Dynamic Preisach modelling of hysteresis for the piezoceramic actuator system

Abstract

Rate-dependent hysteresis property is a common phenomenon in various hysteretic systems including the piezoceramic material system. Dynamic Preisach model is needed to describe the rate-dependent hysteresis. This paper proposes a new dynamic Preisach model by introducing the dependence of the Preisach function on the input variation rate. An input variation rate function was introduced to adjust the relationship of hysteresis loop on the input variation rate for different hysteresis systems. A detailed numerical implementation procedure is also presented. Experiments were conducted to study the hysteresis behavior of piezoceramic actuator system and to verify the proposed model.

Introduction

In smart structures, piezoceramic materials can be used as actuators and as sensors. Among nonlinearities present in piezoceramic material systems, hysteresis has been particularly identified to be sensitive to the varying field conditions. Since piezoceramic materials are ferroelectric, they are fundamentally nonlinear in their response to an applied electric field, exhibiting a hysteresis effect between the electric field and the displacement or the force. Without modelling and incorporating hysteresis in the controller design, the hysteresis will act as an unmodelled phase lag presence, which has the potential to cause instability in a closed-loop system. Hence, reliable modelling and predictions of hysteresis would be a valuable tool when these piezoelectric actuators are employed as part of closed-loop system for purposes of motion control such as an active control and micro-positioning. If hysteresis effects of these material systems can be predicted, then actuator controllers can be designed to correct these effects and the whole controller system can be made to appear as a device with a single valued output function.

There is a vast literature on the hysteresis modelling using Preisach technique, especially in ferromagnetic materials. Recently, the Preisach technique has been applied to the piezoceramic material system. Sreeram and Naganathan [12], [13] carried out the application of classic Preisach model (CPM) to a piezoceramic material system. In their work, Mayergozy's identification and numerical simulation procedures were used to model the hysteresis exhibited by a piezoceramic bimorph. The minor loop patterns and enclosed areas from simulation and experimental data matched well for symmetrical minor hysteresis loop. Hughes and Wen [6] discussed and verified the applicability of CPM to piezoceramic and shape memory alloy systems. The parameter identification based on the input/output data and hysteresis compensation via direct inversion of the identified model was also demonstrated. Freeman and Joshi [4] extended the CPM to a two-input model to handle both the applied electric field and the mechanical stress effects on the output strain. Ge [5] used both the CPM and generalized Preisach models to described the hysteresis of a piezoceramic actuator and developed a tracking control approach by incorporating the hysteresis model in the feed-forward loop to increase the tracking control precision. CPMs are basically static and rate-independent models and do not describe the dynamic hysteresis phenomena. Hysteresis with rate-dependent property has been observed in many experiments. Pokines et al. [10] and Ge [5] showed that hysteresis has rate-dependence properties for piezoceramic materials. Holman's experimental results show that the hysteresis curves become tilted if the frequency of the excitation is increased for a piezo material as shown in Fig. 1. It is considered that the CPM can be used to predict the hysteresis loop of a piezoceramic actuator under harmonic input excitation at frequency well below the resonance frequency of the actuator [5]. Experimental results showed that even in a low frequency range, the hysteresis loop changes with the increase of input frequency. Bertotti's work [1], [2], [3] showed that the hysteresis loop area (the power loss) increases with the increase of magnetizing frequency for soft magnetic materials. Hysteresis curves of ferroelectric liquid crystal displays show that with the increase in frequency, an increase in width in the loop and a decrease in the height of the loop are noticed [11].

Dynamic Preisach models were proposed to deal with such a rate-dependent hysteresis. Two typical models are proposed by Mayergozy [8] and Bertotti [1]. The Preisach function of the dynamic term can be obtained by experiment through measuring the relaxation time. The detailed procedure of the numerical implementation was given by Mayergozy [9].

Bertotti's model is useful in predicting the power loss in a ferromagnetic material. However, it is not so effective in the analysis of the internal magnetic field in the ferromagnetic device [14].

In this paper, a new dynamic Preisach model is proposed to deal with the rate-dependent property for piezoceramic materials by introducing the dependence of the Preisach function on the input variation rate. As a given parameter, the input variation rate is easy to measure and control during the experiments. The output variational speed of the system is dependent on the input variation rate and the hysteresis loop is directly related to the input variation rate. Hence, it is reasonable to use the system input variation rate as a parameter of the dynamic model. The details are discussed in the following sections.