# Active Tendon Control of Stay Cable by a Giant Magnetotrstive Actuator Considering Time-Delay
## 3 Simplification and Verification of Bilinear Controlling Equations
\begin{array}{c}G_{1}\xi_{1}(t)\\ G_{2}\xi_{2}(t)\\ G_{3}\xi_{3}(t)\end{array} \right]=-\mathbf{G}\xi(t) \tag{22}\]  
According to Equations (18) and (20), the following equation can be obtained,  
\[U(t)=-\mathbf{L}^{+}\mathbf{G}\xi(t)=-\mathbf{L}^{+}\mathbf{G}\psi^{-1}\mathbf{ Z} \tag{23}\]  
where  
\[\mathbf{L}=\psi^{-1}\mathbf{B}\text{, }\mathbf{L}^{+}=\left(\mathbf{L}^{\text{T}} \mathbf{L}\right)^{-1}\mathbf{L}^{\text{T}}\]  
where \(\mathbf{L}^{+}\) is the pseudo inverse of \(\mathbf{L}\). To evaluate the efficiency of vibration suppression of the linear simplified model, the simulation analysis of vibration reduction under free vibration and harmonic excitation is carried out. In this paper, the A12 stay cable from the Dongting Lake Bridge [5] is taken as a prototype cable for simulations and experiments. A scaled down cable model with a scale ratio of 1:20 is used in the simulations and experimental studies. The parameters of the cable model are listed in Table 1.  
During the simulations, the weighting matrix \(\mathbf{Q}\) is the identity matrix selected as \(\mathbf{I}_{6\times 6}\) and weighting factor \(R\) is selected as 0.001. Simulations of free vibration without and with active tendon control, including linear LQR control and bilinear control, are conducted with the initial condition of \(\mathbf{Z_{0}}=\) [0.005, 0.005, 0.005, 0, 0, 0]\({}^{\text{T}}\), and the mid-span displacement response of the cable in different control methods is shown in Figure 2a. It can be seen from Figure 2a that the root mean square (RMS) of the mid-span displacement responses under no control, linear LQR control and bilinear control cases are 0.00307 m, 0.00132 m, and 0.00120 m, respectively. The results reflect that the vibrations of the cable are dramatically mitigated by the controller. The displacement responses are reduced about 53.1% and 61.2% with the LQR control and the bilinear control, respectively. Besides, the maximum active forces of the two control algorithms are 15.3 N and 18.5 N, respectively, as shown in Figure 2b. A harmonic excitation force, whose amplitude is around three N and the frequency is set to the fundamental frequency of the cable, is applied at the node of 1/14 span length. The mid-span displacement of the cable of the first 40 s and the steady-state response from 80 s to 100 s are, respectively, shown in Figure 3. It shows the RMS of the mid-span displacement responses under no control, linear LQR control and bilinear control cases are 0.00615 m, 0.00173 m, and 0.00161 m, respectively. The reduction ratios of the two control strategies at the mid-span are 72.13% and 73.82%, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Diameter** & **Length** & **Inclination** & **Static** & **Mass Per** & **Damping** & **Frequency of the** \\
**(mm)** & **(m)** & **(\({}^{\circ}\))** & **Tension** & **Length** & **First Three** & **First Three** \\
**(kg/m)** & **(kg/m)** & **(kg/m)** & **Modes (\%)** & **Modes (Hz)** \\ \hline
1.58 & 6.1 & 0 & 219.1 & 1.335 & 0.6, 0.5, 0.5 & 1.02, 2.05, 3.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main parameters of the scaled cable. Figure 2: Time history of displacement and control force at the mid-span under free vibration. (**a**) Time history of the mid-span displacement. (**b**) Time history of control force. The results indicate that both the linear LQR control and bilinear control algorithms are valid and effective to free vibration and harmonic excitation. Although the former has a slightly worse control effect than the latter, this can still verify that the linear simplified model is feasible and applicable in engineering practices.