Figure 1.
Mass-spring-damper configuration
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March
2019, 9(1): 23-44.
doi: 10.3934/naco.2019003
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.
Keywords:
Dynamical system,
ordinary differential equation,
model order reduction,
Galerkin projection,
asymptotic stability,
Lyapunov equation,
alternating direction implicit method.
Mathematics Subject Classification:
Primary: 65L05, 65F10; Secondary: 34C20, 34D20, 93D20.
Citation:
Roland Pulch. Stability preservation in Galerkin-type projection-based model order reduction. Numerical Algebra, Control & Optimization,
2019, 9
(1)
: 23-44.
doi: 10.3934/naco.2019003
![]()
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. Google Scholar |
| [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.Google Scholar |
| [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.Google Scholar |
| [18] |
MATLAB, version 9. 1. 0. 441655 (R2016b), The Mathworks Inc., Natick, Massachusetts, 2016.Google Scholar |
| [19] |
"MOR Wiki", online document, https://morwiki.mpi-magdeburg.mpg.de/morwiki Cited May 4, 2018.Google Scholar |
| [20] |
P. C. Müller, Modified Lyapunov equations for LTI descriptor systems, J. Braz. Soc. Mech. Sci. & Eng., 28 (2006), 448-452. Google Scholar |
| [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.Google Scholar |
| [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.Google Scholar |
| [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.Google Scholar |
| [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] |
Y. Shmaliy, Continuous-Time Systems, Springer, 2007.Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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.Google Scholar |
| [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.Google Scholar |
| [18] |
MATLAB, version 9. 1. 0. 441655 (R2016b), The Mathworks Inc., Natick, Massachusetts, 2016.Google Scholar |
| [19] |
"MOR Wiki", online document, https://morwiki.mpi-magdeburg.mpg.de/morwiki Cited May 4, 2018.Google Scholar |
| [20] |
P. C. Müller, Modified Lyapunov equations for LTI descriptor systems, J. Braz. Soc. Mech. Sci. & Eng., 28 (2006), 448-452. Google Scholar |
| [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.Google Scholar |
| [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.Google Scholar |
| [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.Google Scholar |
| [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] |
Y. Shmaliy, Continuous-Time Systems, Springer, 2007.Google Scholar |
| [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. Google Scholar |
| [31] |
D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010. |
Figure 2.
Bode plot of stochastic Galerkin system for massspring-damper configuration
Figure 3.
Spectral abscissa of the matrices in the ROMs from conventional system (left) and stabilized system (right)
Figure 4.
Error bound in ${\mathscr{H}_2}$ -norm for the two MOR approaches in mass-spring-damper example
Figure 5.
Schematic of anemometer
Figure 6.
Bode plot of anemometer benchmark
Figure 7.
Output of the anemometer system
Figure 8.
Spectral abscissa of the matrices in the ROMs from conventional system (left) and stabilized system (right)
Figure 9.
Maximum error of ROMs for the output in the time domain concerning anemometer example
Figure 10.
Error bound in ${\mathscr{H}_2}$ -norm for the two MOR approaches in anemometer example
Figure 11.
Condition numbers of reduced mass matrices in anemometer example
Table 1.
Properties of stochastic mass-spring-damper system
| 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 |
Table 2.
Properties of anemometer example
| 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 |
Table 3.
Stability of ROMs for all dimensions $r=1, \ldots, \hat{r}$ using transformation with approximation $M \approx ZZ^\top$ in anemometer example
| 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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