A Robust Dual-loop Control for Finite-dimensional Koopman Model of Nonlinear Dynamical Systems

The Koopman operator is a promising approach for learning nonlinear dynamics using linear operators in high-dimensional function space. However, due to finite-dimensional approximation and imperfect data, model mismatch can arise, resulting in a discrepancy from the actual nonlinear model. As a result, robustness against model mismatch is a critical objective in Koopman-model-based control design. This paper presents a robust dual-loop control scheme for the finite-dimensional Koopman model to address this issue. Firstly, the biased dynamics of the finite-dimensional Koopman model are illustrated by a nonlinear bilinear motor. Multiple trajectory data sets are assumed with measurement noises and used to identify a Koopman model using the extended Dynamic Mode Decomposition (EDMD). The resulting Koopman model is examined to yield biased dynamics from actual nonlinear dynamics. Then, a robust dual-loop control is designed, consisting of an observer-based state-feedback control for the nominal Koopman model and an additional robust loop to improve robustness. The numerical results show that the dual-loop control can improve the robustness of the Koopman operator against model mismatch compared to simply applying nominal control. At low noise levels, both LQG and dual-loop control can regulate the system. However, at higher noise levels, the LQG control strategy fails to regulate the system, but the dual-loop control drives the system to achieve robust performance against the model mismatch.