Koopman-based modeling and control of motion systems with hysteresis dynamics using partial state feedback

Koopman operator theory represents nonlinear dynamical systems as linear systems in an extended state-space. By selecting observable functions composed of derivatives and functions of derivatives derived from the system output, it is possible to model the system without requiring knowledge of its internal states. The Koopman observable functions are iteratively refined to achieve close alignment with the original system dynamics. The resulting linear model in the extended space is then incorporated into a linear quadratic tracker (LQT) framework, enabling the system output to track a desired reference signal. The proposed method is demonstrated on a mechanical motion system with Bouc–Wen hysteresis, where the Koopman-based model and LQT provide robust control of the nonlinear system and can track a smooth trapezoidal step-scan trajectory for wafer scanner machines. The controller was also compared to a traditional PID controller, showing improved performance over the trajectory. Furthermore, simulation results demonstrate the controller's robustness against varying initial conditions, parameter uncertainties, and external disturbances.