Figure 1.
Mass-spring-damper configuration
-
Previous Article
Local smooth representation of solution sets in parametric linear fractional programming problems
- NACO Home
- This Issue
-
Next Article
Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B
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 |
| [1] |
Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1783-1799. doi: 10.3934/jimo.2019029 |
| [2] |
Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 |
| [3] |
Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020040 |
| [4] |
Russell E. Warren, Stanley J. Osher. Hyperspectral unmixing by the alternating direction method of multipliers. Inverse Problems & Imaging, 2015, 9 (3) : 917-933. doi: 10.3934/ipi.2015.9.917 |
| [5] |
Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139 |
| [6] |
Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 |
| [7] |
Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945 |
| [8] |
Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure & Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934/cpaa.2017012 |
| [9] |
Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349 |
| [10] |
Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 |
| [11] |
M. A. Christou, C. I. Christov. Fourier-Galerkin method for localized solutions of the Sixth-Order Generalized Boussinesq Equation. Conference Publications, 2001, 2001 (Special) : 121-130. doi: 10.3934/proc.2001.2001.121 |
| [12] |
Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020030 |
| [13] |
Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067 |
| [14] |
Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172 |
| [15] |
Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008 |
| [16] |
Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147 |
| [17] |
Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268 |
| [18] |
Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 |
| [19] |
Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 |
| [20] |
Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]