arxiv: 2603.27548 · v2 · submitted 2026-03-29 · 🧮 math.OC · cs.SY· eess.SY

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Control Forward-Backward Consistency: Quantifying the Accuracy of Koopman Control Family Models

Masih Haseli , Jorge Cort\'es , Joel W. Burdick

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Pith reviewed 2026-05-14 22:28 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords Koopman control familyforward-backward consistencyerror bounddata-driven modelingcontrol systemsconsistency matrixlifted linear modelbilinear model

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The pith

The relative root-mean-square error of Koopman control family predictors is strictly bounded by the square root of the control consistency index.

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

This paper extends the forward-backward consistency idea to control systems and defines a control consistency matrix for Koopman Control Family models. It proves that the relative root-mean-square prediction error of these models is always less than or equal to the square root of the largest eigenvalue of that matrix. The bound is computable directly from the model matrices and applies without needing separate validation trajectories. Sympathetic readers care because it turns an otherwise opaque accuracy question into a single number that can guide model selection and refinement during data-driven control design.

Core claim

The main result establishes that the relative root-mean-square error of KCF function predictors is strictly bounded by the square root of the control consistency index, defined as the maximum eigenvalue of the consistency matrix. This bound is sharp and closed-form, and it holds for any finite-dimensional KCF representation of a control system.

What carries the argument

The control forward-backward consistency matrix, whose maximum eigenvalue supplies the consistency index that directly upper-bounds predictor error.

Load-bearing premise

The forward-backward regression perspective together with a finite-dimensional Koopman Control Family representation accurately captures the underlying control system.

What would settle it

A concrete counterexample in which the measured relative root-mean-square error of a fitted KCF predictor exceeds the square root of the largest eigenvalue of its computed consistency matrix.

Figures

Figures reproduced from arXiv: 2603.27548 by Joel W. Burdick, Jorge Cort\'es, Masih Haseli.

read the original abstract

This paper extends the forward-backward consistency index, originally introduced in Koopman modeling of systems without input, to the setting of control systems, providing a closed-form computable measure of accuracy for data-driven models associated with the Koopman Control Family (KCF). Building on a forward-backward regression perspective, we introduce the control forward-backward consistency matrix and demonstrate that it possesses several favorable properties. Our main result establishes that the relative root-mean-square error of KCF function predictors is strictly bounded by the square root of the control consistency index, defined as the maximum eigenvalue of the consistency matrix. This provides a sharp, closed-form computable error bound for finite-dimensional KCF models. We further specialize this bound to the widely used lifted linear and bilinear models. We also discuss how the control consistency index can be incorporated into optimization-based modeling and illustrate the methodology via simulations.

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

This paper extends the forward-backward consistency index to controlled systems by defining a control consistency matrix whose largest eigenvalue bounds the relative RMS error of finite-dimensional Koopman control predictors.

read the letter

The main point is that the authors take the earlier unforced consistency idea and build a matrix from the forward-backward regression that directly gives a closed-form error bound for Koopman control family models. They show the matrix is positive semi-definite, derive the bound via a norm inequality, and specialize it to the usual lifted linear and bilinear cases. They also note how the index can be added to optimization-based modeling routines. That is the concrete advance: a computable certificate that stays inside the same data used to fit the model. The approach is clean for the finite-dimensional setting and avoids needing extra validation trajectories for the bound itself. The simulations are meant to show the method in action, though the numbers are not visible here. One soft spot is that everything rests on the finite-dimensional KCF representation and the forward-backward regression being appropriate for the system; the paper does not explore what happens when those modeling choices are poor. The bound is presented as strict, so it would help to see whether it is tight in the examples or tends to be loose. The circularity concern does not appear in the math itself, since the bound follows from the spectral properties rather than from re-using fitted parameters in a tautological way. This work is aimed at people already building and using Koopman lifted models for control, especially those who want an a-priori error measure they can compute from the training data. A reader in data-driven control or operator-theoretic methods would find the explicit matrix construction and the eigenvalue result directly usable. I would send it to peer review because the core derivation is self-contained in the finite-dimensional case and the practical motivation is clear, even if the simulations and assumption checks could be expanded.

Referee Report

0 major / 3 minor

Summary. This manuscript extends the forward-backward consistency index from unforced Koopman models to controlled dynamical systems by introducing the control forward-backward consistency matrix for finite-dimensional Koopman Control Family (KCF) representations. Building on a forward-backward regression perspective, the authors establish that the matrix is positive semi-definite, define the control consistency index as its maximum eigenvalue, and prove that the relative root-mean-square error of KCF function predictors is strictly bounded by the square root of this index. The bound is specialized to lifted linear and bilinear models; the index is further proposed for incorporation into optimization-based modeling procedures, with the overall methodology illustrated through numerical simulations.

Significance. If the central derivation holds, the result supplies a closed-form, a-priori computable error bound for data-driven KCF models that does not require additional validation trajectories. This addresses a practical gap in Koopman-based control, where model fidelity is otherwise assessed only after fitting. The extension to input-affine systems, the explicit specialization to common lifted forms, and the non-circular character of the bound (obtained via direct norm inequalities on the consistency matrix) are genuine strengths that could improve model selection and validation workflows in data-driven control.

minor comments (3)

[Abstract] Abstract: the claim of a 'strict' bound should be accompanied by a brief statement on whether equality is attainable (e.g., for particular choices of observables or data).
[Introduction] The manuscript would benefit from an explicit statement, early in the main text, of the precise finite-dimensional assumption under which the KCF representation is exact; this would clarify the scope without altering the central claim.
[Numerical examples] Simulation section: the description of the test systems, the choice of lifting functions, and the numerical values of the consistency index should be expanded to support reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive review, accurate summary of our contributions, and recommendation for minor revision. The positive assessment of the closed-form error bound and its applicability to data-driven control is appreciated.

Circularity Check

0 steps flagged

No significant circularity; bound derived from independent matrix properties

full rationale

The central result defines the control consistency matrix from the forward-backward regression operators of the finite-dimensional KCF model, proves it is positive semi-definite, and applies a direct norm inequality to bound the relative RMS prediction error by the square root of its maximum eigenvalue. This is a mathematical guarantee that holds for any such model by construction of the definitions and does not reduce the error bound to a fitted parameter, self-citation chain, or tautology. The extension from the input-free case is cited but does not carry the load-bearing step for the control-system bound. The derivation remains self-contained within the stated finite-dimensional setting.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; the ledger is therefore limited to the modeling framework explicitly invoked.

axioms (1)

domain assumption The Koopman Control Family representation is valid for the finite-dimensional lifted models considered.
Invoked when the consistency matrix and error bound are defined for KCF models.

invented entities (1)

control forward-backward consistency matrix no independent evidence
purpose: To provide a closed-form, computable accuracy measure for KCF models
Newly introduced object whose maximum eigenvalue supplies the error bound.

pith-pipeline@v0.9.0 · 5459 in / 1265 out tokens · 45733 ms · 2026-05-14T22:28:24.667272+00:00 · methodology

discussion (0)

Reference graph

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