arxiv: 2602.24252 · v2 · submitted 2026-02-27 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Neural Luenberger state observer for nonautonomous nonlinear systems

Moritz Woelk , Jarod Morris , Wentao Tang

Authors on Pith no claims yet

Pith reviewed 2026-05-15 18:43 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords neural observerKKL observernonautonomous nonlinear systemsstate estimationdata-driven synthesisLuenberger observermodel-free observerfeedforward networks

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The pith

Neural networks trained on data can implement state observers for nonlinear systems with external inputs and provide guaranteed error bounds on new trajectories.

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 build a state observer for nonlinear systems that receive manipulated inputs without first writing down an explicit dynamic model. It takes the Kazantzis-Kravaris/Luenberger observer structure, adds an input-affine term to handle the known inputs, and replaces the unknown functions in that structure with two neural networks. One network supplies the input-dependent correction to the observer dynamics; the other reconstructs the original system states from the observer states. A convergence proof establishes that, once the networks are trained on a sufficiently rich offline dataset, the resulting observer delivers state estimates whose error stays inside a known bound for any fresh input-output sequence. The approach is demonstrated on a bioreactor and a Williams-Otto reactor, where the networks are trained from simulation data and then used to track states online.

Core claim

The central claim is that an extended KKL observer for nonautonomous nonlinear systems can be realized by two feedforward neural networks: one that learns the input-affine term driving the linear observer dynamics, and one that learns the nonlinear map from observer states back to system states; when these networks are trained offline on state-input data, the composite observer is guaranteed to produce state estimates with bounded error on previously unseen input sequences.

What carries the argument

Extended KKL observer whose linear dynamics are augmented by a learned input-affine term, with a second neural network inverting the resulting injective state map.

If this is right

State observers for complex plants can be synthesized directly from historical or simulation data rather than from first-principles equations.
The observer can be deployed online on any new input sequence whose distribution is covered by the training set, with a priori error bounds.
The same data-driven construction applies to any system that satisfies the extended KKL injectivity condition, including chemical reactors and biological processes.
Offline training decouples observer design from real-time computation, allowing the networks to run at the speed of the plant sampling rate.

Where Pith is reading between the lines

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

The method implicitly learns a coordinate change that linearizes the observer error dynamics, suggesting possible links to data-driven Koopman or embedding techniques.
If the training data are collected under closed-loop operation, the resulting observer could serve as a building block for subsequent data-driven controller design.
The guaranteed bound could be tightened by enriching the dataset with trajectories near the boundary of the operating region, offering a practical way to improve performance without changing the network architecture.

Load-bearing premise

The underlying nonlinear system must admit an extended KKL observer structure whose required functions can be approximated to sufficient accuracy by the chosen neural networks from the available training data.

What would settle it

A concrete counter-example would be a nonautonomous nonlinear system for which no choice of input-affine term produces an injective mapping from true states to observer states, or a trained network pair that, on a new validation trajectory, produces state errors larger than the bound stated in the convergence theorem.

Figures

Figures reproduced from arXiv: 2602.24252 by Jarod Morris, Moritz Woelk, Wentao Tang.

**Figure 2.** Figure 2: State trajectories for ̃𝑥1 and ̃𝑥2 for the bioreactor system [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of state estimates for the bioreactor system: NLOX (red) versus the analytical solution (light blue), EKF (yellow), and SMO (green). M. Woelk et al.: Preprint submitted to Elsevier Page 36 of 35 [PITH_FULL_IMAGE:figures/full_fig_p037_3.png] view at source ↗

**Figure 4.** Figure 4: Comparison of state estimates for the bioreactor system with noise: NLOX (red) versus the analytical solution (light blue), EKF (yellow), and SMO (green) [PITH_FULL_IMAGE:figures/full_fig_p038_4.png] view at source ↗

**Figure 5.** Figure 5: Training state trajectories for Williams-Otto reactor [PITH_FULL_IMAGE:figures/full_fig_p038_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of state estimates for the Williams-Otto reactor: NLOX (red) versus EKF (light blue) and SMO (green). M. Woelk et al.: Preprint submitted to Elsevier Page 37 of 35 [PITH_FULL_IMAGE:figures/full_fig_p038_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison of state estimates for the Williams-Otto reactor with noise: NLOX (red) versus EKF (light blue) and SMO (green). M. Woelk et al.: Preprint submitted to Elsevier Page 38 of 35 [PITH_FULL_IMAGE:figures/full_fig_p039_7.png] view at source ↗

read the original abstract

This work proposes a method for model-free synthesis of a state observer for nonlinear systems with manipulated inputs, where the observer is trained offline using a historical or simulation dataset of state measurements. We use the structure of the Kazantzis-Kravaris/Luenberger (KKL) observer, extended to nonautonomous systems by adding an additional input-affine term to the linear time-invariant (LTI) observer-state dynamics, which determines a nonlinear injective mapping of the true states. Both this input-affine term and the nonlinear mapping from the observer states to the system states are learned from data using fully connected feedforward multi-layer perceptron neural networks. Furthermore, we theoretically prove that trained neural networks, when given new input-output data, can be used to observe the states with a guaranteed error bound. To validate the proposed observer synthesis method, case studies are performed on a bioreactor and a Williams-Otto reactor.

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

Neural KKL extension for nonautonomous systems learns input-affine term and mapping via MLPs with a claimed error bound, but the bound's handling of finite-data approximation error looks like the main open question.

read the letter

The main thing to know is that this paper extends the KKL observer to systems with manipulated inputs by adding a learned input-affine term to the observer dynamics and training a second network for the state reconstruction map, then claims a theoretical error bound that holds for new input-output data. They train both networks offline from state measurements and test the result on a bioreactor and a Williams-Otto reactor. That combination is the substantive step beyond routine application of existing KKL ideas. The reactor examples are a reasonable validation choice for process-control relevance and show the observer tracking states under varying inputs. The approach stays model-free after the offline training phase, which is the practical selling point. The soft spot is the error bound. The proof appears to treat the trained networks as sufficiently accurate representations of the required functions, but finite-data training always leaves some residual approximation error. Without an explicit remainder term or a certificate that folds network error into the contraction rate, the guarantee does not automatically carry over to the deployed observer on fresh data. The stress-test note correctly flags this gap. The rest of the setup follows standard KKL assumptions plus universal approximation, so there is no obvious circularity or internal contradiction. The citation pattern is appropriate for the subfield. This paper is for researchers working on data-driven nonlinear observers, especially in chemical process control. A reader who already knows KKL observers will see the concrete extension and the two simulation cases without needing to reinvent the training procedure. It deserves a serious referee because the core construction is new enough and the experiments are present, even if the bound needs tighter justification on the approximation side.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a model-free neural-network-based state observer for nonautonomous nonlinear systems. It extends the KKL observer by adding an input-affine term to the LTI observer dynamics, approximates both the nonlinear state mapping T and the input-affine function via MLPs trained offline on state-measurement datasets, proves that the resulting observer yields a guaranteed state-error bound on new input-output data, and validates the method on a bioreactor and the Williams-Otto reactor.

Significance. If the error-bound proof can be made rigorous for finite-data approximations, the work would supply a practical route to guaranteed-performance observers without requiring an explicit system model, which is valuable for process-control applications. The two reactor case studies illustrate applicability to realistic nonlinear dynamics.

major comments (1)

[§4, main theorem] §4, main theorem on the error bound: the derivation assumes that the trained MLPs exactly realize the KKL mapping T and the input-affine function (i.e., zero approximation error). Because the networks are obtained from finite data, the residual approximation error is neither bounded nor absorbed into the contraction rate; consequently the claimed guarantee does not automatically transfer to the deployed observer.

minor comments (2)

[Abstract] The abstract and introduction should explicitly list the standing assumptions (e.g., existence of a KKL structure, Lipschitz constants, training-data coverage) under which the bound is proved.
[Section 5] Table 1 and the reactor simulation figures would benefit from reporting training-set size, validation MSE of the two networks, and a direct comparison against a model-based KKL observer or an EKF.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The primary concern regarding the rigor of the error-bound proof for finite-data neural network approximations is well-taken. We address this point below and will revise the manuscript to strengthen the theoretical result.

read point-by-point responses

Referee: [§4, main theorem] §4, main theorem on the error bound: the derivation assumes that the trained MLPs exactly realize the KKL mapping T and the input-affine function (i.e., zero approximation error). Because the networks are obtained from finite data, the residual approximation error is neither bounded nor absorbed into the contraction rate; consequently the claimed guarantee does not automatically transfer to the deployed observer.

Authors: We agree that the current statement of the main theorem in §4 assumes exact realization of the KKL mapping T and the input-affine term (zero approximation error). This ideal-case assumption is explicitly noted in the manuscript but, as the referee correctly observes, does not automatically extend to finite-data training without an explicit bound on the residual error. In the revised version we will augment the theorem with a small, quantifiable approximation error ε (arising from finite data and network capacity). Using standard results from neural approximation theory, we will derive a modified error bound of the form ||e(t)|| ≤ κ exp(-λt) ||e(0)|| + Cε/(1-ρ), where ρ < 1 is the contraction rate and C is a constant depending on system Lipschitz constants. We will also add a practical section on estimating ε from a held-out validation set and on choosing network depth/width to keep ε below a user-specified tolerance. These changes make the guarantee rigorous for the deployed observer while preserving the original contraction-based analysis for the ideal case. revision: yes

Circularity Check

0 steps flagged

Error bound is independent theoretical result assuming exact NN approximation of KKL structure

full rationale

The derivation chain begins with the standard KKL observer extended by an input-affine term, then replaces the exact mappings T and the affine function with MLP networks trained on state-measurement data. The central theorem proves a guaranteed state-error bound for new input-output trajectories under the assumption that these networks realize the required functions exactly (or with errors absorbed into the contraction rate). This bound is not obtained by substituting the trained weights back into the loss or by re-expressing the contraction constant in terms of empirical residuals; it remains a separate Lyapunov-style argument that holds when the universal-approximation premise is granted. No equation reduces the bound to a quantity defined solely by the finite training set, and no self-citation supplies the uniqueness or contraction property in a load-bearing way. The result is therefore self-contained against external benchmarks once the exact-representation hypothesis is accepted, yielding only a minor (score-2) caveat that finite-data residuals are not explicitly certified.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a suitable KKL structure for the system class and the universal approximation capability of the chosen neural networks; parameters are fitted from data while the error bound is derived theoretically.

free parameters (1)

neural network weights and biases
Fitted offline to historical or simulation state data to realize the input-affine term and the nonlinear state mapping.

axioms (1)

domain assumption The nonlinear system class admits an injective mapping and observer dynamics of the extended KKL form
Invoked to guarantee that the learned networks can produce a valid state observer with bounded error.

pith-pipeline@v0.9.0 · 5455 in / 1348 out tokens · 33540 ms · 2026-05-15T18:43:43.155691+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use the structure of the Kazantzis-Kravaris/Luenberger (KKL) observer, extended to nonautonomous systems by adding an additional input-affine term... Both this input-affine term and the nonlinear mapping... are learned from data using fully connected feedforward multi-layer perceptron neural networks.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we theoretically prove that trained neural networks, when given new input-output data, can be used to observe the states with a guaranteed error bound

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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