# Active Tendon Control of Stay Cable by a Giant Magnetotrstive Actuator Considering Time-Delay
## 4 The Time-Delay Compensation Method of the Controlled Cable  
When the control system has time delay, the Equation (14) can be written as:  
\[\dot{\mathbf{Z}}=\mathbf{A}\mathbf{Z}+\mathbf{B}_{2}U(t\tau) \tag{24}\]  
where, \(\tau\) is the time delay of the system. For a multi-modal control system, the control force of the \(j\)-th mode without time-delay can be described as:  
\[\mathbf{U}_{mj}(t)=-G_{j1}\xi_{j1}(t)-G_{j2}\xi_{j2}(t) \tag{25}\]  
where, \(\xi_{j1}(t)\) and \(\xi_{j2}(t)\) represent the modal displacement and velocity for the \(j\)-th mode, \(G_{j1}\) and \(G_{j2}\) are the gain coefficients of the modal displacement and velocity without time-delay, respectively. When the time-delay of displacement and velocity in the system are uniformly assumed to be \(\tau\), the Equation (25) can be transformed into the following:  
\[\mathbf{U}_{mj}(t\tau)=-G_{j1}\xi_{j1}(t-\tau)-G_{j2}\xi_{j2}(t-\tau) \tag{26}\]  
Then, modifying the gain coefficient with time-delay compensation, the Equation (26) can be rewritten as:  
\[\mathbf{U}^{{}^{\prime}}_{mj}(t)=-G^{\prime}_{j1}\xi_{j1}(t-\tau)-G^{\prime}_{ j2}\xi_{j2}(t-\tau) \tag{27}\]  
where \(G^{{}^{\prime}}_{j1}\) and \(G^{{}^{\prime}}_{j2}\) are the modified gain coefficients, respectively. In order for the time-delay system equivalent to be ideal, let Equation (25) be equal to Equation (27), namely:  
\[\mathbf{U}_{mj}(t)=\mathbf{U}^{{}^{\prime}}_{mj}(t)=-G_{j1}\xi_{j1}(t)-G_{j2} \xi_{j2}(t)=-G^{\prime}_{j1}\xi_{j1}(t-\tau)-G^{\prime}_{j2}\xi_{j2}(t-\tau) \tag{28}\]  
Equation (28) shows that the delay compensation can be realized by adjusting the gain coefficient. In this paper, the phase shifting method is used to determine the modified gain coefficients. Figure 3: Time history of displacement and force at mid-span under harmonic excitation. (**a**) Time history of displacement of the first 40 s under forced vibration. (**b**) Time history of steady-state displacement. Since there is a time-delay in displacement and velocity during the control process, the phases of displacement and velocity are delayed compared to the state without time-delay, as illustrated in Figure 4.  
When the phase difference caused by the time-delay is considered, the control force of the \(j\)-th modal displacement can be factorized as \(G^{{}^{\prime}}_{j1}\)cos(\(\omega_{j}\tau\)) \(\xi_{j1}\)(t) and -\(G^{{}^{\prime}}_{j1}\)sin(\(\omega_{j}\tau\)) \(\xi_{j2}(t)/\omega_{j}\) in \(\xi_{j1}\) and \(\xi_{j2}\) coordinates, respectively. Similarly, the control force of the \(j\)-th modal velocity can be divided into \(G^{{}^{\prime}}_{j2}\)sin(\(\omega_{j}\tau\))\(\omega_{j}\)\(\xi_{j1}\)(t) and \(G^{{}^{\prime}}_{j2}\)cos(\(\omega_{j}\tau\)) \(\xi_{j2}\)(t). Therefore, the Equation (27) can be written as [33]:  
\[ \begin{split}&\mathbf{U}^{{}^{\prime}}_{mj}(t)=-G^{{}^{\prime}}_{j1} \cos(\omega_{j}\tau)\xi_{j1}(t)+\frac{G^{{}^{\prime}}_{j1}\sin(\omega_{j}\tau)} {\omega_{j}}\xi_{j2}(t)-G^{{}^{\prime}}_{j2}\sin(\omega_{j}\tau)\omega_{j}\xi_ {j1}(t)-G^{{}^{\prime}}_{j2}\cos(\omega_{j}\tau)\xi_{j2}(t)\\ &=-\big{[}G^{{}^{\prime}}_{j1}\cos(\omega_{j}\tau)+G^{{}^{\prime }}_{j2}\sin(\omega_{j}\tau)\omega_{j}\big{]}\xi_{j1}(t)-\bigg{[}-\frac{G^{{}^{ \prime}}_{j1}\sin(\omega_{j}\tau)}{\omega_{j}}+G^{{}^{\prime}}_{j2}\cos(\omega _{j}\tau)\bigg{]}\xi_{j2}(t)\end{split}  \tag{29}\]  
According to Equations (28) and (29), we can obtain:  
\[\mathbf{G}^{{}^{\prime}}_{j}=\left[ \begin{array}{c}G^{{}^{\prime}}_{j1}\\ G^{{}^{\prime}}_{j2}\end{array} \right]=\left[ \begin{array}{cc}\cos\omega_{j} \tau&\omega_{j}\sin\omega_{j}\tau\\ -(1/\omega_{j})\sin\omega_{j}\tau&\cos\omega_{j}\tau\end{array} \right]^{-1} \left[ \begin{array}{c}G_{j1}\\ G_{j2}\end{array} \right]=\mathbf{D}_{j}\bigg{[} \begin{array}{c}G_{j1}\\ G_{j2}\end{array} \bigg{]}=\mathbf{D}_{j}\mathbf{G}_{j} \tag{30}\]  
Considering the first three modes, there is:  
\[\mathbf{G}^{{}^{\prime}}=\mathbf{D}_{e}\mathbf{G} \tag{31}\]  
where  
\[\mathbf{D_{e}}=\left[ \begin{array}{ccc}D_{1}&0&0\\ 0&D_{2}&0\\ 0&0&D_{3}\end{array}