Modified Lyapunov equations for LTI descriptor systems

Müller, Peter C.

doi:10.1590/s1678-58782006000400009

Cited by 9 publications

(8 citation statements)

References 8 publications

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“…The matrix (19) represents a rank-k update of a matrix, where the eigenvalue problem was investigated in [8], for example.…”

Section: Proofmentioning

confidence: 99%

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

Pulch¹

2019

Numerical Algebra, Control &Amp; Optimization

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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 generalise 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.

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“…The matrix (19) represents a rank-k update of a matrix, where the eigenvalue problem was investigated in [8], for example.…”

Section: Proofmentioning

confidence: 99%

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

Pulch¹

2019

Numerical Algebra, Control &Amp; Optimization

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“…Hence we perform the LU-decompositions (26). Efficient numerical methods are available like UMFPACK [11].…”

Section: Numerical Solution Of Linear Systemsmentioning

confidence: 99%

Frequency domain integrals for stability preservation in Galerkin-type projection-based model order reduction

Pulch

2019

International Journal of Control

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We investigate linear dynamical systems consisting of ordinary differential equations with high dimensionality. Model order reduction yields alternative systems of much lower dimensions. However, a reduced system may be unstable, although the original system is asymptotically stable. We consider projection-based model order reduction of Galerkintype. A transformation of the original system ensures that any reduced system is asymptotically stable. This transformation requires the solution of a high-dimensional Lyapunov inequality. We solve this problem using a specific Lyapunov equation. Its solution can be represented as a matrix-valued integral in the frequency domain. Consequently, quadrature rules yield numerical approximations, where large sparse linear systems of algebraic equations have to be solved. We analyse this approach and show a sufficient condition on the error to meet the Lyapunov inequality. Furthermore, this technique is extended to systems of differential-algebraic equations with strictly proper transfer functions by a regularisation. Finally, we present results of numerical computations for high-dimensional examples, which indicate the efficiency of this stability-preserving method.

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“…We apply a regularization of an asymptotically stable DAE system (1), which was also used in [17]. The regularized system matrices read as…”

Section: Regularizationmentioning

confidence: 99%

“…The Lyapunov equations have no solution now. Therefore, we use a regularization technique, which was also employed in [17].…”

Section: Introductionmentioning

confidence: 99%

Stability-preserving model order reduction for linear stochastic Galerkin systems

Pulch

2019

J.Math.Industry

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Mathematical modeling often yields linear dynamical systems in science and engineering. We change physical parameters of the system into random variables to perform an uncertainty quantification. The stochastic Galerkin method yields a larger linear dynamical system, whose solution represents an approximation of stochastic processes. A model order reduction (MOR) of the Galerkin system is advantageous due to the high dimensionality. However, asymptotic stability may be lost in some MOR techniques. In Galerkin-type MOR methods, the stability can be guaranteed by a transformation to a dissipative form. Either the original dynamical system or the stochastic Galerkin system can be transformed. We investigate the two variants of this stability-preserving approach. Both techniques are feasible, while featuring different properties in numerical methods. Results of numerical computations are demonstrated for two test examples modeling a mechanical application and an electric circuit, respectively.

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Modified Lyapunov equations for LTI descriptor systems

Cited by 9 publications

References 8 publications

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

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

Frequency domain integrals for stability preservation in Galerkin-type projection-based model order reduction

Stability-preserving model order reduction for linear stochastic Galerkin systems

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