## 3 Methodology of STST Based on TVCM-VPF
### TVCM-VPF for STST  
STST with the ES corresponding to the top stories is more challenging than the actuator-based RTHS because of the need for velocity and acceleration control. That is, the reliability of the test results heavily depends on the control accuracy of velocity and acceleration rather than the displacement. In contrast to the traditional STT, testing errors in STST can accumulate and propagate in the subsequent steps owing to the numerical solution of the NS. Moreover, the desired targets in STST are the dynamic responses of the structure at the interface; that is, the frequencies of the loading targets are normally not high. In other words, the required bandwidth of the loading control is not the key element in STST; conversely, the control accuracy and stability should be very favorable. To ensure the robustness of the system, appropriate parameters must be chosen. This study aimed to conduct STST with an old shaking table by enhancing its traditional proportional integral derivative (PID) control performance with the TVCM.  
The PID controller is one widely used linear controller and the control parameters are the proportional gain, derivative time, and integral time [46]. The PID controller transforms the error signal to the command signal. The error signal is normally chosen as the difference between the target displacement and measured displacement response of the shaking table, which aims to be reduced with control. The advantage of the PID controller is it is easy to understand and operate. The disadvantage of the method is the control target is solely the displacement regardless of the velocity and acceleration. However, the displacement, velocity, and acceleration are all key elements for the STST. In view of this, the PID controller is not a good choice for the STST and this paper proposes one TVCM-VPF.  
In view of the aforementioned low requirement to control the frequency and high requirement to control the accuracy and stability in STST, this study presents a VPF control rather than the commonly used velocity negative feedback control. In the TVCM-VPF, replacing the \(K_{\mathit{off}}\) in Equations (13)-(15) with \(-K_{\mathit{vf}}\) yields the following:  
\[k_{0}=\frac{K_{\mathit{v}}K_{\mathit{df}}}{1-K_{\mathit{v}}K_{\mathit{vf}}}, \tag{16}\]  
\[\omega=\omega_{0}\sqrt{\left(1-K_{\mathit{v}}K_{\mathit{vf}}\right)}, \tag{17}\]  
\[\xi\omega=0.5K_{\mathit{v}}K_{\mathit{af}}\omega_{0}^{2}+\xi_{0}\omega_{0}. \tag{18}\]  
Clearly, compared with Equation (13), the open-loop gain in Equation (16) substantially rises with the increase in gain \(K_{\mathit{vf}}\). However, the equivalent resonant frequency decreases and the equivalent damping ratio improves with the increase in gain \(K_{\mathit{vf}}\) according to Equations (17) and (18). That is, in view of the control requirements of STST, the TVFC with a positive velocity feedback is expected to perform satisfactorily.  
Similar to the traditional TVCM, the TVC feedforward can be designed to eliminate the slow poles of the controlled system to further improve the response rate, which is not elaborated here. Notably, to obtain a stable response, \(K_{\mathit{vf}}\) should be so small that \(1-K_{\mathit{v}}K_{\mathit{vf}}>0\). Additionally, the controlled system is third order, and its stability condition can be obtained by means of [PERSON]'s criterion, i.e., \(k_{0}<2\xi_{0}\omega_{0}\)