## 3 Parameters of Equivalent Force Control HS Method
### EFC Parameters  
The previous sections focused on the parameters of the structural restoring force model for the numerical substructure, which was applied with asymmetric tension and compression stiffness, as shown in Figure 6, and the non-subdivision strategy was used to determine the inflection point of displacement. This section presents the determination of the EFC parameters such as force-displacement conversion factor \(C_{\text{F}}\), proportional gain \(K_{\text{F}}\), and integral gain \(K_{\text{I}}\). Generally, the force-displacement conversion factor \(C_{\text{F}}\) can be expressed as  
\[C_{\text{F}}=(K_{\text{N}}+K_{\text{PD}}+K_{\text{E}})^{-1} \tag{1}\]  
\[K_{\text{PD}}=\frac{4M}{\Delta t^{2}}+\frac{2C}{\Delta t} \tag{2}\]  
where \(K_{\text{N}}\), \(K_{\text{E}}\) are the stiffness matrices of the numerical and physical substructures, respectively; \(K_{\text{PD}}\) is the pseudo-dynamic stiffness; \(M\), \(C\) are the mass and damping matrices, respectively, and usually taken to be constant. From Equation (2), given a relatively small time interval \(\Delta t\) generally,  
Figure 11: Detailed time history curve of the story drift with different integral steps at 4 s-8 s.  
Figure 12: Comparison of the equivalent force (EF) feedback and EF command on the first floor using the non-subdivision method. \(\mathbf{K}_{\rm PD}\) is considered to be a relatively large component compared to \(\mathbf{K}_{\rm N}\) and \(\mathbf{K}_{\rm E}\). In the above discussion, the proportional gain \(\mathbf{K}_{\rm P}\), and integral gain \(\mathbf{K}_{\rm I}\) are the control parameters, while the force-displacement conversion factor \(\mathbf{C}_{\rm F}\) calculated by the Equations (1) or (2) is used to transform the equivalent force to the displacement of the model. Note that it is difficult to calculate the stiffness of the physical substructure due to the test error. For the sake of convenience, the factor \(\mathbf{C}_{\rm F}\) is always calculated in terms of the initial stiffness matrices of the numerical and experimental substructures. Then, Equation (1) can be adjusted to  
\[\mathbf{C}_{\rm F}=(\mathbf{K}_{\rm N,0}+\mathbf{K}_{\rm PD}+\mathbf{K}_{\rm E,0})^{-1} \tag{3}\]  
where \(\mathbf{K}_{\rm N,0}\), \(\mathbf{K}_{\rm E,0}\) are the initial stiffness matrices of the numerical and experimental substructures, respectively. Because of the asymmetry in the tension and compression stiffness, the following method was used to determine the force-displacement conversion factor \(\mathbf{C}_{\rm F}\): the \(\mathbf{K}_{\rm N,0}\) was used as the practical value while \(\mathbf{K}_{\rm E,0}\) was used as the mean value of the tension and compression stiffness matrices. For the nonlinear test models, the calculation of \(\mathbf{C}_{\rm F}\) was based on how the above method impacted the test results. But the effect of the EFC parameters upon the response performance can be adequately reduced through a reasonable design of the EF controller using a self-adaptive control, sliding mode control [48; 49], etc. The PI controller was used in this study, as shown in Figure 13. The following section presents a detailed determination of the proportional gain \(\mathbf{K}_{\rm P}\) and integral gain \(\mathbf{K}_{\rm I}\). In order to guarantee that the force-displacement conversion factor \(\mathbf{C}_{\rm F}\) in the numerical simulation identically corresponds with the correspondent in the hybrid simulation test, the stiffness of the physical substructure using for the numerical simulation is taken to be the average value of the tested stiffness in the positive and negative directions listed in Table 1. The updated stiffness was listed in Table 4. The elastic nonlinear restoring force model, shown in Figure 6, was adopted for the numerical substructure; at the same time, an elastic linear model was used for the physical substructure.