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arxiv: 2606.07758 · v1 · pith:VSU66YS6new · submitted 2026-06-05 · 📡 eess.SY · cs.SY· math.OC

Koopman meets input-output data: Data-driven output-feedback control of nonlinear systems with closed-loop guarantees

Robin Str\"asser , Julian Berberich , Manuel Schaller , Karl Worthmann , Frank Allg\"ower This is my paper

Pith reviewed 2026-06-27 20:49 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords data-driven controlKoopman operatornonlinear systemsoutput-feedbackinput-output databilinear surrogate modelexponential stability
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The pith

A data-driven output-feedback controller for nonlinear systems achieves exponential stability using only input-output data via a Koopman-based bilinear surrogate model.

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 design controllers for nonlinear systems that relies solely on measured input and output trajectories rather than full state information. It builds an extended state representation from those trajectories and applies Koopman operator theory to produce a bilinear surrogate model. Robust state-feedback design techniques are then used on the surrogate. Observability of the original system transfers exponential stability from the surrogate back to the true nonlinear dynamics.

Core claim

By constructing an extended state from input-output trajectories and lifting the nonlinear system via the Koopman operator, a bilinear surrogate model is obtained directly from data. Robust state-feedback design on this model produces a controller that renders the extended state exponentially stable; observability then ensures the original system state converges exponentially to the origin.

What carries the argument

Extended state representation of the nonlinear system from input-output trajectories, lifted by the Koopman operator into a bilinear surrogate model.

If this is right

  • The resulting controller uses only input-output measurements and requires no direct state access.
  • Exponential stability of the closed loop is guaranteed for the original nonlinear system.
  • Standard robust state-feedback synthesis methods apply directly to the data-derived bilinear surrogate.
  • Numerical simulations confirm that the theoretical guarantees hold in practice.

Where Pith is reading between the lines

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

  • The same lifting approach could be tested on systems where only partial observability holds.
  • The bilinear surrogate structure may simplify extensions to tracking or disturbance-rejection tasks.
  • Similar data-driven lifting might reduce the need for state estimation in other nonlinear control settings.

Load-bearing premise

The nonlinear system must be observable from its input-output trajectories so that stability of the extended state implies stability of the original state.

What would settle it

A simulation or experiment on an observable nonlinear system in which the closed-loop trajectories fail to converge exponentially under the data-driven controller.

read the original abstract

Data-driven control of nonlinear systems from input-output measurements remains a fundamental challenge, as existing approaches with rigorous closed-loop guarantees predominantly require access to full state measurements. In this paper, we address this gap by proposing a data-driven output-feedback controller design method for nonlinear systems that provides provable closed-loop guarantees while operating solely on measured input-output data. Our approach combines Koopman operator theory with an extended state representation of the nonlinear system constructed from input-output trajectories. This allows us to obtain a bilinear surrogate model directly from data, on which robust state-feedback design methods can be applied. By exploiting the observability of the underlying nonlinear system, we establish exponential stability of the extended state, which in turn implies exponential convergence of the original system state to the origin. Finally, we validate our theoretical findings in numerical 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.

Referee Report

2 major / 1 minor

Summary. The paper proposes a data-driven output-feedback controller for nonlinear systems that operates solely on input-output measurements. It combines Koopman operator theory with an extended-state representation constructed from input-output trajectories to obtain a bilinear surrogate model directly from data; robust state-feedback methods are then applied to this surrogate, and observability of the underlying nonlinear system is invoked to transfer exponential stability of the extended state back to exponential convergence of the original state to the origin. The claims are supported by theoretical arguments and numerical simulations.

Significance. If the stability-transfer argument holds with the required uniform bounds, the result would be significant: it extends data-driven Koopman-based control to the output-feedback setting while retaining closed-loop guarantees, without requiring full-state access. The combination of data-driven bilinear lifting with existing robust-design tools is a natural and potentially useful direction.

major comments (2)
  1. [Abstract / theoretical development (stability transfer)] The central stability claim (abstract) that exponential stability of the extended state implies exponential convergence of the original state rests on observability, yet no explicit norm-equivalence inequality of the form ||x|| ≤ C ||z_ext|| with C independent of the data-driven approximation error, trajectory length, and lifting dimension is stated or proved. Without this uniform bound the implication does not close.
  2. [Data-driven model construction] The construction of the bilinear surrogate from input-output data (abstract) is asserted to be direct, but the manuscript provides neither explicit conditions on the richness of the data set nor quantitative error bounds between the true Koopman operator and the identified bilinear model that would be needed to certify the subsequent robust-design step.
minor comments (1)
  1. Notation for the extended state and the observability map should be introduced with a dedicated equation or definition block rather than inline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the stability-transfer argument and the data-driven construction.

read point-by-point responses

  1. Referee: [Abstract / theoretical development (stability transfer)] The central stability claim (abstract) that exponential stability of the extended state implies exponential convergence of the original state rests on observability, yet no explicit norm-equivalence inequality of the form ||x|| ≤ C ||z_ext|| with C independent of the data-driven approximation error, trajectory length, and lifting dimension is stated or proved. Without this uniform bound the implication does not close.

    Authors: We agree that the stability-transfer step requires an explicit uniform norm-equivalence bound independent of approximation error, trajectory length, and lifting dimension. The manuscript invokes observability of the nonlinear system to transfer stability, but does not state the bound in the required form. In the revision we will add a dedicated lemma establishing ||x|| ≤ C ||z_ext|| with C independent of the listed quantities, using the structure of the extended state and the observability assumption. This will make the implication rigorous. revision: yes

  2. Referee: [Data-driven model construction] The construction of the bilinear surrogate from input-output data (abstract) is asserted to be direct, but the manuscript provides neither explicit conditions on the richness of the data set nor quantitative error bounds between the true Koopman operator and the identified bilinear model that would be needed to certify the subsequent robust-design step.

    Authors: The manuscript presents the extended-state construction from input-output trajectories as enabling a direct data-driven bilinear model, but does not supply explicit richness conditions or quantitative error bounds. We will add a section (or appendix) stating persistency-of-excitation-type assumptions on the input-output data set together with quantitative bounds on the approximation error between the true Koopman operator and the identified bilinear surrogate. These additions will support the subsequent robust-design step. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs a bilinear Koopman surrogate from input-output data via an extended state representation, applies existing robust state-feedback methods to the surrogate, and transfers exponential stability to the original state via the observability assumption on the nonlinear system. This transfer is an external implication relying on standard observability properties rather than any self-definitional equivalence, fitted parameter renamed as prediction, or load-bearing self-citation chain. No quoted equations or steps reduce the central claim to its inputs by construction; the derivation remains independent of the data-driven fit itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the observability assumption for transferring stability and on the data-derived bilinear surrogate accurately representing the dynamics for control design. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The underlying nonlinear system is observable from input-output data.
    Invoked to establish that exponential stability of the extended state implies exponential convergence of the original system state.

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discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Koopman operator theory: fundamentals, control, and applications

    eess.SY 2026-07 unverdicted novelty 1.0

    Tutorial on Koopman operator theory, data-driven methods such as EDMD, and their use in controller design for nonlinear systems with provided simulations and code.

Reference graph

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