## 4 Time-Delay Compensation Control Gain Design by GCC Algorithm
After adjusting, \(Q_{4}\), \(Q_{5}\), and \(R\) are taken as 6 \(\times\) 10\({}^{6}\), 3 \(\times\) 10\({}^{2}\), and 1 \(\times\) 10\({}^{-3}\), respectively, and the weight coefficients of the other layers are all defined as one. Three scenarios are analyzed in this part: (1) The variation coefficients of the structure parameters are 0%, and the time-delay is zero; (2) The variation coefficients of the structure parameters are 0%, and the introduced time-varying delays are stochastic numbers (0 to 0.2 s); (3) The variation coefficients of stiffness and mass are \(-\)38.3% and 16.8%, and the introduced time-varying delays are stochastic numbers (0 to 0.2 s). The loading frequency and the peak value of the sinusoidal excitation force are 1 Hz and 45.89 N. Under the above excitation load, the control effects and the AMD parameters of different control systems are listed in Table 3, and the structural responses (includes displacement and acceleration) to the 4 th floor of the different control systems are shown in Figures 15-17. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & **No Time-Delay** & \multicolumn{3}{c}{**GCC Compensation**} & \multicolumn{3}{c}{**Uncertain System with**} \\ \cline{3-10}  & **Index** & \multicolumn{1}{c}{**No Control**} & \multicolumn{1}{c}{**Responses**} & \multicolumn{1}{c}{**Effect (\%)**} & \multicolumn{1}{c}{**Responses**} & \multicolumn{1}{c}{**Effect (\%)**} & \multicolumn{1}{c}{**No Control**} & \multicolumn{1}{c}{**Responses**} & \multicolumn{1}{c}{**Effect (\%)**} \\ \hline Displacement & 2 nd floor & 0.0298 & 0.0205 & 31.0591 & 0.0194 & 34.8642 & 0.0335 & 0.0219 & 34.7817 \\ (m) & 3 rd floor & 0.0432 & 0.0295 & 31.7138 & 0.0277 & 35.8596 & 0.0491 & 0.0316 & 35.6275 \\  & 4 th floor & 0.0497 & 0.0338 & 31.9498 & 0.0320 & 35.6917 & 0.0563 & 0.0358 & 36.4951 \\ \hline Acceleration & 2 nd floor & 0.4028 & 0.1183 & 70.6267 & 0.1375 & 65.8645 & 0.4105 & 0.1332 & 67.5401 \\ (m/s\({}^{2}\)) & 3 rd floor & 0.3082 & 0.1001 & 67.5416 & 0.1172 & 61.9658 & 0.3501 & 0.1412 & 59.6573 \\  & 4 th floor & 0.3097 & 0.0756 & 75.5864 & 0.0875 & 71.7627 & 0.3426 & 0.0970 & 71.7008 \\ \hline AMD control forces (N) & – & 27.8351 & – & 28.1549 & – & – & 28.3361 & – \\ \hline AMD strokes (m) & – & 0.1635 & – & 0.1471 & – & – & 0.1873 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Control effectiveness of structural responses. From Figures 15-17 and Table 3, when the time-delay and the structural parametric uncertainty are not considered, the GCC controller can effectively suppress the structural response. Moreover, when considering the time-delay, the controller can also be used to compensate for the time-varying delay, and the maximum variations of the displacement and acceleration control effects between the systems with and without delays are only \(4.15\%\) and \(-5.58\%\), respectively. Additionally, as the structure parameters have a large uncertainty, the guaranteed cost controller is still valid. Compared with a certain system, the control effects of the displacement and acceleration responses of an uncertain system with GCC compensation only change by \(0.80\%\) and \(-2.31\%\), respectively. Figure 16: Comparison of structural responses to the 4 th floor of the experimental system under scenario 2, (**a**) Displacement; (**b**) Acceleration. Figure 17: Comparison of structural responses to the 4 th floor of the experimental system under scenario 3, (**a**) Displacement; (**b**) Acceleration. Figure 15: Comparison of structural responses to the 4 th floor of the experimental system under scenario 1, (**a**) Displacement; (**b**) Acceleration. Therefore, the guaranteed cost control method can effectively deal with the adverse conditions, and can be used to control the structural response in the ideal range.