## 2 Overview of the Benchmark Problem in RTHS
### Control Plant and Control Problem Statement  
Figure 1b depicts the model of the control plant, where the dynamics of the servo-valve, actuator, PS, and control-structure interaction (CSI, [PERSON] et al. [24]) are individually modeled using transfer functions. Specifically, the servo-valve is represented by a second-order transfer function with parameters of \(\alpha_{1}\) and \(\beta_{0}\) to \(\beta_{2}\); the dynamics of the hydraulic actuator are modeled by a first-order transfer function with parameters of \(\alpha_{3}\); the CSI that caused by the natural velocity feedback loop is modeled by the derivative with parameter \(\alpha_{2}\); the PS is represented by a transfer function with parameters of \(m_{\text{P}}\), \(c_{\text{P}}\), and \(k_{\text{P}}\). The parametric values of the model are identified using experimental data, which are presented in Table 2. Additionally, to validate the robust stability and performance of the designed controller or compensator, model uncertainties were accounted for using a random variable with a normal distribution of each parameter, and are also given in Table 2. Please note that the control plant assigned with the mean values of each parameter are denoted as a nominal plant, whereas that assigned using parameters that differ from the nominal ones are referred to as a perturbed model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Partitioning Configuration** & **Reference Floor Mass (kg)** & **Reference Modal Damping (\%)** \\ \hline Case 1 & 1000 & 5 \\ Case 2 & 1100 & 4 \\ Case 3 & 1300 & 3 \\ Case 4 & 1000 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Partitioning cases. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Component** & **Nominal Value** & **Standard Deviation** & **Units** \\ \hline \(\alpha_{1}\beta_{0}\) & Servo-valve & \(2.13\times 10^{13}\) & - & m\(\cdot\)Pa\(\cdot\)s\({}^{-1}\) \\ \(\alpha_{2}\) & CSI & \(4.23\times 10^{6}\) & - & m Pa \\ \(\alpha_{3}\) & Actuator & 3.3 & 1.3 & 1/s \\ \(\beta_{1}\) & Servo-valve & 425 & 3.3 & Unitiless \\ \(\beta_{2}\) & Servo-valve & \(10\times 10^{4}\) & \(3.3\times 10^{3}\) & 1/s \\ \(m_{\text{P}}\) & PS & \(29.12\) & - & kg \\ \(k_{\text{P}}\) & PS & \(1.19\times 10^{6}\) & \(5\times 10^{4}\) & N/m \\ \(c_{\text{P}}\) & PS & \(114.6\) & - & kg/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parametric values of the control plant. As a result of series, parallel, and feedback connections of those components, the mathematical control plant model, from command displacement to the measured displacement, is given by  
\[G_{\text{CP}}(s)=\frac{A_{0}}{B_{5}s^{5}+B_{4}s^{4}+B_{3}s^{3}+B_{2}s^{2}+B_{1}s+ B_{0}}, \tag{2}\]  
where \(s\) is the Laplace transform variable, \(A_{0}=\alpha_{1}\beta_{0}\), \(B_{5}=m_{\text{P}}\), \(B_{4}=c_{\text{P}}+m_{\text{P}}\alpha_{3}+m_{\text{P}}\beta_{1}\), \(B_{3}=(k_{\text{P}}+c_{\text{P}}\alpha_{3}+\alpha_{2})+(c_{\text{P}}+m_{\text {P}}\alpha_{3})\beta_{1}+m_{\text{P}}\beta_{2}\), \(B_{2}=k_{\text{P}}\alpha_{3}+(k_{\text{P}}+c_{\text{P}}\alpha_{3}+\alpha_{2}) \beta_{1}+(c_{\text{P}}+m_{\text{P}}\alpha_{3})\beta_{2}\), \(B_{1}=k_{\text{P}}\alpha_{3}\beta_{1}+(k_{\text{P}}+c_{\text{P}}\alpha_{3}+ \alpha_{2})\beta_{2}\), \(B_{0}=\alpha_{1}\beta_{0}+k_{\text{P}}\alpha_{3}\beta_{2}\). The magnitude, phase, and time delay of the nominal control plant (Equation (2)) are investigated and presented in Figure 2. In the figure, the time delay is evaluated using \(\tau=\varphi/\omega\). From Figure 2a,b, it can be observed that when the frequency is low, e.g., the frequency \(<\)10 rad/s, both the magnitude and phase are minimal. As the frequency increases, the magnitude decreases while the phase increases, implying that the tracking performance is weakened. Hence, the compatibility and equilibrium cannot be satisfied for all interface boundary DOFs, and the compensation methods for amplitude and phase errors must be employed. Furthermore, the time delay is varying from 25 ms to 8 ms, as shown in Figure 2c.