## 3 Methodology of STST Based on TVCM-VPF  
### Conventional TVCM for SST  
As shown in Figure 1, the conventional TVCM for the shaking table consists of a three-variable feedback control (TVC feedback) and a three-variable feedforward control (TVC feedforward). The TVC feedback aims to assign the poles of the controlled system by feeding back the displacement, velocity, and acceleration of the shaking table. In such a way, a desirable performance in terms of the response rate, table resonant frequency, and damping ratio can be achieved. In Figure 1, \(K_{dr}\), \(K_{cr}\), and \(K_{ar}\) denote the displacement, velocity, and acceleration feedforward gains, respectively. The TVC feedforward control further improves the system performance by eliminating the comparatively slow poles of the system. The open-loop transfer function from the driven voltage on the servo-valve to the shaking table displacement, known as the control plant, can be modeled as a third-order system:  
\[W_{0}(s)=\frac{K_{\mathrm{v}}}{s\Big{(}\frac{s^{2}}{\omega_{0}^{2}}+\frac{2\xi_ {0}}{\omega_{0}}s+1\Big{)}}, \tag{10}\]  
where \(K_{\mathrm{v}}\) denotes the open-loop gain of the hydraulic system, \(s\) means the Laplace operator, and \(\xi_{0}\) and \(\omega_{0}\) represent the natural damping ratio and resonant frequency of the hydraulic system, respectively. By introducing the TVC feedback for the control plant, as shown in Figure 1, the closed-loop transfer function of the system from the input displacement command \(U\) to the displacement response output \(x\) can be written as follows:  
\[W(s)=\frac{1}{K_{df}}\frac{1}{\frac{1+K_{\mathrm{v}}K_{df}}{K_{\mathrm{v}}K_{ df}}\Bigg{[}\frac{1}{\omega_{0}^{2}\big{(}1+K_{\mathrm{v}}K_{ef}\big{)}}s^{2}+ \frac{\frac{2\xi_{0}}{\omega_{0}}+K_{\mathrm{v}}K_{ef}}{1+K_{\mathrm{v}}K_{ef }}s+1\Bigg{]}s+1}, \tag{11}\]  
where \(K_{df}\), \(K_{ef}\), and \(K_{df}\) denote the displacement, velocity, and acceleration feedback gains, respectively. This transfer function can be rearranged as follows:  
\[W(s)=\frac{1}{K_{df}}\frac{1}{\frac{1}{k_{0}}\Big{(}\frac{1}{\omega^{2}}s^{2} +\frac{2\xi}{\omega}s+1\Big{)}s+1}, \tag{12}\]  
with:  
\[k_{0}=\frac{K_{df}}{1+K_{\mathrm{v}}K_{ef}}K_{\mathrm{v}}, \tag{13}\]  
\[\omega=\omega_{0}\sqrt{\big{(}1+K_{\mathrm{v}}K_{ef}\big{)}}, \tag{14}\]  
\[\xi\omega=0.5K_{\mathrm{v}}K_{af}\omega_{0}^{2}+\xi_{0}\omega_{0}, \tag{15}\]  
where \(k_{0}\), \(\omega\), and \(\xi\) are the desired equivalent open-loop gain, frequency, and damping ratio of the system under TVC. Clearly, \(K_{\mathrm{v}}\), \(\xi_{0}\), and \(\omega_{0}\) are determined by the physical properties of the system; it is impossible for users to change them. Conversely, the equivalent characteristics can be realized by tuning the feedback gains, i.e., \(K_{df}\), \(K_{\mathrm{v}f}\), and \(K_{df}\). In a conventional STT, according to Equation (14), with an increase in \(K_{\mathrm{v}f}\), the equivalent resonant frequency increases. The equivalent damping ratio is proportional to \(K_{df}\); hence, a larger \(K_{af}\) indicates the suppression of the response oscillation and the enhancement of stability and robustness. According to Equation (13), the \(K_{df}\) value is normally heightened to suppress the adverse influence of the increase in \(K_{\mathrm{v}f}\) on the response rate. In summary, by increasing \(K_{df}\), \(K_{\mathrm{v}f}\), and \(K_{df}\) in a certain range, the control performance of the shaking table system can be improved. However, a too large \(K_{df}\) often leads to an oscillation response; therefore, the three feedback gains should be carefully designed. Figure 1: Conventional three variable control for shaking table test (STT). A reduction in the velocity gain \(K_{\mathit{off}}\) in a certain range can lead to a small frequency \(\omega\) and large \(k_{0}\) and \(\xi\). Therefore, for the case where a small \(\omega\) is acceptable, the control performance can be enhanced by reducing the velocity gain \(K_{\mathit{off}}\).