Reference-Adaptation Predictive Control Based on a Deep Parallel Model for Piezo-Actuated Stages

The intrinsic hysteresis nonlinearity of piezo-actuated stages (piezo stages) poses a significant challenge for precise trajectory tracking at high speeds. In response, we propose a deep parallel (dPara) model that effectively captures the dynamics of the piezo stage using historical voltage–displacement data over a concise time period. The dPara model, incorporating a parallel combination of a linear block and a feedforward neural network (FNN), exhibits exceptional performance with relative prediction errors ranging between 0.10% and 0.18% on sinusoidal trajectories at frequencies up to 72% of the resonance frequency of the piezo stage. By leveraging this parallel structure, we adapt the reference trajectory for a complex nonlinear model predictive control (MPC), leading to the development of the reference-adaptation MPC (RA-MPC). Furthermore, we design a coordinate ascent algorithm to solve the quadratic programming (QP) problem derived from the RA-MPC at a high frequency of 10 kHz. To assess the superiority of the proposed RA-MPC, comprehensive experiments are conducted under sinusoid, sawtooth, and staircase reference trajectories. Notably, it achieves maximum tracking errors (MTEs) ranging from 0.0263 to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.7136 \; \mu $ </tex-math></inline-formula>m for desired speeds spanning from 40 to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$20\,000 \; \mu $ </tex-math></inline-formula>m/s.