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arxiv: 2604.08018 · v1 · submitted 2026-04-09 · 📡 eess.SY · cs.SY

Data-Driven Unknown Input Reconstruction for MIMO Systems with Convergence Guarantees

Enno Breukelman , Takumi Shinohara , Joowon Lee , Henrik Sandberg This is my paper

Pith reviewed 2026-05-10 18:04 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven reconstructionunknown input estimationMIMO LTI systemsHankel matricesinvariant zerosconstrained least squaresstability guarantees
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The pith

A data-driven estimator reconstructs unknown inputs to LTI MIMO systems and is strictly stable exactly when all invariant zeros lie inside the unit circle.

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 reconstruct unknown inputs in linear time-invariant multiple-input multiple-output systems using only previously recorded input and output data. It formulates an autoregressive estimator as a constrained least-squares problem on Hankel matrices, separating output consistency from matching past inputs. The central result proves that this estimator is strictly stable if and only if the invariant zeros are inside the unit circle, a property checkable directly from the data. This result connects data-driven techniques to classical model-based input reconstruction. Readers care because it enables reliable unknown input estimation without requiring a full system model or initial true inputs.

Core claim

The proposed autoregressive estimator based on constrained least-squares over Hankel matrices reconstructs unknown inputs without needing the true input for initialization. It is strictly stable if and only if all invariant zeros of the trajectory-generating system lie strictly inside the unit circle, and this condition can be verified purely from input and output data. This mirrors model-based results and closes the gap to data-driven settings.

What carries the argument

The constrained least-squares formulation over Hankel matrices that splits the reconstruction into an output-consistency constraint and an input-history-matching objective.

Load-bearing premise

The underlying system must be linear time-invariant and the recorded input-output data must be rich enough to construct representative Hankel matrices.

What would settle it

Simulate the estimator on an LTI system that has at least one invariant zero outside the unit circle and check whether the input reconstruction error remains bounded or grows without bound over time.

Figures

Figures reproduced from arXiv: 2604.08018 by Enno Breukelman, Henrik Sandberg, Joowon Lee, Takumi Shinohara.

Figure 1
Figure 1. Figure 1: With N = 10, estimation converges quickly for a system with only stable invariant zeros. No invariant zeros [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: With N = 10, estimation is exact from the first step for a strongly observable system [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: With N = 10, estimation diverges despite the algorithm finding a suitable g, due to an unstable invariant zero. V. CONCLUSIONS AND FUTURE WORK In this paper, we established a rigorous connection be￾tween model-based and data-driven input reconstruction for MIMO systems, without requiring knowledge of the initial input trajectory. The central result is that our proposed data￾driven estimator inherits the sa… view at source ↗
read the original abstract

In this paper, we consider data-driven reconstruction of unknown inputs to linear time-invariant (LTI) multiple-input multiple-output (MIMO) systems. We propose a novel autoregressive estimator based on a constrained least-squares formulation over Hankel matrices, splitting the problem into an output-consistency constraint and an input-history-matching objective. Our method relies on previously recorded input-output data to represent the system, but does not require knowledge of the true input to initialize the algorithm. We show that the proposed estimator is strictly stable if and only if all the invariant zeros of the trajectory-generating system lie strictly inside the unit circle, which can be verified purely from input and output data. This mirrors existing results from model-based input reconstruction and closes the gap between model-based and data-driven settings. Lastly, we provide numerical examples to demonstrate the theoretical results.

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 autoregressive estimator for reconstructing unknown inputs in LTI MIMO systems. It formulates the problem as a constrained least-squares optimization over Hankel matrices constructed from recorded input-output trajectories, splitting it into an output-consistency constraint and an input-history-matching objective. The central theoretical result is that the estimator is strictly stable if and only if all invariant zeros of the trajectory-generating system lie strictly inside the unit circle, with this condition verifiable purely from input-output data without knowledge of the system model or the true input for initialization. Numerical examples illustrate the approach.

Significance. If the stability result holds, the work meaningfully closes the gap between model-based unknown-input observers and data-driven methods by providing explicit convergence guarantees that mirror classical invariant-zero conditions while remaining verifiable from data alone. This is valuable for applications in fault detection, disturbance estimation, and control where accurate models are unavailable but sufficiently rich I/O trajectories can be collected. The use of the fundamental lemma to preserve the model-based iff structure without introducing fitted parameters is a clear strength.

major comments (2)
  1. [§4.2, Theorem 1] §4.2, Theorem 1: the proof that the data-driven stability condition is equivalent to the model-based invariant-zero condition (and verifiable without the system matrices) is load-bearing for the central claim; the manuscript should make explicit the step that extracts the zero locations directly from the rank properties of the data Hankel matrices under the stated richness assumption.
  2. [§3.2, Eq. (12)–(15)] §3.2, Eq. (12)–(15): the constrained least-squares split into output-consistency and input-history-matching is presented as mirroring the model-based case, but the manuscript must confirm that the constraint set remains non-empty and the solution unique for any sufficiently long trajectory satisfying the fundamental lemma, without additional implicit assumptions on the unknown input.
minor comments (3)
  1. [§2] Notation for the Hankel matrices (e.g., U_p, Y_p) is introduced clearly but the dependence on the window length L and the number of past inputs p should be stated once in a single definition block for readability.
  2. [§5] The numerical examples in §5 would benefit from an explicit statement of the system order, the length of the recorded trajectory, and the noise level used in each case to allow direct reproduction of the stability boundary behavior.
  3. [Abstract and §1] A few minor typographical inconsistencies appear in the abstract and introduction (e.g., “strictly stable” vs. “asymptotically stable”); these should be aligned with the precise statement in Theorem 1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation, the recommendation for minor revision, and the constructive comments that help improve the clarity of our theoretical results. We address each major comment below.

read point-by-point responses

  1. Referee: [§4.2, Theorem 1] §4.2, Theorem 1: the proof that the data-driven stability condition is equivalent to the model-based invariant-zero condition (and verifiable without the system matrices) is load-bearing for the central claim; the manuscript should make explicit the step that extracts the zero locations directly from the rank properties of the data Hankel matrices under the stated richness assumption.

    Authors: We agree that the equivalence proof benefits from an explicit intermediate step. In the revised manuscript we will insert a dedicated lemma immediately before Theorem 1 that shows how the rank properties of the data Hankel matrices (under the persistent-excitation condition of the fundamental lemma) directly encode the locations of the invariant zeros. The lemma will derive the correspondence between rank deficiencies in the appropriate submatrices and the condition that all zeros lie strictly inside the unit circle, using only input-output data and without reference to the system matrices. revision: yes

  2. Referee: [§3.2, Eq. (12)–(15)] §3.2, Eq. (12)–(15): the constrained least-squares split into output-consistency and input-history-matching is presented as mirroring the model-based case, but the manuscript must confirm that the constraint set remains non-empty and the solution unique for any sufficiently long trajectory satisfying the fundamental lemma, without additional implicit assumptions on the unknown input.

    Authors: We will add a short proposition in Section 3.2 confirming the required properties. Under the fundamental-lemma richness assumption (input persistently exciting of order at least equal to the system order plus the estimator horizon), the data Hankel matrix has full row rank. Consequently the output-consistency constraint defines a non-empty affine set, and the quadratic input-history-matching objective is strictly convex, guaranteeing a unique minimizer. These facts hold for any unknown input that is consistent with the recorded trajectories and require no further assumptions on its specific values. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central derivation uses the fundamental lemma to construct Hankel matrices from recorded I/O trajectories for an LTI system representation, then formulates a constrained least-squares estimator whose stability is proven equivalent to the model-based invariant-zero condition. This equivalence is shown directly via the data-driven system matrices without reducing to a fitted parameter or self-referential definition. The claim that the zero condition is verifiable purely from data is an extension of the representation, not a tautology. No load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results appear in the derivation chain. The result is self-contained against the model-based benchmark under the stated LTI and persistence-of-excitation assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from stated elements: the method assumes LTI dynamics and sufficient data richness for Hankel representation; no explicit free parameters or new entities are named, but the constrained least-squares likely involves implicit choices in formulation.

axioms (2)
  • domain assumption The system is linear time-invariant (LTI)
    Stated explicitly as the setting for the MIMO systems under consideration.
  • domain assumption Hankel matrices from input-output data sufficiently represent the system for the estimator
    Central to the constrained least-squares formulation and output-consistency constraint.

pith-pipeline@v0.9.0 · 5449 in / 1459 out tokens · 53795 ms · 2026-05-10T18:04:48.751444+00:00 · methodology

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