Data Informativity under Data Perturbation

arXiv:2505.01641 · math.OC, cs.SY, eess.SY · Submitted 2025-05-03 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "Data Informativity under Data Perturbation".

Rosa: This study introduces "data perturbation" as a novel and generalized noise model to characterize data informativity,

Dev: First, who's behind it and why it matters.

Title and authors: Rosa: So, to kick things off, we're looking at "Data Informativity under Data Perturbation." This paper explores how much information our data actually contains when the system is subject to noise. It’s a foundational piece because it sets up the entire framework.

Dev: The authors are Taira Kaminaga and Hampei Sasahara, and their title immediately signals that they are tackling the core problem of determining if collected data is sufficient for control objectives in data-driven frameworks.

Taro: I’m interested in how this relates to autonomous systems; specifically, if this framework can be applied when the system itself is interacting with an uncertain environment rather than just internal noise.

Rosa: That’s a valid question, Taro; the paper is introducing a generalized noise model called data perturbation that covers both exogenous disturbances and measurement noise subject to linear constraints through quadratic matrix inequalities.

Dev: So it’s essentially unifying different types of prior analyses by defining this new model, which is what makes the title so significant because it provides a comprehensive language for describing data informativity under uncertainty.

Taro: If it unifies those models, does that mean we can use one set of tools to analyze systems with external shocks and sensor errors simultaneously?

Rosa: Precisely, Dev; this unified framework encompasses and extends existing analyses that consider both exogenous disturbances and measurement noise four–six, eight and extends those addressing measurement noise thirteen.

Dev: That unification is what allows them to generalize the energy bound formulation to a broader class of QMI constraints, while also removing restrictive assumptions such as the requirement for a sufficiently large signal-to-noise ratio or SNR.

Taro: Removing that SNR requirement is something I really care about because in real deployments, we rarely have perfect sensors or perfectly clean data streams; this generalization makes the results much more practical for actual engineering applications.

Rosa: It certainly seems broader than what we used to consider before, and they tackle a central challenge arising from the non-convexity of the set of systems consistent with the observed data.

Dev: That non-convexity is exactly where things get difficult because it usually precludes the use of standard tools like the matrix S-procedure.

Taro: So what’s their big methodological move here to deal with that non-convexity problem?

Rosa: They resolve this by developing a novel matrix S-procedure that doesn't rely on the convexity of the system set, instead exploiting geometric properties related to the solution sets of Quadratic Matrix Inequalities.

Dev: That shift in methodology is key because it means we can move past those restrictive assumptions about how well-behaved our underlying system set needs to be for traditional proofs to work.

The paper's summary: Rosa: Now that we’ve talked about the setup, let's look at what they actually found in this paper, "Data Informativity under Data Perturbation." They show how to characterize the set of systems consistent with data under this new perturbation model.

Dev: The core finding here is that they derive conditions to describe this set of consistent systems using a Quadratic Matrix Inequality, but they point out that the equivalence between the classical QMI description and the solution set of a QMI does not hold generally under their proposed data perturbation model.

Taro: So, if it doesn't hold generally, what is their strategy for bridging that gap and establishing when we *can* use a QMI representation?

Rosa: To solve this issue, they provide a sufficient condition under which the set of consistent systems can be equivalently represented via a QMI, and they also reveal that this sufficient condition is necessary when certain conditions on the noise set are met.

Dev: That provides a rigorous way to define exactly what the data is actually telling us about the system dynamics by providing these necessary and sufficient LMI conditions for data informativity under quadratic stabilization.

Taro: So, if we can characterize this set of systems precisely, does that give us a definitive answer on whether our collected data is sufficient to meet a specific control objective?

Rosa: It gives us that characterization by distinguishing between two system sets, R for systems consistent with data and N for systems satisfying specific constraints, and they establish that = R under certain conditions related to the constraint matrix E and twenty-two.

Dev: Characterizing these sets provides a rigorous way to define exactly what the data is actually telling us about the system dynamics, which is pretty powerful because it moves beyond just guessing.

Taro: That rigor allows us to move from qualitative intuition about data sufficiency to a quantitative proof that ties directly into achievable control objectives.

Rosa: This groundwork sets the stage for deeper analysis in subsequent sections, where they extend these results into optimal control and output feedback, moving beyond simple stabilization conditions.

The paper's improvements: Dev: Moving on to the specific improvements they suggest in "Data Informativity under Data Perturbation," we see how this framework enhances previous work. They show how it broadens the scope of applicability by relaxing restrictive assumptions commonly made in prior studies.

Rosa: One major improvement is explicitly removing the requirement for a sufficiently large signal-to-noise ratio, which means their results can be applied even when data quality isn't perfect.

Dev: That’s a big deal because it means we can use these techniques on data streams that are inherently noisy, which is something that applies directly to many industrial and field robotics applications.

Taro: If they've relaxed the SNR assumption, does this imply they can now analyze systems where the noise is dominating the signal, which is a scenario we encounter often in real-world scenarios.

Rosa: Yes, they do; their results generalize previous analyses involving exogenous disturbances four–six, eight and extend those addressing measurement noise thirteen by generalizing the energy bound formulation to a broader class of QMI constraints.

Dev: It extends the analysis beyond just measurement noise, which is great because it covers both process noise and sensor errors in one theoretical structure, unifying analyses previously treated separately.

Taro: That unification is what allows them to tackle mixed noise environments where we have both process noise and sensor errors simultaneously, which is a scenario we encounter often in real-world scenarios.

Rosa: Furthermore, they introduce a framework for structured data perturbation that includes superposition of exogenous disturbance and measurement noise, Hankel-structured perturbation, and element-wise bounded perturbation.

Dev: Dealing with those specific structures is where the co-design strategy comes into play; it’s an outer QMI approximation of the combined noise region that helps find a stabilizing controller K simultaneously.

Taro: That structured approach seems like a practical way to handle complex, realistic noise patterns without having to solve intractable problems in every single specific case separately.

Rosa: And they also provide conditions for achieving H2 and H∞ performance guarantees via state feedback under data perturbation, characterized by LMIs, which is a key extension beyond just basic stabilization.

Dev: Moving toward performance guarantees through these LMIs means we can design controllers that are optimized not just for stability but also for specific error bounds over time, which ties directly into the practical needs of system engineers.

Conclusion: Rosa: So, to wrap up on this paper, the main implications are that they’ve provided a unified noise framework and derived necessary and sufficient conditions for quadratic stabilization using a novel matrix S-procedure.

Dev: They’ve shown we can get robust stability guarantees even when data quality is imperfect by removing the SNR requirement and extending the results to performance metrics like H2 and H∞ bounds.

Taro: I think the biggest practical implication lies in their co-design strategy for structured perturbations, which seems like the most impactful part for pushing these techniques from theoretical papers into deployable, robust control systems.

Rosa: I agree with Taro; it’s about making the math practical enough for actual engineering implementation in complex environments.

Dev: This paper provides a rigorous characterization of data informativity across various noise models, which is a great step toward designing truly resilient AI controllers that can handle messy real-world data.

Taro: It really shows that even when the data structure is messy, there are still mathematically sound ways to ensure the AI system achieves its control objectives.

Rosa: Exactly; we’re looking forward to seeing how this framework translates into tangible results in the next phase of research.

Department of Systems and Control Engineering, Graduate School of Engineering, Institute of Science Tokyo

math.OC, cs.SY, eess.SY

Submitted: 2025-05-03

Updated: 2026-09-30

Comments: 16 pages. Accepted manuscript. Published in IEEE Transactions on Automatic Control (Early Access)

DOI: 10.1109/TAC.2026.3738318

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: This study introduces "data perturbation" as a novel and generalized noise model to characterize data informativity, providing necessary and sufficient conditions formulated as tractable linear

Key concepts

Data Perturbation: A novel and generalized noise model introduced in the paper to characterize data informativity. It covers both exogenous disturbances and measurement noise subject to linear constraints through quadratic matrix inequalities, unifying different prior analyses.
Quadratic Matrix Inequality (QMI): A mathematical tool used in the paper to describe the set of systems consistent with observed data. The authors develop a novel matrix S-procedure that exploits geometric properties related to QMI solution sets instead of relying on system set convexity.
Signal-to-Noise Ratio (SNR) Requirement: A restrictive assumption in prior work that requires a sufficiently large SNR for the results to hold. The new framework removes this requirement, allowing the analysis to be applied even when data quality is imperfect and noise is dominating the signal.

Terminology

Trace — 5× in this paper · explained in 2 episodes
Homogeneity — 4× in this paper · explained in 2 episodes
Signal-to-noise ratio (SNR) — 3× in this paper · explained in 2 episodes
Core — 2× in this paper · explained in 2 episodes
Generalization — 2× in this paper · explained in 13 episodes
Objective Function — 2× in this paper · explained in 2 episodes
Unified Framework — 2× in this paper · explained in 5 episodes
Computational Tractability — 1× in this paper · explained in 2 episodes
Decoupling — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 5 episodes
Verification — 1× in this paper · explained in 2 episodes

Summary

This study introduces data perturbation as a novel and generalized noise model to characterize data informativity, providing necessary and sufficient conditions formulated as tractable linear matrix inequalities (LMIs) for various control objectives under uncertainty. The work is significant because it unifies analyses previously treated separately for exogenous disturbances and measurement noise, extends the applicability of existing frameworks by relaxing restrictive assumptions like the requirement for a sufficiently large signal-to-noise ratio (SNR), and develops novel procedures to handle non-convex system sets arising from data perturbation.

Data Perturbation Framework

The paper introduces data perturbation as a generalized noise model that encompasses both exogenous disturbance and measurement noise. This framework is defined by additive noise to the whole data subject to linear constraints, unifying a wide range of existing noise models. The concept of data informativity is characterized by an inclusion relationship between the set of systems consistent with the observed data and the set of systems for which a particular controller achieves the desired control objective. A central challenge addressed is that under this model, the set of systems consistent with data may be non-convex, thereby precluding the use of the standard matrix S-procedure. To resolve this, a novel matrix S-procedure that does not rely on convexity of the system set is developed by exploiting geometric properties of QMI solution sets.

Characterization of Consistent System Sets

The paper derives conditions to characterize the set of systems consistent with data under data perturbation. Under classical assumptions (exogenous disturbance or measurement noise), this set can be described as the solution set of a QMI, but this equivalence does not generally hold under the proposed model. The authors provide a sufficient condition under which the set of consistent systems can be equivalently represented via a QMI, and reveal that this sufficient condition is also necessary when certain conditions on the noise set are met. This leads to characterizing two system sets, denoted as ΣR (for systems consistent with data) and ΣN (for systems satisfying specific constraints), and establishes that Σ = ΣR under certain conditions related to the constraint matrix E and Φˆ22.

Data Informativity for Quadratic Stabilization

The study derives necessary and sufficient LMI conditions for data informativity with respect to quadratic stabilization via state feedback under data perturbation. These results generalize previous analyses by eliminating restrictive assumptions such as the requirement for a sufficiently large SNR. The framework is extended to include:

Necessary and sufficient conditions for stabilization via state feedback under data perturbation, which are formulated as LMIs equivalent to data informativity.
Conditions for achieving H2 and H∞ performance guarantees via state feedback under data perturbation, also characterized by LMIs.

Extensions to Output Feedback Control and Structured Perturbations

The analysis is extended to more complex control settings:

Stabilization via output feedback under the proposed data perturbation model, where a transformation is developed to reformulate the QMI involving statespace matrices into an equivalent QMI involving coefficient matrices, removing restrictive assumptions on the data.
A framework for structured data perturbation, which includes superposition of exogenous disturbance and measurement noise, Hankel-structured perturbation, and element-wise bounded perturbation. This is approximated by an outer QMI approximation of the combined noise region, leading to a co-design strategy that simultaneously determines the outer approximation matrix Φ and a stabilizing controller K.

Numerical Validation

The theoretical findings are validated through numerical examples. The paper presents visualizations of quadratic stabilization using one-dimensional systems and demonstrates the effectiveness of the co-design approach for structured data perturbation by comparing its performance against a two-step heuristic method, showing that the co-design method is more suitable for stabilizing controller synthesis. Furthermore, it shows that as data length T increases, the H2 performance improves and converges to a limiting value.

Conclusion

The main contributions include introducing the unified noise framework of data perturbation, deriving necessary and sufficient conditions for quadratic stabilization under this model with a novel matrix S-procedure, extending results to optimal control and output feedback, and proposing a computationally tractable co-design method for structured data perturbations that leads to less conservative control design. The derived LMIs provide rigorous characterizations of data informativity across various noise models.

Index Terms

Data-driven control, linear matrix inequality, robust control.

References

[1] Z.-S. Hou and Z. Wang, “From model-based control to data-driven control: Survey, classification and perspective,” Information Sciences, vol. 235, pp. 3–35, 2013.

[2] H. J. van Waarde et al., “Data informativity: A new perspective on data-driven analysis and control,” IEEE Trans. Automat. Contr., vol. 65, no. 11, pp. 4753–4768, 2020.

[3] H. J. Van Waarde et al.

Improvements for AI systems

As an expert researcher, I have analyzed this paper, Data Informativity under Data Perturbation, which provides a rigorous mathematical framework for determining if collected data is sufficient for controlling systems subject to noise (data perturbation).

The primary improvement offered by this paper is a shift from traditional, restrictive assumptions (like requiring high Signal-to-Noise Ratio or assuming system set convexity) to a unified, more general framework.

Here are the specific improvements and what the improved AI system can achieve:

) Improved AI System Capabilities: Robust and Data-Efficient Control Design

The core capability gained is the ability to design controllers that are guaranteed to stabilize or meet performance objectives even when the input/output data used for learning or control synthesis is noisy, corrupted, or structured in complex ways.

Specifically, the improved system can perform the following tasks:

[] Robust Stabilization under General Noise Models: The system can design a state-feedback controller that guarantees stabilization for an unknown linear time-invariant (LTI) system when the input/output data is subject to additive noise constrained by a general Quadratic Matrix Inequality (QMI) or, more powerfully, under the novel data perturbation model.
[] Handling Mixed Noise Environments: The system can simultaneously handle both exogenous disturbances (like process noise or external shocks) and measurement noise (errors in sensor data) within a single theoretical structure, unifying analyses previously treated separately.
[] Robustness Against Structured Data Corruption: The system can be designed to maintain performance when the data perturbation has a specific structure—such as sequential dependencies (Hankel-structured), instantaneous bounds at each time step, or element-wise bounded errors—by using an outer QMI approximation and a tractable co-design strategy.
[] Controller Synthesis from Low-Rank/Rank-Deficient Data: The system can derive stabilizing controllers even when the collected data matrices (e.g., the input/output data matrix) do not possess full row rank, a scenario that often invalidates previous, more restrictive analyses (like those requiring high SNR).
[] Optimal Control and Performance Guarantees: The system can be optimized for specific performance metrics beyond simple stabilization, such as minimizing the H2 norm (energy of the system response) or achieving guaranteed H∞ performance bounds, under these robust data uncertainty conditions.

) Specific Technical Improvements Enabled by the Paper's Framework:

The paper provides several advanced mathematical tools that translate directly into more powerful AI/control algorithms:

[] Novel Matrix S-Procedure for Non-Convex Sets: The system gains a novel matrix S-procedure that does not rely on the convexity of the system set. This allows the controller synthesis to be applied even when data perturbation leads to a non-convex set of consistent systems, which was a major limitation in prior work.
[] Unified LMI Characterization: The paper provides an LMI (specifically, equation 31 for stabilization) that serves as a necessary and sufficient condition for data informativity under the generalized data perturbation model, eliminating the need to verify complex system set inclusions directly.
[] Co-design Strategy for Structured Perturbations: For structured noise (like Hankel perturbations), the paper proposes a computationally tractable co-design approach that simultaneously determines both the best approximation of the noise set and a stabilizing controller via an LMI formulation, transforming what would otherwise be an intractable Bilinear Matrix Inequality (BMI) problem into an equivalent, solvable LMI problem.
[] Relaxed Assumptions for Output Feedback: The paper introduces a transformation to reformulate QMI conditions for output feedback control using coefficient matrices instead of state-space matrices, removing the restrictive full-rank data requirement often imposed in prior studies.

In summary, this paper enables the development of AI systems (specifically data-driven controllers) that are significantly more robust, versatile across different noise types, and capable of operating effectively with less perfect or structured training/control data.

Sources

Controller Synthesis from Noisy-Input Noisy-Output Data