Hindawi Publishing Corporation Smart Materials Research Volume 2012, Article ID 164062, 15 pages doi:10.1155/2012/164062 Research Article Modeling Hysteresis with Inertial-Dependent Prandtl-Ishlinskii Model in Wide-Band Frequency-Operated Piezoelectric Actuator Vahid Hassani, Tegoeh Tjahjowidodo, and Albert D. Soetarto Division of Mechatronics & Design, School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 Correspondence should be addressed to Vahid Hassani, h vhd@yahoo.com Received 12 November 2011; Accepted 3 January 2012 Academic Editor: Tao Li Copyright © 2012 Vahid Hassani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. One of the major problems occurring in many technical applications is the presence of the hysteretic behavior in sensors and actuators, which causes a nonlinear relationship between input and output variables in such devices. Since the nonlinear phenomenon of hysteresis degrades the performance of the piezoelectric materials and piezoelectric drive mechanisms, for example, in positioning control framework, it has to be characterized in order to mitigate the effect of the nonlinearity in the devices. This paper is aimed to characterize and model the hysteresis in typical piezoelectric actuators under load-free and preloaded circumstances incorporating the inertial effect of the system. For this purpose, the piezoelectric actuator is modeled as a mass-spring-damper system, which is expressed in terms of a stop operator as one of the essential yet efficient hysteresis operators in the Prandtl-Ishlinskii (PI) model. The reason of utilizing the stop operator in this study is for the sake of control purposes, as the stop operator plays as the inverse of the play operator in the PI model and can be used in a feed-forward controller scheme to suppress the effect of hysteresis in general control framework. The results reveal that this model exhibits better correspondence to the measurement output compared to that of the classical PI model. 1. Introduction Hysteresis is a nonlinear phenomenon that occurs in some types of materials such as piezoceramics, shape-memory alloys, and magnetostrictive actuators. The word “hysteresis” refers to systems that have memory such that similar loops are repeated in each cycle of operation in every dynamical system and, after removal of the force or electrical field, the system does not return to its original location. In this phenomenon, the current output depends also on the history of the input. The effect of hysteresis nonlinearity can be neglected in some systems. In contrast, ignoring this phenomenon in other types of systems that possess severe hysteresis might create undesirable consequences such as inaccuracy in open loop system, limit cycle, degradation of the tracking perfor- mance, and even instability of closed loop system. To overcome this challenge, some mathematical models, for example, the Preisach model and the PI model, have been proposed to capture the effect of hysteresis in any mechanical systems. Utilization of the PI model is addressed in many works to characterize the hysteresis due to its simplicity compared to the Preisach model. Wang et al. [1] utilized the classical PI model and neural network approximator to design a robust adaptive control, which ensures that the tracking error is bounded in the presence of unknown hysteresis in a closed-loop system. The generalized PI is proposed for modeling an asymmetric hysteresis in piezoce- ramic actuators, where a linear function is defined to modify the threshold of the play operator of the model [2]. Krejci and Kuhnen [3] presented a new method of constructing the inverse of the hysteresis model in an analytical form for the system with hysteresis and creep and used the proposed model in a feed-forward controller to mitigate the nonlinearity arisen from the hysteresis. Al Janaideh et al. [4] presented a method to construct the inverse of the generalized PI model to be used in a feed-forward loop cascaded with an adaptive controller to reduce the effect of hysteresis in a piezoelectric actuator, and in another paper, they utilized the generalized PI model to design a robust