# Research on Variable-Step-Size Adaptive Filter Algorithm with a Momentum Term
## 3 Numerical Simulation and Verification
### Experimental Steps and Results  
In order to verify whether the proposed algorithm has good application effects in engineering, instruments and equipment were built in the aerodynamic elasticity laboratory of CARDC, as shown in Figure 9. By applying the driving voltage to the actuator, the aircraft model is vibrated, and the vibration signals are collected in real-time. In Figure 9a,b, the schematic diagram of the experimental device and the schematic diagram of the measurement and control system are shown respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Estimated Parameters** & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{1}}\) & \(\mathbf{d_{1}}\) \\ \hline True value & \(-0.5\) & \(-0.2\) & \(1.0\) & \(1.5\) & \(-0.8\) \\ Estimated value & \(-0.5158\) & \(-0.2059\) & \(0.9671\) & \(1.4673\) & \(0.0044\) \\ Estimation error & 0.0158 & 0.0059 & 0.0329 & 0.0327 & \(0.8044\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between system identification results and true values. Figure 8: s\({}^{\prime}(t)\) and e\((t)\) time domain and convergence of various parameters. In Table 4, information on the selected measurement and control instruments is shown. The signal processing module uses a series of products from National Instruments (NI) in the United States, the controller is a product from dSPACE in Germany, and the actuator uses piezoelectric stacking materials from Physical Instruments (PI) in Germany. As shown in Figure 10, the real-time data fitting results based on the traditional ARMAX algorithm and N4 SID algorithm are as follows:  
As shown in Figure 11, the real-time data fitting results of the data-driven self-learning algorithms LMS and M-VSNLMS are as follows: \begin{table}
\begin{tabular}{l l l} \hline \hline
**Components** & **Brand** & **Function** \\ \hline Computer operating system & Windows 7 & Program running environment \\ DC stabilized power supply & DP832A & Provide stable power supply for the balance \\ Balance & ADAM-3016 & Measuring force and torque signals \\ Strain signal acquisition module & PXIe-4339 & Signal amplification and filtering \\ Filter & PFI- 28618-M102 & Low pass \\ Controller & dSPACE-Microlabox RTI1202 & A/D conversion, calculation control signal, D/A conversion \\ Power amplifier & E-481K023 & Amplify control signal \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Measurement and control instrument information. Figure 10: Real-time data fitting results of ARMAX and N4 SID algorithms. Figure 9: Schematic diagram of testing instruments. Calculate the correlation coefficient \(\delta_{xy}\) and mean square error \(MSE\) between the output data of the above four algorithms and the expected data. \[\delta_{xy}=\frac{Cov(X,Y)}{D_{x}D_{y}}=\frac{1}{n}\underset{i=1}{\overset{n}{ \underset{i=1}{\sum}}}\frac{X_{i}-\overline{X}}{\sqrt{\underset{i=1}{\overset{n}{ \underset{i=1}{\sum}}}(X_{i}-\overline{X})^{2}}}\frac{Y_{i}-\overline{Y}}{ \underset{i=1}{\overset{n}{\underset{i=1}{\sum}}}(Y_{i}-\overline{Y})^{2}} \tag{16}\]  
\[MSE=\frac{1}{n}\underset{i=1}{\overset{N}{\underset{i=1}{\sum}}}(X_{i}-Y_{i}) ^{2} \tag{17}\]  
Among them, the correlation coefficient \(\delta_{xy}\) is the ratio of the covariance of the output data and the expected data to the product of the expected values of the two. \(X_{i}\), \(Y_{i}\), \(\overline{X}\), and \(\overline{Y}\) respectively represent the sampling and average values of the output data and the expected data, and \(n\) represents the sample data capacity. In Table 5, the experimental results of the four algorithms are compared. From Table 5, it can be seen that the data fitting effect of the data-driven self-learning algorithm is superior to that of traditional algorithms, and the data fitting accuracy of the M-VSNLMS algorithm is higher than that of the LMS algorithm. This is consistent with the simulation results and also indicates that the M-VSNLMS algorithm has a stronger real-time identification ability for physical systems than the LMS algorithm.