## 4 Time-Delay Compensation Control Gain Design by GCC Algorithm  
From Figures 7 and 8, when the frequency of the control voltage is determined, the phase from the input to output remains unchanged, meaning that the time-delay of the system is a certain value. However, as the frequency of the control voltage (affected by external excitation) is changed, the time-delay of the system is always changed. In this section, the time-delay resulted from a  
Figure 11: Phase transfer functions from control voltage to control force with different displacement gains. Figure 12: Phase transfer functions from control voltage to control force with different velocity gains. higher-order CSI effect that can be calculated through numerical analysis. The process is coincident with the actual situation in civil engineering. An El Centro seismic wave (SN direction) in Reference [19] is taken as an example. Three scenarios are analyzed for the seismic wave, namely: (1) Non-CSI effect; (2) Reduced-order CSI effect; and (3) Higher-order CSI effect. Using these systems, comparisons of control voltages and control forces are shown in Figure 13.  
From Figure 13, under the excitation of the Centro El seismic wave, when the non-CSI effect and reduced-order CSI effect are considered, the control force and control voltage are basically in synchronization. Compared to the higher-order CSI effect, the time-delay between the control force and the control voltage is comparatively obvious, where the result is consistent with engineering practices. As the Higher-order CSI effect is considered, the time-varying delays of the experimental system are shown in Figure 14. From these figures, under the excitation of the El Centro seismic wave, the biggest time-delay is 0.11 s. Additionally, after the earthquake, the maximum time-delay (0.11 s) remains unchanged. As the excitation frequency (the frequency of control voltage) is zero, according to Figure 8, the phase is the largest, which means that this time-delay is the longest. Figure 13: Control voltage and control force induced by the El Centro seismic wave, (**a,b**) Non-CSI effect; (**c,d**) Reduced-order CSI effect; (**e,f**) Higher-order CSI effect. Due to an external excitation with a changed frequency, the absolute velocity of the AMD mass is not zero and a higher-order CSI effect exists, which means that a time-varying delay also exists. The time-varying delay includes a constant part \(d=(d_{1}+d_{2}+d_{3}+d_{4}+d_{5})\) and a time-varying part \(d_{6}(t)\). The control force considering time-varying delay is:  
\[u_{d}(t)=-GZ[t-d(t)] \tag{31}\]  
where \(d(t)\) is the time-varying delay. Structural parameters \(M^{\prime}\) and \(K^{\prime}\), considering uncertainty, are:  
\[M^{\prime}=M+\Delta M=(1+\delta_{M})M,\ K^{\prime}=K+\Delta K=(1+\delta_{K})K. \tag{32}\]  
where \(\delta_{M}\) and \(\delta_{K}\) are the variation coefficients of the structural mass and stiffness, respectively. The closed-loop system with uncertain parameters and a time-varying delay can be described as:  
\[\dot{Z}(t)=(A+\Delta A)Z(t)+(B_{2}+\Delta B_{2})u_{d}(t)+(B_{2}+\Delta B_{2})u(t) \tag{33}\]  
Supposing \(\Delta\overline{A}=-\Delta\overline{B}_{2}G\), system (33) is:  
\[\dot{Z}(t)=(A+\Delta A)Z(t)+\big{(}\overline{A}+\Delta\overline{A}\big{)}Z[t -d(t)]+(B_{2}+\Delta B_{2})u(t) \tag{34}\]  
The perturbation matrix is assumed follow the form described in Reference [33]. \[\Big{[} \begin{array}{cc}\Delta A&\Delta B_{2}&\Delta\overline{A}\end{array}  \Big{]}=DF\Big{[} \begin{array}{cc}E_{1}&E_{2}&E_{d}\end{array} \Big{]} \tag{35}\]  
where \(F\) is an uncertain matrix satisfying \(F=\overline{\delta}I\), \(|\overline{\delta}|<1\) and \(\overline{\delta}\) is an uncertain constant. Moreover:  
\[D=\left[ \begin{array}{cc}0&0\\ -\frac{K}{M}&-\frac{C}{M}\end{array} \right]\!,\ E_{1}=\left[ \begin{array}{cc} \big{[}\frac{(\delta_{K}+1)}{(\delta_{M}+1)}-1\big{]}I&0\\ 0&\big{[}\big{(}\delta_{M}+1\big{)}^{-1}-1\big{]}I\end{array} \right]\!,\ E_{2}=E_{d}=\left[