## 4 Experimental Verification
### Parameter Identification Results  
In subsequent experiments, the normalized root mean square error (NRMSE) and normalized maximum error (NME) are selected as unified performance evaluation indicators. Among them, \(y_{ai}\) and \(y_{i}\) represent the actual angle and predicted angle of the FSM system, respectively, and \(M\) denotes the number of samples. \[NME=\frac{\max\left|y_{ai}-y_{i}\right|}{\max(y_{i})-\min(y_{i})}\times 100\% \tag{10}\]  
\[NRMSE=\frac{\sqrt{\frac{1}{M}\sum\left(y_{ai}-y_{i}\right)^{2}}}{\max(y_{i})- \min(y_{i})}\times 100\% \tag{11}\]  
Given that the MHL model consists of three components, i.e., a time delay, a rate-dependent model, and a linear dynamic term, the parameter identification process of the model is divided into three distinct steps in sequence. Step 1: A step signal is input into the FSM, and the output of the system is compared with the input signal to measure the experimental system delay, which is 68 \(\upmu\)s, as shown in Figure 5.  
Step 2: When a low-frequency signal is used as input for the FSM, its dynamic behavior is not obvious. Therefore, a variable-amplitude signal \(v(t)=\frac{U}{2}\left[\sin\left(2\pi t-\frac{\pi}{2}\right)+1\right]\) with \(U\in[0,100]V\) is utilized to excite the FSM, and its output is obtained through the sensor. By using hysteresis curve description methods [19; 20] and employing the least square method, the parameters of major hysteresis curves, as shown in Table 1, are determined. Subsequently, a rate-independent model \(H_{\text{S}}(v(t))\) and its inverse model are established. Figure 9 shows the prediction results of the established rate-independent model. It can be observed that the identified model predicts the actual angle output of the FSM very well, with an NME of 0.96% and an NRMSE of 0.25%. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Model** & **Inverse Model** \\ \hline \((p_{5},q_{5})\) & \((-0.4624,0.9457)\) & \((0.4523,-0.04333)\) \\ \((p_{4},q_{4})\) & \((1.788,-3.005)\) & \((-1.422,0.3898)\) \\ \((p_{3},q_{3})\) & \((-2.398,3.248)\) & \((1.7,-0.5418)\) \\ \((p_{2},q_{2})\) & \((2.037,-2.083)\) & \((-1.125,0.4221)\) \\ \((p_{1},q_{1})\) & \((2.228,4.104)\) & \((1.397,0.7696)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model and inverse model parameters of \(f_{0}(x)\) and \(g_{0}(x)\). Figure 8: Experimental setup. To capture the dynamic behavior of the FSM within a certain frequency range, a full-amplitude frequency sweep signal \(v(t)=50\big{[}\sin\bigl{(}2\pi ft-\frac{\pi}{2}\bigr{)}+1\big{]}\) with frequencies \(f\in[0,100]\) Hz is employed to stimulate the FSM. By performing a differential operation between the output \(H_{\text{S}}(v(t))\) of the rate-independent model and the system output \(y(t)\), the output \(y(t)-H_{\text{S}}(v(t))\) of the rate correction term is acquired. By using the least square method, the parameters of the rate-dependent model and its inverse model are identified based on \(y(t)-H_{\text{S}}(v(t))\) and the input signal, as shown in Table 2. The established model is based on the phenomenological approach, where its modeling parameters bear no direct relation to the physical nature of hysteresis. Consequently, the characterization of the rate-dependent hysteresis behavior can be achieved through experimental observation data [7]. Step 3: Signals with a frequency range of \(f\in[0,4000]\) Hz are applied to stimulate the FSM, and the system output \(y(t)\) and the rate-dependent model output \(w(t)\) are obtained. By utilizing the amplitude and phase relationship between signals \(y(t)\) and \(w(t)\), the parameters of the linear dynamic term \(G(s)\) are further identified, as shown in Equation (12). The order and coefficients of the identified linear dynamic term \(G(s)\) are determined based on the input-output data of the system and the fitting accuracy of the identification method, without necessarily providing physical insights into the modeling problems [5].