## 3 Simulation Results and Empirical Results Verifications  
### Simulation Results  
In the simulation, the maximum input voltage V\({}_{p}\) was set to 1000 mV. The directly measured displacement \(\mathbf{x}_{meas}(t)\) and current \(\mathbf{i}_{meas}(t)\) were used to conduct analysis in the experiment case. The values of the input parameters and units are given in Table 1. The fitting equation of the nonlinear parameters was defined as Equation (7). The values initial guesses of parameters were initially inferred based on \(\mathbf{w}\). In this case study, the initial guesses of parameters can be randomly done in the suitable range. To explain the suitable range of determined results, Table 2 examines the influence of three sets of initial guesses on optimization convergence. All of initial guesses are generated with random numbers. The range of the initial guesses is assumed around 10 or 1/100 times the exact parameters from Table 1. With regard to the correctness and calculation efficiency of the method proposed in this study, the following random voltage V\({}_{p}\) and angular frequency \(\omega_{f}\) dependent sinus signal was used:  
\[e(t_{n})=\text{Random}(\text{V}_{p})\cdot\sin\left(\text{Random}(\omega_{f} )t_{n}\right), \tag{24}\]  
where \(t_{n}=0\sim t_{N}\) and \(\omega_{f}=50\) Hz to 10,000 Hz. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Values** \\ \hline \(M_{m}\) & 0.049 \(\times 10^{-3}\) (kg) \\ \(R_{e}\) & 4.560 (Ohm) \\ \(Bl(\mathbf{x})\) & 0.391 \(\exp\left(\frac{-(x-0.017)^{2}}{0.319^{2}}\right)\) (N/A) \\ \(R_{m}(\mathbf{x})\) & 0.079 \(\exp\left(\frac{(x-0.012)^{2}}{0.224^{2}}\right)\) (kg/s) \\ \(L_{e}(\mathbf{x})\) & 0.010 \(\times 10^{-3}\exp\left(\frac{-(x-0.017)^{2}}{0.575^{2}}\right)\) (H) \\ \(K_{m}(\mathbf{x})\) & 1010.218 \(\exp\left(\frac{(x-0.010)^{2}}{0.876^{2}}\right)\) (N/m) \\ \hline \end{tabular}
\end{table} 
Table 1: Exact linear and nonlinear electroacoustic parameters. The reason for a conventional distortion measurement of displacement and current is because the harmonic distortions usually occur in the low frequencies range (as 10 Hz to 300 Hz) caused by the nonlinear force factor \(Bl(\mathbf{x})\), stiffness \(K_{m}(\mathbf{x})\) and damping factor \(R_{m}(\mathbf{x})\). However, the Inductance \(L_{e}(\mathbf{x})\) (magnetic AC-field varies with coil position) increases the impedance at higher frequencies (after 500 Hz). Therefore, the distortions related to the nonlinear parameters should be measured by exciting the miniature transducer with a broadband signal, such as pseudo-random noise and by applying the proposed method to the input and output signal. The efficiency and feasibility of the proposed method are demonstrated using simulation tests. First, a set of exact parameters is given. Second, the numerical solver is used to solve the exact solutions \(\mathbf{x}_{sim}(t)\) and \(\mathbf{i}_{sim}(t)\). Subsequently, the above computational procedure is used to obtain the predict results. As a result in Table 3, the objective function in each case is smaller than 1 \(\times\)\(10^{-7}\), it indicates that accurate parameters can be obtained by the proposed method. It is observed that the iteration numbers would be increased if the initial guesses are not close to the exact parameters. Therefore, it can be determined that the assumed initial guess value is appropriate by confirming the number of iterations and the convergence function. Following the nonlinear optimization procedure that was proposed in the previous section, the nonlinear parameters, impedance curves, displacement \(\mathbf{x}(t)\), and current \(\mathbf{i}(t)\) were obtained using the proposed algorithms BFGS, BFGS-Brent, and BFGS with constraints that were subsequently compared. A comparison of the positive and estimated values is shown in Figure 3a-d.