# Active Tendon Control of Stay Cable by a Giant Magnetotrstive Actuator Considering Time-Delay
## 8 Conclusions  
In this paper, the simulation and experimental research on the active control of cables with time-delay is carried out. The main work includes the motion equation for the active control of cable, active control algorithm, GMA actuator and its mechanical model, time delay compensation algorithm, and mitigating performance evaluation of cable. Some conclusions are obtained as follows.  
When the axial control force is used to control the cable lateral vibration, the controlled state equation is a bilinear equation due to the influence of the sag. In order to apply the time delay compensation method of the linear control system, the nonlinear term in the controlled state equation is ignored and the controlled equation is linearized. The simulation results show that, for the cable with small sag, the mitigating effect obtained  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **The Time-Delay** \\ **(s)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Mitigating Ratio** \\ **without Time-Delay** \\ **(\%)** \\ \end{tabular} } & \
\begin{tabular}{c} **Mitigating Ratio** \\ **without Time-Delay** \\ **Compensation with** \\ **Time-Delay** \\ **(\%)** \\ \end{tabular} &
\begin{tabular}{c} **Mitigating Ratio** \\ **with Time-Delay** \\ **Compensation** \\ **(\%)** \\ \end{tabular} \\ \hline
0.00 & 67.80 & & \\
0.05 & & 65.24 & 63.47 \\
0.07 & & 60.89 & 61.55 \\
0.09 & & 58.65 & 62.97 \\
0.11 & & 47.83 & 62.68 \\
0.13 & & \(-\)18.26 & 60.72 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Experimental response reductions at mid-span under harmonic excitation.  
Figure 12: The experimental displacements at mid-span with 0.11 s time-delay under random excitation.  
by using the linearized control equation is only slightly lower than that by the bilinear controlled equation, which indicates that the linearization method is feasible.  
A GMA was fabricated with Tb0.3 Dy0.7 Fe1.95 alloy magnetostrictive materials, which provides axial control force for cables. The relationship between input voltage and output force of GMA is obtained by experiment and applied to the control experiment. Based on the phase-shifting method, a multi-mode time-delay compensation algorithm is proposed, and the time-delay compensation matrix based on state feedback control is obtained.  
Considering four different cases, simulations and experiments are conducted to investigate the mitigating performance under free vibration, harmonic excitation and random excitation. The results show that, with the increase in time delay, the mitigating ratios without time-delay compensation decrease significantly, while the excellent mitigating effect can be achieved by using the proposed time-delay compensation method.  
Conceptualization, X.W. and X.H.; methodology, W.W. and R.H.; software, W.W. and R.H.; validation, W.W., Z.Y. and R.H.; investigation, W.W.; writing--original draft preparation, W.W., Z.Y. and R.H.; writing--review and editing, R.H.; supervision, X.W. and X.H.; funding acquisition, W.W., X.W. and X.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key Research and Development Program of China, grant number 2019 YFC1511101, the National Science Fund for Distinguished Young Scholars, grant number 52025082, the National Natural Foundation of China, grant number 51908210, and the Natural Foundation of Hunan Province, grant number 2020 JJ5074.  
Not applicable.  
The financial supports from the National Key Research and Development Program of China (grant number 2019 YFC1511101), the National Science Fund for Distinguished Young Scholars (grant number 52025082), the National Natural Foundation of China (grant number 51908210), and the Natural Foundation of Hunan Province (grant number 2020 JJ5074) are greatly appreciated.  
The authors declare no conflict of interest.