## 5 Numerical Validation  
Virtual RTHSs were carried out with three different ground acceleration records, and the four partitioning cases shown in Table 1 were considered for each input. Moreover, one nominal vRTHSs and 20 perturbed vRTHSs were included to consider the modeling uncertainties for the control plant in each partitioning configuration. A total of 252 vRTHSs (12 nominal vRTHSs and 240 perturbed vRTHSs) were carried out in this study. Figure 8: RMSEs for different situations. (**a**) Effect of \(\mathbf{P}_{0}\); (**b**) effect of \(R\). Figure 10: Effects of R on the convergence values of system model parameters. Figure 9: Effects of \(\mathbf{P}_{0}\) on the convergence values of system model parameters. In this section, a four-parameter discrete model of the control plant is adopted, namely  
\[d_{k+1}^{\mathrm{c}}=x_{1}d_{k+1}^{\mathrm{m}}+x_{2}d_{k}^{\mathrm{m}}+x_{3}d_{k -1}^{\mathrm{m}}+x_{4}d_{k-2}^{\mathrm{m}}, \tag{25}\]  
and the corresponding feedforward controller is  
\[d_{k+1}^{\mathrm{c}}=x_{1}d_{k+1}+x_{2}d_{k}+x_{3}d_{k-1}+x_{4}d_{k-2}. \tag{26}\]  
To determine the initial values of \(\mathbf{x}_{0}\), \(\mathbf{P}_{0}\), and \(R\), a virtual RTHS (vRTHS) without any compensation strategy was carried out, where the El Centro ground acceleration is scaled to 0.2 g. By employing the least-square method, the initial value \(\mathbf{x}_{0}\) reaches [11.1688 -20.4226 13.6773 -3.4339]1. The measurement noise variance \(R\) is set to \(1\times 10^{-4}\), which is determined by the characteristics of the noise level. Similar to that used in the last subsection, the initial covariance matrix is given by \(\mathbf{P}_{0}=\text{diag}([11.1688^{2}\ 20.4226^{2}\ 13.6773^{2}\ 3.4339^{2}])\). Footnote 1: The \(\mathbf{x}_{0}\) and \(\mathbf{x}_{1}\) are the same as the \(\mathbf{x}_{0}\) and \(\mathbf{x}_{1}\). The displacement time history of the first floor from nominal virtual RTHS for Case 1 under the El Centro earthquake record is shown in Figure 11, with the reference response provided. From the figure, it can be seen that the displacement of the virtual RTHS is almost identical to the reference one, which indicates a good tracking performance of the proposed compensation strategy. The evolutions of the estimated system model parameters are given in Figure 12, correspondingly. It can be seen that once the dynamics of the control plant are excited and there are differences between the desired and measurement displacements, the KF responds rapidly to adjust the system model parameter, and then the parameters converge rapidly to a relatively steady value. It may be noted that the convergence trend of \(x_{3}\) and \(x_{4}\) is not very evident, but the compensated control plant exhibits a nearly perfect tracking performance, as shown in Figure 11b. This is because the parameters \(x_{3}\) and \(x_{4}\) are related to the dynamics of the control plant in the high-frequency range, which are not excited completely. Furthermore, it is seen that there are few large fluctuations during the loading period, implying the stable performance of the parameter estimation method. The quantitative indicators that were introduced earlier for the nominal vRTHSs under El Centro, Kobe, and Morgan Hill ground motions are presented in Table 5. In the table, it is shown that the maximum value of \(J_{1}\), which is the calculated time delay, is 0.2 ms, which is much less than the critical time delay (approximately 4 ms) in all cases. This indicates that the proposed KF-ADC method is efficient in terms of providing a stable RTHS considering different partitioning schemes. Moreover, the maximum value of \(J_{2}\) and \(J_{3}\) is less than 2%, implying that the KF-ADC method also achieves good global agreement between the desired and measured displacements in all cases. Figure 11: Displacement time histories of the first floor for Case 1. (**a**) Overview; (**b**) Enlarged view.