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arxiv: 2506.10118 · v2 · pith:L4LXPRA5new · submitted 2025-06-11 · 🧮 math.NA · cs.NA· cs.SY· eess.SY· math.DS· math.OC

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

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

Pith reviewed 2026-05-25 07:56 UTC · model grok-4.3

classification 🧮 math.NA cs.NAcs.SYeess.SYmath.DSmath.OC
keywords data-driven model reductionbalanced truncationsecond-order systemsgeneralized proportional dampingquadrature-based methodsstructured reduced-order modelingmodel order reductionsystem identification
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The pith

A data-driven quadrature procedure reformulates position-velocity balanced truncation for second-order systems while preserving generalized proportional damping.

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

The paper develops a method to build reduced-order models for linear systems with second-order time derivatives using only input-output data. It reformulates position-velocity balanced truncation via quadrature and ensures the resulting surrogates keep a generalized proportional damping structure, in which the damping matrix is a linear combination of the mass and stiffness matrices. A separate least-squares step then recovers the two scalar damping coefficients directly from the data. This structured approach matters for control-system design because the reduced models remain physically interpretable and inexpensive to simulate. The work extends an existing quadrature technique from unstructured first-order systems to this second-order setting with the added damping constraint.

Core claim

The central claim is that a quadrature-based data-driven procedure computes position-velocity balanced truncations for second-order systems, yielding surrogates that encode generalized proportional damping; the two damping coefficients can then be recovered by minimizing a least-squares error over those coefficients.

What carries the argument

The quadrature-based data-driven reformulation of position-velocity balanced truncation, which produces reduced models that encode generalized proportional damping and permits least-squares recovery of the damping coefficients.

If this is right

  • The computed reduced models retain both the second-order differential structure and the generalized proportional damping form.
  • Damping coefficients can be inferred from input-output data alone without access to the full system matrices.
  • The procedure generalizes the quadrature-based balanced truncation method from first-order to second-order systems.
  • The resulting surrogates remain suitable for computer-aided control-system design where physical interpretability is required.

Where Pith is reading between the lines

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

  • The same data-driven quadrature step could be adapted to other linear structures if an analogous algebraic constraint on the system matrices exists.
  • Recovering the damping coefficients separately may allow the reduced model to be updated when material properties change without recomputing the entire reduction.
  • Because the method works from data, it could be paired with experimental frequency-response measurements to build structured models directly from hardware tests.

Load-bearing premise

The reduced-order models produced by the quadrature procedure will satisfy or closely approximate the generalized proportional damping structure so that the subsequent least-squares fit for the coefficients remains meaningful and stable.

What would settle it

Numerical tests in which the least-squares residual for the damping coefficients stays large across increasing numbers of quadrature nodes, or in which the reduced models fail to match the original system's input-output map once the damping structure is enforced.

read the original abstract

Structured reduced-order modeling is a central component in the computer-aided design of control systems in which cheap-to-evaluate low-dimensional models with physically meaningful internal structures are computed. In this work, we develop a new approach for the structured data-driven surrogate modeling of linear dynamical systems described by second-order time derivatives via balanced truncation model-order reduction. The proposed method is a data-driven reformulation of position-velocity balanced truncation for second-order systems and generalizes the quadrature-based balanced truncation for unstructured first-order systems to the second-order case. The computed surrogates encode a generalized proportional damping structure, and we propose a computational procedure for inferring the damping coefficients from data by minimizing a least-squares error over the coefficients. Several numerical examples demonstrate the effectiveness of the proposed method.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a data-driven quadrature-based reformulation of position-velocity balanced truncation for second-order systems obeying generalized proportional damping (C = αM + βK). It generalizes the unstructured first-order quadrature BT method to the structured second-order setting, assembles reduced matrices directly from frequency-response samples, asserts that the resulting surrogates preserve the damping structure, and supplies a post-processing least-squares procedure to recover the scalar coefficients α and β from the reduced matrices. Numerical examples are used to illustrate performance.

Significance. If the structure-preservation property holds, the approach would supply a practical, data-driven route to structure-preserving reduced models for second-order systems arising in structural dynamics and vibration control, extending existing quadrature BT techniques while retaining physical interpretability of the damping terms.

major comments (3)
  1. [Abstract, §3] Abstract and §3 (quadrature construction): the claim that the computed surrogates 'encode a generalized proportional damping structure' is load-bearing for the subsequent least-squares inference step, yet the quadrature formulas for the reduced M_r, C_r, K_r are assembled from independent frequency-response samples without an explicit projection or algebraic identity that enforces C_r = α M_r + β K_r when the original system satisfies the relation. No derivation shows exact preservation or quantifies the residual.
  2. [§4] §4 (least-squares coefficient recovery): the normal equations for α, β are formed from the reduced matrices; when the quadrature approximation deviates from exact linear dependence, the Gram matrix can become ill-conditioned. No conditioning analysis, residual bounds, or numerical diagnostics are supplied to confirm that the fit remains stable and meaningful for the data-driven (non-projected) reduced matrices.
  3. [§5] §5 (numerical examples): the reported error tables compare reduced-model frequency responses but do not tabulate the least-squares residual ||C_r - α M_r - β K_r|| or the condition number of the coefficient problem, leaving open whether the inferred coefficients are reliable or merely artifacts of an approximate structure.
minor comments (2)
  1. [§2] Notation for the position-velocity Gramians and the quadrature nodes/weights should be introduced with explicit definitions before their first use in the algorithmic description.
  2. [§5] Figure captions for the frequency-response plots should state the number of quadrature nodes and the frequency range used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate planned revisions to incorporate the requested clarifications and diagnostics.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (quadrature construction): the claim that the computed surrogates 'encode a generalized proportional damping structure' is load-bearing for the subsequent least-squares inference step, yet the quadrature formulas for the reduced M_r, C_r, K_r are assembled from independent frequency-response samples without an explicit projection or algebraic identity that enforces C_r = α M_r + β K_r when the original system satisfies the relation. No derivation shows exact preservation or quantifies the residual.

    Authors: We agree that the quadrature-based assembly does not enforce exact algebraic preservation of C_r = α M_r + β K_r via an explicit identity, as the reduced matrices are formed independently from frequency samples. The approximate structure arises because the original system's frequency response satisfies the GPD relation, and the quadrature inherits this dependence to the accuracy of the rule. In the revision we will add a derivation relating the residual to the quadrature error and supply an a-priori bound on ||C_r − α M_r − β K_r||. revision: partial

  2. Referee: [§4] §4 (least-squares coefficient recovery): the normal equations for α, β are formed from the reduced matrices; when the quadrature approximation deviates from exact linear dependence, the Gram matrix can become ill-conditioned. No conditioning analysis, residual bounds, or numerical diagnostics are supplied to confirm that the fit remains stable and meaningful for the data-driven (non-projected) reduced matrices.

    Authors: We acknowledge that a conditioning analysis of the Gram matrix and residual bounds are missing. The revised manuscript will include a short subsection deriving a condition-number bound in terms of the smallest singular value of the reduced matrices and the quadrature accuracy, together with explicit residual estimates for the least-squares problem. revision: yes

  3. Referee: [§5] §5 (numerical examples): the reported error tables compare reduced-model frequency responses but do not tabulate the least-squares residual ||C_r - α M_r - β K_r|| or the condition number of the coefficient problem, leaving open whether the inferred coefficients are reliable or merely artifacts of an approximate structure.

    Authors: We agree that these quantities should be reported. We will augment the numerical examples with a table (or additional columns) listing, for each test case, the least-squares residual norm and the condition number of the normal-equation matrix, thereby confirming that the recovered coefficients are meaningful. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained.

full rationale

The paper describes a data-driven quadrature-based reformulation of position-velocity balanced truncation for second-order systems that generalizes unstructured first-order quadrature BT. The reduced matrices are assembled directly from frequency-response samples or quadrature weights. A separate post-processing least-squares minimization is then used to infer the two scalar damping coefficients α and β. No equation or step defines the reduced-order model itself in terms of those fitted coefficients, nor does any central claim reduce by construction to a fit or to a self-citation chain. The least-squares step is presented as inference on an already-computed surrogate rather than a definitional identity, and the abstract and method description treat the structure preservation as a property of the construction rather than an assumption that forces the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the target systems admit a generalized proportional damping structure that can be recovered from data via least-squares; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Reduced models obtained by the data-driven quadrature procedure will admit a generalized proportional damping structure (damping matrix is linear combination of mass and stiffness matrices).
    This premise is required for the least-squares inference step to be well-posed and is stated as the encoding property of the surrogates.

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discussion (0)

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