A Behavioral Approach to Data-Driven Control With Noisy Input–Output Data

This article deals with data-driven stability analysis and feedback stabilization of linear input–output systems in autoregressive (AR) form. We assume that noisy input–output data on a finite time-interval have been obtained from some unknown AR system. Data-based tests are then developed to analyze whether the unknown system is stable, or to verify whether a stabilizing dynamic feedback controller exists. If so, stabilizing controllers are computed using the data. In order to do this, we employ the behavioral approach to systems and control, meaning a departure from existing methods in data driven control. Our results heavily rely on a characterization of asymptotic stability of systems in AR form using the notion of quadratic difference form as a natural framework for Lyapunov functions of autonomous AR systems. We introduce the concepts of informative data for quadratic stability and quadratic stabilization in the context of input–output AR systems and establish necessary and sufficient conditions for these properties to hold. In addition, this article will build on results on quadratic matrix inequalities and a matrix version of Yakubovich's
$S$
-lemma.