I. Introduction
In recent years, piezoactuators have been important devices for microscopy applications, such as scanning tunneling microscopy [1], scanning near field optical microscopy [2], [3], and high-frequency vibration control [4]. However, since the materials of piezoactuators are ferroelectric, they fundamentally exhibit hysteresis behavior in response to an applied electric field [5], [6]. This behavior usually leads to problems of severe inaccuracy [4]–[6] and deteriorated tracking performance [7]–[10] when the piezoactuators are operated in an open-loop mode. Therefore, for manipulation of piezoactuators, many studies in modeling hysteresis behavior by mathematical functions or equations have been published in [11]–[17], such as the Preisach functions [11]–[14] and the nonlinear piecewise circuits [15]–[17]. However, these proposed modeling techniques always have a complicated mechanism. This often leads to difficulty in model-based control design for systems. Moreover, modeling techniques that were for positioning control or hysteresis compensation of piezoelectric devices have been presented in [8], [9], [18]–[21]. In [8], the hysteresis behavior described by the Preisach model was compensated by the proportional–integral–derivative (PID) feedback controller incorporated with the feedforward loop. From their experimental tracking responses of the piezoactuator to sinusoidal trajectories, it seems that the hysteresis effect is reduced due to compensation. However, the controller design was independent of the hysteresis dynamics represented by the Preisach model. Furthermore, due to the complicated computation and iterative integrals, using the Preisach model leads to an increasing difficulty in analysis of transient response and system stability. In [9] and [18], the model consisting of several elasto-slide elements with the massless blocks and massless springs subjected to Coulomb friction were used to approximately describe the motion dynamics of piezoactuators with hysteresis behavior, and the PID plus feedback linearization controller with the repetitive controller was designated. Although the model with hysteresis phenomenon can be used to approximately describe the dynamics of piezoactuators, the system parameters employed in this model, such as the number of the massless blocks and springs, break force, and stiffness, need to be determined by several experienced experiments. In [19], the hysteresis behavior was modeled by a set of hysteresis operators that included a gain and an input-dependent lag, and the tracking controller was completed by a traditional proportional–integral (PI) controller. Unfortunately, there were many restrictions and assumptions in this method, such as the rate of change in control input voltages, the decision of input-dependent lag, and the restrictions on some parameters. Moreover, the design of the traditional PI tracking controller was independent of the adopted model of the controlled piezoactuators; that is, the modeling of the piezoactuators in [19] seems to be unnecessary for their controller design. In [20] and [21], the mechanical models consisting of the mass-spring-damper systems were investigated. In these models, the differential equations related to the mechanical motion dynamics with the specified nonlinear terms were used to describe the motion dynamics of piezoelectric mechanisms with hysteresis behavior. Although these dynamic models can approximately represent the motion dynamics of piezoelectric devices or mechanisms, the information about how to design a high-performance positioning controller was not provided.