## 3 Boundary Control Design  
In this section, \(H_{\infty}\)ILC-based MOL scheme was considered in the presence of string system parameters uncertainties and external disturbances, aiming to eliminate the unwanted perturbations, prevent any the distributed and boundary disturbances, and stabilize the string tip payload to equilibrium. The following \(H_{\infty}\)ILBC law is designed for uncertain vibrating string system:  
\[Uc^{k+1}=Uc^{k}+B_{d1}^{a}F^{k}+V^{k} \tag{13}\]  
\[Y^{k}=(H+\Delta H)Uc^{k}+D_{21}^{a}F^{k} \tag{14}\]  
where  
\[V^{k}=C(\omega)E^{k} \tag{15}\]  
\[C(\omega)=Q^{-1}(\omega)L(\omega) \tag{16}\]  
where \(Q_{N_{H}}\) is a filter parameter, \(L_{N_{H}}\) is a learning function parameter, and \(\omega\) is a delay operator as shown:  
\[Q(\omega)=Q_{N_{H}}\omega^{N_{H}}+Q_{(N_{H}-1)}\omega^{(N_{H}-1)}+\ldots+Q_{1} \omega+Q_{0} \tag{17}\]  
\[L(\omega)=L_{N_{H}}\omega^{N_{H}}+L_{(N_{H}-1)}\omega^{(N_{H}-1)}+\ldots+L_{1} \omega+L_{0} \tag{18}\]  
\[E^{k}=R^{k}-Y_{N}^{k}(N_{\tau}) \tag{19}\]  
where \(R^{k}\) is the desired boundary deflection of the string, \(Y_{N}^{k}(N_{\tau})\) is the actual boundary deflection, \(k\) means the iteration through the iteration axis. Define  
\[Uc^{k}=\left[ \begin{array}{cccccccccccc}0&0&\cdots&0&0&\cdots&0&\cdots&0&0& \cdots&0&0&U_{\mathbb{C}_{N}}^{k}(0)&U_{\mathbb{C}_{N}}^{k}(1)&\cdots&U_{ \mathbb{C}_{N}}^{k}(N_{\tau}-1)\end{array} \right]^{T},\]  
\[F^{k}=\left[ \begin{array}{cccccccccccc}F_{1}^{k}(1)&F_{1}^{k}(2)&\cdots&F_{ 1}^{k}(N_{t})&F_{2}^{k}(1)&F_{2}^{k}(2)&\cdots&F_{2}^{k}(N_{t})&\cdots&F_{N-1}^ {k}(1)&F_{N-1}^{k}(2)&\cdots&F_{N-1}^{k}(N_{t})&D_{N}^{k}(1)&D_{N}^{k}(2)&\cdots &D_{N}^{k}(N_{t})\end{array} \right]^{T},\]  
\[v^{k}=\left[ \begin{array}{cccccccccccc}v_{1}^{k}(1)&V_{1}^{k}(2)&\cdots&V_{ 1}^{k}(N_{t})&V_{2}^{k}(1)&V_{2}^{k}(2)&\cdots&V_{2}^{k}(N_{t})&\cdots&V_{N-1}^ {k}(1)&V_{N-1}^{k}(2)&\cdots&V_{N-1}^{k}(N_{t})&V_{N}^{k}(1)&V_{N}^{k}(2)&\cdots &V_{N}^{k}(N_{t})\end{array} \right]^{T}\]  
\[\gamma^{k}=\left[ \begin{array}{cccccccccccc}\gamma_{1}^{k}(1)&\gamma_{1}^ {k}(2)&\cdots&\gamma_{1}^{k}(N_{t})&\gamma_{2}^{k}(1)&\gamma_{2}^{k}(2)&\cdots &\gamma_{2}^{k}(N_{t})&\cdots&\gamma_{N-1}^{k}(1)&\gamma_{N-1}^{k}(2)&\cdots& \gamma_{N-1}^{k}(N_{t})&\gamma_{N}^{k}(1)&\gamma_{N}^{k}(2)&\cdots&\gamma_{N}^ {k}(N_{t})\end{array} \right]^{T}\]  
\[E^{k}=\left[ \begin{array}{cccccccc}0&0&\cdots&0&0&0&\cdots&0&\cdots&0&0&\cdots &0&E_{N}^{k}(1)&E_{N}^{k}(2)&\cdots&E_{N}^{k}(N_{\tau})\end{array} \right].\]  
For iteration \(k\), \(Uc^{k}\) denotes the boundary control input. \(k\), \(F^{k}\) denotes the iteration-varying distributed and boundary disturbance, \(B_{d1}^{a}=\left[ \begin{array}{cc}0&\psi I&0\end{array} \right]\), \(\Delta H\) is the iteration-varying uncertainties of the string system, and \(D_{21}^{a}=\left[ \begin{array}{cc}\sqrt{\varphi}\Delta H&0&\psi I\end{array} \right]\). The matrix \(H\) is a Toeplitz matrix whose components are Markov parameters [20], \(\left[ \begin{array}{cc}N_{\tau}\times N&N_{\tau}\times N\end{array} \right]= size(H)=\left[ \begin{array}{cc}N_{H}&N_{H}\end{array} \right]\), \(\varphi=1\), and \(\psi=0.1\), \(N_{\tau}\) is the control horizon indicating the number of components utilized to release the upcoming control trajectories, and \[H=\left[