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arxiv: 2505.01641 · v1 · submitted 2025-05-03 · 🧮 math.OC · cs.SY· eess.SY

Data Informativity under Data Perturbation

Taira Kaminaga , Hampei Sasahara This is my paper

Pith reviewed 2026-05-22 17:09 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords data informativitydata perturbationquadratic matrix inequalitieslinear matrix inequalitiesstate feedback stabilizationoutput feedbackdata-driven controlnoise models
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The pith

A data perturbation noise model characterized by quadratic matrix inequalities allows necessary and sufficient conditions for data informativity to be expressed as tractable linear matrix inequalities.

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

The paper develops a generalized noise model called data perturbation to analyze whether collected data suffice for control objectives in the presence of noise. It derives linear matrix inequality conditions that are necessary and sufficient for the data to be informative for state-feedback stabilization, for performance guarantees under state feedback, and for output-feedback stabilization. The approach covers existing noise models for disturbances and measurement noise while relaxing prior restrictive assumptions. The central technical step is a new matrix S-procedure that certifies these conditions even though the set of systems consistent with the data is generally non-convex.

Core claim

Under the data perturbation model, in which noise satisfies quadratic matrix inequalities, the data informativity problem for stabilization and performance objectives reduces to feasibility of linear matrix inequalities. This reduction is obtained by a novel matrix S-procedure that exploits geometric properties of the quadratic-matrix-inequality solution sets and therefore does not require the set of consistent systems to be convex. The resulting conditions are both necessary and sufficient and subsume several earlier analyses as special cases.

What carries the argument

Novel matrix S-procedure for non-convex quadratic-matrix-inequality solution sets, which converts data informativity into linear-matrix-inequality feasibility without convexity assumptions.

If this is right

  • Existing results on data informativity under exogenous disturbances or measurement noise become special cases.
  • Several restrictive assumptions common in prior data-driven control literature are removed.
  • Sufficient conditions for multiple simultaneous noise sources follow by approximating their combined effect inside the quadratic-matrix-inequality framework.
  • The same linear-matrix-inequality tests apply to any noise model whose constraints can be written as quadratic matrix inequalities.

Where Pith is reading between the lines

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

  • The framework could be used to certify data sets for real-time controller adaptation under complex sensor and actuator noise.
  • Numerical tests on standard control benchmarks with synthetic perturbation noise would directly validate the linear-matrix-inequality conditions.
  • Similar geometric arguments might extend data-informativity analysis to other objectives such as optimal regulation or robust tracking.

Load-bearing premise

The geometric properties of quadratic-matrix-inequality solution sets permit a matrix S-procedure that works even when the set of consistent systems is non-convex.

What would settle it

A concrete data set, perturbation bound, and system class for which the proposed linear matrix inequalities are feasible yet the data fail to guarantee stabilization, or vice versa.

Figures

Figures reproduced from arXiv: 2505.01641 by Hampei Sasahara, Taira Kaminaga.

Figure 1
Figure 1. Figure 1: The controller K stabilizes all system in Σ. generated as follows: the elements of the data matrices X and U are independently sampled from the standard normal distribution. The matrix X+ is then generated using the true system dynamics. Data perturbations are sampled from a uniform distribution over the prescribed perturbation set D or Dstr, depending on the scenario. We employ the Metropolis algorithm [2… view at source ↗
Figure 2
Figure 2. Figure 2: H2 performance. distribution, while setting U2 to be the zero matrix. Then, the LMI (31) is feasible and we obtain the stabilizing controller K =  0.192 0.188 −0.514 0 0 0  . The singularity of N22 results in the second row of K being zero. It can be confirmed that the image of K = V−Y P −1 is included in the image of V− given as a submatrix of V , which is spanned by orthogonal bases of im N22. This res… view at source ↗
Figure 4
Figure 4. Figure 4: Proportion of feasible datasets. 1) Visualized Example: Similar to Sec. VII-A, we visu￾alize the set of systems with the one-dimensional system (A∗ , B∗ ) = (1.2, 0.6). We collect data (X+, X, U) under element-wise bounded noise represented as Dstr = nP2n+m i=1 PT j=1 ei,2n+mδij ej,T ⊤ [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

Data informativity provides a theoretical foundation for determining whether collected data are sufficiently informative to achieve specific control objectives in data-driven control frameworks. In this study, we investigate the data informativity subject to noise characterized by quadratic matrix inequalities (QMIs), which describe constraints through matrix-valued quadratic functions. We introduce a generalized noise model, referred to as data perturbation, under which we derive necessary and sufficient conditions formulated as tractable linear matrix inequalities for data informativity with respect to stabilization and performance guarantees via state feedback, as well as stabilization via output feedback. Our proposed framework encompasses and extends existing analyses that consider exogenous disturbances and measurement noise, while also relaxing several restrictive assumptions commonly made in prior work. A central challenge in the data perturbation setting arises from the non-convexity of the set of systems consistent with the data, which renders standard matrix S-procedure techniques inapplicable. To resolve this issue, we develop a novel matrix S-procedure that does not rely on convexity of the system set by exploiting geometric properties of QMI solution sets. Furthermore, we derive sufficient conditions for data informativity in the presence of multiple noise sources by approximating the combined noise effect through the QMI framework. The proposed results are broadly applicable to a wide class of noise models and subsume several existing methodologies as special cases.

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

1 major / 2 minor

Summary. The manuscript introduces a generalized 'data perturbation' noise model characterized by quadratic matrix inequalities (QMIs) and derives necessary-and-sufficient LMI conditions for data informativity with respect to stabilization and performance guarantees via state feedback as well as stabilization via output feedback. It develops a novel matrix S-procedure that exploits geometric properties of QMI solution sets to handle non-convexity of the consistent-system set without relying on convexity assumptions, while also providing sufficient conditions for multiple noise sources; the framework extends prior work on exogenous disturbances and measurement noise and subsumes several existing methodologies as special cases.

Significance. If the geometric argument in the novel S-procedure is shown to be lossless and to preserve necessity for non-convex QMI sets, the results would meaningfully advance data-driven control by relaxing restrictive assumptions common in the literature and supplying tractable LMI conditions applicable to a broad class of noise models. The explicit treatment of both state-feedback and output-feedback cases strengthens the contribution.

major comments (1)
  1. [Abstract] Abstract (paragraph on central challenge): the claim that the novel matrix S-procedure 'does not rely on convexity of the system set by exploiting geometric properties of QMI solution sets' is load-bearing for the necessary-and-sufficient LMI conditions. The manuscript must state the precise geometric assumptions and supply an explicit derivation (or counter-example verification) showing that the reformulation remains exact for non-convex sets; otherwise necessity may fail and the conditions become conservative.
minor comments (2)
  1. Clarify the precise definition of the data-perturbation noise model (e.g., the quadratic form and the set of admissible perturbations) early in the introduction so that the subsequent geometric argument is immediately interpretable.
  2. Add a short remark on how the derived LMIs reduce to known results when the perturbation set is restricted to the special cases of exogenous disturbance or measurement noise.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point directly below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on central challenge): the claim that the novel matrix S-procedure 'does not rely on convexity of the system set by exploiting geometric properties of QMI solution sets' is load-bearing for the necessary-and-sufficient LMI conditions. The manuscript must state the precise geometric assumptions and supply an explicit derivation (or counter-example verification) showing that the reformulation remains exact for non-convex sets; otherwise necessity may fail and the conditions become conservative.

    Authors: We appreciate the referee highlighting that the exactness of the novel matrix S-procedure is central to the necessity claims. The derivation in the manuscript exploits the geometric structure of QMI solution sets—specifically, that any pair of matrices satisfying the defining quadratic matrix inequality can be separated via an auxiliary multiplier matrix whose existence is equivalent to the original set membership, without invoking convexity of the overall consistent-system set. This equivalence follows from the range properties of the data matrices and the fact that the QMI defines a closed set whose boundary permits a lossless reformulation. To make this fully transparent, we will revise the manuscript by adding an explicit statement of the geometric assumptions (compactness of the perturbation set and full row rank of the data matrix) together with a self-contained derivation in Section 3 (or a new appendix) that demonstrates necessity is preserved for non-convex cases. We are also prepared to include a brief remark on the conditions under which the result would become conservative. Should the referee have a concrete counter-example in mind, we would welcome the opportunity to examine it. revision: yes

Circularity Check

0 steps flagged

No circularity: LMI conditions derived via novel geometric S-procedure from first principles

full rationale

The derivation starts from the data perturbation model expressed as QMIs, identifies the non-convex consistent-system set as the central obstacle, and introduces a new matrix S-procedure that exploits geometric properties of QMI solution sets to obtain necessary and sufficient LMIs. This reformulation is presented as an independent technical step rather than a renaming, fit, or self-referential definition of the target informativity conditions. No equations reduce the final LMI statements to quantities fitted from the same data, and the framework is shown to subsume prior results as special cases without load-bearing self-citation chains. The central claim therefore retains independent content outside its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard convex-optimization tools plus a domain-specific geometric assumption about QMI solution sets; no free parameters or new physical entities are introduced.

axioms (2)
  • standard math Quadratic matrix inequalities and linear matrix inequalities obey the usual algebraic and duality properties used in robust control
    Invoked throughout the derivation of the LMI conditions.
  • domain assumption Geometric properties of the solution sets of quadratic matrix inequalities permit a matrix S-procedure even when the consistent-system set is non-convex
    This is the key premise that replaces the convexity requirement of the classical S-procedure and is stated as the resolution to the central challenge.
invented entities (1)
  • Data perturbation noise model no independent evidence
    purpose: Generalized description of noise via quadratic matrix inequalities that subsumes exogenous disturbances and measurement noise
    Introduced as the central modeling extension; no independent falsifiable prediction outside the paper is supplied.

pith-pipeline@v0.9.0 · 5760 in / 1731 out tokens · 62593 ms · 2026-05-22T17:09:34.790185+00:00 · methodology

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
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    Relation between the paper passage and the cited Recognition theorem.

    A central challenge ... non-convexity of the set of systems consistent with the data ... novel matrix S-procedure that does not rely on convexity of the system set by exploiting geometric properties of QMI solution sets.

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Forward citations

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  2. Scalable Outer Approximation of Minkowski Sums of Matrix Ellipsoids for Data-Driven Control

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  3. Adversarial Destabilization Attacks to Direct Data-Driven Control

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    Small adversarial perturbations to training data can destabilize data-driven LQR controllers, and the paper proposes DGSM attacks plus regularization and robust control defenses.

Reference graph

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