Integrating Uncertainties for Koopman-Based Stabilization

Abstract: Over the past decades, the Koopman operator has been widely applied in data-driven control, yet its theoretical foundations remain underexplored. This paper establishes a unified framework to address the robust stabilization problem in data-driven control via the Koopman operator, fully accounting for three uncertainties: projection error, estimation error, and process disturbance. It comprehensively investigates both direct and indirect data-driven control approaches, facilitating flexible methodology selection for analysis and control. For the direct approach, considering process disturbances, the lifted-state feedback controller, designed via a linear matrix inequality (LMI), robustly stabilizes all lifted bilinear systems consistent with noisy data. For the indirect approach requiring system identification, the feedback controller, designed using a nonlinear matrix inequality convertible to an LMI, ensures closed-loop stability under worst-case process disturbances. Numerical simulations via cross-validation validate the effectiveness of both approaches, highlighting their theoretical significance and practical utility.