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Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B
Stability preservation in Galerkin-type projection-based model order reduction
Institute of Mathematics and Computer Science, University of Greifswald, Walther-Rathenau-Str. 47, 17489 Greifswald, Germany |
We consider linear dynamical systems consisting of ordinary differential equations with high dimensionality. The aim of model order reduction is to construct an approximating system of a much lower dimension. Therein, the reduced system may be unstable, even though the original system is asymptotically stable. We focus on projection-based model order reduction of Galerkin-type. A transformation of the original system guarantees an asymptotically stable reduced system. This transformation requires the numerical solution of a high-dimensional Lyapunov equation. We specify an approximation of the solution, which allows for an efficient iterative treatment of the Lyapunov equation under a certain assumption. Furthermore, we generalize this strategy to preserve the asymptotic stability of stationary solutions in model order reduction of nonlinear dynamical systems. Numerical results for high-dimensional examples confirm the computational feasibility of the stability-preserving approach.
References:
[1] |
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM Publications, 2005.
doi: 10.1137/1.9780898718713. |
[2] |
Z. Bai and R. Freund,
A partial Padé-via-Lanczos method for reduced order modeling, Linear Algebra Appl., 332-334 (2001), 139-164.
doi: 10.1016/S0024-3795(00)00291-3. |
[3] |
P. Benner, S. Gugercin and K. Willcox,
A survey of projection-based model order reduction methods for parametric dynamical systems, SIAM Review, 57 (2015), 483-531.
doi: 10.1137/130932715. |
[4] |
P. Benner, V. Mehrmann and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, Lecture Notes in Compuational Science and Engineering, Vol. 45, Springer, 2005.
doi: 10.1007/3-540-27909-1. |
[5] |
M. Braun, Differential Equations and Their Applications, 3rd edition, Springer, 1983. |
[6] |
A. Castagnotto, M. Cruz Varona, L. Jeschek and B. Lohmann,
sss & sssMOR: Analysis and reduction of large-scale dynamic systems in MATLAB, Automatisierungstechnik, 65 (2017), 134-150.
|
[7] |
R. Castañé Selga, B. Lohmann and R. Eid,
Stability preservation in projection-based model order reduction of large scale systems, Eur. J. Control, 18 (2012), 122-132.
doi: 10.3166/ejc.18.122-132. |
[8] |
J. Ding and G. Yao,
The eigenvalue problem of a specially updated matrix, Appl. Math. Comput., 185 (2007), 415-420.
doi: 10.1016/j.amc.2006.07.040. |
[9] |
R. Freund,
Model reduction methods based on Krylov subspaces, Acta Numerica, 12 (2003), 267-319.
doi: 10.1017/S0962492902000120. |
[10] |
M. I. Gil', Explicit Stability Conditions for Continuous Systems: A Functional Analytic Approach, Springer, 2005.
doi: 10.1007/b99808. |
[11] |
S. Gugercin and A. C. Antoulas,
A survey of model reduction by balanced truncation and some new results, Int. J. Control, 77 (2004), 748-766.
doi: 10.1080/00207170410001713448. |
[12] |
J. K. Hale, E. F. Infante and F. S. P. Tsen,
Stability in linear delay systems, J. Math. Anal. Appl., 115 (1985), 533-555.
doi: 10.1016/0022-247X(85)90068-X. |
[13] |
S. J. Hammarling,
Numerical solution of stable non-negative definite Lyapunov equation, IMA J. Numer. Anal., 2 (1982), 303-323.
doi: 10.1093/imanum/2.3.303. |
[14] |
T. C. Ionescu and A. Astolfi, On moment matching with preservation of passivity and stability, in: 49th IEEE Conference on Decision and Control, (2010), 6189-6194. |
[15] |
B. Kramer and J. R. Singler,
A POD projection method for large-scale algebraic Riccati equations, Numer. Algebra Contr. Optim., 6 (2016), 413-435.
doi: 10.3934/naco.2016018. |
[16] |
J.-R. Li and J. White,
Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. & Appl., 24 (2002), 260-280.
doi: 10.1137/S0895479801384937. |
[17] |
B. Lohmann and R. Eid, Efficient order reduction of parametric and nonlinear models by superposition of locally reduced models in: Methoden und Anwendungen der Regelungstechnik (eds. G. Roppenecker and B. Lohmann), Shaker, 2009. |
[18] |
MATLAB, version 9. 1. 0. 441655 (R2016b), The Mathworks Inc., Natick, Massachusetts, 2016. |
[19] |
"MOR Wiki", online document, https://morwiki.mpi-magdeburg.mpg.de/morwiki Cited May 4, 2018. |
[20] |
P. C. Müller,
Modified Lyapunov equations for LTI descriptor systems, J. Braz. Soc. Mech. Sci. & Eng., 28 (2006), 448-452.
|
[21] |
T. Penzl, LYAPACK: A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems, Users' Guide (Version 1. 0), 1999. |
[22] |
S. Prajna, A. van der Schaft and G. Meinsma,
An LMI approach to stabilization of linear port-controlled Hamiltonian systems, Systems & Control Letters, 45 (2002), 371-385.
doi: 10.1016/S0167-6911(01)00195-5. |
[23] |
S. Prajna, POD model reduction with stability guarantee, in: Proceedings of 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, December 2003, 5254-5258. |
[24] |
R. Pulch,
Model order reduction and low-dimensional representations for random linear dynamical systems, Math. Comput. Simulat., 144 (2018), 1-20.
doi: 10.1016/j.matcom.2017.05.007. |
[25] |
R. Pulch and F. Augustin, Stability preservation in stochastic Galerkin projections of dynamical systems preprint, arXiv: 1708:00958. |
[26] |
W. H. A. Schilders, M. A. van der Vorst and J. Rommes (eds. ), Model Order Reduction: Theory, Research Aspects and Applications, Mathematics in Industry, Vol. 13, Springer, 2008.
doi: 10.1007/978-3-540-78841-6_1. |
[27] |
R. Seydel, Practical Bifurcation and Stability Analysis, 3rd edition, Springer, 2010.
doi: 10.1007/978-1-4419-1740-9. |
[28] | |
[29] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd edition, Springer, New York, 2002.
doi: 10.1007/978-0-387-21738-3. |
[30] |
T. Wolf, H. Panzer and B. Lohmann,
Model order reduction by approximate balanced truncation: a unifying framework, Automatisierungstechnik, 61 (2013), 545-556.
|
[31] |
D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010. |
show all references
References:
[1] |
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM Publications, 2005.
doi: 10.1137/1.9780898718713. |
[2] |
Z. Bai and R. Freund,
A partial Padé-via-Lanczos method for reduced order modeling, Linear Algebra Appl., 332-334 (2001), 139-164.
doi: 10.1016/S0024-3795(00)00291-3. |
[3] |
P. Benner, S. Gugercin and K. Willcox,
A survey of projection-based model order reduction methods for parametric dynamical systems, SIAM Review, 57 (2015), 483-531.
doi: 10.1137/130932715. |
[4] |
P. Benner, V. Mehrmann and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, Lecture Notes in Compuational Science and Engineering, Vol. 45, Springer, 2005.
doi: 10.1007/3-540-27909-1. |
[5] |
M. Braun, Differential Equations and Their Applications, 3rd edition, Springer, 1983. |
[6] |
A. Castagnotto, M. Cruz Varona, L. Jeschek and B. Lohmann,
sss & sssMOR: Analysis and reduction of large-scale dynamic systems in MATLAB, Automatisierungstechnik, 65 (2017), 134-150.
|
[7] |
R. Castañé Selga, B. Lohmann and R. Eid,
Stability preservation in projection-based model order reduction of large scale systems, Eur. J. Control, 18 (2012), 122-132.
doi: 10.3166/ejc.18.122-132. |
[8] |
J. Ding and G. Yao,
The eigenvalue problem of a specially updated matrix, Appl. Math. Comput., 185 (2007), 415-420.
doi: 10.1016/j.amc.2006.07.040. |
[9] |
R. Freund,
Model reduction methods based on Krylov subspaces, Acta Numerica, 12 (2003), 267-319.
doi: 10.1017/S0962492902000120. |
[10] |
M. I. Gil', Explicit Stability Conditions for Continuous Systems: A Functional Analytic Approach, Springer, 2005.
doi: 10.1007/b99808. |
[11] |
S. Gugercin and A. C. Antoulas,
A survey of model reduction by balanced truncation and some new results, Int. J. Control, 77 (2004), 748-766.
doi: 10.1080/00207170410001713448. |
[12] |
J. K. Hale, E. F. Infante and F. S. P. Tsen,
Stability in linear delay systems, J. Math. Anal. Appl., 115 (1985), 533-555.
doi: 10.1016/0022-247X(85)90068-X. |
[13] |
S. J. Hammarling,
Numerical solution of stable non-negative definite Lyapunov equation, IMA J. Numer. Anal., 2 (1982), 303-323.
doi: 10.1093/imanum/2.3.303. |
[14] |
T. C. Ionescu and A. Astolfi, On moment matching with preservation of passivity and stability, in: 49th IEEE Conference on Decision and Control, (2010), 6189-6194. |
[15] |
B. Kramer and J. R. Singler,
A POD projection method for large-scale algebraic Riccati equations, Numer. Algebra Contr. Optim., 6 (2016), 413-435.
doi: 10.3934/naco.2016018. |
[16] |
J.-R. Li and J. White,
Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. & Appl., 24 (2002), 260-280.
doi: 10.1137/S0895479801384937. |
[17] |
B. Lohmann and R. Eid, Efficient order reduction of parametric and nonlinear models by superposition of locally reduced models in: Methoden und Anwendungen der Regelungstechnik (eds. G. Roppenecker and B. Lohmann), Shaker, 2009. |
[18] |
MATLAB, version 9. 1. 0. 441655 (R2016b), The Mathworks Inc., Natick, Massachusetts, 2016. |
[19] |
"MOR Wiki", online document, https://morwiki.mpi-magdeburg.mpg.de/morwiki Cited May 4, 2018. |
[20] |
P. C. Müller,
Modified Lyapunov equations for LTI descriptor systems, J. Braz. Soc. Mech. Sci. & Eng., 28 (2006), 448-452.
|
[21] |
T. Penzl, LYAPACK: A MATLAB Toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems, Users' Guide (Version 1. 0), 1999. |
[22] |
S. Prajna, A. van der Schaft and G. Meinsma,
An LMI approach to stabilization of linear port-controlled Hamiltonian systems, Systems & Control Letters, 45 (2002), 371-385.
doi: 10.1016/S0167-6911(01)00195-5. |
[23] |
S. Prajna, POD model reduction with stability guarantee, in: Proceedings of 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, December 2003, 5254-5258. |
[24] |
R. Pulch,
Model order reduction and low-dimensional representations for random linear dynamical systems, Math. Comput. Simulat., 144 (2018), 1-20.
doi: 10.1016/j.matcom.2017.05.007. |
[25] |
R. Pulch and F. Augustin, Stability preservation in stochastic Galerkin projections of dynamical systems preprint, arXiv: 1708:00958. |
[26] |
W. H. A. Schilders, M. A. van der Vorst and J. Rommes (eds. ), Model Order Reduction: Theory, Research Aspects and Applications, Mathematics in Industry, Vol. 13, Springer, 2008.
doi: 10.1007/978-3-540-78841-6_1. |
[27] |
R. Seydel, Practical Bifurcation and Stability Analysis, 3rd edition, Springer, 2010.
doi: 10.1007/978-1-4419-1740-9. |
[28] | |
[29] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd edition, Springer, New York, 2002.
doi: 10.1007/978-0-387-21738-3. |
[30] |
T. Wolf, H. Panzer and B. Lohmann,
Model order reduction by approximate balanced truncation: a unifying framework, Automatisierungstechnik, 61 (2013), 545-556.
|
[31] |
D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010. |











dimension n | 5440 |
# non-zero entries in A | 25120 |
# non-zero entries in E | 6400 |
spectral abscissa α(E-1A) | -0.0048 |
dimension n | 5440 |
# non-zero entries in A | 25120 |
# non-zero entries in E | 6400 |
spectral abscissa α(E-1A) | -0.0048 |
dimension n | 29008 |
# non-zero entries in A | 201622 |
# non-zero entries in E | 29008 |
spectral abscissa α(E-1A) | -146.3 |
dimension n | 29008 |
# non-zero entries in A | 201622 |
# non-zero entries in E | 29008 |
spectral abscissa α(E-1A) | -146.3 |
input rank qF | number of iterations nit | maximum dimension |
40 | 10 | 11 |
40 | 28 | 12 |
60 | 10 | 20 |
60 | 20 | 18 |
input rank qF | number of iterations nit | maximum dimension |
40 | 10 | 11 |
40 | 28 | 12 |
60 | 10 | 20 |
60 | 20 | 18 |
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