Integrated Rate and Inertial Dependent Prandtl- Ishlinskii Model for Piezoelectric Actuator V. Hassani Nanyang Technological University School of Mechanical and Aerospace Engineering Division of Mechatronics and Design 50 Nanyang Avenue, Singapore-639798 E-mail: vahi0001@e.ntu.edu.sg T. Tjahjowidodo Nanyang Technological University School of Mechanical and Aerospace Engineering Division of Mechatronics and Design 50 Nanyang Avenue, Singapore-639798 E-mail: ttegoeh@ntu.edu.sg Abstract— Piezoelectric drive mechanisms consist of piezoelectric materials that actuate the mechanism at different ranges of frequency, as the piezoelectric actuators are subjected to a nonlinear phenomenon of hysteresis which is sensitive to frequency changes. Designing an adaptive hysteretic model facilitates the operation of such mechanisms in general control framework. In this paper, an Integrated Rate and Inertial dependent Prandtl-Ishlinskii model using an exponential function for varying damping factor is proposed for positioning control of a piezoelectric actuator at frequency range between 1 Hz to 200 Hz. Keywords-piezoelectric actuator; Prandtl-Ishlinskii Model; Hysteresis Loop; Rate-Dependent Model; Inertial-Dependent Model I. INTRODUCTION In recent years, applications of piezoelectric actuators in piezoelectric motors and mechanisms have been widely explored [1-6]. The apparent characteristics of the piezoelectric actuators such as fast frequency response and high resolution down to nanometer level have encouraged researchers as well as industries to develop micro-positioning devices. In spite of the advantage for micro-positioning application, the piezoelectric actuators possess severe nonlinearity due to a hysteresis relation between the voltage input and the displacement output. This phenomenon degrades the performance of the actuator and the mechanism in positioning control applications as it causes inaccuracy in the open loop system, inadvertent oscillations of the system and even instability of a closed loop system [7, 8]. Moreover, the hysteresis phenomenon in piezoelectric actuators is dependent on the rate of applied input and aging of the actuators [9]. In other words, the intensity of hysteresis increases with the increase in operating frequency and vice versa. As a result, developing a frequency-dependent or rate-dependent model of hysteresis has attracted significant attentions of researchers. Considering the works done in this area, readers may refer to Al Janaideh [10], who proposed a rate-dependent Prandtl- Ishlinskii model by presenting a threshold value and density function, which appear as a function of varying input rate to characterize the hysteresis in the piezoelectric actuator at frequency range between 0.1 Hz to 200 Hz. A different approach was proposed by Ang [11] to model the hysteresis at low frequencies up to 50 Hz utilized a linear function relating the slope of the hysteresis loop to input rate applied to the actuator. Application of different periodical inputs motivated Al Janaideh [12] to consider an initial rate-dependent model at higher frequency range between 0.1 Hz to 500 Hz. The relation between threshold value and the rate of input applied to piezoelectric actuator was considered to gain a better approach to develop rate-dependent Prandtl-Ishlinskii model [13]. The inverse rate-dependent model was proposed by Al Janaideh [14] to be used as a feed forward controller in a control loop for positioning control of an actuator. Since the hysteresis loops sometimes also exhibit asymmetrical shape, a generalized rate- dependent hysteresis model was proposed further to capture the asymmetric shape of the hysteresis in a rate-dependent Prandtl- Ishlinskii model [15]. Ang [16] designed a feedforward controller through utilization of inverse of the rate-dependent model that formerly presented in [11]. In this paper, an inertial-dependent Prandtl-Ishlinskii model is proposed in term of the stop operator which is one of the two essential yet well-known operators of the Prandtl-Ishlinskii model. Using an exponential function that expresses relation between damping factor and the changes in frequency directs us to convert the initial rate-independent model into a rate- dependent model of the hysteresis at frequency range between 1 Hz to 200 Hz for the applied input voltages up to 80V. II. CLASSICAL PRANDTL-ISHLINSKII MODEL This model exploits two well-known operators i.e., a play operator and a stop operator. The stop operator plays as an inverse of the play operator and can be solely used in a feed forward controller scheme to mitigate the hysteretic effects in piezoelectric actuators. The properties of these operators are discussed in this section. A. Stop Operator Suppose [0, ] m E C t is a space of piecewise monotone continuous functions. For any input () [0, ] m E vt C t ∈ , the stop operator is defined by the following expressions, () min( , max( , )) r e v r rv = - (1) 1 1 [; ](0) ( (0) ) r r E vw e v w - - = - (2) 2011 2nd International Conference on Instrumentation, Control and Automation 15-17 November 2011, Bandung, Indonesia 978-1-4577-1460-3/11/$26.00 ©2011 IEEE 152