Koopman-based controller design for nonlinear systems with closed-loop guarantees

Koopman operator theory has emerged as a powerful tool for data-driven analysis and control of nonlinear dynamical systems, enabling (bi)linear representations of complex dynamics. Its practical application requires finite-dimensional approximations, which inevitably introduce model errors. Existing approaches often neglect these errors, limiting the reliability of Koopman-based control. This thesis closes this gap by presenting a systematic control framework for nonlinear systems that connects Koopman theory and data-driven modeling to robust control design with rigorous closed-loop guarantees. The framework consists of two main components. First, bilinear surrogate models are derived from data using Koopman operator theory, providing explicit residual error bounds that account for finite-data and approximation effects. Second, these surrogate models form the basis for designing robust state-feedback controllers that explicitly incorporate the model uncertainties, ensuring desired closed-loop properties of the underlying nonlinear system. The framework in this thesis provides a rigorous and systematic foundation for Koopman-based control, establishing both numerical effectiveness and theoretical closed-loop guarantees, such as exponential stability and quadratic performance, via linear matrix inequalities and sum-of-squares relaxations.