# High-Performance Flux Tracking Controller for Reluctance Actuator
## 1 Introduction  
The wafer stage and reticle stage, shown in Figure 1, are important units used in the lithography machine. In practice, the desired IC pattern is optically transferred onto a silicon wafer via a reticle mask that contains a slit for exposing light on the wafer. Since one wafer can have many ICs, the wafer needs to be repositioned from exposure to exposure. The exposure itself takes place during a scanning motion of the wafer and the reticle [1, 2]. As shown in Figure 2, the reticle stage and wafer stage consist of a long-stroke stage and a short-stroke stage [3, 4], in which the long stroke achieves the large stroke and coarse positioning. Meanwhile, the short-stroke stage is responsible for fine positioning accuracy. The voice coil motor is widely used in high-precision positioning systems due to the high linearity argued in [5, 6]. However, with the increasing demand for throughput, the reticle and wafer stages must be operated at higher speeds and accelerations. Voice coil motors (VCM), however, are near their physical limit and cannot provide large enough thrust for the required high acceleration in the photolithography. Thus, a novel reluctance actuator is proposed in [7, 8, 9]. Compared with VCM, the reluctance motor has the advantages of high output density andlow energy consumption, but the strong hysteresis nonlinearity restricts the control accuracy of the reluctance actuator and limits its application in the high-precision positioning system [10; 11; 12; 13], especially when hysteresis is coupled with the displacement effect and eddy current effect, which lead to displacement-dependent and rate-dependent hysteresis in the reluctance actuator. For this reason, pursuing a way of alleviating displacement-dependent and rate-dependent hysteresis in the reluctance actuator is crucial in the high precision control of the reluctance actuator. According to the survey, one of the most general ways to suppress hysteresis is with flux feedback in a reluctance actuator. For example, in [10], the Hall element is used to measure the flux signal in the air gap, then the signal is used as a feedback variable. Whereas the method which uses the Hall element to measure flux signal in [10] sacrifices the travel of the reluctance actuator, which limits the application in the minimum air gap, thus reducing the motor efficiency. In order to tackle this problem and improve the efficiency of the reluctance motor, in [11], a sense coil sensor was used to measure the flux signal, and a shear hysteresis model, which considers the effect of displacement on the hysteresis loop, was proposed. However, the model is complex, which makes the controller design very difficult, so its application is limited. Thus, up to now, the reluctance actuator has not been adequately investigated, which gives rise to the motivation of our research. It is worth pointing out that the control methods of hysteresis nonlinearity, which alleviates the rate-dependent hysteresis, are affluent in the Piezoelectric actuator. For example, feedforward control [14; 15; 16; 17], namely inverse model compensation control, was widely used. In application, the inverse of the hysteresis is obtained first; then, to utilize the inverse hysteresis as the input of the system was proposed to eliminate the hysteresis  
Figure 1: Diagram of a lithography scanner. Figure 2: 1-DoF model of reticle stage. effect as well as feedback control methods, which include the PI controller [18], adaptive control method [19], sliding mode controller, and so on [20; 21]. Specifically, according to the hysteresis model, the nonlinear feedback controller is designed directly, which drives the object to keep consistent with the desired trajectory signal, considering the complexity of the hysteresis model, which makes the design of the feedback controller difficult. Under the framework of inverse model compensation, a series of hysteresis modeling methods are concerned, such as the Bouc-Wen model [22], Prandtl-Ishlinskii model [23; 24], and Preisach [25], and so on.