.1137/040618102. |
[21] |
S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods,, SIAM J. Sci. Comput., 35 (2013).
doi: 10.1137/130906635. |
[22] |
M. Heinkenschloss, D. C. Sorensen and K. Sun, Balanced truncation model reduction for a class of descriptor systems with applications to the Oseen equations,, SIAM J. Sci. Comput., 30 (2008), 1038.
doi: 10.1137/070681910. |
[23] |
P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation,, in Finite Element Methods in Flow Problems (eds. J. T. Oden, (1974), 121. Google Scholar |
[24] |
P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution,, Textbooks in Mathematics, (2006).
doi: 10.4171/017. |
[25] |
J. R. Li and J. White, Low rank solution of Lyapunov equations,, SIAM J. Matrix Anal. Appl., 24 (2002), 260.
doi: 10.1137/S0895479801384937. |
[26] |
T. Penzl, A cyclic low rank Smith method for large sparse Lyapunov equations,, SIAM J. Sci. Comput., 21 (2000), 1401.
doi: 10.1137/S1064827598347666. |
[27] |
Y. Saad, Numerical Methods for Large Eigenvalue Problems,, Manchester University Press, (1992).
|
[28] |
W. H. A. Schilders, H. A. van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications,, Springer-Verlag, (2008).
doi: 10.1007/978-3-540-78841-6. |
[29] |
V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations,, SIAM J. Sci. Comput., 29 (2007), 1268.
doi: 10.1137/06066120X. |
[30] |
E. D. Sontag, Mathematical Control Theory,, 2nd edition, (1998).
doi: 10.1007/978-1-4612-0577-7. |
[31] |
T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation,, Linear Algebra Appl., 415 (2006), 262.
doi: 10.1016/j.laa.2004.01.015. |
[32] |
T. Stykel, Gramian-based model reduction for descriptor systems,, Math. Control Signals Systems, 16 (2004), 297.
doi: 10.1007/s00498-004-0141-4. |
[33] |
M. S. Tombs and I. Postlethwaite, Truncated balanced realization of a stable nonminimal state-space system,, Internat. J. Control, 46 (1987), 1319.
doi: 10.1080/00207178708933971. |
[34] |
B. Vandereycken and S. Vandewalle, A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations,, SIAM J. Matrix Anal. Appl., 31 (2010), 2553.
doi: 10.1137/090764566. |
[35] |
T. Wolf, H. K. F. Panzer and B. Lohmann, ADI iteration for Lyapunov equations: a tangential approach and adaptive shift selection,, arXiv e-prints 1312.1142v1, (1312). Google Scholar |
[36] |
K. Zhou, G. Salomon and E. Wu, Balanced realization and model reduction for unstable systems,, Internat. J. Robust and Nonlinear Cont., 9 (1999), 183.
doi: 10.1002/(SICI)1099-1239(199903)9:3<121::AID-RNC395>3.0.CO;2-1. |
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