# Active Tendon Control of Stay Cable by a Giant Magnetotrstive Actuator Considering Time-Delay
## 7 Active Control Experiment of Cable with the Time-Delay  
### Experimental Set-Up  
The cable model used in the experiment is the same as the simulation model, as shown in Figure 9a. The cable is made by steel wire with a diameter of 1.58 mm and a length of 6.1 m. To achieve the identical dynamic parameters of the practical cable, some additional mass blocks are uniformly arranged along the span of the cable model, and each mass block is 0.192 kg. The other detailed parameters are shown in Table 1. The active axial control force is generated by GMA (the required current and voltage of the actuator are 0.6 A and 15 V, respectively, and therefore the power consumption is 9 W) as shown in Figure 9b. In addition, there are two laser displacement sensors (model: optoNCDT14, which is manufactured by \(\mu\)e Company with the sampling frequency 1000 Hz) set at 1/2 and 1/4 span length, respectively, to measure the displacement responses and a vibration exciter (model: KDJ-20, manufactured by Yangzhoukedong Technology Electronics Company in Yangzhou, China) connected at the point that is 1/14 span length away from the fixed support to provide harmonic and random excitation. The control system is shown in Figure 9c, where the active control can be come true by Simulink (developed by the MathWorks Company in Massachusetts, USA) and dSPACE (developed by the German dSPACE Company in Paderborn, German), and the amount of time-delay is adjusted by the Simulink module illustrated in Figure 9d. The module \"Transport Delay\" can adjust the amount of the time-delay, and the control force (the module \"kongzhili\") is outputted based on the LQR algorithm considering the time-delay compensation. According to the relationship between force and voltage, the output voltage (the module \"dianya\") is computed and applied to the cable by the actuator. The excitation in the experiment included three cases: free vibration, harmonic excitation and random excitation. For the free vibration experiment, the initial displacement of 0.011 m was applied at the middle of the span. In the harmonic excitation experiment, a harmonic excitation force with the frequency of first modal of the cable was provided by a vibration exciter connected at the point of around 1/14 span length. For the random excitation experiment, a Gaussian white noise with RMS value 0.81N and peak value 2.48N was applied at 1/14 span length of the cable. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**The Time-Delay (s)**} & \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 & 63.37 & & \\
0.05 & & 62.47 & 60.86 \\
0.07 & & 60.66 & 60.49 \\
0.9 & & 58.70 & 59.31 \\
0.11 & & \(-\)15.12 & 59.12 \\
0.13 & & divergence & 57.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Response reduction at mid-span of three control strategies under random excitation. For the active control of the cable with mode-based LQR algorithm, it is necessary to measure the cable response to determine all modal responses and then to determine the real-time state variables. However, the modal response cannot be directly measured in the practical control process, and it is very difficult to obtain all the modal responses of the cable accurately with limited sensors. As the displacement response of the cable is mainly the first mode component, the influence of the higher-order mode is ignored in this paper, and only the first-order modal response is considered. When \(n=1\), Equation (1) is simplified as follows:  
\[w(x,t)=\phi_{1}(x)q_{1}(t) \tag{35}\]  
when \(x=L/2\), we have,  
\[w(L/2,t)=q_{1}(t) \tag{36}\]  
From the Equations (29) and (30), the displacement at the mid-span of the cable is equivalent to the first-order modal displacement.