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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

Received  November 2017 Revised  May 2018 Published  October 2018

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.

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. [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] Y. Shmaliy, Continuous-Time Systems, Springer, 2007. [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] Y. Shmaliy, Continuous-Time Systems, Springer, 2007. [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.
Mass-spring-damper configuration
Bode plot of stochastic Galerkin system for massspring-damper configuration
Spectral abscissa of the matrices in the ROMs from conventional system (left) and stabilized system (right)
Error bound in ${\mathscr{H}_2}$-norm for the two MOR approaches in mass-spring-damper example
Schematic of anemometer
Bode plot of anemometer benchmark
Output of the anemometer system
Spectral abscissa of the matrices in the ROMs from conventional system (left) and stabilized system (right)
Maximum error of ROMs for the output in the time domain concerning anemometer example
Error bound in ${\mathscr{H}_2}$-norm for the two MOR approaches in anemometer example
Condition numbers of reduced mass matrices in anemometer example
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
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
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 $\hat r$ 40 10 11 40 28 12 60 10 20 60 20 18
 input rank qF number of iterations nit maximum dimension $\hat r$ 40 10 11 40 28 12 60 10 20 60 20 18
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