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Data Informativity under Data Perturbation

Taira Kaminaga, Hampei Sasahara

TL;DR

This work addresses data informativity for control when data are perturbed by general noise modeled via QMIs. It introduces data perturbation as a unifying framework that captures exogenous disturbances and measurement noise, and develops a novel matrix S-procedure to handle nonconvex sets of systems consistent with data. The authors derive necessary and sufficient LMIs for quadratic stabilization, and extend the analysis to H2/H∞ performance and output-feedback control, including structured perturbations with outer-QMI approximations and a co-design method that couples noise-approximation and controller synthesis. The approach yields broadly applicable, computationally tractable conditions that relax strong data assumptions and are supported by diverse numerical examples demonstrating improved feasibility and performance.

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.

Data Informativity under Data Perturbation

TL;DR

This work addresses data informativity for control when data are perturbed by general noise modeled via QMIs. It introduces data perturbation as a unifying framework that captures exogenous disturbances and measurement noise, and develops a novel matrix S-procedure to handle nonconvex sets of systems consistent with data. The authors derive necessary and sufficient LMIs for quadratic stabilization, and extend the analysis to H2/H∞ performance and output-feedback control, including structured perturbations with outer-QMI approximations and a co-design method that couples noise-approximation and controller synthesis. The approach yields broadly applicable, computationally tractable conditions that relax strong data assumptions and are supported by diverse numerical examples demonstrating improved feasibility and performance.

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.
Paper Structure (26 sections, 22 theorems, 108 equations, 4 figures)

This paper contains 26 sections, 22 theorems, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Data Informativity for Quadratic Stabilization under Data Perturbations
  4. Data Perturbation
  5. Problem Formulation under Data Perturbation
  6. QMI Characterization of Consistent System Set
  7. LMI Characterization of Data Informativity
  8. Data Informativity for Unbounded System Set
  9. Data Informativity for Optimal Control under Data Perturbation
  10. ${\mathcal{H}_2}$ control
  11. ${\mathcal{H}_\infty}$ control
  12. Data Informativity with Output Feedback Control under Data Perturbation
  13. Problem Formulation
  14. LMI Characterization of Data Informativity
  15. Data Informativity under Structured Data Perturbation
  16. ...and 11 more sections

Key Result

Proposition II.1

Let $M, N\in\mathbb{S}^{q+r}$. Assume that $N\in\Pi_{q, r}$ and $M_{22}\preceq 0$. Then, $\mathcal{Z}_{q,r}(N)\subseteq\mathcal{Z}_{q,r}^+(M)$ holds if and only if there exist scalars $\alpha\geq 0$ and $\beta>0$ such that

Figures (4)

  • Figure 1: The controller $K$ stabilizes all system in $\Sigma$.
  • Figure 2: $\mathcal{H}_2$ performance.
  • Figure 3: Comparison of the sets of systems.
  • Figure 4: Proportion of feasible datasets.

Theorems & Definitions (44)

  • Proposition II.1: DDCQMI:Waarde2023_siam_qmi
  • Definition III.2
  • Lemma III.3
  • proof
  • Lemma III.4
  • proof
  • Theorem III.5
  • proof
  • Theorem III.6
  • Lemma III.7
  • ...and 34 more