Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Data $H(jω_i)$ to Balanced Truncation Family: A Projection-based Non-intrusive Approach

Umair Zulfiqar

TL;DR

The paper tackles data-driven model order reduction for the balanced truncation (BT) family by developing a projection-based framework that uses transfer function samples on the imaginary axis to implicitly approximate Gramians. It introduces rational interpolation with pole/zero/shifted-pole placement and leverages it to construct non-intrusive, data-driven realizations for standard BT and eight generalizations (including FLBT, TLBT, SWBT, LQG BT, H∞ BT, PRBT, BRBT, and BST) without requiring spectral factor measurements. The key contributions are (i) projection-based Gramian factorization $\mathcal{P} \approx (V\tilde{Z}_p)(V\tilde{Z}_p)^*$ and $\mathcal{Q} \approx (W\tilde{Z}_q)(W\tilde{Z}_q)^*$, (ii) a unified shifted-pole/zero-placement scheme that yields ROMs with prescribed spectral properties, and (iii) numerical evidence on a high-order RLC model showing data-driven ROMs achieve accuracy comparable to intrusive BT across a range of orders and variants, with dominant Hankel-like singular values well captured. This framework eliminates the need for spectral-factor data and explicit Lyapunov/Riccati solves in the reduced models, enabling practical, non-intrusive MOR in control and systems applications.

Abstract

This paper presents data-driven implementations of balanced truncation and several of its generalizations that rely exclusively on transfer function samples on the imaginary axis. Rather than implicitly approximating the Gramians via numerical quadrature, the proposed approach approximates them implicitly through projection. This enables multiple members of the balanced truncation family to be implemented non-intrusively using practically measurable data, without requiring spectral factorizations. Using this projection-based framework, data-driven implementations are developed for standard balanced truncation, frequency-limited balanced truncation, time-limited balanced truncation, self-weighted balanced truncation, LQG balanced truncation, H-infinity balanced truncation, positive-real balanced truncation, bounded-real balanced truncation, and stochastic balanced truncation. Numerical results demonstrate that the proposed non-intrusive implementations achieve performance comparable to their intrusive counterparts and accurately capture the dominant Hankel singular values.

From Data $H(jω_i)$ to Balanced Truncation Family: A Projection-based Non-intrusive Approach

TL;DR

The paper tackles data-driven model order reduction for the balanced truncation (BT) family by developing a projection-based framework that uses transfer function samples on the imaginary axis to implicitly approximate Gramians. It introduces rational interpolation with pole/zero/shifted-pole placement and leverages it to construct non-intrusive, data-driven realizations for standard BT and eight generalizations (including FLBT, TLBT, SWBT, LQG BT, H∞ BT, PRBT, BRBT, and BST) without requiring spectral factor measurements. The key contributions are (i) projection-based Gramian factorization and , (ii) a unified shifted-pole/zero-placement scheme that yields ROMs with prescribed spectral properties, and (iii) numerical evidence on a high-order RLC model showing data-driven ROMs achieve accuracy comparable to intrusive BT across a range of orders and variants, with dominant Hankel-like singular values well captured. This framework eliminates the need for spectral-factor data and explicit Lyapunov/Riccati solves in the reduced models, enabling practical, non-intrusive MOR in control and systems applications.

Abstract

This paper presents data-driven implementations of balanced truncation and several of its generalizations that rely exclusively on transfer function samples on the imaginary axis. Rather than implicitly approximating the Gramians via numerical quadrature, the proposed approach approximates them implicitly through projection. This enables multiple members of the balanced truncation family to be implemented non-intrusively using practically measurable data, without requiring spectral factorizations. Using this projection-based framework, data-driven implementations are developed for standard balanced truncation, frequency-limited balanced truncation, time-limited balanced truncation, self-weighted balanced truncation, LQG balanced truncation, H-infinity balanced truncation, positive-real balanced truncation, bounded-real balanced truncation, and stochastic balanced truncation. Numerical results demonstrate that the proposed non-intrusive implementations achieve performance comparable to their intrusive counterparts and accurately capture the dominant Hankel singular values.
Paper Structure (20 sections, 7 theorems, 156 equations, 10 figures)

This paper contains 20 sections, 7 theorems, 156 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Loewner Framework mayo2007framework
  3. Quadrature-based Balanced Truncation goseaQuad
  4. Unresolved Issues in Data-driven Balanced Truncation Family
  5. Main Work
  6. Rational Interpolation with Desired Properties
  7. Pole Placement
  8. Zero Placement
  9. Shifted Pole Placement
  10. Balanced Truncation
  11. Frequency-limited Balanced Truncation
  12. Time-limited Balanced Truncation
  13. Self-weighted Balanced Truncation
  14. LQG Balanced Truncation
  15. $\mathcal{H}_\infty$ Balanced Truncation
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $(\sigma_1, \dots, \sigma_{n_s})$ be $n_s$ distinct interpolation points, and let $(\lambda_1, \dots, \lambda_{n_s})$ be $n_s$ distinct desired poles such that the two sets have no elements in common. Define Further, assume that the pairs $(-S_v, L_v)$ and $(-S_p, L_v)$ are observable. Let $X_p$ be the unique solution of the Sylvester equation If $V$ is defined as in Kry_V, then the interpol

Figures (10)

  • Figure 1: Performance Comparison between Intrusive and Non-intrusive BT
  • Figure 2: Performance Comparison between Intrusive and Non-intrusive TLBT
  • Figure 3: Performance Comparison between Intrusive and Non-intrusive SWBT
  • Figure 4: Performance Comparison between Intrusive and Non-intrusive LQGBT
  • Figure 5: Performance Comparison between Intrusive and Non-intrusive $\mathcal{H}_\infty$ BT
  • ...and 5 more figures

Theorems & Definitions (17)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 7 more