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清华大学学报(自然科学版)  2023, Vol. 63 Issue (10): 1640-1649    DOI: 10.16511/j.cnki.qhdxxb.2023.26.002
  机械工程 本期目录 | 过刊浏览 | 高级检索 |
刘涛, 杨开明, 朱煜
清华大学 机械工程系, 摩擦学国家重点实验室, 精密超精密制造装备及控制北京市重点实验室, 北京 100084
Parameter tuning of the wafer stage compensation feedforward controller of the lithography machine
LIU Tao, YANG Kaiming, ZHU Yu
Beijing Key Laboratory of Precision/Ultra-Precision Manufacturing Equipment and Control, State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
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摘要 前馈控制器是光刻机工件台在高加减速工况下实现纳米级运动精度的关键环节。该文针对传统情况下四阶前馈拟合逆模型能力较差、难以完全消除参考轨迹导致重复性误差的问题,提出了一种以四阶前馈为基础,外加有理分式补偿器的前馈控制架构,并针对该有理分式补偿器控制参数整定的问题,提出了一种数据驱动的参数整定方法。该方法利用系统辨识的相关规则,将前馈补偿器参数整定过程的非凸优化问题转化为凸优化问题,进而给出了全局最优参数整定方法以及参数迭代过程中梯度、Hessian矩阵的无偏估计方法;通过光刻机工件台的实验验证了所提参数整定方法具有收敛性。实验结果表明:所提出的补偿前馈能够有效消除四阶前馈未消除的残余误差。
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关键词 前馈控制有理前馈控制器补偿前馈数据驱动参数整定工件台    
Abstract:[Objective] 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. [Methods] 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. [Results] 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. [Conclusions] 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.
Key wordsfeedforward controller    rational feedforward controller    compensation feedforward    data-driven    parameter tuning    wafer stage
收稿日期: 2022-06-24      出版日期: 2023-09-01
通讯作者: 杨开明,副研究员,     E-mail:
作者简介: 刘涛(1996-),男,博士研究生。
刘涛, 杨开明, 朱煜. 光刻机工件台前馈补偿器参数整定方法[J]. 清华大学学报(自然科学版), 2023, 63(10): 1640-1649.
LIU Tao, YANG Kaiming, ZHU Yu. Parameter tuning of the wafer stage compensation feedforward controller of the lithography machine. Journal of Tsinghua University(Science and Technology), 2023, 63(10): 1640-1649.
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