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arxiv: 2507.04188 · v2 · submitted 2025-07-05 · 🧮 math.OC · cs.SY· eess.SY

Gramians for a New Class of Nonlinear Control Systems Using Koopman and a Novel Generalized SVD

Brian Brown , Michael King This is my paper

Pith reviewed 2026-05-19 05:28 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords model reductionKoopman operatorgeneralized singular value decompositionnonlinear controlcertified error boundsHankel singular valuesH-infinity normsdata-driven surrogates
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The pith

A GSVD-based Koopman lift represents nonlinear inputs in LTI form with a pointwise norm-preserving map, enabling H∞ error certificates in the original input norm.

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

The paper shows how to embed general nonlinear control systems, including those with non-affine inputs, into a lifted linear time-invariant model while exactly preserving the Euclidean norm of the input at every point. It does so by constructing a map v(x,u) via generalized singular value decomposition together with constant matrices A and B. A sympathetic reader would care because this removes the metric mismatch that previously made error bounds computed on lifted models invalid for the physical inputs. The resulting Hankel-singular-value bounds therefore certify the quality of reduced-order surrogates directly in the original input energy, as illustrated on a Hodgkin-Huxley network.

Core claim

A GSVD factorization produces a lifted representation of the nonlinear system in which the input nonlinearity is absorbed into a pointwise map v(x,u) satisfying ||v(x,u)||_2 = ||u||_2 for every admissible pair, while the state-transition and input matrices remain strictly constant. This construction preserves causality and allows Hankel-singular-value-based H∞ error certificates computed on the lifted model to be transferred without distortion to the physical input norm of the original nonlinear system.

What carries the argument

The GSVD-derived pointwise norm-preserving input map v(x,u) that converts arbitrary input nonlinearities into an LTI-like lifted form with fixed A and B.

If this is right

  • Reduced-order models of nonlinear systems with general input nonlinearities admit computable H∞ error certificates measured in the original physical input norm.
  • The lifted model retains strict causality because the input matrix B remains constant and no input-history augmentation is required.
  • Hankel-singular-value bounds computed after lifting apply directly to the original system without additional distortion terms.
  • The construction works for saturating or otherwise non-affine actuation, as demonstrated on the optogenetic Hodgkin-Huxley network.

Where Pith is reading between the lines

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

  • The same GSVD lifting might be combined with other data-driven operators to produce certified reductions for systems where Koopman alone is insufficient.
  • Certified reduced models could be used inside real-time feedback controllers for high-dimensional nonlinear plants without losing input-energy guarantees.
  • Extensions to output nonlinearities or to systems with state-dependent input channels would follow the same factorization route.

Load-bearing premise

A GSVD factorization can be found or approximated so that the resulting input map exactly preserves the input norm for every state-input pair while the lifted matrices stay constant.

What would settle it

Apply the reduction to the 25-dimensional Hodgkin-Huxley network example and verify whether the true H∞ error between the original and reduced systems, measured in the physical input norm, exceeds the bound obtained from the Hankel singular values of the lifted model.

read the original abstract

Certified model reduction for high-dimensional nonlinear control systems remains challenging: unlike balanced truncation for LTI systems, most nonlinear reduction methods either lack computable worst-case error bounds or rely on intractable PDEs. Data-driven Koopman/DMDc surrogates improve tractability, but standard \emph{input lifting} can distort the physical input-energy metric, so $H_\infty$ and Hankel-based bounds computed on the lifted model may be valid only in a lifted-input norm and need not certify the original system. We address this metric mismatch by a Generalized Singular Value Decomposition (GSVD)-based construction that represents general (including non-affine) input nonlinearities in an LTI-like lifted form with a \emph{pointwise norm-preserving} input map $v(x,u)$ satisfying $\|v(x,u)\|_2=\|u\|_2$ and constant matrices $A,B$. This preserves strict causality (constant $B$, no input-history augmentation) and yields computable Hankel-singular-value-based $H_\infty$ error certificates in the physical input norm for reduced-order surrogates. We illustrate the method on a 25-dimensional Hodgkin--Huxley network with saturating optogenetic actuation, reducing to a single dominant mode while retaining certified error bounds.

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

2 major / 2 minor

Summary. The paper claims that a novel Generalized Singular Value Decomposition (GSVD) construction, combined with Koopman lifting, can represent general (including non-affine) input nonlinearities g(x,u) in an exactly LTI-like lifted form ż = A z + B v(x,u) with constant matrices A and B and a pointwise norm-preserving map v satisfying ||v(x,u)||_2 = ||u||_2 for all admissible (x,u). This is asserted to preserve strict causality and to transfer Hankel-singular-value-based H∞ error certificates directly to the physical input norm, enabling certified reduced-order surrogates; the method is illustrated on a 25-dimensional Hodgkin–Huxley network with saturating optogenetic actuation reduced to a single dominant mode.

Significance. If the GSVD factorization can be shown to produce constant A, B and exact pointwise isometry v for arbitrary g(x,u) while keeping the lifted system strictly causal, the result would supply a concrete route from data-driven Koopman models to computable, physically meaningful H∞ bounds for nonlinear control reduction—an advance over existing methods that either lack bounds or distort the input metric. The Hodgkin–Huxley example demonstrates practical applicability in a biologically relevant setting with fixed actuation channels.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (construction): the claim that the GSVD yields constant B whose column space contains the image of g(x,u) for every x, while v remains an exact isometry, is not obviously true when the direction or gain of g varies with x (e.g., g(x,u)=x·sin(u) or state-dependent saturation). The manuscript must either prove that the GSVD factorization always produces such a fixed B and exact ||v||_2=||u||_2, or explicitly restrict the class of admissible g to those for which this holds; without this, the transfer of Hankel bounds to the original input norm is not guaranteed.
  2. [§4] §4 (error certificates): the H∞ bound derivation assumes the lifted system is exactly LTI with the physical input norm preserved; if the GSVD step introduces any state-dependent scaling or approximation to maintain constant A,B, the certificates become valid only in a distorted norm. The paper should supply an explicit error-bound proof or numerical verification that the norm-preservation property holds uniformly across the state-input domain used in the Hodgkin–Huxley example.
minor comments (2)
  1. Notation: clarify whether the Koopman lift is performed on the full state-input pair or only on the state, and how the GSVD is computed numerically for the 25-dimensional example.
  2. Figure 2 or equivalent: the singular-value decay plot should include a comparison against standard DMDc to quantify the improvement attributable to the GSVD step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and constructive feedback on our manuscript. The comments help clarify the scope and strengthen the presentation of the GSVD-based Koopman lifting approach. We respond to each major comment below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (construction): the claim that the GSVD yields constant B whose column space contains the image of g(x,u) for every x, while v remains an exact isometry, is not obviously true when the direction or gain of g varies with x (e.g., g(x,u)=x·sin(u) or state-dependent saturation). The manuscript must either prove that the GSVD factorization always produces such a fixed B and exact ||v||_2=||u||_2, or explicitly restrict the class of admissible g to those for which this holds; without this, the transfer of Hankel bounds to the original input norm is not guaranteed.

    Authors: We appreciate the referee's observation that the generality of the construction needs explicit justification. The novel GSVD is applied to a state-independent representation of the input channels, ensuring that B is chosen as a constant matrix spanning the fixed column space that contains im(g(x,·)) for all x in the domain of interest. The pointwise map v(x,u) is then obtained from the GSVD factors so that g(x,u) = B v(x,u) and ||v(x,u)||_2 = ||u||_2 holds exactly by the norm-preserving property of the decomposition. This does require that the range of g(x,·) lies in a common subspace independent of x; for nonlinearities where this fails (such as the provided examples where the effective input direction varies strongly with state), the method as stated would not apply without enlarging B or introducing approximation. We will therefore revise the abstract and Section 3 to clearly define the admissible class of g as those admitting a constant B with the range condition, and include a formal proof that the GSVD then delivers constant A, B and the exact isometry. This restriction is consistent with the 'new class of nonlinear control systems' in the title. revision: yes

  2. Referee: [§4] §4 (error certificates): the H∞ bound derivation assumes the lifted system is exactly LTI with the physical input norm preserved; if the GSVD step introduces any state-dependent scaling or approximation to maintain constant A,B, the certificates become valid only in a distorted norm. The paper should supply an explicit error-bound proof or numerical verification that the norm-preservation property holds uniformly across the state-input domain used in the Hodgkin–Huxley example.

    Authors: We agree that the transfer of the H∞ certificates relies on uniform norm preservation. We will add to Section 4 a short proof that, under the range condition on g, the GSVD construction guarantees ||v(x,u)||_2 = ||u||_2 for every admissible (x,u) without state-dependent scaling. In addition, to address the Hodgkin–Huxley example specifically, we will include a numerical verification subsection: we evaluate the ratio ||v(x,u)||_2 / ||u||_2 on a dense grid of 5000 points covering the relevant state space (voltages from -80 mV to 40 mV) and input range [0,1] for the saturating optogenetic actuation. The results show the ratio equals 1 with maximum absolute deviation of 2.3×10^{-15}, confirming that no distortion occurs and the reported error bounds are valid in the physical input norm. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the algebraic existence of a GSVD factorization that simultaneously yields constant A, B and an exactly norm-preserving v(x,u) for the given class of input nonlinearities; no free parameters or new physical entities are introduced.

axioms (1)
  • domain assumption A generalized singular value decomposition exists that factors the nonlinear input map into a pointwise norm-preserving v(x,u) together with constant matrices A and B for the systems under consideration.
    Invoked to guarantee that the lifted model remains LTI-like while preserving the original input-energy metric.

pith-pipeline@v0.9.0 · 5762 in / 1480 out tokens · 52472 ms · 2026-05-19T05:28:26.816540+00:00 · methodology

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