I. Introduction
Recently, due to the requirement of the micrometer or nanometer resolution in displacement, high stiffness, and fast frequency response, piezoelectric actuators often are used in many precision positioning applications, such as scanning tunneling microscopy [1], near-field optical scanning microscopy [2], [3], and vibration control [4]. However, because the materials of piezoelectric actuators are ferroelectric, they fundamentally exhibit hysteresis behavior in response to an applied electric field [5], [6]. Unfortunately, this behavior usually leads to problems with severe inaccuracy [4]–[6] and deteriorated tracking performance [7]–[10] when the piezoelectric are operated in an open-loop mode. Therefore, there have been many modeling techniques in hysteresis behavior [11]–[17] by mathematical functions or equations, such as the Preisach function [11]–[14] and the nonlinear, piecewise circuit description [15]–[17]. However, these modeling techniques were either frequency dependent and complicated or nonsystematic. From the viewpoint of system control engineering, these cases lead to a large difficulty in model-based controller design that is often used for system control engineers.