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$\mathcal{H}_\infty$ model order reduction for quadratic output systems

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Pith's one-line read The paper introduces an H-infinity norm for LTIQO systems as the sum of linear and quadratic transfer-function suprema, proves it is a norm and bounds the L2-to-L2 gain, and gives an optimization-based, nonintrusive model reduction…

desk verdict Genuinely new H-infinity norm for quadratic-output systems and a careful MOR algorithm, but the numerical victory lap is over the norm it optimizes; time-domain results are mixed. read the letter →

arxiv 2505.12529 v2 pith:ZYLTOEWD submitted 2025-05-18 math.OC math.DS

Birgit Hillebrecht , Benjamin Unger This is my paper · ORCID
classification math.OCmath.DS MSC 37J0637M9965P1093A1593B15
keywords H-infinitynormquadraticoutputsystemsmodelorderreductionport-HamiltonianleveledleastsquaresbalancedtruncationH2-optimaltransferfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes a meaningful $\mathcal{H}_\infty$ norm for linear time-invariant systems with quadratic output (LTIQO systems), where the output depends linearly and quadratically on the state and input. The norm is defined as the sum of the largest singular values of the linear transfer function and of the quadratic transfer function, and it is proved to be a norm, to reduce to the classical $\mathcal{H}_\infty$ norm when the quadratic terms vanish, and to bound the $L_2$-to-$L_2$ output gain. On top of this norm, the paper builds a nonintrusive optimization-based model order reduction algorithm that minimizes an $\mathcal{H}_\infty$ error measure between the full-order and reduced-order models, and reports that the resulting reduced models beat balanced truncation and an $\mathcal{H}_2$-focused two-sided iterative algorithm in $\mathcal{H}_\infty$ error. A structure-preserving variant for port-Hamiltonian systems is also given. A sympathetic reader would care because this supplies a worst-case, frequency-domain quality measure for a model class that arises in port-Hamiltonian simulation, optimal control, and stochastic problems, where previously only $\mathcal{H}_2$-type methods were available.

What carries the argument

The central object is the $\mathcal{H}_\infty$ norm of Definition 2.1, written as the sum of two suprema: the largest singular value of the linear transfer function $G_1(s)=C(sI-A)^{-1}B+D$ and the largest singular value of the quadratic transfer function $G_2(s_1,s_2)=\sum_{j=1}^p e_j\otimes \operatorname{vec}(K_j(s_1,s_2))^T$ with $K_j(s_1,s_2)=B^T(s_2I-A)^{-T}M_j(s_1I-A)^{-1}B+P_j$. Theorem 2.2 shows it is a norm, that it reduces to the standard $\mathcal{H}_\infty$ norm for LTI systems, and that it bounds the $L_2$-to-$L_2$ gain. The optimization machinery replaces the nonsmooth problem of minimizing $\|\Sigma-\hat{\Sigma}\|_{\mathcal{H}_\infty}$ by a leveled least squares objective over a parametrized reduced model: for sampled frequency grids, each singular value above a level $\gamma$ is penalized through a ReLU term, with $\gamma$ updated by bisection and frequencies adapted during the iteration. Gradients are derived from singular-value derivatives and, for the quadratic part, from derivatives of $\|K_e(s_1,s_2)\|_F^2$. A condensed parametrization for port-Hamiltonian systems, based on the factorization $\hat{A}=(\hat{J}-\hat{R})\hat{Q}$ and Proposition 3.3, reduces the degrees of freedom and preserves the Hamiltonian structure in the reduced model.

What would settle it

Construct a stable LTIQO system and two reduced models with the same, or smaller, new $\mathcal{H}_\infty$ error but different realized $L_2$ output errors for a concrete input; if the model with smaller $\mathcal{H}_\infty$ error has larger actual $L_2$ error, then minimizing this norm does not necessarily minimize output error. Concretely, compute for a fixed input $u$ the ratio $\|y - \hat{y}\|_2 / (\|u\|_2 + \|u\|_2^2)$ and compare it with $\|\Sigma - \hat{\Sigma}\|_{\mathcal{H}_\infty}$ on a benchmark where the bound is known to be loose; a systematic high-frequency or chirp input sweep would show whether the $\mathcal{H}_\infty$ advantage translates to time-domain performance, since the paper's Figure 4 already reports one case where TSIA has lower $L_2$ error.

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Extended reading notes

Core claim

The paper's central claim is that LTIQO systems admit an $\mathcal{H}_\infty$ norm that is a genuine norm, generalizes the classical LTI norm, and controls time-domain output error, and that this norm can be minimized by a gradient-based algorithm. In detail, Definition 2.1 sets $\|\Sigma\|_{\mathcal{H}_\infty} = \sup_{s\in\mathbb{C}^+} \sigma_{\max}(G_1(s)) + \sup_{s_1,s_2\in\mathbb{C}^+} \sigma_{\max}(G_2(s_1,s_2))$; Theorem 2.2 proves the norm properties and the bound $\|y\|_2 \leq \|G_1\|_{\mathcal{H}_\infty}\|u\|_2 + \|G_2\|_{\mathcal{H}_\infty}\|u\|_2^2$, and Corollary 2.3 transfers the bound to the error system. The optimization algorithm (Algorithm 2) minimizes a leveled least squares surrogate of $\|\Sigma - \hat{\Sigma}\|_{\mathcal{H}_\infty}$, updating the level $\gamma$ by bisection and sampling frequencies adaptively. The authors report numerical results on a mass-spring-damper benchmark in which the method 'clearly outperform[s]' balanced truncation and TSIA in $\mathcal{H}_\infty$ error, including a structure-preserving port-Hamiltonian variant.

Load-bearing premise

The load-bearing premise is that the $\mathcal{H}_\infty$ norm should be the plain sum of the linear and quadratic transfer function peaks; the paper notes in a footnote that no rigorous rationale currently justifies any particular split of the two contributions.

Editorial extensions

If this is right

  • Theorem 2.2 gives every stable LTIQO system a scalar worst-case measure that controls output error, so model reduction results for this class can be compared quantitatively.
  • Algorithm 2 produces reduced models that, on the tested benchmark, achieve smaller $\mathcal{H}_\infty$ error than balanced truncation and TSIA, so frequency-domain worst-case quality is directly optimized rather than approximated.
  • The nonintrusive formulation requires only transfer function evaluations, so the method can be applied to data-driven or black-box settings where full state matrices are unavailable.
  • The condensed port-Hamiltonian parametrization reduces degrees of freedom and, in the numerical example, produces better-conditioned system matrices, making structure-preserving $\mathcal{H}_\infty$ reduction computationally cheaper.
  • Corollary 2.3 implies that a small $\mathcal{H}_\infty$ error guarantees a small output error in an $L_2$ sense up to constants, giving an error certificate for the reduced model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the equal-weight additive norm is a modeling choice; weighting the linear and quadratic parts differently would likely improve time-domain performance on inputs whose energy sits in one frequency regime, a direction the paper explicitly leaves open.
  • The paper's own Figure 4 shows TSIA has lower $L_2$ error for high-frequency and chirp inputs; an editor would infer the claimed superiority is specific to the $\mathcal{H}_\infty$ measure, and the practical gain depends on input spectrum.
  • Testable extension: apply the same norm and algorithm to stochastic variance outputs or linear-quadratic cost outputs, two classes the paper names as natural LTIQO applications; no additional theory appears needed beyond the single-quadratic-output derivative formulas.
  • Editorial inference: because the bound in Theorem 2.2 is not tight, two models with the same $\mathcal{H}_\infty$ norm can have different true $L_2$-to-$L_2$ gains; a data-driven gain computation could rank such models where the norm cannot.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the H-infinity norm and bounds are derived from first principles; the numerical comparison is partly objective-aligned but qualified, and the self-citations are technical background.

full rationale

The core derivation is self-contained. Definition 2.1 defines the LTIQO H-infinity norm as ||G1|| + ||G2|| from the linear and quadratic transfer functions; Theorem 2.2 proves the norm properties, the imaginary-axis reduction, and the L2-gain bound using standard maximum-modulus and Parseval arguments, without assuming the target MOR result. Corollary 2.3 then bounds the output error by the H-infinity error, and Algorithm 2 minimizes a leveled least-squares surrogate built from the same two transfer-function contributions. This means the favorable H-infinity comparison in Figure 1 is partly evaluation on the algorithm's own objective, as the paper itself says ('which is the expected result as we directly minimize the H-infinity norm'), but it is not a definitional equivalence: the surrogate uses relu, finite frequency samples, and an adapted level, while the reported error is the full H-infinity norm. The independent time-domain check in Figure 4 shows TSIA actually wins for high-frequency and chirp inputs, so the superiority claim is not forced. The Section 3.1 footnote admits that no rigorous rationale justifies the specific linear/quadratic partitioning, which is a heuristic limitation, not circularity. The only self-citations ([6], [25]) provide background on pH-Hamiltonian output representation and dissipative-Hamiltonian stability factorization; they are not used as an external uniqueness theorem and do not carry the central claim. No equation was found to reduce to its own input by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The paper's contribution rests on a small set of mathematical assumptions, of which the most notable are the unproven convolutional representation from [11], the standard stability and analyticity assumptions, and the explicitly heuristic additive definition of the H-infinity norm. The free parameters are algorithm hyperparameters chosen for the numerical example, not physical constants. No new physical entities are introduced.

free parameters (3)
  • Level gamma bisection initial range and tolerance = gamma_u,init=100, gamma_l,init=0, epsilon_gamma=0.01
    Hand-chosen for the mass-spring-damper numerical example; controls the target H-infinity error level and termination of the bisection, and therefore the ROM quality.
  • Initial frequency sampling grids Omega1 and Omega2 = Omega2 grid over [-10^2,10^2]^2 with positive frequencies, Omega1 not specified precisely
    Chosen by hand and updated via the adaptive sampling routine from [21]; the finite sample set determines which frequency regions drive the optimization.
  • Weighting of linear versus quadratic contributions in f_lls = Equal level gamma for both contributions
    The paper states no rigorous rationale for this equal partitioning; an alternative weighting would lead to different ROMs and is left for future work.
assumptions (7)
  • domain assumption The system matrix A has all eigenvalues in the open left-half complex plane (asymptotic stability).
    Assumed after equation (1) so that H-infinity norms and gains are finite; standard for H-infinity model order reduction.
  • domain assumption The input-output behavior of the LTIQO system is exactly described by the convolutional kernel representation (7) with H1 and H2 as given, taken from [11].
    The paper builds G2 on this representation without re-deriving it; if the representation omits cross terms involving x and u, the transfer function G2 would be incomplete.
  • ad hoc to paper The H-infinity norm of the LTIQO system is defined as the sum of the maxima of the linear and quadratic transfer functions (Definition 2.1); no rigorous rationale is given for the additivity or the equal weighting.
    This is the central modeling choice of the paper; it is load-bearing for the algorithm's objective and is explicitly flagged as lacking justification in the footnote in Section 3.1.
  • domain assumption For the structure-preserving parametrization, every asymptotically stable matrix can be factored as (J-R)Q with J skew-symmetric and R,Q positive semidefinite, and equivalently the condensed form (22) with Q=I is attainable.
    Taken from [26, Lem. 2] and [6, Sec. 4]; the paper does not prove the reduction to Q=I in full detail and refers to [6].
  • standard math Parseval's theorem and the maximum modulus principle justify evaluating the transfer functions on the imaginary axis and bounding time-domain L2 norms by H-infinity norms.
    Used in the proof of Theorem 2.2 items 2 and 3.
  • standard math The singular value derivative formula (14) from [23] applies to the singular values sigma_i appearing in the gradients.
    Used to derive all gradient formulas; requires nonzero and simple singular values, a mild regularity condition not checked in the experiments.
  • domain assumption The leveled least squares objective (12) with a single scalar level gamma is a valid proxy for H-infinity norm minimization; the LTIQO extension inherits this from SOBMOR [5] and the properties in Section 3.1 are relaxed.
    The paper shows that for LTIQO systems either property 1 or property 2 of the LTI case must be weakened; the resulting relation between f_lls = 0 and the H-infinity error is only approximate.

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Cite this review

Pith. "Pith review of $\mathcal{H}_\infty$ model order reduction for quadratic output systems." pith.science (2026). https://pith.science/paper/ZYLTOEWD

@misc{pith2026250512529,
  author       = {Pith},
  title        = {Pith review of: $\mathcalH_\infty$ model order reduction for quadratic output systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYLTOEWD}},
  note         = {Machine review of arXiv:2505.12529}
}
abstract

Linear time-invariant quadratic output (LTIQO) systems generalize linear time-invariant systems to nonlinear regimes. Problems of this class occur in multiple applications naturally, such as port-Hamiltonian systems, optimal control, and stochastical problems. We introduce an $\mathcal{H}_\infty$-norm for LTIQO systems with one or multiple outputs and propose an algorithm to optimize a reduced order model (ROM) to be close in the $\mathcal{H}_\infty$-norm to a given full order model. We illustrate the applicability and the performance with an established numerical example and compare the resulting ROMs with results from balanced truncation and $\mathcal{H}_2$-focussed algorithms.

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