Data-Driven Control Synthesis Using Koopman Operator: A Robust Approach

This paper presents a data-driven control design method for nonlinear systems using the Koopman operator framework. The Koopman operator lifts nonlinear dynamics to a higher-dimensional space, where observable functions evolve linearly. We initially consider an approximate linear time-invariant (LTI) lifted representation of the nonlinear system. To address residual errors, we calculate an approximation of the
$\ell_{2}$
-gain of the error system based on data. Subsequently, we synthesize a robust dynamic feedback controller, relying solely on the LTI frequency response, providing closed-loop guarantees. Additionally, we consider linear parameter-varying (LPV) lifted models to further minimize the error systems
$\ell_{2}$
-gain. In that case, we propose control design based on robustly stabilising the LTI dynamics and compensating for the parameter-varying part. The proposed strategy ensures internal stability under the assumption that the parameter-varying dynamics are open-loop BIBO stable. Moreover, it provides nominal performance guarantees for specific input-output channels for which the parameter-varying dynamics are fully cancelled.