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# Stability preservation in Galerkin-type projection-based model order reduction

• 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.

Mathematics Subject Classification: Primary: 65L05, 65F10; Secondary: 34C20, 34D20, 93D20.

 Citation: • • Figure 1.  Mass-spring-damper configuration

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

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

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 $\hat r$ 40 10 11 40 28 12 60 10 20 60 20 18
• Figures(11)

Tables(3)

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