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Interpolatory model reduction of dynamical systems with root mean squared error

Sean Reiter, Steffen W. R. Werner

TL;DR

This paper addresses efficient reduced-order modeling for the root mean squared error (RMS) of large-scale dynamical systems by formulating RMS as a quadratic-output observable. It develops an interpolatory projection framework for linear quadratic-output (QO) systems in the frequency domain, deriving Hermite and tangential interpolation conditions and practical construction via projection spaces. Numerical experiments on a plate with tuned vibration absorbers demonstrate that interpolation-based QO MOR, particularly with full nonlinear output information, significantly improves accuracy over traditional linear-output reductions, though convergence and TVA-frequency challenges remain. The work advances RMS-focused surrogates with potential impact on structural dynamics, vibro-acoustics, and quadratic regulator problems by enabling accurate, low-cost RMS predictions from reduced-order models.

Abstract

The root mean squared error is an important measure used in a variety of applications such as structural dynamics and acoustics to model averaged deviations from standard behavior. For large-scale systems, simulations of this quantity quickly become computationally prohibitive. Classical model order reduction techniques attempt to resolve this issue via the construction of surrogate models that emulate the root mean squared error measure using an intermediate linear system. However, this approach requires a potentially large number of linear outputs, which can be disadvantageous in the design of reduced-order models. In this work, we consider directly the root mean squared error as the quantity of interest using the concept of quadratic-output models and propose several new model reduction techniques for the construction of appropriate surrogates. We test the proposed methods on a model for the vibrational response of a plate with tuned vibration absorbers.

Interpolatory model reduction of dynamical systems with root mean squared error

TL;DR

This paper addresses efficient reduced-order modeling for the root mean squared error (RMS) of large-scale dynamical systems by formulating RMS as a quadratic-output observable. It develops an interpolatory projection framework for linear quadratic-output (QO) systems in the frequency domain, deriving Hermite and tangential interpolation conditions and practical construction via projection spaces. Numerical experiments on a plate with tuned vibration absorbers demonstrate that interpolation-based QO MOR, particularly with full nonlinear output information, significantly improves accuracy over traditional linear-output reductions, though convergence and TVA-frequency challenges remain. The work advances RMS-focused surrogates with potential impact on structural dynamics, vibro-acoustics, and quadratic regulator problems by enabling accurate, low-cost RMS predictions from reduced-order models.

Abstract

The root mean squared error is an important measure used in a variety of applications such as structural dynamics and acoustics to model averaged deviations from standard behavior. For large-scale systems, simulations of this quantity quickly become computationally prohibitive. Classical model order reduction techniques attempt to resolve this issue via the construction of surrogate models that emulate the root mean squared error measure using an intermediate linear system. However, this approach requires a potentially large number of linear outputs, which can be disadvantageous in the design of reduced-order models. In this work, we consider directly the root mean squared error as the quantity of interest using the concept of quadratic-output models and propose several new model reduction techniques for the construction of appropriate surrogates. We test the proposed methods on a model for the vibrational response of a plate with tuned vibration absorbers.
Paper Structure (11 sections, 1 theorem, 39 equations, 2 figures, 2 tables)

This paper contains 11 sections, 1 theorem, 39 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Vibrations of a plate with tuned vibration absorbers
  3. Quadratic-output systems in time domain
  4. Interpolation of linear quadratic-output systems
  5. Projection-based model order reduction
  6. Interpolatory model reduction for linear systems
  7. Interpolation in the quadratic-output setting
  8. Numerical experiments
  9. Experimental setup
  10. Discussion
  11. Conclusions

Key Result

Theorem 1

Let $H\colon \mathfrak{i}\mathbb{R} \to \mathbb{C}$ be the transfer function eqn:qotf and $\skew2\widehat{H}\colon \mathfrak{i}\mathbb{R} \to \mathbb{C}$ be the reduced-order transfer function eqn:qotf_red obtained via projection eqn:qosys_red, both restricted to the imaginary axis. Consider interpo for $i = 1, \ldots, k$. Then, for $i = 1, \ldots, k$, the following interpolation conditions hold:

Figures (2)

  • Figure 1: Visual sketch of a plate equipped with TVAs AumW23.
  • Figure 2: Frequency response results for reduced-order models of orders $r = 50$ and $r=75$: All proposed methods provide reasonably accurate approximations of the original system behavior. The interpolation methods that take the output equation into account provide visibly more accurate results than the classical Galerkin-based methods. $\mathsf{LQO\hbox{-}IRKA}$ performs multiple orders of magnitude worse for $r=75$ compared to the other methods due to the method not converging in the prescribed amount of iteration steps.

Theorems & Definitions (2)

  • Theorem 1
  • proof