## 1. Introduction

Piezoelectric actuators are one of the new-generation actuators that function on the principle referred to as the inverse piezoelectric effect. They have been attracting great attention both from researchers and the practitioners for decades owing to their attractive features, such as high positioning resolution and rapid response. A comparative example of the new-generation actuator is the magnetostrictive actuator [

1]. Magnetostrictive actuator utilizes a material which exhibits magnetostriction-physical phenomenon of certain material that shows elastic deformation under magnetic field. Both piezoelectric and magnetostrictive materials exhibit inverse phenomenon, and they can be used not only as actuators but also as sensors. Piezoelectric actuators tend to be used in small size applications whereas the magnetostrictive actuators can be applied to large scale target in which kW order output is necessary. Both exhibit similar mechanical characteristics if their sizes are similar. Piezoelectric actuator is basically a source of linear vibration whereas magnetostrictive actuators can also be used as a power source of planar motion. Recent literature revealed that piezoelectricity was found in organic biomaterials [

2] that were successfully applied to several biomedical devices.

Many applications of the piezoelectric actuators can be found in recent literature. Stefanski et al. [

3] used the piezo stack ring actuator to control hydraulic valves. Liu and Guo [

4] applied a stack type piezoelectric actuator for the position control of laser focusing equipment. Piezo bimorph actuators were successfully implemented in the field of micro-manipulation. El-Sayed et al. [

5] developed a micro-gripper using a piezo bimorph actuator and evaluated its characteristics both theoretically and experimentally. Jain et al. [

6] developed a mobile micro-manipulation system for peg-in-hole assembly in which a compliant piezoelectric actuator was used. They demonstrated that the developed gripper could perform the assigned task. However, they also pointed out that the transient motion paths of the actuator for obtaining the desired strain for manipulation differed depending on the driving voltage history determined by the PD feedback control law. The observed phenomenon was caused by the hysteresis nonlinearity of the piezoelectric actuator. The tracking accuracy would severely deteriorate unless the hysteresis was properly compensated.

Significant efforts have been devoted to the mathematical modeling and hysteresis compensation for piezoelectric actuators. Historically, hysteresis nonlinearity has been treated as an uncertainty of a nominal linear element, and several robust control techniques have been applied. Tsai and Chen [

7] applied the

${H}_{\infty}$ control to compensate for the uncertainty. Alternative and intensively studied approaches in the literature include the usage of phenomenological hysteresis models [

8]. Examples include the Prandtl–Ishlinskii (PI) model [

9], the Preisach model [

10], the play and stop models [

11,

12], the Bouc–Wen model [

13], and the Duhem model [

14]. These phenomenological models are able to capture hysteretic behaviors accurately. The use of these models for the compensation of hysteresis generally requires the calculation of their inverse hysteresis models, and the results are used as feed-forward controllers for the actuator.

Real-world piezoelectric actuators exhibit rate/frequency-dependent hysteresis. The inverse hysteresis model solution for the compensation of hysteresis nonlinearity requires the development of rate/frequency-dependent hysteresis models. Al Janaideh et al. [

15] introduced a rate-dependent threshold in the play operator in their PI model to capture the increased dominance of the hysteretic behavior on the increase of driving velocity or frequency. Yang et al. [

9,

16] introduced the envelope function to the play operator of the PI model to capture the increasing amplitude of the hysteresis loop, as the frequency of the driving signal increases. Xiao and Li [

10] proposed the modified inverse Preisach model that uses the weighted sum of the distribution functions, each of which is identified by a pure sinusoidal input for compensating frequency-dependent hysteresis.

We have been working for several years on the modeling and compensation of frequency-dependent hysteresis of the bimorph piezoelectric actuator (PZBA-00030, FDK Co., Tokyo, Japan) shown in

Figure 1. Its bandwidth is comparably lower than that of stack type piezoelectric actuators. However, it exhibits very complex frequency-dependent hysteresis in its response, which captivated our interest. There are several driving frequencies whose responses include large odd harmonic oscillation. It even shows the interleaved hysteresis in which both clockwise and counterclockwise loops are included in a single period of response at some driving frequency.

Li et al. [

17] stated that the odd harmonic oscillation of a piezoelectric actuator is caused by the hysteresis nonlinearity. They regarded the odd harmonic component of the response as a disturbance and synthesized a repetitive controller for its attenuation. We previously proposed an enhanced Bouc–Wen model for capturing odd harmonic oscillation induced by a pure sinusoidal input at some driving frequency and proposed a corresponding compensator [

13] based on the direct inverse multiplication proposed by Rakotondrabe [

18].

Regarding the interleaved hysteresis, Alatawneh and Pillay [

19] recently showed that interleaved hysteresis could be captured by the Preisach model by relaxing certain constraints in its distribution function. To the best of our knowledge, only a few results exist on the modeling of interleaved hysteresis, but no result disclosing its compensation methods can be found in the literature.

We recently reported a phenomenological model of hysteresis for the bimorph actuator that covers (a) hysteretic behavior including odd harmonic oscillation at a lower frequency range, (b) interleaved hysteresis over its mechanical resonance, and (c) highly asymmetric large hysteresis loop at much higher frequencies with a single mathematical structure [

20]. However, we have not evaluated the use of the proposed model for hysteresis compensation. This fact motivates our current research, which attempts to synthesize a hysteresis compensator using the modified version of our model in [

20].

The present article proposes a phenomenological hysteresis model for the bimorph piezoelectric actuator based on the modified play model; its use in the compensation of hysteresis that can be observed over the available bandwidth (1–110 Hz) of the actuator. The modified play model which captures frequency-dependent hysteresis is proposed in

Section 2. The development of the model for the refined treatment of the interleaved hysteresis, and odd harmonic oscillation is continued in

Section 3. The compensator design based on the developed model is explained in

Section 4. The results of the experiments for evaluating the modeling accuracy and hysteresis compensation performance is reported in

Section 5. The summarizing conclusion, along with some future work implications, is given in

Section 6.