Scenario Approach with Post-Design Certification of User-Specified Properties
This work addresses the need for certifying post-design properties in data-driven control systems, offering an incremental extension to the scenario approach.
The paper tackles the problem of guaranteeing additional user-specified properties after design in the scenario approach, providing distribution-free upper bounds on the risk of failing these properties without needing extra test data, with examples in H2 and pole-placement problems.
The scenario approach is an established data-driven design framework that comes equipped with a powerful theory linking design complexity to generalization properties. In this approach, data are simultaneously used both for design and for certifying the design's reliability, without resorting to a separate test dataset. This paper takes a step further by guaranteeing additional properties, useful in post-design usage but not considered during the design phase. To this end, we introduce a two-level framework of appropriateness: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation. We provide distribution-free upper bounds on the risk of failing to meet the post-design appropriateness; these bounds are computable without using any additional test data. Under additional assumptions, lower bounds are also derived. As part of an effort to demonstrate the usefulness of the proposed methodology, the paper presents two practical examples in H2 and pole-placement problems. Moreover, a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.