## 2 Experimental Model Considering CSI Effect
### Experimental System and Its Theoretical Model  
The experimental model shown in Figure 3 is a four-storey shearing type structure made of steel [6]. The columns and the floors of the test structure are made of two thin steel plates, which are 8 mm thick and 30 mm width. The length of the floors is 1400 mm, and the interstory height of the structure is 800 mm. In this experiment, four accelerometers are installed in the beam-column joints of each structural floor to measure the structural response, respectively. Accelerometers with a type of GT02 and  
Figure 1: Model of an AMD control system. Figure 3: Pictures of the steel frame structure, (**a**) Practicality; (**b**) Exhibition. Figure 2: A Flow diagram of an AMD control system showing CSI. three Micro-Epsilon laser displacement sensors are used to measure the horizontal acceleration and displacement of each floor, along the minor-axis. Here, the acceleration measurement is used as the feedback signal to calculate the real-time control forces, and then, an EtherCAT bus system can be used for transmitting the forces to a servo motor. The displacement measurement of the second, third, and fourth floors are used for verifying the control effectiveness. The loading system is composed of a reducer with an eccentric mass and an inverter. The frequency range of the inverter is 0-50 Hz, and the speed range is 0-120 rpm. The AMD control system shown in Figure 3b mainly includes a servo motor from a local company, an EtherCAT bus system, a dSPACE with a type of DS1103, and a computer. During the tests, the chassis of the AMD control device is fixed on the fourth floor of the structure and all of the motion signals are gathered by the dSPACE. An EL3008 input terminal, an EC1100 coupling terminal, an EL4034 output terminal, and a servo controller with a type of AX5000 constitute the EtherCAT bus system, which can be used to ensure the accuracy of signal transmission. After calculating, its structural frequencies are listed in Table 1.  
Regarding the simplified model of the experimental system, as shown in Figure 4, the control force is exerted on the top floor. Therefore, the force equilibrium of the control system is:  
\[M\ddot{X}(t)+C\dot{X}(t)+KX(t)=B_{gs}\ddot{x}_{g}(t)+B_{s}u(t) \tag{1}\]  
where \(M\), \(C\), and \(K\) are the mass matrix, damping matrix, and stiffness matrix of the system, respectively. \(\ddot{X}\), \(\dot{X}\), and \(X\) are the acceleration, velocity, and displacement vectors of the system relative to the ground, respectively. \(\ddot{x_{g}}\) is the external excitation. \(B_{s}\) and \(B_{gs}\) are the position matrices of the control force and earthquake, respectively. \(u\) is the active, controllable magnetic force of the AMD system. The experimental structure is modeled as an in-plane lumped-mass shear structure, and the matrices \(M\), \(C\), \(K\), \(X\), \(B_{s}\), and \(B_{gs}\) are expressed as: \begin{table}
\begin{tabular}{c c} \hline \hline
**Order** & **Frequency (Hz)** \\ \hline
1 & 0.162 \\
2 & 0.589 \\
3 & 1.093 \\
4 & 1.852 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Natural frequencies of the experimental system. Figure 4: The simplified model of an AMD system. \[M=\left[ \begin{array}{cccccc}m_{1}&0&0&0&0\\ 0&m_{2}&0&0&0\\ 0&0&m_{3}&0&0\\ 0&0&0&m_{4}&0\\ 0&0&0&0&m_{a}\end{array} \right]_{5\times 5},\ C=\left[ \begin{array}{cccccc}c_{1}+c_{2}&-c_{2 }&0&0&0\\ -c_{2}&c_{2}+c_{3}&-c_{3}&0&0\\ 0&-c_{3}&c_{3}+c_{4}&-c_{4}&0\\ 0&0&-c_{4}&c_{4}+c_{a}&-c_{a}\\ 0&0&0&-c_{a}&c_{a}\end{array} \right]_{5\times 5}, \tag{2}\] \[K=\left[ \begin{array}{cccccc}k_{1}+k_{2}&-k_{2}&0&0&0\\ -k_{2}&k_{2}+k_{3}&-k_{3}&0&0\\ 0&-k_{3}&k_{3}+k_{4}&-k_{4}&0\\ 0&0&-k_{4}&k_{4}+k_{a}&-k_{a}\\ 0&0&0&-k_{a}&k_{a}\end{array} \right]_{5\times 5},\ X=\left[ \begin{array}{c}x_{ 1}\\ x_{ 2}\\ x_{ 3}\\ x_{ 4}\\ x_{ 5\times 1}\end{array} \right]_{5\times 1},\ B_{g}=\left[ \begin{array}{c}0\\ 0\\ -1\\ 1\end{array}