**1. Introduction**

Piezoelectric materials are capable of undergoing reversible phase transitions as a result of voltage and pressure. Due to these special material properties, piezoelectric materials have become increasingly popular in sensors and actuators. Piezoelectric ceramics actuators (PCAs) have been employed in precision positioning systems for their large force generation, high stiffness, high resolution and fast response. However, they commonly exhibit strong hysteresis behaviors, which greatly degrade the overall positioning accuracy.

According to whether the rate of the input is considered or not, hysteresis behaviors can be divided into rate-dependent and rate-independent hysteresis behaviors. The corresponding hysteresis models can be classified into rate-dependent and rate-independent hysteresis models. Over the past few decades, grea<sup>t</sup> efforts have been devoted to developing hysteresis models such as the Prandtl–Ishlinskii model [1–3], Preisach model [4–6], Maxwell-Slip model [7,8], Duhem model [9,10], Polynomial-based hysteresis model [11,12], and Bouc–Wen model [13,14]. For modeling of rate-independent hysteresis, it mainly focuses on the nonlinear relationship between the amplitude of input voltage and output displacement at low input frequency or rate. However, for modeling of rate-dependent hysteresis, it needs an analysis of the nonlinear relationship between the rate of input voltage and output displacement at high input frequency or rate. Comparisons of rate-dependent and rate-independent hysteresis models reveal that the rate-independent model is just a special kind of rate-dependent model when the rate of input is low enough. Therefore, it is more difficult to develop the rate-dependent

model relatively. Overall, most related literatures focused on developing rate-independent hysteresis models and few literatures paid attention to modeling of rate-dependent hysteresis.

Due to its differential equations and ability to capture an analytical form, the Bouc–Wen model has been widely applied in hysteresis modeling and compensation for piezoelectric ceramics actuators. Based on the classical Bouc–Wen (CB–W) model, Zhu and Wang [15] added a non-symmetrical formula to describe non-symmetrical hysteresis and the corresponding experiments demonstrated its validity. Fujii et al. [16] proposed an extended Bouc–Wen model by introducing a velocity sign sensitivity. To eliminate the influence of nonlinear hysteresis, Li et al. [17] presented an adaptive sliding mode control with perturbation estimation (SMCPE) based on the classical Bouc–Wen model. In addition, Liu et al. [18] proposed an adaptive neural output-feedback control based on a modified Bouc–Wen model. Lin and Yang [19] used a Bouc–Wen model to describe the hysteresis behavior and designed a hysteresis-observer based control to compensate for the piezoelectric actuator.

It should be noted that the piezoelectric actuator possesses a non-symmetrical hysteresis according to a lot of experimental research [20,21]. When the input rate is high, the non-symmetrical characteristic of the piezoelectric actuator is more serious. However, the classical Bouc–Wen model is used to describe a symmetrical hysteresis. When the input frequency or rate is low, the modeling error of the classical Bouc–Wen model is not large. However, when the input frequency or rate is high, its modeling error is large. Therefore, it can be found that the classical Bouc–Wen model is mainly used to characterize the rate-independent hysteresis behavior and modeling, but cannot characterize the rate-dependent hysteresis behavior precisely though it is a rate-dependent hysteresis model according to traditional classifications. The modeling accuracies of published hysteresis models are not high enough. Furthermore, due to the existence of many parameters and differential equations, it is a hard task to identify the parameters of hysteresis models.

In our previous work [22], we have developed an enhanced Bouc–Wen model by introducing input frequency. But there is a limitation that the developed model cannot be applied when the input frequency is unknown. To solve the problems above, this paper proposed a generalized Bouc–Wen (GB–W) model by introducing relaxation functions in the classical Bouc–Wen model, which can characterize both rate-independent and rate-dependent hysteresis behavior for piezoelectric ceramics actuators precisely. A lot of experiments are conducted in advance to characterize hysteresis behaviors and subsequently the relaxation functions are determined cautiously based on these experimental characteristics. The generalized Bouc–Wen (GB–W) model doesn't have the aforementioned limitation and can be widely applied. In addition, the nonlinear least squares method through MATLAB/Simulink is used to identify the corresponding parameters of hysteresis models. Both simulations and experiments finally demonstrate the validity of the developed model. Therein, the classical Bouc–Wen model is set as a comparison. The rest of this paper is arranged as follows: Section 2 gives the descriptions of the classical Bouc–Wen model. In Section 3, the generalized Bouc–Wen model is presented. Section 4 gives the experimental validation of results and discussion. Finally, conclusions are drawn in Section 5.

#### **2. Classical Bouc–Wen Model**

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The Bouc–Wen model was initially applied for nonlinear vibrational mechanics. With the rapid development of smart actuators, it was gradually used to describe nonlinear hysteresis for piezoelectric actuators. The hysteresis curve can be considered as the superposition of a linear component *X*(*t*) and a hysteretic component *h*(*t*). The classical hysteretic Bouc–Wen model as a nonlinear system is described as follows:

$$y(t) = X(t) + h(t) = k \cdot u(t) + h(t) \tag{1}$$

$$\dot{h}(t) = a\dot{u}(t) - \beta \dot{u}(t)|h(t)|^n - \gamma |\dot{u}(t)| |h(t)|^{n-1} h(t) \tag{2}$$

where *u*(*t*) is the input voltage and *y*(*t*) is the output displacement. *k*, *α*, *β*, *γ* and *n* are the model parameters, which decide the shape of hysteresis curves. In order to simplify the model, *n* is usually set as 1 and the hysteretic components is expressed by

$$h(t) = a\dot{u}(t) - \beta \dot{u}(t)|h(t)| - \gamma \left|\dot{u}(t)\right| h(t) \tag{3}$$

#### **3. Generalized Bouc–Wen Model**

To analyze the performance of the classical Bouc–Wen model in detail, some efforts were devoted to research on the characteristics of its parameters. The variations of its parameters *k* and *α* at different frequencies of the input are shown in Figures 1 and 2, respectively. Table 1 gives the detailed values of parameters of the classical Bouc–Wen model at different frequencies. The parameters were identified by the nonlinear least squares method through MATLAB/Simulink, which will be introduced in detail in the next part. The identified results based on experimental data clearly reveal that the parameters *k* and *α* both decrease with the increase in frequency. Such frequency dependence of the parameters *k* and *α* cannot be described by the classical Bouc–Wen model. The parameters of the classical Bouc–Wen model are fixed constants, which cannot characterize their change trend with the increase in frequency. To some degree, thus, it can be concluded that the classical Bouc–Wen model cannot describe rate-dependent hysteresis behaviors.

**Figure 1.** Variations of *k* under *u*(*t*) = 5 sin(<sup>2</sup>*π f t*) + 5 at different frequencies.

**Figure 2.** Variations of *α* under *u*(*t*) = 5 sin(<sup>2</sup>*π f t*) + 5 at different frequencies.


**Table 1.** Identified parameters of the classical Bouc–Wen (CB–W) model at different frequencies.

#### *3.1. Formulation of the Generalized Bouc–Wen Model*

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The results above show that the classical Bouc–Wen model cannot describe rate-dependent hysteresis behaviors precisely. It is resulted by a non-symmetrical hysteresis of piezoelectric actuators while the classical Bouc–Wen model is a symmetrical model. Therefore, the classical Bouc–Wen model should be redefined to solve this problem. However, it should be noted that the redefined model should possess the capacity to describe both rate-independent and rate-dependent hysteresis, which is the purpose of this work.

How to formulate a generalized hysteresis model of piezoelectric actuators is still a hard task and some researchers have made some contributions, such as, Al Janaideh et al. [23] presented a generalized P-I model with relaxation functions to characterize rate-dependent hysteresis behaviors. Mayergoyz [24] proposed a generalized Preisach model of hysteresis by introducing a generalized density function. Based on the experimental dynamic characteristics and researchers' experiences above, the generalized Bouc–Wen model is thus formulated upon integrating relaxation functions *k*(*v*(*t*), .*v*(*t*)) and *<sup>α</sup>*(*v*(*t*), .*<sup>v</sup>*(*t*)), such as

$$y(t) = X(t) + h(t) = k(u(t), \dot{u}(t)) \cdot \dot{u}(t) + h(t) \tag{4}$$

$$\dot{h}(t) = a(u(t), \dot{u}(t)) \cdot \dot{u}(t) - \beta \dot{u}(t)|h(t)| - \gamma |\dot{u}(t)|h(t) \tag{5}$$

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where *k*(*u*(*t*), .*u*(*t*)) and *<sup>α</sup>*(*u*(*t*), .*u*(*t*)) are relaxation functions of current input *u*(*t*) and its rate of .*u*(*t*) and *k*(*u*(*t*), .*u*(*t*)) is a positive function. It should be noted that the generalized Bouc–Wen model could describe rate-independent hysteresis behaviors at low input frequency. Therefore, *k*(*u*(*t*), .*u*(*t*)) and *<sup>α</sup>*(*u*(*t*), .*u*(*t*)) should converge to fixed constants when the input frequency is low. Based on the characteristics above, the relaxation functions can be defined by inducing the exponential function, such as

$$k(u(t), \dot{u}(t)) = p e^{-q\dot{\hat{u}}(t)} \tag{6}$$

$$\kappa(\dot{u}(t), \dot{u}(t)) = \varepsilon e^{\delta |\dot{\hat{u}}(t)|} \tag{7}$$

where *p* ≥ 0, *q* ≥ 0, *ε*, *δ*, *β* and *γ* are constants. From the expressions of the relaxation functions, it can be found that when the input frequency is very low, such that .*u*(*t*) ∼= 0, the relaxation functions converge to fixed constants, such that *k*(*u*(*t*), .*u*(*t*)) ∼= *p* and *<sup>α</sup>*(*u*(*t*), .*u*(*t*)) ∼= *ε*. Thus, it can describe the rate-independent hysteresis model the same as the classical Bouc–Wen model.

Compared with the classical Bouc–Wen model, the parameters *k*(*u*(*t*), .*u*(*t*)) and *<sup>α</sup>*(*u*(*t*), .*u*(*t*)) of the proposed model values vary with the rate of input, which are not fixed constants any more. Furthermore, the parameters *k*(*u*(*t*), .*u*(*t*)) in the rising hysteresis curves at the same input are different from that in the decreasing hysteresis curves. The characteristics above form the non-symmetrical hysteresis of the proposed model.

#### *3.2. Properties of the Generalized Bouc–Wen Model*

This section will focus on analyses of the properties of the generalized Bouc–Wen model. First, the characteristics of the hysteretic component *h*(*t*) should be analyzed based on simulations. Figure 3 shows the relationship between *h*(*t*) and input frequency *f* under input voltage *u*(*t*) = 5 sin(<sup>2</sup>*πf t*) + 5. The corresponding parameters of the generalized Bouc–Wen model are set as *p* = 0.2107, *q* = 1.189 × <sup>10</sup>−5, *ε*= −0.1331, *δ* = 5.2622 × <sup>10</sup>−4, *β* = 5.3743 and *γ* = 6.4698. The results clearly reveal that the width of the hysteretic component *h*(*t*) increases monotonically with the increase in the rate of input.

**Figure 3.** Variations of the component *h*(*t*) under *u*(*t*) = 5 sin(<sup>2</sup>*π f t*) + 5: (**a**) *f* = 10 Hz; (**b**) *f* = 40 Hz; (**c**) *f* = 80 Hz.

Figure 4 shows the relationship between the component *X*(*t*) and input frequency *f* . The results reveal that the component *X*(*t*) is still nearly linear with input voltages at low frequencies. But the curves of the component *X*(*t*) have a hysteresis loop with the increase in frequencies, which shows non-symmetrical characteristics.

**Figure 4.** Variations of the component *X*(*t*) under *u*(*t*) = 5 sin(<sup>2</sup>*π f t*) + 5: (**a**) *f* = 10 Hz; (**b**) *f* = 40 Hz; (**c**) *f* = 80 Hz.
