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A modified AAA algorithm for learning stable reduced-order models from data

Tommaso Bradde, Stefano Grivet-Talocia, Quirin Aumann, Ion Victor Gosea

TL;DR

This work introduces stabAAA, a stability-certified variant of the Adaptive AAA algorithm for data-driven, reduced-order modeling of large-scale LTI systems. By deriving an algebraic link between the barycentric denominator D(s) and stability, the authors constrain the AAA optimization to ensure asymptotic stability, using a convex SDP-based relaxation that embeds the stability requirements into the iterative process. The resulting approach retains AAA’s automatic order selection and interpolation capabilities while guaranteeing that the reduced transfer function hat{H}(s) has poles strictly in the left half-plane; stability is enforced either exactly or via a convex surrogate when needed. Numerical experiments on MOR benchmarks from PCB interconnects, ISS 1R models, and acoustic absorbers show stabAAA delivering competitive or superior accuracy compared to Vector Fitting and other AAA-based methods, with the added benefit of built-in stability guarantees. The work also discusses limitations, such as potential accuracy degradation in highly constrained cases and the practical need to adapt model order or tightening tolerances to achieve desired errors in the presence of non-ideal data.

Abstract

In recent years, the Adaptive Antoulas-Anderson AAA algorithm has established itself as the method of choice for solving rational approximation problems. Data-driven Model Order Reduction (MOR) of large-scale Linear Time-Invariant (LTI) systems represents one of the many applications in which this algorithm has proven to be successful since it typically generates reduced-order models (ROMs) efficiently and in an automated way. Despite its effectiveness and numerical reliability, the classical AAA algorithm is not guaranteed to return a ROM that retains the same structural features of the underlying dynamical system, such as the stability of the dynamics. In this paper, we propose a novel algebraic characterization for the stability of ROMs with transfer function obeying the AAA barycentric structure. We use this characterization to formulate a set of convex constraints on the free coefficients of the AAA model that, whenever verified, guarantee by construction the asymptotic stability of the resulting ROM. We suggest how to embed such constraints within the AAA optimization routine, and we validate experimentally the effectiveness of the resulting algorithm, named stabAAA, over a set of relevant MOR applications.

A modified AAA algorithm for learning stable reduced-order models from data

TL;DR

This work introduces stabAAA, a stability-certified variant of the Adaptive AAA algorithm for data-driven, reduced-order modeling of large-scale LTI systems. By deriving an algebraic link between the barycentric denominator D(s) and stability, the authors constrain the AAA optimization to ensure asymptotic stability, using a convex SDP-based relaxation that embeds the stability requirements into the iterative process. The resulting approach retains AAA’s automatic order selection and interpolation capabilities while guaranteeing that the reduced transfer function hat{H}(s) has poles strictly in the left half-plane; stability is enforced either exactly or via a convex surrogate when needed. Numerical experiments on MOR benchmarks from PCB interconnects, ISS 1R models, and acoustic absorbers show stabAAA delivering competitive or superior accuracy compared to Vector Fitting and other AAA-based methods, with the added benefit of built-in stability guarantees. The work also discusses limitations, such as potential accuracy degradation in highly constrained cases and the practical need to adapt model order or tightening tolerances to achieve desired errors in the presence of non-ideal data.

Abstract

In recent years, the Adaptive Antoulas-Anderson AAA algorithm has established itself as the method of choice for solving rational approximation problems. Data-driven Model Order Reduction (MOR) of large-scale Linear Time-Invariant (LTI) systems represents one of the many applications in which this algorithm has proven to be successful since it typically generates reduced-order models (ROMs) efficiently and in an automated way. Despite its effectiveness and numerical reliability, the classical AAA algorithm is not guaranteed to return a ROM that retains the same structural features of the underlying dynamical system, such as the stability of the dynamics. In this paper, we propose a novel algebraic characterization for the stability of ROMs with transfer function obeying the AAA barycentric structure. We use this characterization to formulate a set of convex constraints on the free coefficients of the AAA model that, whenever verified, guarantee by construction the asymptotic stability of the resulting ROM. We suggest how to embed such constraints within the AAA optimization routine, and we validate experimentally the effectiveness of the resulting algorithm, named stabAAA, over a set of relevant MOR applications.
Paper Structure (27 sections, 6 theorems, 67 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 27 sections, 6 theorems, 67 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and problem statement
  3. Transfer function interpolation, barycentric forms, and the AAA algorithm
  4. The Loewner framework
  5. The Adaptive Antoulas-Anderson (AAA) algorithm
  6. AAA model structure and realization
  7. The real-valued AAA algorithm
  8. Stability of AAA models
  9. Stability characterization and enforcement for AAA models
  10. Stability characterization
  11. Exact stability enforcement
  12. Stability enforcement based on convex optimization
  13. Problem relaxation
  14. The cost function of the relaxed convex problem
  15. A real-valued AAA algorithm with guaranteed stability
  16. ...and 12 more sections

Key Result

Lemma 3.1

As adapted from gosea2020rational, a realization for the transfer function $\hat{H}(s)$ in eq:AAAReal is given by i.e., its transfer function, given by $\tilde{H}(s) = \tilde{{\textbf{C}}} (s \tilde{{\textbf{E}}} - \tilde{{\textbf{A}}})^{-1} \tilde{{\textbf{B}}}$, satisfies the equality $\tilde{H}(s) = \hat{H}(s)$.

Figures (4)

  • Figure 1: Graphical illustration of the various relations between Minimum-Phase (MP), Almost Strictly Positive Real (ASPR), Positive Real (PR), and Strictly Positive Real (SPR) functions. The intersection between PR and the set of Not-MP contains PR functions having zeros on $j\mathbb{R}$.
  • Figure 2: High Speed PCB link test case. Top panel: The plot reports a zoom on the dominant poles identified by the standard AAA algorithm (hollow black circles) and by stabAAA (red asterisks). Middle panel: Magnitude plot of the reference data against the response of the model obtained with stabAAA. Bottom Panel: Comparison between the residual error magnitude shown by ROMs obtained applying different approaches.
  • Figure 3: International Space Station test case. Top panel: zoom on the dominant poles identified by the standard AAA algorithm (hollow black circles) and by the stable AAA (red asterisks). Middle panel: magnitude plot of the reference data against the response of the stabAAA model. Bottom panel: comparison between the residual error magnitude obtained by applying different approaches.
  • Figure 4: Acoustic absorber test case. Top panel: zoom on the dominant poles identified by the standard AAA algorithm (hollow black circles) and by the stable AAA (red asterisks). Middle panel: magnitude plot of the reference data against the response of the stabAAA model. Bottom panel: comparison between the residual error of the ROMs.

Theorems & Definitions (13)

  • Lemma 3.1
  • Definition 4.1: Relative Degree
  • Definition 4.2: Minimum Phase ilchmann1993non
  • Proposition 4.1
  • proof
  • Definition 4.3: Positive Real (PR) function brogliato2007dissipative
  • Definition 4.4: Strictly Positive Real (SPR) function brogliato2007dissipative
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • ...and 3 more