Data Informativity under Data Perturbation

TL;DR

This paper develops a comprehensive theoretical framework for assessing data informativity in control systems under quadratic matrix inequality-based noise models, extending existing methods and addressing non-convexity challenges.

Contribution

It introduces a generalized noise model called data perturbation, with tractable LMI conditions for data informativity, and develops a novel matrix S-procedure to handle non-convex system sets.

Findings

Derived necessary and sufficient conditions for data informativity under data perturbation.

Extended existing analyses to include multiple noise sources and relaxed assumptions.

Developed a novel matrix S-procedure exploiting geometric properties of QMI solutions.

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.