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Linearization of Recurrent-Neural-Network- Based Models for Predictive Control of Nano-Positioning Systems Using Data-Driven Koopman Operators

Recent studies have shown that the nonlinear dynamics of nano-positioning systems (e.g., piezo-electric actuators (PEAs)) can be accurately captured by recurrent neural networks (RNNs). One direct application of this technique is PEA system control for precision positioning: linearize the nonlinear RNN model and then apply model predictive control (MPC). However, due to the linearization approach commonly used (e.g., Taylor series), the control bandwidth and the control performance are quite limited as the obtained linear system is only guaranteed to be accurate within small neighborhood of the linearization point. To address this issue, we propose a Koopman operator-based approach for linearization and then use the obtained linear parameter-varying model for predictive control. This linearization scheme can significantly decrease the overall approximation error within the MPC prediction horizon, and thus, lead to improved tracking performance. The proposed approach was validated through two applications—trajectory tracking of PEA, and deformation control of polymers during atomic force microscope nano-indentation.