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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
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##### References:
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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