Integrating Uncertainties for Koopman-Based Stabilization

TL;DR

This paper develops a unified framework for robust stabilization in data-driven control using the Koopman operator, explicitly handling uncertainties like projection, estimation, and process disturbances, and proposes controllers for both direct and indirect methods.

Contribution

It introduces a comprehensive approach that accounts for multiple uncertainties in Koopman-based control, with new LMI-based controllers for robust stabilization in noisy data scenarios.

Findings

LMI-based controller stabilizes systems with process disturbances.

Framework effectively handles projection and estimation errors.

Numerical simulations confirm robustness and practical utility.

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.