## 3 A Compensation Strategy Using an \(\mathbf{H_{\infty}}\) Control Law
### Numerical and Experimental Verification  
Three systems are built for the four-story frame (Figure 2): a time-delay system without compensation, the system with pole-assignment (PA) compensation [18] and the system with H\({}_{\infty}\) compensation. The latter two systems include multiple time delays. In PA compensation method, by appropriately changing the compensation gains, the poles of the time-delay system are located in a target area. Conservatively, the multiple time delays are defined as 0.4 s (\(\tau_{1}\)) and 0.6 s (\(\tau_{2}\)) since one of the time delays in the experimental system is about 0.2\(\sim\)0.3 s after testing. In the experimental system, a control-force signal is provided to the servo motor, and the actual system time delay is the time required to apply the control forces to the structure. For engineering applications, it is difficult to measure the displacement and velocity responses of a building. In the study, a state observer using structural accelerations is built from the reference [31]. Finally, the simulink diagram of the experimental system is shown in Figure 10.  
Under an excitation load, the numerical and measured results are listed in Tables 3 and 4. The control effects and the relevant control parameters are shown in Figures 11 and 12. From Figures 11 and 12 and Tables 3 and 4, the maximum structural responses calculated by the numerical method are basically consistent with the measured results, which are closely related to the accuracy of the sensor and numerical model. The structural responses of the numerical results follow the sine law under a sinusoidal excitation load. On the contrary, the structural responses of the measured results do not fully follow the sine law due to the interaction effect between the controller and the frame, and the coupling effect of the structural horizontal and vertical vibrations. The strokes of the numerical results illustrated in Figure 11f are obviously different from those of the measured results illustrated in Figure 12f. There are three adverse aspects: the environmental interference signals, the inhomogeneous magnetic field between the rotor and stator and the unsmooth support track of the motor. In addition, due to the mechanical construction of the servo motor, the frequency of the control forces in the measured test is inconsistent with the numerical simulation. As a result, there are significant differences between the control forces of the numerical results and those of the measured results, which are shown in Figures 11e and 12e, respectively. The control system without compensation has a negative impact on reduction of the structural responses. By contrast, the control systems with the multiple time delays tend to reduce the dynamic responses, and the efficiency of PA compensation system is slightly worse than that of H\({}_{\infty}\) compensation system, especially the acceleration control effects. The reason is that the calculation process of the former system excludes the optimal control-gain \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Index**} & \multirow{2}{*}{**No Control**} & \multicolumn{2}{c}{**No Compensation**} & \multicolumn{2}{c}{**PA Compensation**} & \multicolumn{2}{c}{**H\({}_{\infty}\) Compensation**} \\ \cline{3-8}  & & \multirow{2}{*}{**Responses**} & **Effect (\%)** & **Responses** & **Effect (\%)** & **Responses** & **Effect (\%)** \\ \hline Displacement & 2 nd floor & 1.58 & 1.62 & \(-\)2.53 & 1.11 & 29.75 & 1.01 & 36.08 \\ (cm) & 4 th floor & 2.67 & 2.62 & 1.87 & 1.86 & 30.34 & 1.69 & 36.70 \\ Acceleration & 2 nd floor & 14.95 & 16.81 & \(-\)12.44 & 5.86 & 60.80 & 6.09 & 59.26 \\ (cm/s\({}^{2}\)) & 4 th floor & 11.74 & 16.03 & \(-\)36.54 & 4.29 & 63.46 & 3.84 & 67.29 \\ Control force (N) & — & 14.90 & — & 14.19 & — & 14.12 & — \\ Stroke (cm) & — & 17.55 & — & 15.13 & — & 11.82 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The control effects under the measured results (mean square values). Figure 10: The Simulink diagram of the experimental system.