## 2 System Model and Methodology
### Structural Model  
Figure 2 shows the model diagram of the hybrid isolation device. When establishing its dynamic model, the system's spatial mass distribution can typically be neglected, simplifying it into a mass point. Similarly, the spatial distribution of the isolation device can be ignored, reducing it to a spring-damper system. The dynamic equation of the hybrid isolation device is expressed as (1):  
\[M\ddot{x}+C\dot{x}+Kx=-M\ddot{u}+F_{PID} \tag{1}\]  
In this equation, \(M\) is the mass, \(C\) is the damping coefficient, \(K\) is the stiffness, \(\ddot{u}\) is the seismic acceleration, \(\mathrm{F_{PID}}\) is the PID control force, \(\mathrm{x}\), \(\dot{x}\), and \(\ddot{x}\), respectively, represent the displacement, velocity, and acceleration. When the impact of control force on the substructure is not considered, the dynamic equation is given as shown in Equation (2):  
\[ \begin{split}&\left[\begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array}\right]\left[\begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\end{array}\right]+\left[\begin{array}{cc}c_{1}+c_{2}&-c_{2}\\ -c_{2}&c_{2}\end{array}\right]\left[\begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array}\right]+\left[\begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{array}\right]\left[\begin{array}{c}x_{1}\\ x_{2}\end{array}\right]=\\ &-\left[\begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array}\right]\ddot{u}+\left[\begin{array}{c}0\\ F_{PID}\end{array}\right]\end{split}  \tag{2}\]  
Figure 2: PID hybrid seismic isolation system. Expanding Equation (2) yields the detailed equation, as shown in Equation (3):  
\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+(c_{1}+c_{2})\dot{x}_{1}-c_{1}\dot{x}_{ 2}+(k_{1}+k_{2})x_{1}-k_{1}x_{2}=-m_{1}\ddot{u}(t)\\ m_{2}\ddot{x}_{2}-c_{2}(\dot{x}_{1}-\dot{x}_{2})-k_{2}(x_{1}-x_{2})=-m_{2}\ddot{u }(t)+F_{PID}\end{array} \right. \tag{3}\]  
When the impact of control force on the substructure is considered, the dynamic equation is given as shown in Equation (4):  
\[\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]\left[ \begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\end{array} \right]+\left[ \begin{array}{cc}c_{1}+c_{2}&-c_{2}\\ -c_{2}&c_{2}\end{array} \right]\left[ \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right]+\left[ \begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]= \tag{4}\] \[-\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]\ddot{u}+\left[ \begin{array}{c}-F_{PID}\\ F_{PID}\end{array} \right]\]  
Expanding Equation (4) yields the detailed equation, as shown in Equation (5):  
\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+(c_{1}+c_{2})\dot{x}_{1}-c_{1}\dot {x}_{2}+(k_{1}+k_{2})x_{1}-k_{1}x_{2}=-m_{1}\ddot{u}(t)-F_{PID}\\ m_{2}\ddot{x}_{2}-c_{2}(\dot{x}_{1}-\dot{x}_{2})-k_{2}(x_{1}-x_{2})=-m_{2}\ddot{ u}(t)+F_{PID}\end{array} \right. \tag{5}\]  
When the control force on the substructure is considered, the transfer rate of the hybrid isolation device is studied. By applying the Laplace transform, the time-domain equation is converted into a complex frequency-domain equation, as shown in Equation (6):  
\[\left\{ \begin{array}{l}m_{1}s^{2}X_{1}(s)+(c_{1}+c_{2})sX_{1}(s)-c_{2}sX_{2 }(s)+(k_{1}+k_{2})X_{1}(s)-k_{2}X_{2}(s)=\\ -m_{1}s^{2}u(s)-F_{PID}(s)\\ m_{2}s^{2}X_{2}(s)-c_{2}s(X_{1}(s)-X_{2}(s))-k_{2}(X_{1}(s)-X_{2}(s))=\\ -m_{2}s^{2}u(s)+F_{PID}(s)\end{array} \right. \tag{6}\]  
By designing the hybrid isolation system using a bandpass method, the transfer functions of the superstructure and the substructure are obtained, as shown in Equation (7):  
\[H(s)=\frac{X_{2}(s)}{X_{1}(s)} \tag{7}\]  
By combining Equations (6) and (7), the transfer rate equation can be derived.