## 1 Introduction
A type of ball screw preload structure using PEAs was introduced in [15], which achieved real-time adjustment of preload force, but the response was not fast enough due to the control strategy used. In the field of high-accuracy control, PEAs are used extensively and have a promising development application prospect. Due to the characteristics of fast response, large mass ratio, and high stiffness [16], the PEAs have been widely applied in micro robots [17], scanning tunneling microscopes [18], etc. Consequently, preload force control with rapid response speed can be achieved through PEAs. Meanwhile, the inherent nonlinear property of PEAs, coming from the creep, hysteresis and vibration of the piezoelectric [19; 20], greatly decreases the control precision of PEAs. Among the three characteristics mentioned above, hysteresis has the predominant influence on control performance. Hysteresis, which has a major impact on the nonlinear characteristic of PEAs, describes the dependence of the state of a system on its history, as occurs in shape-memory alloys and piezoelectric material. Due to their hysteretic character, both current and historical control inputs affect the output of PEAs. For decades, researchers have made efforts to eliminate the nonlinearity of PEAs. There have been many theoretical studies on hysteresis, and the most popular method for compensating the hysteresis error is inversion-based feedforward control. Before applying this method, a model of hysteresis is needed. Simply speaking, two typical categories of hysteresis models, namely, physics-based models and phenomenological models [21], have been proposed in the literature. Among physics-based models, the Jiles-Atherton model is a well-known model [22], and the Domain Wall Model [23] is based on this. The inversion model of these two models is available for linear controller design. One non-negligible limitation of physics-based models is that they are generally quite complicated. With respect to phenomenological models, the classical Preisach model is described by combining an infinite number of Preisach hysteresis operators, and it is widely considered to be a base toolkit of hysteresis modeling due to its general structure [24]. The nonlinear Preisach model has a broader area of application over the classical one, and makes it possible to fit experimentally measured data using first- and second-order transition curves [25]. Another widely used model is the Prandtl-Ishlinskii (P-I) model, where hysteresis is modeled by the elementary hysteresis operator called Play or Stop. The analytical inverse of P-I model is available. Consequently, it is more efficient for real-time applications [26]. Furthermore, there exist many other models and they are also pertinent for use in hysteresis. All in all, the aforementioned models should be inverted without exception so that they can compensate the hysteresis error. In this paper, a variable preload force control device using three PEAs is developed, and a P-I model-based control strategy for realizing the precise control of the output force is applied. PEAs are used to increase the stability of feed drive system, considering the application of PEAs on preload system, the force control is relatively more simple and convenient way. Three PEAs are installed between the double-nut to exert preload force on the ball screw and eliminate the backlash. Furthermore, the main idea of the control method is to use a feedforward compensator associated with a PID controller. Thereby, the inverse of the P-I model can be readily applied as the compensator. After conducting a series of experiments, the feedforward compensator along with the PID controller attained better performance compared with the one using a traditional PID controller. This paper focuses on the increased stability of the feed drive system in ultraprecision machining with the application of PEAs and the P-I model-based control strategy to increase the stiffness of the system at a high response rate.