Model Reduction for Second‐Order Dynamical Systems Revisited

Beddig, Rebekka S.; Benner, Peter; Dorschky, Ines; Reis, Timo; Schwerdtner, Paul; Voigt, Matthias; Werner, Steffen W. R.

doi:10.1002/pamm.201900224

Cited by 9 publications

(10 citation statements)

References 22 publications

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“…As the third and final strategy, we will consider a greedy selection of interpolation points based on the H ∞ -norm. Here, we will make use of the ideas developed in [41,42] to use the large-scale sparse H ∞ -norm computation methods from [43][44][45]. The main idea is to choose the next interpolation point, during the iteration, as the frequency where the H ∞ -norm of the error system is attained, i.e., the frequency for which the approximation error attains the maximum.…”

Section: Heuristics For Selecting Interpolation Pointsmentioning

confidence: 99%

Structure-preserving interpolation of bilinear control systems

2021

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In this paper, we extendthe structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output structured bilinear systems. Then, we extend these results to the case of multi-input multi-output structured bilinear systems by matrix interpolation. The effectiveness of our structure-preserving approach is illustrated by means of various numerical examples.

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Section: Heuristics For Selecting Interpolation Pointsmentioning

confidence: 99%

Structure-preserving interpolation of bilinear control systems

2021

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“…Theorem 3.1. Let θ 0 ∈ R n θ be given and assume that G ∈ H nu×nu ∞ and G pH (•, θ) ∈ RH nu×nu ∞ as in (7). Suppose further that for given s 0 ∈ C + , the maximum singular value of G(s 0 ) − G pH (s 0 , θ) is simple and let u ∈ C nu and v ∈ C nu be the corresponding left and right singular vectors, respectively.…”

Section: Parametrized Port-hamiltonian Systemsmentioning

confidence: 99%

“…The fast greedy structure preserving MOR has been presented in [7]. The main idea is to combine the method for H ∞ norm computation from [1,43] with interpolatory MOR, i. e. iteratively compute the H ∞ norm of the difference between the FOM and the ROM and update the ROM such that it interpolates the FOM at the point on the imaginary axis, where the H ∞ norm is attained.…”

Section: Initializationmentioning

confidence: 99%

SOBMOR: Structured Optimization-Based Model Order Reduction

Schwerdtner¹,

Voigt²

2020

Preprint

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Model order reduction (MOR) methods that are designed to preserve structural features of a given full order model (FOM) often suffer from a lower accuracy when compared to their non structure preserving counterparts. In this paper, we present a framework for MOR based on direct parameter optimization. This means that the elements of the system matrices are iteratively varied to minimize an objective functional that measures the difference between the FOM and the reduced order model (ROM). Structural constraints are encoded in the parametrization of the ROM. The method only depends on frequency response data and can thus be applied to a wide range of dynamical systems.We illustrate the effectiveness of our method on a port-Hamiltonian and on a symmetric second order system in a comparison with other structure preserving MOR algorithms. * This work is supported by the German Research Foundation (DFG) within the project VO2243/2-1: "Interpolationsbasierte numerische Algorithmen in der robusten Regelung" and the DFG Cluster of Excellence MATH+ within the project AA4-5: "Energy-based modeling, simulation, and optimization of power systems under uncertainty".

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“…The initialization step is based on a greedy interpolation strategy presented in Beddig et al (2019). In particular, the initial ROM is constructed by iteratively adding imaginary interpolation points at which the H ∞ norm of the current error transfer function is attained.…”

Section: Preliminariesmentioning

confidence: 99%

Adaptive Sampling for Structure Preserving Model Order Reduction of Port-Hamiltonian Systems

Schwerdtner¹,

Voigt²

2021

Preprint

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We present an adaptive sampling strategy for the optimization-based structure preserving model order reduction (MOR) algorithm developed in [Schwerdtner, P. and Voigt, M. (2020). Structure preserving model order reduction by parameter optimization, Preprint arXiv:2011.07567]. This strategy reduces the computational demand and the required a priori knowledge about the given full order model, while at the same time retaining a high accuracy compared to other structure preserving but also unstructured MOR algorithms. A numerical study with a port-Hamiltonian benchmark system demonstrates the effectiveness of our method combined with its new adaptive sampling strategy. We also investigate the distribution of the sample points.

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Model Reduction for Second‐Order Dynamical Systems Revisited

Cited by 9 publications

References 22 publications

Structure-preserving interpolation of bilinear control systems

Structure-preserving interpolation of bilinear control systems

SOBMOR: Structured Optimization-Based Model Order Reduction

Adaptive Sampling for Structure Preserving Model Order Reduction of Port-Hamiltonian Systems

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