Optimal feedforward control with a parametric structure applied to a wafer stage

To meet the ever-increasing requirements for accurate manufacturing equipment, feedforward control has been widely regarded as an effective method. A nominal feedforward controller equals to the inverse model, which conventional model-based approaches could not acquire accurately due to the inevitable model error, the stability of the inverse model and so on. Therefore, a data-based feedforward control based on a parametric structure is proposed. The structure takes acceleration and snap set-points as signal inputs and both paths equip finite impulse response filters. Each finite impulse response filter is parameterized by a series of coefficients, assuming that the difference between the actual output and nominal output is an affine function of these coefficients. The coefficients are obtained from a gradient and Hessian approximation-based algorithm and optimized by minimizing a quadratic objective function. Two methods are proposed to approximate the gradient: the direct approach and the Toeplitz matrix approach. Finally, the proposed algorithm is assessed on a developed wafer stage. The results show that the proposed parametric structure improves the scanning tracking performance and provides a more desirable way to deal with the model with several resonances in low frequency.