Parameter tuning of the wafer stage compensation feedforward controller of the lithography machine

The feedforward controller is crucial to achieving nano-level motion accuracy for the lithography wafer stage under high acceleration and deceleration conditions. Traditional 4-order feedforward is widely used to control precision motion systems because of its intuitive physical meaning and simple parameter tuning. However, its capacity to fit the inverse model is inadequate, and it is difficult to eliminate the repetitive error caused by the input trajectory. Therefore, a feedforward control architecture using the 4-order feedforward and an extra rational fraction compensator is proposed.

In this study, the input signal of the compensator is the higher-order derivative of the reference trajectory, and the numerator and denominator of the compensator use the delay unit as the basis function. Therefore, obtaining the unknown parameters of the basis function is crucial to the design. This paper proposes a data-driven iterative parameter tuning strategy for the compensation controller. The difficulty is that the tuning problem is a nonconvex optimization problem, making global parameter optimization challenging. This paper uses the relevant rules of system identification to address the issue at hand. The purpose of adding compensatory feedforward is to eliminate the residual error after using the 4-order feedforward, which is equivalent to achieving a zero-generalized error. Since the generalized error has a linear connection with the compensator parameters, the original nonconvex optimization problem is successfully transformed into a convex problem by minimizing the 2-norm of the generalized error. Through the above transformation, the global optimal point is obtained by the Gauss—Newton method, and the step size condition for ensuring iterative convergence is provided. In addition, the gradient and Hessian matrix of the objective function need to be incorporated into the parameter updating law, even though their exact values are difficult to obtain. This paper derives their unbiased estimates using two impulse response experiments and 2 trajectory tracking experiments.

The proposed method was applied to the wafer stage of the lithography machine, and the experiment showed the following results: (1) Using the proposed method to tune three compensation controllers with different orders, their error 2-norm almost converged after five iterations. (2) After adding compensation feedforward, the acceleration and deceleration phase errors were reduced from ±35 nm to ±10 nm; the constant velocity phase error was almost equal to the positioning error, and its trajectory tracking effect was very close to that of iterative learning control (ILC) compensation. (3) Compared with the existing compensation controller parameter tuning method, the maximum moving average and moving standard deviation at velocity phase of the proposed method were smaller, and the lower the compensator order, the more obvious the advantage. (4) After changing trajectory, the proposed compensator could still achieve a better control effect than ILC compensation.

The above experiments verify the convergence performance of the proposed parameter tuning algorithm. It is shown that the proposed feedforward compensation architecture can effectively eliminate the residual repetition error of the 4-order feedforward; simultaneously, it can adapt to variable trajectories. In addition, compared to the current compensator tuning result, this method can achieve a superior trajectory tracking control effect while using a low-order compensation controller.