*3.3. Parameters Identification*

Due to the existence of derivation and more parameters, it is not easy to identify the Bouc–Wen model for most researchers. In this study, the objective function *F* is expressed by

$$F = \operatorname{Min} \sum\_{i=1}^{n} f^2(u) \tag{8}$$

with

$$f(\mu) = y\_i - y\_i^{HM} \tag{9}$$

$$
\dot{y}\_i{}^{HM} = X(iT) + h(iT) = p e^{-q\dot{u}(iT)}\dot{u}(iT) + h(iT) \tag{10}
$$

$$\dot{h}(\ddot{x}T) = \varepsilon \varepsilon^{\delta \left| \dot{u}(\ddot{x}T) \right|} \cdot \dot{u}(\dddot{x}T) - \beta \dot{u}(\dddot{x}T)|h(\dddot{x}T)| - \gamma \left| \dot{u}(\dddot{x}T) \right| h(\dddot{x}T) \tag{11}$$

where *n* is the total number of samples, *T* is the sampling period, *i* = 1, 2, 3 ··· *n* is the *i*th sampling period, *yiHM* and *yi* are the predicted output by hysteresis model and experimental displacements of piezoelectric actuators, respectively. According to these equations above, it can be concluded that *f*(*u*) is a nonlinear function of the parameters *p*, *q*, *<sup>ε</sup>*,*δ*,*β* and *γ*. It is a nonlinear least squares problem. It is

better to choose the nonlinear least squares method instead of the least squares method to identify parameters of the proposed model in our study.

In this paper, the nonlinear least squares method using the Trust-Region-Reflective algorithm is presented to identify the parameters of classical and generalized Bouc–Wen models. This method uses the nonlinear least squares function for optimization through the MATLAB/Simulink Optimization Toolbox. The corresponding identification steps of the nonlinear least squares method is carried out offline as follows:


**Figure 5.** Classical Bouc–Wen (CB–W) model implemented with Matlab/Simulink.

**Figure 6.** Generalized Bouc–Wen (GB–W) model implemented with MATLAB/Simulink.

In traditional identification methods, the identified procedure is usually a long and complex work with many equations and steps. For example, in reference [15], Zhu and Wang added a non-symmetrical formula based on the classical Bouc–Wen model to describe non-symmetrical hysteresis. The parameters for the linear and hysteresis components are separately identified by utilizing the final value theorem of the Laplace transform and the least squares method, respectively. However, the nonlinear least squares method using the Trust-Region-Reflective algorithm in this study is presented to identify all parameters for the linear and hysteresis components at the same time, which can simplify the identification procedure. In addition, all operations are conducted by using the MATLAB/Simlink optimization tools and there are no complex algorithms needed to write, which can simplify the identification procedure further. Furthermore, there are just four steps above and the whole computing time is controlled in several minutes. Therefore, the nonlinear least squares method can quickly and simply identify the parameters of hysteresis models. Last but not least, this method can be expanded to apply to other complex model identifications, which is useful and meaningful undoubtedly.

#### **4. Experimental Validation Results and Discussion**
