# Hybrid Adaptive Controller Design with Hysteresis Compensator for a Piezo-Actuated Stage
## 1 Introduction  
Piezo-actuated stage (P-AS) driven by piezoelectric ceramics are broadly applied to projection lithography lenses, scanning probe microscopy, astronomical telescopes, modern optics, other advanced optoelectronic devices, and ultraprecision systems [1, 2, 3] owing to high resolutions and response speeds and the impregnability of the magnetic fields [4, 5]. However, the intrinsic rate-dependent (RD) hysteresis of piezoelectric ceramic materials significantly affects the control and positioning accuracies of P-AS [6, 7, 8]. The response of RD hysteresis characteristic exhibits a special multi-value mapping relationship, where the P-AS output depends on the current input and the previous output of P-AS [9]. In addition, the RD characteristic indicates that the frequency of the P-AS input is one of the important factors affecting their output [10]. Therefore, current research focuses on developing reasonable control strategies to depress the hysteresis characteristic of the P-AS [11, 12, 13]. At present, the methods for eliminating hysteresis are mainly divided into two categories: feedforward compensation control methods based on the hysteretic inverse model and feedback control methods. The feedforward compensation control method needs to establish the hysteretic model first, then obtain the inverse model of the system via resolution or identification of the hysteretic model, and finally construct the feedforwardcontroller based on the inverse model [14; 15]. Common models for characterizing hysteresis include the Duhem model [16], backlash-like model [17], Bouc-Wen (BW) model [18; 19; 20], Preisach model [21; 22], Prandtl-Ishlinskii model [23; 24; 25], and Krasnoselskii-Pokrovskii model [26; 27; 28]. In addition, there are various other models in the literature to depict the complex hysteresis of piezoelectric motors [29; 30; 31]. The feedforward compensation control method proposed in the above literature can effectively suppress the effect of hysteresis on system accuracy and achieve accurate tracking control. However, the accuracy of modeling seriously affects the performance of feedforward compensation control methods. The control effect of feedforward controllers is significantly weakened if modeling errors and uncertain noise exist. In addition, the feedforward controller has poor robustness to resist the interference of external disturbances. To overcome the shortcomings of feedforward compensation control methods, feedback control methods have been proposed [32; 33; 34]. The common feedback control methods include PID control [35], sliding mode control [36; 37], iterative learning control [38; 39], adaptive control [40], robust control [41; 42], model predictive control [43; 44], and modified repetitive control [45]. These control methods can suppress external disturbances and increase the control accuracy compared with the feedforward compensation control methods [46]. Adaptive control is a feedback control method that can compensate for the modeling error and external uncertain noise of the feedforward compensation control methods [47; 48; 49]. Exploiting this property, we propose a hybrid adaptive control (HAC) strategy based on a hysteresis controller to decrease the influence of the hysteresis characteristic for P-AS and realize precise positioning. First, a Hammerstein model is proposed consisting of a BW hysteresis sub-model and dynamic linear sub-model to depict the RD hysteresis characteristic for P-AS. An improved differential evolution algorithm and a least-squares algorithm are employed to identify the two sub-models, respectively. Then, the inverse BW model is identified by the improved differential evolution algorithm as a compensator to handle the RD hysteresis characteristic of P-AS. To reduce the effects of modeling errors and uncertain noise, a hybrid adaptive controller is proposed, and the stability of the raised approach is certified by the Lyapunov theory.