Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data

Sean Reiter; Steffen W. R. Werner

arxiv: 2606.00298 · v1 · pith:OEGZZ4MSnew · submitted 2026-05-29 · 🧮 math.NA · cs.LG· cs.NA· cs.SY· eess.SY· math.DS· math.OC

Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data

Sean Reiter , Steffen W. R. Werner This is my paper

Pith reviewed 2026-06-28 21:02 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NAcs.SYeess.SYmath.DSmath.OC

keywords balanced truncationHermite interpolationquadraturereduced-order modelinglinear dynamical systemsstability preservationderivative datatransfer function

0 comments

The pith

A symmetric Hermite formulation of quadrature-based balanced truncation builds reduced-order models from transfer function values and derivatives while preserving state-space Hermiticity and asymptotic stability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a new version of quadrature-based balanced truncation that incorporates Hermite interpolation to use both transfer function evaluations and their derivatives. It shows that this symmetric formulation inherits state-space Hermiticity from the data-generating system. Because Hermiticity implies asymptotic stability for linear systems, the reduced models remain stable. The approach targets data-driven reduction for control design where only frequency response data is available rather than the full system matrices. A sympathetic reader would care because stability preservation removes the need for post-processing fixes that can degrade approximation quality.

Core claim

The symmetric Hermite formulation of the quadrature-based balanced truncation algorithm constructs linear reduced-order models from evaluations of the full-order system's transfer function and its derivative at chosen points, and this formulation preserves state-space Hermiticity and consequently asymptotic stability of the system that generated the data.

What carries the argument

Symmetric Hermite quadrature-based balanced truncation, which performs balanced truncation on a quadrature approximation built from symmetric interpolation of both the transfer function and its first derivative.

If this is right

Reduced models obtained this way inherit asymptotic stability directly from the data source without additional enforcement steps.
The method applies to any Hermitian linear system for which transfer function and derivative samples can be computed or measured.
Quadrature weights and nodes can be chosen independently of the system matrices, allowing purely data-driven construction.
The resulting reduced models remain suitable for downstream control tasks that require stability guarantees.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Hermite symmetry idea could be tested on other interpolation-based reduction techniques to check whether stability inheritance generalizes beyond quadrature-based truncation.
Because derivative data improves local approximation quality, the method may yield lower-order models of comparable accuracy compared with value-only sampling for the same computational budget.
Engineering workflows that already collect frequency-response data with automatic differentiation or finite differences could adopt this approach to obtain stable reduced models without extra simulation cost.

Load-bearing premise

The supplied data consists of exact transfer function and derivative values at the chosen points, and the underlying full-order system is Hermitian.

What would settle it

A concrete counter-example in which the reduced model produced by the symmetric Hermite method has at least one eigenvalue with positive real part while the full-order system is asymptotically stable.

read the original abstract

Data-driven reduced-order modeling is an essential component in the computer-aided design of control systems. In this work, we present a novel symmetric Hermite formulation of the quadrature-based balanced truncation algorithm that constructs linear reduced-order models from evaluations of the full-order system's transfer function and its derivative. Significantly, the Hermite formulation preserves desirable qualitative properties of the system used to generate the data, such as state-space Hermiticity and, consequently, asymptotic stability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Symmetric Hermite extension to quadrature balanced truncation claims to inherit stability from exact transfer-function and derivative samples, but the preservation step needs the full derivations to confirm.

read the letter

The core contribution is a symmetric Hermite version of quadrature-based balanced truncation that takes both function values and derivatives of the transfer function and produces a reduced model that stays Hermitian (hence stable) when the original system is. That is the one concrete advance over the earlier quadrature framework.

The paper does a clean job of stating the data setting and the target property. Framing the method around exact samples and structure preservation is straightforward and matches what control engineers often need.

The soft spot is that the abstract alone gives no equations, error analysis, or numerical checks, so it is impossible to verify whether the symmetry step actually enforces Hermiticity under the stated conditions or whether quadrature errors break it. The method also requires the full-order system to be Hermitian and stable to begin with; that is an explicit assumption rather than a hidden flaw, but it narrows the practical scope. No obvious circularity or invented entities appear.

This is aimed at researchers in systems and control who already work with frequency-domain data and balanced truncation. A reader who needs stable reduced models from derivative samples could find the technique useful once the proofs are checked.

It deserves a serious referee to examine the derivations and any experiments. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a symmetric Hermite formulation of the quadrature-based balanced truncation algorithm for constructing reduced-order linear dynamical systems from exact evaluations of the full-order transfer function and its derivative. The central claim is that this symmetric formulation preserves state-space Hermiticity (and thus asymptotic stability) of the underlying system used to generate the data.

Significance. If the preservation property holds under the stated data conditions, the result would be significant for data-driven model reduction in control applications, where inheriting stability from the generating system is a desirable qualitative guarantee not always provided by standard quadrature-based methods. The approach extends existing balanced truncation techniques to derivative data while aiming for structure preservation.

major comments (1)

The manuscript consists only of the abstract; no derivation of the symmetric Hermite quadrature step, no error analysis, and no numerical evidence are provided to support the claim that Hermiticity and stability are preserved. This is load-bearing for the central claim, as the abstract asserts the property without showing the mechanism or conditions under which it holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for explicit support of the central preservation claims. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The manuscript consists only of the abstract; no derivation of the symmetric Hermite quadrature step, no error analysis, and no numerical evidence are provided to support the claim that Hermiticity and stability are preserved. This is load-bearing for the central claim, as the abstract asserts the property without showing the mechanism or conditions under which it holds.

Authors: We agree with this assessment of the current manuscript version. The provided text is limited to the abstract and does not contain the requested derivation of the symmetric Hermite quadrature step, error analysis, or numerical evidence. In the revised manuscript we will add a dedicated section deriving the symmetric formulation and proving that it preserves state-space Hermiticity (hence asymptotic stability) when the data are exact evaluations of a Hermitian full-order transfer function and its derivative. We will also include an a priori error analysis relating the reduced-order model to the full-order system under the stated data conditions, together with numerical experiments on standard benchmark systems that demonstrate the preservation property in practice. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and available description present a symmetric Hermite quadrature-based balanced truncation method that constructs reduced models from exact transfer-function and derivative evaluations while inheriting state-space Hermiticity and stability from the generating system. No equations, fitted parameters renamed as predictions, self-citations used as load-bearing uniqueness theorems, or ansatzes smuggled via prior work are visible. The central claim is conditional on the input data and system satisfying the stated prerequisites (exact samples, Hermitian/stable full-order system), which are external to the derivation rather than self-referential. This is a self-contained algorithmic construction with no reduction of outputs to inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is provided, so the ledger is populated from the stated claims; the method rests on the assumption that transfer-function and derivative data are available and that the full-order system belongs to the Hermitian class.

axioms (2)

domain assumption The full-order system is linear time-invariant and its transfer function evaluations plus derivatives are available at chosen frequencies or points.
Stated in the abstract as the input data source for the algorithm.
domain assumption The underlying system satisfies state-space Hermiticity so that the reduced model can inherit it.
Invoked when the abstract claims preservation of Hermiticity and consequent stability.

pith-pipeline@v0.9.1-grok · 5616 in / 1315 out tokens · 19178 ms · 2026-06-28T21:02:58.152701+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 9 canonical work pages · 1 internal anchor

[1]

A. C. Antoulas.Approximation of Large-Scale Dynamical Systems, volume 6 ofAdv. Des. Control. SIAM, Philadelphia, PA, 2005.doi:10.1137/1.9780898718713

work page doi:10.1137/1.9780898718713 2005
[2]

A. C. Antoulas, C. A. Beattie, and S. Gugercin.Interpolatory Methods for Model Reduction. Computational Science & Engineering. SIAM, Philadelphia, PA, 2020. doi:10.1137/1.9781611976083

work page doi:10.1137/1.9781611976083 2020
[3]

Benner and J

P. Benner and J. Saak. Linear-quadratic regulator design for optimal cooling of steel pro- files. Technical Report SFB393/05-05, Sonderforschungsbereich 393Parallele Numerische Simulation f¨ ur Physik und Kontinuumsmechanik, TU Chemnitz, D-09107 Chemnitz (Ger- many), 2005. URL:http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597

2005
[4]

Benner, S

P. Benner, W. Schilders, S. Grivet-Talocia, A. Quarteroni, G. Rozza, and L. M. Silveira. Model Order Reduction. Volume 1: System- and Data-Driven Methods and Algorithms. De Gruyter, Berlin, Boston, 2021.doi:10.1515/9783110498967. Preprint. 2026-05-29 S. Reiter, S. W. R. Werner: Symmetric Hermite quadrature-based balanced truncation14

work page doi:10.1515/9783110498967 2021
[5]

D. Billger. The butterfly gyro. In P. Benner, V. Mehrmann, and D. C. Sorensen, editors, Dimension Reduction of Large-Scale Systems, volume 45 ofLect. Notes Comput. Sci. Eng., pages 349–352. Springer, Berlin, Heidelberg, 2005.doi:10.1007/3-540-27909-1_ 18

work page doi:10.1007/3-540-27909-1_ 2005
[6]

G. H. Golub and C. F. Van Loan.Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, fourth edition, 2013

2013
[7]

I. V. Gosea, S. Gugercin, and C. Beattie. Data-driven balancing of linear dynamical systems.SIAM J. Sci. Comput., 44(1):A554–A582, 2022.doi:10.1137/21M1411081

work page doi:10.1137/21m1411081 2022
[8]

B. C. Moore. Principal component analysis in linear systems: controllability, observ- ability, and model reduction.IEEE Trans. Autom. Control, AC–26(1):17–32, 1981. doi:10.1109/TAC.1981.1102568

work page doi:10.1109/tac.1981.1102568 1981
[9]

C. T. Mullis and R. A. Roberts. Synthesis of minimum roundoff noise fixed point dig- ital filters.IEEE Trans. Circuits Syst., 23(9):551–562, 1976.doi:10.1109/TCS.1976. 1084254

work page doi:10.1109/tcs.1976 1976
[10]

Butterfly gyroscope

Oberwolfach Benchmark Collection. Butterfly gyroscope. hosted at MORwiki – Model Order Reduction Wiki, 2004. URL:http://modelreduction.org/index.php/ Butterfly_Gyroscope

2004
[11]

Steel profile

Oberwolfach Benchmark Collection. Steel profile. hosted at MORwiki – Model Order Reduction Wiki, 2005. URL:https://modelreduction.org/morwiki/Steel_Profile

2005
[12]

Data-driven balanced truncation for second-order systems with generalized proportional damping

S. Reiter and S. W. R. Werner. Data-driven balanced truncation for second-order systems with generalized proportional damping. e-print 2506.10118, arXiv, 2025. Numerical Analysis (math.NA).doi:10.48550/arXiv.2506.10118

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2506.10118 2025
[13]

Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data

S. Reiter and S. W. R. Werner. Code, data and results for numerical experiments in “Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data” (version 1.0), May 2026.doi:10.5281/zenodo.20384015. Preprint. 2026-05-29

work page doi:10.5281/zenodo.20384015 2026

[1] [1]

A. C. Antoulas.Approximation of Large-Scale Dynamical Systems, volume 6 ofAdv. Des. Control. SIAM, Philadelphia, PA, 2005.doi:10.1137/1.9780898718713

work page doi:10.1137/1.9780898718713 2005

[2] [2]

A. C. Antoulas, C. A. Beattie, and S. Gugercin.Interpolatory Methods for Model Reduction. Computational Science & Engineering. SIAM, Philadelphia, PA, 2020. doi:10.1137/1.9781611976083

work page doi:10.1137/1.9781611976083 2020

[3] [3]

Benner and J

P. Benner and J. Saak. Linear-quadratic regulator design for optimal cooling of steel pro- files. Technical Report SFB393/05-05, Sonderforschungsbereich 393Parallele Numerische Simulation f¨ ur Physik und Kontinuumsmechanik, TU Chemnitz, D-09107 Chemnitz (Ger- many), 2005. URL:http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597

2005

[4] [4]

Benner, S

P. Benner, W. Schilders, S. Grivet-Talocia, A. Quarteroni, G. Rozza, and L. M. Silveira. Model Order Reduction. Volume 1: System- and Data-Driven Methods and Algorithms. De Gruyter, Berlin, Boston, 2021.doi:10.1515/9783110498967. Preprint. 2026-05-29 S. Reiter, S. W. R. Werner: Symmetric Hermite quadrature-based balanced truncation14

work page doi:10.1515/9783110498967 2021

[5] [5]

D. Billger. The butterfly gyro. In P. Benner, V. Mehrmann, and D. C. Sorensen, editors, Dimension Reduction of Large-Scale Systems, volume 45 ofLect. Notes Comput. Sci. Eng., pages 349–352. Springer, Berlin, Heidelberg, 2005.doi:10.1007/3-540-27909-1_ 18

work page doi:10.1007/3-540-27909-1_ 2005

[6] [6]

G. H. Golub and C. F. Van Loan.Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, fourth edition, 2013

2013

[7] [7]

I. V. Gosea, S. Gugercin, and C. Beattie. Data-driven balancing of linear dynamical systems.SIAM J. Sci. Comput., 44(1):A554–A582, 2022.doi:10.1137/21M1411081

work page doi:10.1137/21m1411081 2022

[8] [8]

B. C. Moore. Principal component analysis in linear systems: controllability, observ- ability, and model reduction.IEEE Trans. Autom. Control, AC–26(1):17–32, 1981. doi:10.1109/TAC.1981.1102568

work page doi:10.1109/tac.1981.1102568 1981

[9] [9]

C. T. Mullis and R. A. Roberts. Synthesis of minimum roundoff noise fixed point dig- ital filters.IEEE Trans. Circuits Syst., 23(9):551–562, 1976.doi:10.1109/TCS.1976. 1084254

work page doi:10.1109/tcs.1976 1976

[10] [10]

Butterfly gyroscope

Oberwolfach Benchmark Collection. Butterfly gyroscope. hosted at MORwiki – Model Order Reduction Wiki, 2004. URL:http://modelreduction.org/index.php/ Butterfly_Gyroscope

2004

[11] [11]

Steel profile

Oberwolfach Benchmark Collection. Steel profile. hosted at MORwiki – Model Order Reduction Wiki, 2005. URL:https://modelreduction.org/morwiki/Steel_Profile

2005

[12] [12]

Data-driven balanced truncation for second-order systems with generalized proportional damping

S. Reiter and S. W. R. Werner. Data-driven balanced truncation for second-order systems with generalized proportional damping. e-print 2506.10118, arXiv, 2025. Numerical Analysis (math.NA).doi:10.48550/arXiv.2506.10118

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2506.10118 2025

[13] [13]

Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data

S. Reiter and S. W. R. Werner. Code, data and results for numerical experiments in “Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data” (version 1.0), May 2026.doi:10.5281/zenodo.20384015. Preprint. 2026-05-29

work page doi:10.5281/zenodo.20384015 2026