Development of direct data-driven control methods with stability guarantee

The Direct Data-Driven Control (DDDC) is an approach to controller design which solely relies on data collected though an experiment on the plant. This approach is mainly advantageous when it is difficult to obtain an accurate model for the system, or when the dynamics of the plant is known to be nonlinear and, in general, very complex. The DDDC approach is very flexible, because it can adapt to changes and can be more robust to uncertainties rather than classical control techniques. This thesis explores the main techniques developed within the framework of DDDC, pointing out both their strengths and limitations. Based on the model-matching formulation, we present an alternative approach that aims to ensure that the controller designed with our method results to be stable and robust to uncertainties, even in the presence of noise in the data. The objective of this work is to improve the potential applicability of DDDC in real-world contexts, since actual methods in literature often exploit strong assumptions, such as the availability of infinite data, which is not feasible in practice. Therefore, the proposed approach aims to make data-based control more reliable and useful in realistic scenarios. In particular, we focus on the design of two types of controllers: fully parameterized and linearly parameterized controllers. The thesis makes two main contributions. First, it suggests a method to design a fully parameterized controller that guarantees the closed-loop stability of the system when only a finite amount of data is available, in the noise-free case, by means of polynomial optimization techniques solved with semidefinite positive relaxation methods. Second, it develops a method to design a linearly parameterized controller that ensures the closed-loop stability of the system, by formulating an optimization problem whose constraints include information on the maximum magnitude of the noise to enhance the approach’s robustness in the presence of noise affecting the data. Finally, the proposed methods address controller sparsity by proposing an algorithm based on l1-norm regularization to optimally select the most relevant terms required for design.