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arxiv: 2604.08328 · v1 · submitted 2026-04-09 · 📡 eess.SY · cs.SY

Data-Driven Moving Horizon Estimators for Linear Systems with Sample Complexity Analysis

Peihu Duan , Jiabao He , Yuezu Lv , Guanghui Wen This is my paper

Pith reviewed 2026-05-10 17:16 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven estimationmoving horizon estimationlinear systemssample complexitystate estimationGaussian noisepersistent excitation
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The pith

A data-driven moving horizon estimator keeps the expected state estimation error ultimately bounded for unknown linear systems.

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

The paper designs a data-driven moving horizon estimator that solves an optimization problem built from both offline and online measurements to recover the state of a linear system whose matrices are unknown. It proves that the expected 2-norm of the resulting estimation error remains ultimately bounded and supplies an explicit upper bound in terms of the Gaussian noise covariances. Sample complexity analysis then shows that longer offline data records tighten this bound, while also quantifying how close the data-driven estimator comes to the performance of a conventional moving horizon estimator that knows the exact system matrices.

Core claim

By formulating and solving an optimization problem that incorporates both offline and online system data, a novel data-driven moving horizon estimator (DDMHE) is designed. The expected 2-norm of the estimation error is ultimately bounded, with an explicit relationship to the system noise covariances. Sample complexity analysis establishes how the length of the offline data affects the estimation error, and the performance gap to the traditional moving horizon estimator with known matrices is quantified.

What carries the argument

The data-driven moving horizon estimator (DDMHE), an optimization problem that replaces explicit system matrices with data matrices assembled from offline and online measurements to produce the state estimate.

If this is right

  • The expected 2-norm of the estimation error decreases as the length of the offline data increases, following the derived sample complexity bounds.
  • The performance difference between the data-driven estimator and the model-based moving horizon estimator is explicitly bounded by terms involving noise covariances and data quality.
  • The ultimate boundedness result continues to hold for any fixed noise level once the offline data length exceeds a threshold determined by the sample complexity analysis.

Where Pith is reading between the lines

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

  • The same data-driven construction could be tested on systems whose parameters drift slowly, provided fresh offline batches are collected periodically.
  • In closed-loop control settings the bounded-error guarantee may allow the estimator to be placed directly inside a feedback loop without a separate identification step.

Load-bearing premise

The offline data must be rich enough to satisfy persistent excitation so that the data-driven formulation can stand in for the unknown system dynamics.

What would settle it

Numerical or experimental trials in which the expected estimation error grows unbounded as the offline data record is lengthened under persistent excitation would contradict the claimed ultimate boundedness.

Figures

Figures reproduced from arXiv: 2604.08328 by Guanghui Wen, Jiabao He, Peihu Duan, Yuezu Lv.

Figure 1
Figure 1. Figure 1: The pre-collected state-input-output system trajectory, where yellow, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The true state trajectories and the corresponding estimates of the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The values of MSE of 200 Monte Carlo trials using different estimation [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

This paper investigates the state estimation problem for linear systems subject to Gaussian noise, where the model parameters are unknown. By formulating and solving an optimization problem that incorporates both offline and online system data, a novel data-driven moving horizon estimator (DDMHE) is designed. We prove that the expected 2-norm of the estimation error of the proposed DDMHE is ultimately bounded. Further, we establish an explicit relationship between the system noise covariances and the estimation error of the proposed DDMHE. Moreover, through a sample complexity analysis, we show how the length of the offline data affects the estimation error of the proposed DDMHE. We also quantify the performance gap between the proposed DDMHE using noisy data and the traditional moving horizon estimator with known system matrices. Finally, the theoretical results are validated through 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 / 3 minor

Summary. The paper proposes a data-driven moving horizon estimator (DDMHE) for linear systems with unknown parameters subject to Gaussian noise. It replaces unknown system matrices in the MHE optimization with Hankel matrices built from offline input-output trajectories, proves that the expected 2-norm of the estimation error is ultimately bounded with an explicit dependence on noise covariances, derives a sample-complexity result linking offline data length to the error bound, quantifies the performance gap to the model-based MHE, and validates the claims via numerical simulations.

Significance. If the central bounds hold, the work provides a useful advance in data-driven state estimation by supplying rigorous error guarantees and concrete data-length requirements. The sample-complexity analysis and explicit gap quantification are particular strengths, offering practical guidance for applications where system matrices are unavailable. These elements could influence adaptive control and identification tasks.

major comments (2)
  1. [Proof of the main bounded-error theorem] The proof of the ultimate boundedness result combines the standard MHE error recursion with a data-driven perturbation term controlled by offline data length. The step establishing that the expected error remains ultimately bounded requires verifying that the perturbation bound is uniform in time and that the expectation can be interchanged with the limit; this justification is not fully detailed and is load-bearing for the central claim.
  2. [Sample complexity section] The sample-complexity analysis supplies the minimal offline trajectory length to make the representation error smaller than a tolerance. However, the explicit dependence of the high-probability statement on both the data length and the noise variance is not stated in closed form, which affects how the ultimate error bound can be computed a priori.
minor comments (3)
  1. [DDMHE formulation] The definition and dimensions of the Hankel matrices should be recalled explicitly at the start of the DDMHE formulation section for readers unfamiliar with data-driven methods.
  2. [Numerical examples] The numerical simulations would be strengthened by a direct comparison of observed error decay rates against the predicted sample-complexity bound.
  3. [Introduction and references] A few recent references on noisy extensions of Willems' fundamental lemma would help contextualize the data-driven representation step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments on our manuscript. We address each major comment point by point below, providing clarifications where needed and indicating the revisions we will make to strengthen the presentation of the proofs and results.

read point-by-point responses

  1. Referee: [Proof of the main bounded-error theorem] The proof of the ultimate boundedness result combines the standard MHE error recursion with a data-driven perturbation term controlled by offline data length. The step establishing that the expected error remains ultimately bounded requires verifying that the perturbation bound is uniform in time and that the expectation can be interchanged with the limit; this justification is not fully detailed and is load-bearing for the central claim.

    Authors: We appreciate the referee pointing out the need for additional detail on this step. The perturbation term is generated by the fixed offline data matrices and is therefore independent of the online time index, yielding a uniform bound over all t. For the interchange, the error recursion satisfies a uniform contraction property with an integrable dominating function (arising from the Gaussian noise assumption), so the limsup of the expectation equals the expectation of the limsup by the dominated convergence theorem. In the revised manuscript we will insert a short lemma immediately after the main theorem that explicitly states these two facts and verifies the conditions. revision: yes

  2. Referee: [Sample complexity section] The sample-complexity analysis supplies the minimal offline trajectory length to make the representation error smaller than a tolerance. However, the explicit dependence of the high-probability statement on both the data length and the noise variance is not stated in closed form, which affects how the ultimate error bound can be computed a priori.

    Authors: We agree that an explicit closed-form expression would improve usability. The high-probability bound on the Hankel-matrix representation error is obtained via a matrix concentration inequality and takes the form P(‖E_N‖ > ε) ≤ 2 exp(−c N ε² / σ²), where N is the offline length, σ² is the noise variance, and c is a universal constant depending only on the system dimension. We will add this explicit expression to the sample-complexity theorem and substitute it directly into the ultimate error bound, thereby making the a-priori computation transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain.

full rationale

The central derivation replaces unknown system matrices with Hankel matrices constructed from offline input-output trajectories under an explicit persistent-excitation hypothesis, then obtains the ultimate bound on expected 2-norm error by adding a perturbation term to the standard MHE recursion and controlling the perturbation size via sample-complexity bounds on the data length. The performance gap to the known-matrix MHE is quantified by direct comparison of the two optimization problems and their respective error recursions. No equation reduces the claimed bound to a fitted parameter renamed as a prediction, no load-bearing premise rests solely on a self-citation, and the persistent-excitation and Gaussian-noise assumptions are stated as external hypotheses rather than derived from the result itself. The analysis is therefore self-contained against the stated assumptions and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions for linear systems and Gaussian noise plus the existence of sufficiently informative offline data; no new entities are invented and no free parameters are explicitly fitted beyond typical horizon length choices.

axioms (2)
  • domain assumption The system is linear time-invariant with additive Gaussian noise.
    Stated in the abstract as the setting for the state estimation problem.
  • domain assumption Offline data is available and can be used to replace unknown system matrices in the estimator.
    Core to the data-driven formulation.

pith-pipeline@v0.9.0 · 5444 in / 1368 out tokens · 40327 ms · 2026-05-10T17:16:59.043962+00:00 · methodology

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