## 3 Parameters of Equivalent Force Control HS Method
### EFC Parameters
Figure 14 shows the time histories of the inter-story displacement with a varying \(\mathbf{K}_{\rm I}\) of 0.001, 0.0065, and 0.01, and a fixed \(\mathbf{K}_{\rm P}\) of 0.05 (only the result of the first floor is shown here, because there were similar results for the other floors). The reference results are obtained using the fourth-order single-step integration method and these can serve as exact solutions (W). Note that with increase of the \(\mathbf{K}_{\rm I}\), the results based on the PI controller begin to approach the theoretical solution. This indicates that a moderate increase in \(\mathbf{K}_{\rm I}\) can enhance the accuracy of the numerical integration method. However, the comparison between the EF feedback and EF command, shown in Figure 15, suggests that the overshoot (Figure 15c) or undershoot (Figure 15a) phenomenon occurs when \(\mathbf{K}_{\rm I}\) when is too large or too small. In other words, a moderate integral gain can enhance both the EF response speed and the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Floors** & **1** & **2** & **3** & **4–12** \\ \hline Initial stiffness in the positive direction (\(\rm kN/mm\)) & 446 & 327 & 327 & 295 \\ Initial stiffness in the negative direction (\(\rm kN/mm\)) & 446 & 327 & 327 & 359 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculated parameters for test model. Figure 13: Equivalent force controller. accuracy of the integration algorithm, while an excessive increment will lead to an overestimation and thereby compromise the results. It is important to point out that increasing the frequency of the EF feedback can relieve some of these accuracy issues. However, this will also dramatically increase the testing duration and potentially lead to a greater rate of test error. Figure 14: Time history curves of story drift with different integral gains. Figure 15: Comparison of the EF feedback and EF command: (**a**) \(K_{\rm P}=0.05\), \(K_{\rm I}=0.001\); (**b**) \(K_{\rm P}=0.05\), \(K_{\rm I}=0.005\); (**c**) \(K_{\rm P}=0.05\), \(K_{\rm I}=0.01\). The time histories of the inter-storey displacement with a varying \(\mathbf{K_{P}}\) of 0.05, 0.01, 0.90 and 1.00, and a fixed \(\mathbf{K_{I}}\) of 0.0065 are shown in Figure 16. As the proportional gain \(\mathbf{K_{I}}\) increases, the results from the PI controller deviates from the theoretical results--something similar can be seen in the comparison of the EF feedback and EF command as shown in Figure 17. The excessive actuating displacement may arise due to an excessive proportional gain--which can cause the overshooting and oscillation of the EF feedback that produces unrealistic hysteretic responses. [PERSON] et al. [49] analyzed the EFC method for real-time substructure testing and found that an extensively small proportional gain often led to a time-lag problem of actuator response. This, in turn, introduced negative damping into the numerical substructure model and affected the test results. Increasing the proportional gain can accelerate the EF response, which reduces the time lag in the EF response as well as the negative damping. In contrast, an oversized proportional gain can easily lead to oscillation phenomenon in the EF feedback, and distorted results. In conclusion, the \(\mathbf{K_{P}}\) = 0.05, \(\mathbf{K_{I}}\) = 0.0065 was adopted in this study.