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arxiv: 2602.13957 · v3 · pith:AMXXGA4Wnew · submitted 2026-02-15 · 📡 eess.SY · cs.SY

Learning-based data-enabled moving horizon estimation with application to membrane-based biological wastewater treatment process

Xiaojie Li , Xunyuan Yin This is my paper

Pith reviewed 2026-05-21 13:15 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-enabled moving horizon estimationKoopman operator theoryWillems fundamental lemmaneural network learningnonlinear systemsstate estimationwastewater treatment
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The pith

Data-learned Koopman surrogates enable convex moving horizon estimation for nonlinear systems without explicit models.

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

The paper proposes lifting nonlinear systems to a linear parameter-varying Koopman surrogate using neural networks trained on data. Willems fundamental lemma constructs a trajectory representation that avoids identifying system matrices explicitly. This leads to a convex optimization-based moving horizon estimator that provides real-time surrogate state estimates and reconstructs the original states. Sufficient conditions ensure the estimation error remains stable. The method is shown effective on a simulated biological wastewater treatment process, offering a model-free alternative for state estimation in complex nonlinear dynamics.

Core claim

By lifting the nonlinear system to a linear parameter-varying Koopman surrogate with lifting functions and scheduling mappings learned by neural networks from data, and employing Willems fundamental lemma for a trajectory-based representation, the authors formulate a convex data-enabled moving horizon estimation design that estimates the Koopman surrogate states in real time and reconstructs the original system states, along with sufficient conditions for estimation error stability.

What carries the argument

The convex data-enabled MHE formulated from the trajectory representation of the learned Koopman surrogate via Willems' fundamental lemma.

If this is right

  • Real-time estimates of Koopman surrogate states are computed via convex optimization.
  • Original nonlinear states are reconstructed from the surrogate estimates.
  • Stability of the estimation error is ensured under the derived sufficient conditions.
  • The technique applies to nonlinear processes like membrane-based biological wastewater treatment without explicit modeling.

Where Pith is reading between the lines

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

  • The approach may enable online learning updates for the neural networks to handle time-varying systems.
  • It could integrate with data-driven predictive control for fully model-free operation.
  • Similar lifting techniques might apply to fault detection or monitoring in other data-rich industrial settings.

Load-bearing premise

The nonlinear system admits an accurate lift to a linear parameter-varying Koopman surrogate with lifting functions and scheduling mappings learnable from data by neural networks such that the trajectory representation via Willems fundamental lemma is valid for the convex MHE.

What would settle it

Demonstrating divergence of the estimation error in the wastewater treatment simulation when the learned Koopman surrogate is used, or violation of the stability conditions in practice, would falsify the claim.

read the original abstract

In this paper, we propose a data-enabled moving horizon estimation (MHE) approach for a class of nonlinear systems without explicit modeling, by leveraging Koopman operator theory and Willems fundamental lemma. Specifically, the nonlinear system is lifted to a linear parameter-varying Koopman surrogate, in which the lifting functions and scheduling mappings are learned directly from data using neural networks. Willems fundamental lemma is then employed to construct a trajectory-based representation of the Koopman surrogate, which bypasses the explicit identification of the matrices of the Koopman surrogate. Based on this representation, we formulate a convex data-enabled MHE design, which provides real-time estimates of the Koopman surrogate states, from which the states of the original nonlinear system are reconstructed. Sufficient conditions are derived to ensure the stability of the estimation error. The effectiveness of the proposed method is illustrated using a simulated membrane-based biological wastewater treatment process.

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 proposes a learning-based data-enabled moving horizon estimation (MHE) approach for nonlinear systems. The nonlinear dynamics are lifted to a linear parameter-varying (LPV) Koopman surrogate whose lifting functions and scheduling mappings are learned from data via neural networks. Willems' fundamental lemma is invoked to obtain a trajectory-based data-driven representation that avoids explicit identification of the surrogate matrices. This representation is used to formulate a convex MHE that estimates the lifted states in real time, from which the original nonlinear states are reconstructed. Sufficient conditions are derived to guarantee stability of the estimation error. The method is demonstrated on a simulated membrane-based biological wastewater treatment process.

Significance. If the central construction is rigorous, the work offers a model-free, convex optimization route to stable state estimation for nonlinear systems by combining Koopman lifting, neural-network approximation, and Willems' lemma inside an MHE framework. This could be practically useful for process-control applications where first-principles models are unavailable, such as biological wastewater treatment. The explicit derivation of stability conditions and the convexity of the resulting program are potential strengths, provided the approximation errors do not invalidate the trajectory representation or the stability guarantees.

major comments (2)
  1. [Trajectory representation and MHE formulation] The invocation of Willems' fundamental lemma on the neural-network-approximated LPV Koopman surrogate (section describing the data-driven trajectory representation) is load-bearing for both the convex MHE formulation and the subsequent stability claims. Standard statements of the lemma (and its LPV extensions) require exact linear-in-parameters dynamics; because the scheduling map is only approximated by an NN, the Hankel-matrix representation is not guaranteed to be exact, and the paper does not provide a quantitative bound on the resulting representation error or show that the convex program remains faithful to the true lifted dynamics.
  2. [Stability analysis] The sufficient stability conditions for the estimation error (section deriving the stability guarantees) appear to be stated for the exact lifted LPV system. It is unclear whether these conditions remain valid once the neural-network approximation errors in the lifting functions and scheduling mappings are taken into account; a robustness margin or an explicit error-propagation argument is needed to support the claim that the reconstructed nonlinear-state error converges.
minor comments (2)
  1. [Numerical example / wastewater treatment application] The numerical example would benefit from reporting the training/validation split, the NN architectures (number of layers, neurons, activation functions), and quantitative metrics (e.g., prediction error on held-out trajectories) for the learned lifting functions and scheduling map.
  2. [Preliminaries and notation] Notation for the scheduling variable and its embedding inside the data matrices should be made fully explicit so that readers can verify how the LPV structure is preserved in the Hankel matrices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and for recognizing the potential utility of the proposed data-enabled MHE approach in process-control settings. We address the two major comments point by point below, indicating the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Trajectory representation and MHE formulation] The invocation of Willems' fundamental lemma on the neural-network-approximated LPV Koopman surrogate (section describing the data-driven trajectory representation) is load-bearing for both the convex MHE formulation and the subsequent stability claims. Standard statements of the lemma (and its LPV extensions) require exact linear-in-parameters dynamics; because the scheduling map is only approximated by an NN, the Hankel-matrix representation is not guaranteed to be exact, and the paper does not provide a quantitative bound on the resulting representation error or show that the convex program remains faithful to the true lifted dynamics.

    Authors: We agree that Willems' lemma and its LPV extensions are stated for exact linear-in-parameters dynamics, and that the neural-network approximation of the lifting functions and scheduling map therefore introduces a representation error. The manuscript constructs the trajectory-based representation from data lifted by the trained networks and formulates the convex MHE directly on that representation; the stability claims are likewise derived for the resulting surrogate. To make this explicit, the revised manuscript will add a dedicated paragraph in the data-driven representation section that (i) states the approximation assumption, (ii) recalls a standard uniform approximation bound for feed-forward networks on compact sets, and (iii) shows that the resulting Hankel-matrix mismatch remains bounded by a term proportional to the training residual. This bound will be used to argue that the convex program remains a faithful relaxation of the true lifted dynamics up to a known additive perturbation. revision: yes

  2. Referee: [Stability analysis] The sufficient stability conditions for the estimation error (section deriving the stability guarantees) appear to be stated for the exact lifted LPV system. It is unclear whether these conditions remain valid once the neural-network approximation errors in the lifting functions and scheduling mappings are taken into account; a robustness margin or an explicit error-propagation argument is needed to support the claim that the reconstructed nonlinear-state error converges.

    Authors: The stability theorem is proved for the lifted LPV surrogate under the assumption that the surrogate exactly represents the lifted dynamics. Because the networks are trained on data from the original nonlinear system, the approximation error appears as a bounded disturbance in the lifted state and scheduling equations. In the revision we will augment the stability section with a short robustness lemma that propagates this bounded disturbance through the MHE error dynamics, yielding an ultimate bound on the nonlinear-state estimation error that is linear in the network approximation error. The original LMI conditions remain sufficient for the nominal (zero-error) case; the additional term supplies the requested robustness margin. The wastewater example will be re-run with the explicit error bound reported to illustrate the practical size of the margin. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper learns lifting functions and scheduling mappings for an LPV Koopman surrogate via neural networks from data, applies Willems' fundamental lemma to obtain a trajectory representation bypassing explicit matrix identification, formulates a convex data-enabled MHE, and derives stability conditions for the estimation error. No step in this chain reduces a claimed result or prediction to its own inputs by construction, such as a fitted parameter being renamed as a prediction or a stability condition holding only by self-definition. The approach relies on external data, standard lemmas, and learned approximations rather than self-referential loops or load-bearing self-citations that collapse the central claims. The derivation remains independent and self-contained against the provided benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the assumption that a nonlinear system can be accurately represented by a data-learned Koopman LPV surrogate and that Willems' lemma applies directly to the lifted trajectories. Neural-network parameters are fitted from data and constitute the main free parameters.

free parameters (1)
  • Neural-network weights for lifting functions and scheduling mappings
    Learned directly from input-output data to define the Koopman surrogate.
axioms (1)
  • domain assumption The nonlinear system admits a lift to a linear parameter-varying Koopman surrogate whose dynamics can be represented via Willems' fundamental lemma
    Invoked to enable the trajectory-based convex MHE without explicit matrix identification.

pith-pipeline@v0.9.0 · 5682 in / 1408 out tokens · 65328 ms · 2026-05-21T13:15:16.572896+00:00 · methodology

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