**1. Introduction**

Piezoelectric ceramic, a new type of functional material, plays an important role in many real-world applications due to its superior performance in converting electrical energy into mechanical energy [1–4]. Piezoelectric ceramic actuators (PCAs) are ones of the most important applications of the piezoelectric ceramic material owing to their small size, high accuracy, and fast response. Hysteresis behavior is an inherent characteristic of PCAs and has already become a bottleneck in developing the applications of the PCAs. Therefore, it is of grea<sup>t</sup> significance to develop more precise hysteresis models for characterizing hysteresis behaviors.

The existing models of PCAs can be generally divided into the rate-independent and rate-dependent hysteresis models [5]. The rate-independent hysteresis models include the Preisach model [6,7], Prandtle–Ishlinskii model [8,9], Maxwell-slip model [10,11] and polynomial-based hysteresis model [12,13], and can be used to describe the nonlinear relationship between the input voltage and the output displacement of PCAs. However, the input rates of these models are generally lower than that of the rate-dependent hysteresis models, such as the Bouc–Wen model [14,15] and Dahl model [16–18]. That is mainly because, unlike the rate-independent hysteresis model, the rate-dependent one can describe the dynamic relationship between the input rate and the output. However, the existing rate-dependent hysteresis models generally have low prediction precision and high complexity of model equations. Therefore, how to construct a new, simpler rate-dependent hysteresis model is an urgen<sup>t</sup> and challenging issue of grea<sup>t</sup> significance.

Due to its differential equations, the Duhem model has been used to describe and compensate piezoelectric hysteresis behaviors [19–22]. For example, C.-J. Lin and P.-T. Lin [23] combined the Bouc–Wen model, Dahl model and Duhem model as a modified Duhem model and presented a feedforward controller. Wang et al. [24] identified the Duhem model by neural network methods and designed a robust adaptive controller to compensate hysteresis behaviors. Xie et al. [25] presented an observer-based adaptive controller based on the Duhem model for piezoelectric actuators.

The classical Duhem model only characterizes symmetrical hysteresis loops while the actual hysteresis loops of piezoelectric actuators are non-symmetrical. It is worth mentioning the fact that the higher the frequency or the amplitude of the input excitation is, the more serious the hysteresis behaviors are [26,27]. When the frequency or amplitude of input excitation signal is increasing, the non-symmetrical of hysteresis loops is more serious. Therefore, the classical Duhem model already cannot precisely describe rate-dependent hysteresis behaviors at high-frequency and high-amplitude excitations. Thus, Oh and Bernstein [28] proposed the rate-independent and rate-dependent semilinear Duhem models without analyzing the modeling errors in detail at high-frequency and high-amplitude excitations by using a complex model. So far, few e fforts have been devoted to developing new hysteresis models based on Duhem model.

Motivated by the aforementioned discussions, this paper proposes a modified Duhem model to describe rate-dependent hysteresis behaviors by introducing trigonometric functions. The proposed model has a simple expression and can detailly characterize rate-dependent hysteresis behaviors precisely at high-frequencies and high-amplitude input excitations. The parameters of models can be easily identified by the nonlinear least squares method. The validity of the proposed model is demonstrated via simulation experiments. The rest of this article is organized as follows: In Section 2, the hysteresis system is constructed to introduce the expression of the classical Duhem model. Section 3 introduces the proposed model and the identification of corresponding parameters. Section 4 aims to verify the validity of the established model, and compares it with the classical Duhem model. The results of the analysis are obtained immediately. The conclusion of this paper is placed in Section 5.

#### **2. Classical Duhem Model (CDM)**

In 1986, Coleman and Hodgdon [29] proposed a hysteresis model for ferromagnetic materials, which describes the relationship between the magnetic field *H(t)* and magnetic flux *B(t)* as follows:

$$\dot{B}(t) = a \left| \dot{H}(t) \right| \cdot \left[ \beta (H(t)) - B(t) \right] + \gamma \dot{H}(t) \tag{1}$$

where α, β and γ are the parameters controlling the shape and size of the hysteresis loop. According to the relationship between single-input and single-output [28], the hysteresis system is given by

$$
\dot{\mathbf{x}}(t) = f(\mathbf{x}(t), \mathbf{u}(t), \dot{\mathbf{u}}(t)), \mathbf{x}(0) = \mathbf{x}\_0, t \ge 0,\tag{2}
$$

$$y(t) = h(\mathbf{x}(t), \mathbf{u}(t))\tag{3}$$

where *u(t)* is the input, *y(t)* is the output and *x(t)* is a part of it. When this hysteresis model is used to describe hysteresis system of PCAs, the classical Duhem model (CDM) is proposed and expressed as follows: 

⎪⎪⎪⎪⎨

$$\begin{cases} \mathcal{Y}(t) = X(t) - h(t) \\ X(t) = k \cdot u(t) \\ \dot{h}(t) = a\dot{u}(t) - \beta \Big| \dot{u}(t) \Big| h(t) + \gamma \Big| \dot{u}(t) \Big| u(t) \end{cases} \tag{4}$$

where *X*(*t*) is the linear component and *h*(*t*) is the hysteretic component, *u*(*t*) is the input voltage and . *u*(*t*) is the derivative of voltage, and *k*, α, β and γ are the model parameters.

To evaluate the performance of the CDM, two groups of experiments were conducted. First of all, a sinusoidal input signal *ua*(*t*) = sin(<sup>2</sup>π · *t*)+1 with the frequency of 1 Hz was taken as a reference signal to identify the CDM parameters. The corresponding CDM parameters were identified by utilizing the nonlinear least squares method as *k* = 0.261, α = 0.062, β = 0.131 and γ = 0.001. The first group of experiments, called Exp-a, adopted an input excitation signal *ua*(*t*) = 8 sin(<sup>2</sup>π · *f*1*<sup>t</sup>*) +6 sin(<sup>2</sup>π · *f*2*<sup>t</sup>*) +14 with *f* 1 = 15 Hz and *f* 2 = 40 Hz. Figure 1 shows the corresponding comparison between the experimental and simulation results. The results reveal that the error of one point of the CDM (the dotted line) is nearly 2 μm (20% of the displacement range). The maximum modeling error is about 2 μm, which is undoubtedly big. In the second group of experiments, called Exp-b, an input excitation signal

*ua*(*t*) = 20 sin(<sup>2</sup>π · *t*) + 20 was used. Figure 2 shows the corresponding comparison between the experimental and simulation results of the CDM. The maximum error is about 0.5 μm. It should also be noticed that the point with the maximum modeling error is the point with .*u*(*t*) → 0. These actual experimental results demonstrate that the CDM cannot precisely describe the rate-dependent hysteresis behaviors at high-frequency and high-amplitude excitation signals.

**Figure 1.** Comparison of the output displacements between experimental data (Exp-a) and simulation of the classical Duhem model (CDM).

**Figure 2.** Comparison of the output displacements between experimental data (Exp-b) and simulation of CDM.

#### **3. Modified Duhem Model (MDM)**

The main components of CDM are linear component *X*(*t*) and hysteretic component *h*(*t*), the former having large influence on the output. Thus, the optimization based on the linear component *X*(*t*) is an important research hotspot [30,31]. It should be noted that the main structures of CDM are kept, which can still describe the fundamental characteristics of hysteresis behaviors. With respect to the direct relationship between the input and output, there is an important variation .*<sup>u</sup>*(*t*), which has large influences on the whole model when the input frequency is high. The special points . *u*(*t*) = 0 are the demarcation points where the input voltage curves go up and down. These important points decide the final shape of hysteresis loops, which has been demonstrated in the previous literature [12,13]. The CDM only characterizes symmetrical hysteresis loops while the actual hysteresis loops of piezoelectric actuators are non-symmetrical. When the frequency or amplitude of input excitation signal is increasing, the non-symmetrical of hysteresis loops is more serious, Therefore, the corresponding errors of the CDM are higher at high-frequency excitations, especially the special points

. *u*(*t*) = 0. Figure 2 also demonstrates that the points with .*u*(*t*) → 0 have bigger errors. The actual output displacement of PCA varies little when the value of .*u* varies greatly at high-frequency excitations. The trigonometric function also has the similar characteristics. Its output varies little when the input varies greatly. Therefore, the trigonometric function as a periodic function has the special points where their derivatives are zero, which can be easily used to compensate the bigger errors of the special points . *u*(*t*) = 0. Furthermore, it has a simple expression. Thus, it is a good try to introduce the trigonometric function based on the CDM. Lastly, a modified Duhem model (MDM) based on CDM is proposed and expressed as follows:

$$\begin{cases} Y(t) = X(t) - h(t) \\ X(t) = k \cdot u(t) + p \cdot u(t) \cdot \cos[\left| \dot{u}(t) \right|] + q \cdot \dot{u}(t) \\ \dot{h}(t) = \alpha \cdot \dot{u}(t) - \beta \cdot \left| \dot{u}(t) \right| \cdot h(t) + \gamma \cdot u(t) \cdot \left| \dot{u}(t) \right| + \varepsilon \cdot \dot{u}(t) \cdot \sin[\left| \dot{u}(t) \right|] \end{cases} \tag{5}$$

where *p*, *q*, ε, *k*, α, β and γ are constants. It must be noticed that when .*u*(*t*) → 0, there is cos .*u*(*t*) → 1 and *p* · *u*(*t*) · cos .*u*(*t*) → *p* · *<sup>u</sup>*(*t*), which can be perfectly used to compensate for the bigger errors of the special points .*u*(*t*) → 0.

Over the past decade, several methods for parameters identification of models [19,24,32] have been developed, but their identification processes are generally complex. In our previous work [15,33], the nonlinear least squares method is proposed to identify Bouc–Wen model. The nonlinear least squares method adopts the trust-region-reflective algorithm and take the nonlinear least squares function for optimization through the MATLAB/Simulink Optimization Toolbox. Compared with the previous methods, the method is much simpler and can be more easily applied to identify other models. In this paper, the nonlinear least squares method is adopted to identify the MDM and CDM. The objective function *F* is defined as follows:

$$F = \operatorname{Min} \sum\_{i=1}^{n} f^2(u) \tag{6}$$

$$f(u) = \mathcal{Y}\_i - \mathcal{Y}\_i^{\text{HM}} \tag{7}$$

In CDM, there is

$$\begin{cases} Y\_i^{\text{HM}}(iT) = \mathcal{X}(iT) - h(iT) \\ \mathcal{X}(iT) = k \cdot \boldsymbol{u}(iT) \\ \dot{h}(iT) = \boldsymbol{\alpha} \cdot \dot{\boldsymbol{u}}(iT) - \boldsymbol{\beta} \cdot \left| \dot{\boldsymbol{u}}(iT) \right| \cdot h(iT) + \boldsymbol{\gamma} \cdot \boldsymbol{u}(iT) \cdot \left| \dot{\boldsymbol{u}}(iT) \right| \end{cases} \tag{8}$$

In MDM, there is ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

$$\begin{cases} \begin{aligned} \mathcal{Y}\_{i}^{HM}(iT) &= X(iT) - h(iT) \\ X(iT) &= k \cdot u(iT) + p \cdot u(iT) \cdot \cos\left[\left|\dot{u}(iT)\right|\right] + q \cdot \dot{u}(iT) \\ \dot{h}(iT) &= a \cdot \dot{u}(iT) - \beta \cdot \left|\dot{u}(iT)\right| \cdot h(iT) + \gamma \cdot u(iT) \cdot \left|\dot{u}(iT)\right| + \varepsilon \cdot \dot{u}(iT) \cdot \sin\left[\left|\dot{u}(iT)\right|\right] \end{aligned} \tag{9}$$

where *i* = 1, 2, 3, ··· , *n* is the number of sample experiments, *T* is the period of a sample, *Yi* is the *i*-th output displacement of the PCAs obtained from experiments, and *YHMi* is the *i*-th output simulated by the hysteresis model. The corresponding identification steps of the nonlinear least squares method were carried out offline as follows:


*YHMi* (*iT*) predicted by the CDM or MDM. Equations (8) and (9) are expressed using MATLAB/Simulink blocks.


**Figure 3.** Classical Duhem model implemented with Matlab/Simulink.

**Figure 4.** Modified Duhem model implemented with Matlab/Simulink.
