## 2 Establishment of the Multi-Time-Delay Control System
### Impact Analysis of Multi-Time Delays  
The force equilibrium equation is,  
\[M\ddot{X}(t)+C\dot{X}(t)+KX(t)=B_{w}w(t)+\sum_{i=1}^{m}B_{si}u_{i}(t), \tag{10}\]  
where \(M\), \(C\) and \(K\) indicate the mass, damping and stiffness matrix. \(u_{i}\) is the force of the \(i\)th controller. \(B_{s}\) indicates the location matrices of control forces, and \(B_{w}\) indicates the location matrices of external excitations. \(X\) is the displacement of the system. The \(i\)th time-delay is \(\tau_{i}\), and Equation (1) is expressed as,  
\[\dot{Z}(t)=AZ(t)+\sum_{i=1}^{m}B_{i}u_{i}(t-\tau_{i})+Ew(t) \tag{11}\]  
where \(Z\) indicates the state vector, which includes structural displacement and velocity responses. \(A\) is the state matrix, \(B_{i}\) is the control matrix of the \(i\)th force and \(E\) is the excitation matrix,  
\[A=\left[ \begin{array}{cc}0&I\\ -M^{-1}K&-M^{-1}C\\ \end{array} \right],B_{i}=\left[ \begin{array}{cc}0\\ M^{-1}B_{si}\\ \end{array} \right],E=\left[ \begin{array}{cc}0\\ M^{-1}B_{w}\\ \end{array} \right]. \tag{12}\]  
To clarify the difference between the system with a single-controller (System1) and the system with multi-controllers (System2 which includes two controllers: AMD1 and AMD2), a four-story frame along its minor-axis is used as an example to compare and analyze these above methods, which is shown in Figure 2. The above frame is constructed by steel with a damping ratio of 0.02. The important parameters are revealed in Table 1. More details and the dynamic properties of the structure are found in the reference [32]. Three cases are considered in this paper: the control systems without time delays, the systems with equal time delays and the systems with unequal time delays. Under the SN direction of the El-Centro seismic wave, the equal time delays are assumed to be 0.05 s (\(\tau_{1}=\tau_{2}=0.05\) s), and the time-delay difference is set as 0.02 s, which means that \(\tau_{1}=0.04\) s and \(\tau_{2}=0.04+0.02=0.06\) s. Table 2 shows the acceleration control effects, and the control parameters are shown in Figures 3-8. The ratio between the reduction of structural responses and the responses without control is defines as Control effect. The control forces and strokes constitute AMD parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **System1** & **System2** \\ \hline The weight of the auxiliary mass (kg) & \(20\times 1\) & \(10\times 2\) \\ The effective stroke (m) & \(\pm 0.4\) & \(\pm 0.4\) \\ The maximum driving force (N) & \(\pm 1000\times 1\) & \(\pm 500\times 2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The important parameters of different AMD systems. Figure 3: The AMD parameters of the systems without time delays, (**a**) the forces of System1, (**b**) the total forces of System2, (**c**) the strokes of System1, (**d**) the strokes to AMD1 of System2. Figure 2: The experimental frame: (**a**) its picture and (**b**) its schematic diagram. Figure 4: The deviations to the control parameters of System2 without time delays, (**a**) the forces and (**b**) the strokes. Figure 5: The control parameters of the systems with equal time delays, (**a**) the forces of System1, (**b**) the total forces of System2, (**c**) the strokes of System1, (**d**) the strokes to AMD1 of System2. Figure 6: The deviations to the control parameters of System2 with equal time delays, (**a**) the forces and (**b**) the strokes. From Table 2, the multi-controllers can basically ensure synchronization in the systems without time delays. Therefore, the control effects of System2 are close to those of System1. When the equal time delays are considered, the acceleration control effects to the 8 th floor of System1 and System2 decrease by 47.41% and 52.05%, respectively. This is due to the fact that the structural equivalent damping becomes smaller as the time delays increase. The energy dissipation capacity of the control system is obviously reduced. This leads to an increase in the acceleration responses and a significant decrease in the control effects.