## 4 Experimental Validation of STST Based on TVCM-VPF
### Results of STST Based on TVCM-VPF  
Traditional STT with predefined loading targets and STST with the TVCM-VPF were conducted to validate the effectiveness of the proposed method and determine the advantages of the proposed method over the traditional PID controller. Figure 7 presents the experimental results of sine wave target (command) tests with an amplitude of 2 mm and a frequency of 2 Hz. It can be observed that the results of the TVCM-VPF match the command displacement much better than those of the PID controller. Loading delays associated with Figure 7a,b were 48.41 and 7.53 ms, meaning a decrease of 84.45% owing to the proposed scheme. The corresponding error analysis results are listed in Table 2. Compared with the PID controller, the correlation coefficient of the displacement response for the 2 Hz sine wave tests increased from 83.81% to 99.36% with the contribution of the TVCM-VPF; the amplitude error reduced from 11.4% to 0.53%. Compared with the traditional PID controller, in STST with the TVCM-VPF, the correlation coefficient of the displacement response for the 4 Hz sine wave tests increased from 65.32% to 99.21%; the amplitude error reduced from 33.7% to 0.83%. The results indicate the advantage of the TVCM-VPF over the traditional PID controller, especially for moderately high frequencies. STST with the TVCM-VPF was conducted to validate the advantages of the proposed method over the traditional PID controller. The experimental results of the displacement and acceleration responses on the top floor of the frame structure subjected to the El Centro (NS, 1940) earthquake wave record with a peak ground acceleration of 0.35 m/s\({}^{2}\) are presented in Figures 8 and 9 for the two control methods. In these figures, the command means the loading target of the shaking table, and the response represents the measured response of the shaking table. It can be observed from Figure 8 that under the conventional PID control, the displacement response nearly overlaps the displacement command of the shaking table; however, there is an obvious error between the acceleration response and the acceleration command of the shaking table. Conversely, from Figure 9 of the proposed TVCM-VPF, good matching of the responses and commands in terms of displacement and acceleration can be observed. With the contribution of the TVCM-VPF, the time delay between the displacement response and command is clearly reduced. The method can not only address the problem of displacement delay in the shaking table but can also eliminate the displacement overshoot caused by velocity feedforward compensation. The response rate and control accuracy were clearly improved. The corresponding control error indexes are listed in Table 3. From this table, it can be observed that the accuracy of the acceleration and displacement waveform reconstruction under the TVCM is significantly improved compared with that of PID control; in other words, the acceleration correlation coefficient increases from 67.73% to 90.22% with the contribution of the TVCM-VPF. By fine-tuning the control parameters, the TVCM-VPF has improved the performance in terms of accuracy and robustness. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sine Wave (Hz)**} & \multicolumn{2}{c}{**Amplitude Deviation (\%)**} & \multicolumn{2}{c}{**Correlation Coefficient (\%)**} \\ \cline{2-5}  & **PID** & **TVCM-VPF** & **PID** & **TVCM-VPF** \\ \hline
0.5 & 0.44 & 0.37 & 99.34 & 99.98 \\
1 & 0.73 & 0.55 & 98.95 & 99.84 \\
2 & 11.4 & 0.53 & 83.81 & 99.36 \\
4 & 33.7 & 0.83 & 65.32 & 99.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error comparison of the sine wave excitation tests. Figure 7: Experimental results of the sine wave target test. Figure 8: Shaking table substructure testing (STST) results with conventional proportional integral derivative (PID) control.