A Deep Serial Model and Predictive Control for Piezo-Actuated Positioning Stages

The inherent hysteresis nonlinearity of piezo-actuated positioning stages (piezo-stages) is very difficult to deal with due to the amplitude- and frequency-dependent characteristics, which severely limits the tracking accuracy for high speed trajectories. In this article, we first develop a deep serial model to describe the dynamics of the piezo-stage by using historical voltage-displacement data over a period of time. It achieves relative prediction errors less than 0.16% on sinusoidal trajectories with frequencies greater than 72% of the resonance frequency of the piezo-stage through an elaborately designed network structure that includes direct connections between the input layer and the output layer. Then, we design an integral model predictive control (iMPC) and build a feedforward neural network (FNN) to learn its optimal solution offline. This forms the proposed FNN-iMPC and ensures the feasibility of evaluating the control law within the sampling time of 0.1 ms. It achieves a maximum positioning error of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.02 \; \mu$</tex-math></inline-formula>m for a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm 32 \; \mu$</tex-math></inline-formula>m staircase reference signal and a maximum tracking error of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.19 \; \mu$</tex-math></inline-formula>m for a sinusoidal reference signal with a range of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10 \; \mu$</tex-math></inline-formula>m and a frequency of 500 Hz in real experiments.