Data-based Moving Horizon Estimation under Irregularly Measured Data

Isabelle Krauss; Matthias A. M\"uller; Tobias M. Wolff; Victor G. Lopez

arxiv: 2512.20259 · v2 · submitted 2025-12-23 · 📡 eess.SY · cs.SY

Data-based Moving Horizon Estimation under Irregularly Measured Data

Tobias M. Wolff , Isabelle Krauss , Victor G. Lopez , Matthias A. M\"uller This is my paper

Pith reviewed 2026-05-16 20:39 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords moving horizon estimationdata-based methodsirregular measurementsstate estimationrobust stabilitylinear systemsimplicit representationspractical stability

0 comments

The pith

A moving horizon estimator achieves practical robust exponential stability for linear systems using only irregular measurements and implicit data representations.

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

The paper develops a moving horizon estimation scheme for linear systems that relies only on a limited number of irregularly timed output measurements. Instead of requiring a full mathematical model of the system, it uses an implicit representation derived directly from collected data trajectories. The authors establish that this estimator is practically robustly exponentially stable in a sample-based sense under mild conditions. This matters because it allows accurate state estimation in settings where continuous or regular measurements are impractical or expensive, such as in medical or industrial monitoring. The method is illustrated through application to a model of drug absorption in the gastrointestinal tract.

Core claim

The central claim is that for linear systems a moving horizon estimator can be formulated using only measured data to implicitly represent the system dynamics, and that this estimator exhibits sample-based practical robust exponential stability when the data satisfies certain conditions.

What carries the argument

The key mechanism is the data-based implicit representation of the linear system from measured input-output trajectories, which replaces the need for an explicit state-space model in the moving horizon optimization problem.

If this is right

State estimates remain close to true values even when measurements arrive at irregular intervals.
The approach applies to any linear system for which sufficient trajectory data can be collected to form the implicit representation.
Robustness to bounded disturbances is guaranteed alongside the practical stability.
The framework can be implemented in applications where building an explicit model is difficult or costly.

Where Pith is reading between the lines

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

This data-driven method might reduce the need for dense sampling in real-time monitoring by tolerating irregular data arrival.
It could be combined with partial prior knowledge of the system to handle cases where data alone is insufficient.
Scalability checks on higher-dimensional examples beyond the gastrointestinal case would clarify computational limits in practice.

Load-bearing premise

The collected data trajectories must allow construction of an implicit representation that fully captures the system's behavior for the purposes of the stability analysis.

What would settle it

Running the estimator on a linear system with irregular measurements where the error between estimated and true states fails to converge to a small neighborhood despite satisfying the data conditions would disprove the stability claim.

Figures

Figures reproduced from arXiv: 2512.20259 by Isabelle Krauss, Matthias A. M\"uller, Tobias M. Wolff, Victor G. Lopez.

**Figure 1.** Figure 1: Simulation results of system (17) for four different settings. In the first plot, we consider that all output measurements are available to the state estimation scheme (benchmark case). In the following three plots, we consider that the state estimation scheme has only 48, 19, and 9 random measurement available. MSE Benchmark 2.40 · 10−3 48 measurements (on average every second) 5.90 · 10−3 19 measurements… view at source ↗

read the original abstract

In this work, we introduce a sample- and data-based moving horizon estimation framework for linear systems. We perform state estimation in a sample-based fashion in the sense that we assume to have only few, irregular output measurements available. This setting is encountered in applications where measuring is expensive or time-consuming. Furthermore, the state estimation framework does not rely on a standard mathematical model, but on an implicit system representation based on measured data. We prove sample-based practical robust exponential stability of the proposed estimator under mild assumptions. Furthermore, we apply the proposed scheme to estimate the states of a gastrointestinal tract absorption system.

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

The paper gives a data-driven MHE for linear systems with irregular measurements and a stability claim, but the implicit representation needs explicit rank checks to support the proof.

read the letter

This paper develops a moving horizon estimator that works from sparse, irregular output measurements by building an implicit data representation instead of using an explicit linear model. The approach targets cases where full trajectories are unavailable because measurements are expensive or slow. They prove sample-based practical robust exponential stability under mild assumptions and test it on a gastrointestinal absorption example. The combination of data-driven implicit models with MHE for irregular sampling is the main new element; prior work on data-based control usually assumes regular sampling or full trajectories. The framework is laid out clearly and the application example makes the method concrete for readers who need to estimate states with limited data. The stability result is presented as a proof rather than just simulation, which strengthens the contribution. The main soft spot is whether the implicit representation remains faithful when sampling is irregular. Data-based methods rely on the collected trajectories satisfying a persistency-of-excitation rank condition so the data matrix can replace the model. Irregular measurements can produce incomplete or rank-deficient data unless the mild assumptions explicitly ensure sufficient coverage or bound the resulting error. The paper should verify this condition or show how the proof tolerates deviations. This work is aimed at control researchers and practitioners who already use moving horizon estimation and want to move to data-driven versions when models are hard to obtain. A reader working on data-driven control or limited-sensor applications would find it useful. It deserves peer review because the core idea addresses a practical gap with some theoretical backing, even if the assumptions around the data matrix need tightening in revision.

Referee Report

2 major / 2 minor

Summary. The paper introduces a data-based moving horizon estimation (MHE) framework for linear systems that uses only a small number of irregularly sampled output measurements. It constructs an implicit system representation directly from the measured trajectories rather than relying on an explicit state-space model, and claims to prove sample-based practical robust exponential stability of the resulting estimator under mild assumptions. The approach is illustrated on a gastrointestinal tract absorption system.

Significance. If the stability result holds with the irregular-sampling case properly handled, the contribution would be notable for data-driven estimation in resource-constrained sensing scenarios. It extends ideas from the fundamental lemma and data-driven control to the MHE setting with incomplete trajectories, which is relevant for applications where full-state or regular measurements are impractical.

major comments (2)

[Abstract / Stability Analysis] Abstract and stability section: the central claim of sample-based practical robust exponential stability rests on an implicit data-based representation replacing the explicit model. The manuscript must explicitly state the mild assumptions and verify (or bound the error from) the persistency-of-excitation rank condition on the data matrix assembled from irregularly sampled trajectories; without this, the stability guarantee does not necessarily carry over, as rank deficiency can occur under sparse sampling patterns.
[Stability Proof] The proof sketch provided does not include an explicit rank condition or error bound for the irregular-sampling case. If the data matrix is only assumed to satisfy the required rank without verification or relaxation, the transfer from the data-driven representation to the practical robust exponential stability bound is not load-bearing.

minor comments (2)

[Preliminaries] Notation for the implicit representation and the moving-horizon cost should be introduced with a clear table or diagram showing how the irregular samples are assembled into the data matrix.
[Numerical Example] The numerical example on the gastrointestinal system would benefit from a table reporting estimation error norms under different sampling densities to illustrate robustness to irregularity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments highlight important aspects of the stability analysis under irregular sampling, which we address point by point below. We agree that greater explicitness is needed regarding the assumptions and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / Stability Analysis] Abstract and stability section: the central claim of sample-based practical robust exponential stability rests on an implicit data-based representation replacing the explicit model. The manuscript must explicitly state the mild assumptions and verify (or bound the error from) the persistency-of-excitation rank condition on the data matrix assembled from irregularly sampled trajectories; without this, the stability guarantee does not necessarily carry over, as rank deficiency can occur under sparse sampling patterns.

Authors: We agree that the mild assumptions require more explicit statement. The current manuscript invokes them in the stability theorem but does not list them clearly in the abstract or early in the stability section. In the revision we will add an explicit list of assumptions, including that the data matrix assembled from the irregularly sampled trajectories satisfies the persistency-of-excitation rank condition (full row rank). This condition is part of the 'mild assumptions' already used to guarantee that the implicit representation is faithful; we will also insert a short remark noting that rank deficiency may arise only under pathological sampling patterns that violate the richness of the collected data. revision: yes
Referee: [Stability Proof] The proof sketch provided does not include an explicit rank condition or error bound for the irregular-sampling case. If the data matrix is only assumed to satisfy the required rank without verification or relaxation, the transfer from the data-driven representation to the practical robust exponential stability bound is not load-bearing.

Authors: The proof proceeds by showing that, once the rank condition holds, the data-based implicit representation is algebraically equivalent to the underlying linear system, allowing the standard MHE stability arguments to carry over directly. We will revise the proof to insert an explicit invocation of the rank condition at the step where the implicit representation is introduced and to clarify that the practical robust exponential stability bound follows from this equivalence without requiring an additional error term. No separate verification procedure for the rank is provided because the condition is an assumption on the collected data (standard in data-driven literature); if the referee believes a quantitative bound on the distance to rank deficiency would strengthen the result, we are prepared to add a brief discussion of this point. revision: yes

Circularity Check

0 steps flagged

Stability proof remains independent of data fits

full rationale

The paper's core result is a mathematical proof of sample-based practical robust exponential stability for the moving-horizon estimator. It replaces an explicit model with an implicit data-based representation constructed from measured trajectories and states mild assumptions under which the proof holds. No equation or step reduces the claimed stability guarantee to a parameter fitted from the same data used for validation, nor does any load-bearing premise collapse to a self-citation chain. The gastrointestinal-tract application is presented separately after the proof, so the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a data-based implicit representation for linear systems and mild assumptions enabling the stability proof; no free parameters or invented entities are mentioned.

axioms (2)

domain assumption The underlying system is linear and the collected data provides a sufficient implicit representation to replace an explicit model.
Stated directly in the abstract as the basis for the data-based framework.
ad hoc to paper Mild assumptions hold that enable sample-based practical robust exponential stability.
Invoked in the abstract to support the stability proof but not detailed.

pith-pipeline@v0.9.0 · 5400 in / 1187 out tokens · 22549 ms · 2026-05-16T20:39:59.308444+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 represent the system dynamics by means of an implicit system representation based on the so-called fundamental lemma (Willems et al., 2005). ... u[0,N−1] is persistently exciting of order L+n.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

We prove sample-based practical robust exponential stability ... under mild assumptions.

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

3 extracted references · 3 canonical work pages

[1]

and Wakasa, Y

Adachi, R. and Wakasa, Y. (2021). Dual system representation and prediction method for data-driven estimation. In2021 60th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE), 1245–1250. IEEE. Allan, D.A., Rawlings, J., and Teel, A.R. (2021). Nonlinear de- tectability and incremental input/output-to-state stability.SIAM Jo...

work page arXiv 2021
[2]

Schiller, J.D., Muntwiler, S., K¨ ohler, J., Zeilinger, M.N., and M¨ uller, M.A

Nob Hill Publishing Madison, WI. Schiller, J.D., Muntwiler, S., K¨ ohler, J., Zeilinger, M.N., and M¨ uller, M.A. (2023). A Lyapunov function for robust stability of moving horizon estimation.IEEE Transactions on Automatic Control, 68(12), 7466–7481. Turan, M.S. and Ferrari-Trecate, G. (2022). Data-driven unknown- input observers and state estimation.IEEE...

work page arXiv 2023
[3]

Thus, we can boundα(t) as defined in (Wolff et al., 2024b, Eq

Hence, there exist au max and an xmax so that||u(t)|| ≤u max and||x(t)|| ≤x max ∀t∈I ≥0. Thus, we can boundα(t) as defined in (Wolff et al., 2024b, Eq. (23)) by ||α(t)|| ≤H ux p Lumax +x max =: αmax (A.14) with Hux = max 0≤t≤L HLt(ud [0,N−1]) H1(xd [0,N−L t−1]) !† (A.15) andσ x as defined in (Wolff et al., 2024b, Eq. (25)) by ||σx [−Lt,0](t)||2 ≤n(L t + 1...

work page 2023

[1] [1]

and Wakasa, Y

Adachi, R. and Wakasa, Y. (2021). Dual system representation and prediction method for data-driven estimation. In2021 60th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE), 1245–1250. IEEE. Allan, D.A., Rawlings, J., and Teel, A.R. (2021). Nonlinear de- tectability and incremental input/output-to-state stability.SIAM Jo...

work page arXiv 2021

[2] [2]

Schiller, J.D., Muntwiler, S., K¨ ohler, J., Zeilinger, M.N., and M¨ uller, M.A

Nob Hill Publishing Madison, WI. Schiller, J.D., Muntwiler, S., K¨ ohler, J., Zeilinger, M.N., and M¨ uller, M.A. (2023). A Lyapunov function for robust stability of moving horizon estimation.IEEE Transactions on Automatic Control, 68(12), 7466–7481. Turan, M.S. and Ferrari-Trecate, G. (2022). Data-driven unknown- input observers and state estimation.IEEE...

work page arXiv 2023

[3] [3]

Thus, we can boundα(t) as defined in (Wolff et al., 2024b, Eq

Hence, there exist au max and an xmax so that||u(t)|| ≤u max and||x(t)|| ≤x max ∀t∈I ≥0. Thus, we can boundα(t) as defined in (Wolff et al., 2024b, Eq. (23)) by ||α(t)|| ≤H ux p Lumax +x max =: αmax (A.14) with Hux = max 0≤t≤L HLt(ud [0,N−1]) H1(xd [0,N−L t−1]) !† (A.15) andσ x as defined in (Wolff et al., 2024b, Eq. (25)) by ||σx [−Lt,0](t)||2 ≤n(L t + 1...

work page 2023