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Efficient Reduction of Interconnected Subsystem Models using Abstracted Environments

Luuk Poort, Bart Besselink, Rob H. B. Fey, Nathan van de Wouw

TL;DR

The paper tackles the computational bottleneck of structure-preserving MOR for interconnected subsystems by introducing two abstracted reduction frameworks that use low-order environment abstractions to preserve external IO behavior. It develops a robust-performance analysis that relates high-level accuracy requirements on the reduced interconnected map $\mathcal{F}_l(S,\hat{\Sigma}_B)$ to low-level bounds on environment/subsystem abstraction and reduction errors, ensuring stability and a prescribed $\mathcal{H}_\infty$-norm bound on the reduction error $\Lambda_C$. Two concrete variants are proposed: (i) environment-abstracted reduction (RAR-$E$) and (ii) subsystem-abstracted reduction (RAR-$\Sigma$), with a practical framework to optimize the low-level specifications via LMIs and convex optimization, aided by weighting functions $V$ and $W$. A wafer-stage structural-dynamics benchmark demonstrates that RAR-$E$ achieves the largest reduction with competitive accuracy relative to the state-of-the-art RSS method, while RAR-$\Sigma$ provides maximal accuracy at higher reduced orders; the results underscore the trade-offs between modularity, reduction extent, and accuracy, governed by the chosen weighting and abstraction strategy.

Abstract

We present two frameworks for structure-preserving model order reduction of interconnected subsystems, improving tractability of the reduction methods while ensuring stability and accuracy bounds of the reduced interconnected model. Instead of reducing each subsystem independently, we take a low-order abstraction of its environment into account to better capture the dynamics relevant to the external input-output behaviour of the interconnected system, thereby increasing accuracy of the reduced interconnected model. This approach significantly reduces the computational costs of reduction by abstracting instead of fully retaining the environment. The two frameworks differ in how they generate these abstracted environments: one abstracts the environment as a whole, whereas the other abstracts each individual subsystem. By relating low-level errors introduced by reduction and abstraction to the resulting high-level error on the interconnected system, we are able to translate high-level accuracy requirements (on the reduced interconnected system) to low-level specifications (on abstraction and reduction errors) using techniques from robust performance analysis. By adhering to these low-level specifications, restricting the introduced low-level errors, both frameworks automatically guarantee the accuracy and stability of the reduced interconnected system. We demonstrate the effectiveness of both frameworks by applying them to a structural dynamics model of a two-stroke wafer stage, achieving improved accuracy and/or greater reduction compared to an existing method from literature.

Efficient Reduction of Interconnected Subsystem Models using Abstracted Environments

TL;DR

The paper tackles the computational bottleneck of structure-preserving MOR for interconnected subsystems by introducing two abstracted reduction frameworks that use low-order environment abstractions to preserve external IO behavior. It develops a robust-performance analysis that relates high-level accuracy requirements on the reduced interconnected map to low-level bounds on environment/subsystem abstraction and reduction errors, ensuring stability and a prescribed -norm bound on the reduction error . Two concrete variants are proposed: (i) environment-abstracted reduction (RAR-) and (ii) subsystem-abstracted reduction (RAR-), with a practical framework to optimize the low-level specifications via LMIs and convex optimization, aided by weighting functions and . A wafer-stage structural-dynamics benchmark demonstrates that RAR- achieves the largest reduction with competitive accuracy relative to the state-of-the-art RSS method, while RAR- provides maximal accuracy at higher reduced orders; the results underscore the trade-offs between modularity, reduction extent, and accuracy, governed by the chosen weighting and abstraction strategy.

Abstract

We present two frameworks for structure-preserving model order reduction of interconnected subsystems, improving tractability of the reduction methods while ensuring stability and accuracy bounds of the reduced interconnected model. Instead of reducing each subsystem independently, we take a low-order abstraction of its environment into account to better capture the dynamics relevant to the external input-output behaviour of the interconnected system, thereby increasing accuracy of the reduced interconnected model. This approach significantly reduces the computational costs of reduction by abstracting instead of fully retaining the environment. The two frameworks differ in how they generate these abstracted environments: one abstracts the environment as a whole, whereas the other abstracts each individual subsystem. By relating low-level errors introduced by reduction and abstraction to the resulting high-level error on the interconnected system, we are able to translate high-level accuracy requirements (on the reduced interconnected system) to low-level specifications (on abstraction and reduction errors) using techniques from robust performance analysis. By adhering to these low-level specifications, restricting the introduced low-level errors, both frameworks automatically guarantee the accuracy and stability of the reduced interconnected system. We demonstrate the effectiveness of both frameworks by applying them to a structural dynamics model of a two-stroke wafer stage, achieving improved accuracy and/or greater reduction compared to an existing method from literature.
Paper Structure (22 sections, 7 theorems, 53 equations, 12 figures, 2 tables)

This paper contains 22 sections, 7 theorems, 53 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Interconnected systems
  4. Structure-preserving model reduction
  5. Abstracted reduction
  6. Problem statement
  7. Abstracted reduction of interconnected systems
  8. Error analysis and the relation of error bounds
  9. Error relations
  10. Direct environment abstraction
  11. Subsystem abstraction
  12. A robust performance perspective on error analysis
  13. Optimizing accuracy specifications for robust abstracted reduction
  14. Optimized low-level accuracy specifications
  15. Robust abstracted reduction
  16. ...and 7 more sections

Key Result

Lemma 1

Consider the $p_j\times m_j$ transfer matrix $\Sigma^{j}$ and $(p_C+m_j)\times(m_C+p_j)$ transfer matrix $\check{E}^{j}$, such that $\mathcal{F}_l(\check{E}^{j},\Sigma^{j})$ is well-posed and internally stable. Then, for any $\check{F}^{j}$ as in eq:F_defs, with weighting matrices $G_u^{j}\in\mathbb

Figures (12)

  • Figure 1: Approaches to interconnected model reduction: (a) closed-loop reduction of the interconnected model, (b) open-loop reduction of independent subsystem models and (c) structure-preserving reduction.
  • Figure 2: (a) Lower LFT of $S(s)$ and $\Sigma_B(s)$, constituting the interconnected model $\mathcal{F}_l(S,\Sigma_B)$ and (b) lower LFT of $S(s)$ and $\hat{\Sigma}_B(s)$, constituting the reduced, interconnected model $\mathcal{F}_l(S,\hat{\Sigma}_B)$.
  • Figure 3: (a) Lower LFT of $E^{j}$ and $\Sigma^{j}$, constituting the interconnected model $\mathcal{F}_l(E^{j},\Sigma^{j}) = \mathcal{F}_l(S,\Sigma_B)$ and (b) lower LFT of $S$ and all $\Sigma^{l}$ for $l \in \{1,\dots,k\}\backslash \{j\}$, constituting $E^{j}$.
  • Figure 4: Abstracted reduction using (a) environment abstraction (\ref{['alg:env_absred']}), and (b) using subsystems abstraction (\ref{['alg:ss_absred']}).
  • Figure 5: Schematic representation of $\hat{\Sigma}_B\coloneqq \mathop{\mathrm{diag}}\nolimits(\hat{\Sigma}^{1},\dots,\hat{\Sigma}^{k})$, as in \ref{['eq:Eclem_sysrb']}.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 10 more