Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data-driven Implementations of Various Generalizations of Balanced Truncation

Umair Zulfiqar, Qiu-Yan Song, Zhi-Hua Xiao, Victor Sreeram

TL;DR

The paper develops non-intrusive, data-driven extensions of balanced truncation to a broad family of BT generalizations by leveraging interpolation-based Loewner and RADI/ADI techniques. It shows how transfer-function samples alone can yield LQG-BT, H∞-BT, PR-BT, BR-BT, SW-BT, and BST ROMs that preserve key properties (positive-realness, bounded-realness, minimum-phase) and can be used for controller design without access to the original state-space realization. Central contributions include theoretical results linking DD interpolation to RADI/Lyapunov-based Gramian approximations, practical non-intrusive implementations, and numerical validation on a high-order benchmark demonstrating accuracy on par with intrusive methods. The work provides a practical pathway for data-driven MOR and controller synthesis for unknown plants using only frequency-domain measurements or impulse responses. Overall, it extends the reach of non-intrusive BT to a broad set of system classes with validated performance gains.

Abstract

There exist two main approaches for non-intrusive implementations of approximate balanced truncation within the Loewner framework: the quadrature-based method [1] and the Alternating Direction Implicit (ADI)-based method [2]. Both approaches rely solely on samples of the transfer function to construct truncated balanced models, eliminating the need for access to the original model's statespace realization. Recently, the quadrature-based approach has been extended to various generalizations of balanced truncation, including positive-real balanced truncation, bounded-real balanced truncation, and balanced stochastic truncation. While this extension [3] is theoretically non-intrusive-meaning it does not require the original state-space realization-it depends on samples of spectral factorizations of the transfer function. Since practical methods for obtaining such samples are currently unavailable, this extension remains largely a theoretical contribution. In this work, we present a non-intrusive ADI-type framework for these generalized balanced truncation methods that requires only samples of the original transfer function for implementation.

Data-driven Implementations of Various Generalizations of Balanced Truncation

TL;DR

The paper develops non-intrusive, data-driven extensions of balanced truncation to a broad family of BT generalizations by leveraging interpolation-based Loewner and RADI/ADI techniques. It shows how transfer-function samples alone can yield LQG-BT, H∞-BT, PR-BT, BR-BT, SW-BT, and BST ROMs that preserve key properties (positive-realness, bounded-realness, minimum-phase) and can be used for controller design without access to the original state-space realization. Central contributions include theoretical results linking DD interpolation to RADI/Lyapunov-based Gramian approximations, practical non-intrusive implementations, and numerical validation on a high-order benchmark demonstrating accuracy on par with intrusive methods. The work provides a practical pathway for data-driven MOR and controller synthesis for unknown plants using only frequency-domain measurements or impulse responses. Overall, it extends the reach of non-intrusive BT to a broad set of system classes with validated performance gains.

Abstract

There exist two main approaches for non-intrusive implementations of approximate balanced truncation within the Loewner framework: the quadrature-based method [1] and the Alternating Direction Implicit (ADI)-based method [2]. Both approaches rely solely on samples of the transfer function to construct truncated balanced models, eliminating the need for access to the original model's statespace realization. Recently, the quadrature-based approach has been extended to various generalizations of balanced truncation, including positive-real balanced truncation, bounded-real balanced truncation, and balanced stochastic truncation. While this extension [3] is theoretically non-intrusive-meaning it does not require the original state-space realization-it depends on samples of spectral factorizations of the transfer function. Since practical methods for obtaining such samples are currently unavailable, this extension remains largely a theoretical contribution. In this work, we present a non-intrusive ADI-type framework for these generalized balanced truncation methods that requires only samples of the original transfer function for implementation.

Paper Structure

This paper contains 17 sections, 7 theorems, 118 equations, 6 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Interpolation Theory beattie2017model
  4. Interpolatory Loewner framework mayo2007framework
  5. Pseudo-optimal Rational Krylov Algorithm (PORK) wolf2014h
  6. Balanced Truncation (BT) Family
  7. Quadrature-based Balanced Truncation (QuadBT) goseaQuad
  8. Data-driven ADI-based Balanced Truncation (DD-ADI-BT) zulfiqar2025non
  9. Main Work
  10. RADI-based Non-intrusive Implementation of LQG-BT
  11. RADI-based Non-intrusive Implementation of PR-BT
  12. RADI-based Non-intrusive Implementation of BR-BT
  13. ADI-based Non-intrusive Implementation of SW-BT
  14. ADI/RADI-based Non-intrusive Implementation of BST
  15. Practical Considerations in Implementation
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $\hat{V}$ be as defined in (Kry_V), with all interpolation points $(\sigma_1, \dots, \sigma_v)$ located in the right half of the $s$-plane. Assume further that the pair $(S_v, L_v)$ is observable and that $Q_v > 0$ uniquely solves the Lyapunov equation: Then, the ROM $\hat{H}(s)$, defined by which interpolates $H(s)$ at $(\sigma_1, \dots, \sigma_v)$, satisfies the following properties:

Figures (6)

  • Figure 1: Performance Comparison between LQG-BT and DD-LQG-BT
  • Figure 2: Performance Comparison between $\mathcal{H}_\infty$-BT and DD-$\mathcal{H}_\infty$-BT
  • Figure 3: Performance Comparison between PR-BT and DD-PR-BT
  • Figure 4: Performance Comparison between BR-BT and DD-BR-BT
  • Figure 5: Performance Comparison between SW-BT and DD-SW-BT
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 1
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 5 more