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.

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 performance, 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. [

Various rate-dependent PI models have been proposed to capture the hysteresis in different rate of input or different frequency values of input applied to the piezoelectric actuators. Al Janaideh et al. [

In this study, an inertial-dependent model is proposed to overcome the large amount of errors imposed in the modeling with the classical PI model, in particular in capturing the hysteresis at high frequencies as well as high amplitudes of inputs applied to a piezoelectric actuator under load-free and preloaded conditions. As a rate-independent model, it is able to capture the hysteresis at different input magnitudes and frequencies even in the presence of preloading force.

One of the well-known operator-based models, which are used to characterize the hysteresis in piezoceramics and other types of materials, is the classical PI model that utilizes two essential hysteresis operators, namely, the play operator and the stop operator.

The rate-independent stop operator is illustrated in Figure

Classical stop operator.

The rate-independent play operator is illustrated in Figure

Classical play operator.

In order to determine the dynamic characteristics of the piezoelectric actuators, the linear constitutive equation has been proposed [

This model takes into consideration two important terms of inertial and damping effects which are shown by

A dedicated set up was built to characterize a piezoelectric actuator experimentally. In this work, the piezoceramic actuator (P-887.90, from PHYSIK Instrument Co.) was used. This actuator provides maximum displacement of

Configuration of the experimental setup.

In order to characterize the hysteretic behavior of the piezoelectric actuator, two sets of sinusoidal inputs for low-amplitude voltage signal (20 V amplitude with 20 V bias) and high-amplitude voltage signal (40 V amplitude with 40 V bias) were applied to the piezoelectric actuator in load-free and preloading conditions. These sets of input signals were applied at varying frequency ranging from 10 Hz to 200 Hz and different preloading forces of 4 N, 8 N, and 12 N. The reason of selecting different amplitudes besides different frequencies is to analyze the hysteresis characteristic in the piezoelectric actuator in terms of amplitude and frequency variations. Different preload condition on the piezoelectric actuator is also expected to affect the hysteresis characteristics. This motivates the experiment to be conducted at different preloading forces.

Applying two sets of low-amplitude and high-amplitude inputs at different frequencies besides different values of preloading force implies three important aspects in the hysteretic behavior. The first thing, which is observed from this experiment, is attributed to the rate dependency of the hysteresis curve; that is, when the rate of the input voltage is increased, the width of the hysteresis curve is also increasing. Figure

Width and slope of the hysteresis curve at different frequencies and input amplitudes for load-free piezoelectric actuator.

Figure

Width and slope of the hysteresis curve at different frequencies and input amplitudes for preloaded piezoelectric actuator.

The second finding of this experiment is regarding the relationship between the maximum stroke of the actuator and maximum input voltage applied to the actuator. For the case of 40 V input, the maximum stroke of 10.12

The third finding observed from this experiment is regarding a tight relationship between amplitude and frequency in one hand and asymmetrical hysteresis shape and high slope of the hysteresis curve in the other hand. As illustrated in Figures

Asymmetric hysteresis curve at different frequencies and input amplitudes for load-free piezoelectric actuator.

In order to capture the symmetrical hysteresis shapes, which are recorded up to 200 Hz with low-amplitude and high-amplitude voltages, a stop operator was utilized. As described in Section

The classical PI model was implemented to characterize the hysteresis in the piezoelectric actuator by utilizing four classical stop operators, in the cases, where the symmetric hysteresis curve takes place. This modeling approach is parameterized by four threshold values of

As can be seen in Figures

The results of modeling with the classical PI for load-free piezoelectric actuator for different input voltage and frequency: (a) 40 V; 10 Hz, (b) 80 V; 10 Hz, (c) 40 V; 25 Hz, (d) 80 V; 25 Hz, (e) 40 V; 200 Hz, (f) 80 V; 200 Hz.

The results of modeling with the classical PI for preloaded piezoelectric actuator under 12 N force, for different input voltage and frequency: (a) 40 V and 10 Hz, (b) 80 V and 10 Hz, (c) 40 V and 25 Hz, and (d) 80 V and 25 Hz.

In order to overcome the problem, as discussed in the previous section, the inertial-dependent model is introduced, where two important dynamic terms of inertial and damping effects, which are stated by

The results of modeling with Inertial-dependent PI for load-free actuator with different input voltage and frequency: (a) 40 V and 10 Hz, (b) 80 V and 10 Hz, (c) 40 V and 200 Hz, and (d) 80 V and 200 Hz.

One of the interesting points appears from the characterization of the hysteresis at frequency of 200 Hz with high-amplitude voltage, where the hysteresis curve starts to skew resulting in an asymmetrical hysteresis. The error in this case seems to be larger compared to the cases at lower frequencies (see Figure

Figure

The results of modeling with the inertial-dependent PI with different input voltage, frequency, and preload: (a) 40 V; 200 Hz; 4 N, (b) 40 V; 200 Hz; 8 N, (c) 40 V; 200; 12 N, (d) 80 V; 100 Hz; 4 N, (e) 80 V; 100 Hz; 8 N, and (f) 80 V; 100 Hz; 12 N.

The result shows that the introduction of the inertial and damping effect to the classical PI model improves the performance of the model in wide range of operation. As can be seen in Figures

On the other hand, from the simulation at different frequencies with different amplitudes in both preloading and load-free circumstances, the parameter of damping effect is varying as illustrated in Figure

Damping factor variations at different frequencies with different amplitudes (▲, low-amplitude voltage, 40 V; ●, high-amplitude voltage, 80 V).

In this study, the performance of the classical PI model was evaluated using sets of data which incorporate two different magnitudes of the input voltage and varying frequency from 10 Hz to 200 Hz for different preloading conditions. It is shown that the classical Prandtl-Ishlinskii model is unable to characterize the hysteresis in piezoelectric actuators at wide range of frequency and amplitude.

In order to tackle this problem, as the physical aspect of the piezoelectric actuator will manifest itself in the dynamic behavior, the inertial-dependent PI model is proposed to capture the hysteresis at different input amplitudes and frequencies. The proposed model exhibits better performance compared to that of the classical one as implied by the modeling error for wide range of applications. However, moving toward high-frequency applications, the performance of the proposed model is degrading. Incorporating the rate-dependent parameters, for example, as presented by Al-Janaideh et al. [