Learning and optimal control of uncertain systems via data-driven methods with stability guarantees

With the advent of largely available computational power and datasets, a broad number of new algorithms for learning and control of dynamical systems has been proposed; however, the complexity of the considered frameworks has outpaced the theoretical analysis of these new techniques. To reduce this gap, this thesis focuses on the development of theoretical tools and control algorithms for the learning and optimal control of uncertain systems. The main challenges in these frameworks regard the problems of i) guaranteeing informative systems trajectories, ii) extracting the information from measurable quantities and iii) designing stabilizing and optimal control laws from gathered data. We start by addressing the problem of giving Persistency of Excitation (PE) guarantees in the context of Linear Time-Invariant (LTI) systems, finding necessary and sufficient conditions to obtain this property via an input signal. Our results are developed within a notation which underlines the perfect analogies between the continuous- and discrete-time frameworks. Next, we study data-driven approaches for the stabilization of LTI systems. We start by addressing the design of model-free observers that extract full-state information from output data, and we proceed with the design of controllers via LMIs when state measurements and derivatives are unavailable. Next, we consider the Linear Quadratic Regulator (LQR) problem, and we propose a nonlinear on-policy controller which globally converges to the optimal feedback preserving the stability of the interconnection during the transient. Finally, leaving the linear framework, we design a model-free optimal control algorithm which, differently from other techniques, takes into account the problem of safety whilst performing the necessary exploration. A distinctive feature of this thesis is its retrospective look at classical techniques and concepts from the adaptive and linear multivariable control field, which we repropose in combination with more recent approaches and which we believe hold the answers to many current questions.