Data-based representations and control of linear and nonlinear systems

The objective of this thesis is the development of data-based representations and control design techniques for linear and nonlinear systems. This framework relies only on collected data and avoids explicit model identification, yet provides strong theoretical guarantees on stability and robustness in the presence of uncertainties. Crucial to many recent works on data-based analysis and control is that the data-generating system is linear and that sufficiently long (noise-free) measurements are available. To address these restrictions, our contributions in this thesis are organized into two main parts: (I) the development and advancement of data-based representation and control methods for linear systems (Chapters 3-5), and (II) the extension of such techniques to classes of nonlinear systems (Chapters 6-7). In particular, we develop data-based representations of linear time-invariant (LTI) systems using only short, irregularly measured, noisy input-output data. We additionally provide a novel uncertainty quantification technique based on the measured noisy data. Building on these results, we propose a robust, sample and computationally efficient data-driven predictive control scheme. For this scheme, we provide strong theoretical guarantees on recursive feasibility and stability of the resulting closed-loop system. Lastly, we propose a new non-minimal state-space representation which enables data-driven analysis and control for general multivariable LTI systems based only on (noisy) input-output data. In the second part of the thesis, we focus on the development of data-based representations and control design techniques for classes of nonlinear systems. Specifically, for the wide class of feedback linearizable systems, we propose a data-based representation of all its finite-length trajectories by leveraging the system's linear structure in transformed coordinates. Such representations use only input-output data together with a set of user-defined basis functions. In the presence of uncertainties, we provide error bounds on the difference between the true and approximate system trajectories. These results are then used to develop a purely output-feedback data-driven nonlinear predictive control scheme with stability and robustness guarantees. For more general classes of nonlinear systems, we propose a novel notion of descriptor embedding, which is a new lifting procedure wherein a stabilizing controller is designed for a higher-dimensional linear parameter varying descriptor system and then shown to be stabilizing for the original nonlinear system. Notably, the proposed approach allows for obtaining new (global) data-driven stabilization techniques for nonlinear systems which are not based on nonlinearity cancellation, thus allowing for better closed-loop performance. Finally, throughout the thesis, the developed results are illustrated with numerous simulation examples that highlight their advantages and applicability.