# High-Performance Flux Tracking Controller for Reluctance Actuator
## 1 Introduction
However, the classical hysteresis model can only express the static hysteresis relationship between input and output and cannot accurately establish the rate-dependent hysteresis model of the system. Hence, to perfect the control accuracy in a Piezoelectric actuator, one way is to develop a particularly accurate rate-dependent hysteresis model, such as a neural network [26], improved J-A model [27], and Modular model [28; 29; 30] and so on, then use the inverse model for feedforward control. The goal of establishing the inverse model compensation in this article is to connect the static hysteresis inverse model in series in front of the controlled object, to mitigate or even eliminate the effect of static hysteresis nonlinearity in the system, and to realize the conversion of the complex hysteresis nonlinear system into a linear system with certain deviations, which is easy for the design of the controller. Considering the complexity of the hysteresis nonlinearity of the reluctance motor, which includes displacement-dependent and rate-dependent, multivalued mapping, and memorability, the hysteresis nonlinearity model is hard to establish. Currently, there is no suitable modeling method and control strategy to achieve the control accuracy of reluctance motors. Inspired by [31; 32], to suppress the hysteresis nonlinearity of the reluctance motor effectively, the hysteresis model will be divided into two parts: the rate-dependent hysteresis model and the displacement-dependent hysteresis model. In application, following the work of [33; 34; 35], we use the Hammerstein model, a widely used rate-dependent modeling method due to its simple structure and convenient controller design. Subsequently, the effect of displacement on the hysteresis model, namely the displacement displacement-dependent hysteresis model, is considered as the disturbance of the Hammerstein model. In practice, the inverse model of the nonlinear part in the Hammerstein model is obtained first, then the inverse model is obtained to eliminate hysteresis nonlinearity and transform the nonlinear system into a linear system with inversion bias and disturbance. It is worth pointing out that the disturbance includes not only the effect of displacement on the hysteresis model but also other disturbances of the system. Last but not least, a reliable controller is designed to achieve high-precision control of the reluctance actuator. More specifically, in this article, the Hammerstein structure was applied, where the Bouc-Wen model describes the nonlinear characteristics and linear dynamics represents the rate-dependent characteristics of the hysteresis loop, as well as the disturbance term indicating the influence of the displacement on the model and the disturbance caused by modeling error. Thus, the rate-dependent nonlinearity of the reluctance actuator was described. Additionally, considering the system disturbance and model error caused by displacement changes, a new compound control method is designed, which consists of inverse model compensation and the discrete sliding mode controller based on the disturbance observer (DO-DSMC). It is worth noting that the inversion based on the Bouc-Wen model is obtained to compensate for the system's nonlinearity, and the nonlinear system is transformed into a linear system with inversion bias and disturbance. What is more, the sliding mode controller based on the disturbance observer, which is proposed in [36], and was widely used in practical engineering, such as aerospace, industrial control, robot fields, and so on, was adopted due to the advantages of the inherent order reduction properties of the sliding mode variable structure and the invariance of the sliding modes to the parameter uptake and external disturbances of the controlled system, high response speed, strong robustness, and low chattering effect.