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Abstracted Model Reduction: A General Framework for Efficient Interconnected System Reduction

Luuk Poort, Lars A. L. Janssen, Bart Besselink, Rob H. B. Fey, Nathan van de Wouw

TL;DR

The concept of abstracted model reduction is introduced: a framework to improve the tractability of structure-preserving methods for the complexity reduction of interconnected system models and a systematic approach is formulated to automatically determine sufficient abstraction and reduction orders to preserve stability and guarantee a given frequency-dependent error specification.

Abstract

This paper introduces the concept of abstracted model reduction: a framework to improve the tractability of structure-preserving methods for the complexity reduction of interconnected system models. To effectively reduce high-order, interconnected models, it is usually not sufficient to consider the subsystems separately. Instead, structure-preserving reduction methods should be employed, which consider the interconnected dynamics to select which subsystem dynamics to retain in reduction. However, structure-preserving methods are often not computationally tractable. To overcome this issue, we propose to connect each subsystem model to a low-order abstraction of its environment to reduce it both effectively and efficiently. By means of a high-fidelity structural-dynamics model from the lithography industry, we show, on the one hand, significantly increased accuracy with respect to standard subsystem reduction and, on the other hand, similar accuracy to direct application of expensive structure-preserving methods, while significantly reducing computational cost. Furthermore, we formulate a systematic approach to automatically determine sufficient abstraction and reduction orders to preserve stability and guarantee a given frequency-dependent error specification. We apply this approach to the lithography equipment use case and show that the environment model can indeed be reduced by over 80\% without significant loss in the accuracy of the reduced interconnected model.

Abstracted Model Reduction: A General Framework for Efficient Interconnected System Reduction

TL;DR

The concept of abstracted model reduction is introduced: a framework to improve the tractability of structure-preserving methods for the complexity reduction of interconnected system models and a systematic approach is formulated to automatically determine sufficient abstraction and reduction orders to preserve stability and guarantee a given frequency-dependent error specification.

Abstract

This paper introduces the concept of abstracted model reduction: a framework to improve the tractability of structure-preserving methods for the complexity reduction of interconnected system models. To effectively reduce high-order, interconnected models, it is usually not sufficient to consider the subsystems separately. Instead, structure-preserving reduction methods should be employed, which consider the interconnected dynamics to select which subsystem dynamics to retain in reduction. However, structure-preserving methods are often not computationally tractable. To overcome this issue, we propose to connect each subsystem model to a low-order abstraction of its environment to reduce it both effectively and efficiently. By means of a high-fidelity structural-dynamics model from the lithography industry, we show, on the one hand, significantly increased accuracy with respect to standard subsystem reduction and, on the other hand, similar accuracy to direct application of expensive structure-preserving methods, while significantly reducing computational cost. Furthermore, we formulate a systematic approach to automatically determine sufficient abstraction and reduction orders to preserve stability and guarantee a given frequency-dependent error specification. We apply this approach to the lithography equipment use case and show that the environment model can indeed be reduced by over 80\% without significant loss in the accuracy of the reduced interconnected model.

Paper Structure

This paper contains 21 sections, 7 theorems, 44 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Interconnected system representation
  4. Structure-preserving model reduction
  5. Problem statement
  6. Abstracted reduction framework
  7. The abstracted reduction algorithm
  8. Computational benefits and applicability
  9. Error analysis and the relation of bounds
  10. Error relations
  11. A robust performance perspective
  12. A priori specifications for abstracted reduction
  13. The robust abstracted reduction framework for stability and accuracy guarantees
  14. Frequency-based robust abstracted reduction
  15. Computational considerations and tuning parameters
  16. ...and 6 more sections

Key Result

Lemma 1

Consider the $p\times m$ transfer matrix $\Sigma(s)$ and $(p_C+m)\times(m_C+p)$ transfer matrix $\hat{E}(s)$, such that $\mathcal{F}_l(\hat{E},\Sigma)$ is well-posed and internally stable. Then, for any $\hat{F}(s)$ in eq:F_defs, with weighting matrices $G_y\in\mathbb{C}^{p\times p}$ and $G_u\in\mat

Figures (13)

  • Figure 1: The interconnected system as a) an interconnection of three subsystems, and b) an interconnection of a single system and its environment.
  • Figure 2: Three modular approaches for the reduction of a system within a dynamic environment: a) Open-loop (independent) reduction of the system model, b) Closed-loop (structure-preserving) reduction of the interconnection of the system and its environment, and c) Abstracted model reduction, where the system is reduced in connection to an abstracted environment model.
  • Figure 3: (a) Lower LFT of $E(s)$ and $\Sigma(s)$, constituting the interconnected model $\mathcal{F}_l(E,\Sigma)$ and (b) lower LFT of $E(s)$ and $\hat{\Sigma}(s)$, constituting the reduced, interconnected model $\mathcal{F}_l(E,\hat{\Sigma})$.
  • Figure 4: Schematic representation of the steps of abstracted reduction.
  • Figure 5: Computational cost reduction per the amount of reduction of $E(s)$, assuming negligible abstraction cost and cubic scaling of the reduction's cost. The results are given for three different relative orders of $E(s)$ and $\Sigma(s)$.
  • ...and 8 more figures

Theorems & Definitions (24)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 14 more