## 3 Boundary Control Design
### Stability Analysis  
In this part, the robust stability for the close-loop string system is assured with the proposed \(H_{\infty}\)ILC formulas as given in Equations (13) and (14). From Equations (11) and (12), the discrete-state space string system without uncertainties and external disturbances is expressed as  
\[\left[ \begin{array}{cccccccc}Z_{1}(t_{d}+1)&Z_{2}(t_{d}+1)&\cdots&Z_{(N-1)}(t _{d}+1)&Z_{2}(t_{d}+1)\end{array} \right]=A_{d}\left[ \begin{array}{cccccccc}Z_{ 1}(t_{d})&Z_{2}(t_{d})&\cdots&Z_{(N-1)}(t_{d})&Z_{2}(t_{d})\end{array} \right]+ B_{d}\left[ \begin{array}{cccccccc}0&0&\cdots&0&u_{\infty}(t_{d})\end{array} \right] \tag{22}\]  
\[\left[ \begin{array}{cccccccc}Y_{1}(t_{d})&Y_{2}(t_{d})&\cdots&Y_{(N-1)}(t_{d} )&Y_{N}(t_{d})\end{array} \right]=C_{d}\left[ \begin{array}{cccccccc}Z_{1}(t_{ d})&Z_{2}(t_{d})&\cdots&Z_{(N-1)}(t_{d})&Z_{N}(t_{d})\end{array} \right] \tag{23}\]  
where \(t_{d}\) denote the discrete-time through the time axis. Reformulating Equations (22) and (23) into a super-vector iterative learning control (SVILC) form by taking \(Z\)-transformer for Equation (23) and define the resulting string system by \(G(z)\) as follows:  
\[\left[ \begin{array}{cccccccc}Y_{1}^{k}(z)&Y_{2}^{k}(z)&\cdots&Y_{(N_{H}-1)}^{k }(z)&Y_{N_{H}}^{k}(z)\end{array} \right]=G(z)\!\left[ \begin{array}{cccccccc}0& 0&\cdots&0&U_{C_{N_{H}}}^{k}(z)\end{array} \right] \tag{24}\]  
where \(G(z)=C_{d}(zI-A_{d})^{-1}B_{d}+D_{d}\), then Equation (24) is expressed by  
\[\left[ \begin{array}{cccccccc}Y_{1}^{k}&Y_{2}^{k}&\cdots&Y_{(N_{H}-1)}^{k}&Y _{N_{H}}^{k}\end{array} \right]=H\!\left[ \begin{array}{cccccccc}0&0&\cdots &0&U_{C_{N_{H}}}^{k}\end{array} \right]. \tag{25}\]  
From Equation (25), after introducing a delay operator \(\omega\) through the iteration axis, we obtain  
\[\left[ \begin{array}{cccccccc}Y_{1}^{k}(\omega)&Y_{2}^{k}(\omega)&\cdots&Y_{ (N_{H}-1)}^{k}(\omega)&Y_{N_{H}}^{k}(\omega)\end{array} \right]=H\!\left[ \begin{array}{cccccccc}0&0&\cdots&0&U_{C_{N_{H}}}^{k}(\omega)\end{array}  \right]. \tag{26}\]  
Considering the iterative learning boundary control algorithms as follows  
\[ \begin{array}{ll}U_{C_{N_{H}}}^{k+1}&=(I-Q_{N_{H}-1})U_{C_{N_{H}}}^{k}+(Q_{ N_{H}-1}-Q_{N_{H}-2})U_{C_{N_{H}}}^{k-1}\\ &+\ldots+(Q_{1}-Q_{0})U_{C_{N_{H}}}^{k-n+1}+Q_{0}U_{C_{N_{H}}}^{k-n}\\ &+L_{N_{H}}E_{N_{H}}{}^{k}+L_{N_{H}-1}E_{N_{H}}{}^{k-1}+\ldots+L_{0}E_{N_{H}}{ }^{k-n}\end{array}  \tag{27}\]  
Applying \(\omega\)-transform to Equation (27), one obtains  
\[(\omega-1)Q(\omega)Uc_{N_{H}}=L(\omega)E_{N_{H}}(\omega). \tag{28}\]  
From Equation (28), one obtains  
\[Uc_{N_{H}}=\frac{I}{(\omega-1)}C(\omega)E_{N_{H}}(\omega). \tag{29}\]  
The integrator \((\omega-1)\) in Equation (29) assures that the \(E_{N_{H}}(\omega)\to 0\)  during ILBC converges. After considering the string system opposed to parameter uncertainties, varying-iteration distributed disturbance \(F_{N-1}^{k}\) (the distributed disturbance is a function in time, position and iteration), and varying-iteration boundary disturbance \(D_{N}^{k}\) (the distributed disturbance is a function in time and iteration), it has become an \(H_{\infty}\) problem. Equations (22) and (23) are transformed to  
\[\left[ \begin{array}{cccccccc}Z_{1}^{k}(t_{d}+1)&Z_{2}^{k}(t_{d}+1)&\cdots&Z _{(N-1)}^{k}(t_{d}+1)&Z_{N}^{k}(t_{d}+1)\end{array} \right] =(A_{d}+\Delta A_{d})\left[ \begin{array}{cccccccc}Z_{1}^{k}(t_{d})&Z_{2}^ {k}(t_{d})&\cdots&Z_{(N-1)}^{k}(t_{d})&Z_{N}^{k}(t_{d})\end{array} \right] \tag{30}\] \[+(B_{\Omega}+\Delta B_{\Omega})\left[ \begin{array}{cccccccc}F_{1 }^{k}(t_{d})&F_{2}^{k}(t_{d})&\cdots&F_{N-1}^{k}(t_{d})&D_{N}^{k}(t_{d}) \end{array} \right]\] \[+(B_{\Omega}+\Delta B_{\Omega})\left[ \begin{array}{cccccccc}0&0& \cdots&0&U_{C_{N}}^{k}(t_{d})\end{array} \right]\] \[\left[ \begin{array}{cccccccc}Y_{1}^{k}(t_{d})&Y_{2}^{k}(t_{d})& \cdots&Y_{(N-1)}^{k}(t_{d})&Y_{N}^{k}(t_{d})\end{array} \right]=C_{d}\!\left[ \begin{array}{cccccccc}Z_{1}^{k}(t_{d})&Z_{2}^{k}(t_{d})&\cdots&Z_{(N-1)}^{k }(t_{d})&Z_{N}^{k}(t_{d})\end{array}