## 3 Boundary Control Design
### Stability Analysis
\right] \tag{31}\]  
where \(\Delta A_{d},\Delta B_{\Omega},\) and \(\Delta B_{\Omega 2}\) denote the system parameter uncertainties matrices for string system. Therefore, the \(kth\) iteration of Equation (26) with varying uncertainties and with iteration-dependent distributed and boundary disturbances is expressed by  
\[Y^{k}=(H+\Delta H)U^{k}+B_{H}F^{k} \tag{32}\]  
where  
\[B_{H}=\left[ \begin{array}{cc}(B_{d1}+\Delta B_{d1})&0\\ 0&0\end{array} \right].\]Then, the resulting ILBC system is integrated into the \(H_{\infty}\) framework such that \(\|E^{k}\|_{2^{\infty}}\) is minimum and the closed loop string system is stabilizable through \(H=H+\Delta H\), with \(\|\Delta H\|_{\infty^{\infty}}<\phi_{H}\) as follows  
\[\|P_{EF}\|_{\infty^{\infty}}=\sup_{F^{k}\in\mathcal{L}_{2^{\infty}}}\frac{\|E^ {k}\|_{2^{\infty}}}{\|F^{k}\|_{2^{\infty}}} \tag{33}\]  
such that  
\[\|P_{EF}\|_{\infty^{\infty}}<\varphi \tag{34}\]  
where  
\[\|E^{k}\|_{2^{\infty}}=\sum_{k=0}^{\infty}E^{kT}E^{k} \tag{35}\]  
\[\|F^{k}\|_{2^{\infty}}=\sum_{k=0}^{\infty}F^{kT}F^{k}. \tag{36}\]  
Equation (34) guarantees that the \(H_{\infty}\) norm is minimized from \(F^{k}\) to \(E^{k}\). For minimizing \(E^{k}\) and sensitivity matrices Equations (40) and (41), the \(H_{\infty}\) algebraic approach has been used for robust stability of the uncertain vibrating string system as follows:  
\[Z^{k+1^{\gamma}}=A_{1}Z^{k}+B_{1}F^{k}+B_{2}U^{k}, \tag{37}\]  
\[X^{k}=C_{d1}Z^{k}+D_{11}F^{k}+D_{12}U^{k}, \tag{38}\]  
\[\Upsilon^{k}=C_{d2}Z^{K}+D_{21}F^{k}. \tag{39}\]  
The sensitivity matrices from \(F^{k}\) to \(E^{k}_{N}\) and from \(D^{k}\) to \(E^{k}_{N}\) are expressed as follows  
\[S_{F}=C_{s}(\omega)H_{s}(\omega)(I+C_{s}(\omega)H_{s}(\omega))^{-1}+(\omega I -I)^{-1}(I+C_{s}(\omega)H_{s}(\omega))^{-1}, \tag{40}\]  
\[S_{D}=C_{s}(\omega)(I+C_{s}(\omega)H_{s}(\omega))^{-1}+C_{s}(\omega)(\omega I -I)^{-1}(I+C_{s}(\omega)H_{s}(\omega))^{-1}, \tag{41}\]  
where  
\[H_{s}(\omega)=(\omega I-I)^{-1}(H+\Delta H). \tag{42}\]  
Define  
\[ \begin{array}{l}Z^{k}=\left[\begin{array}{cccccccc}[PERSON]^{k}_{1}(1)&[PERSON]^{k}_{1 }(2)&\cdots&Z^{k}_{1}(N_{tao})&\cdots&Z^{k}_{N}(1)&[PERSON]^{k}_{N}(2)&\cdots&Z^{k}_{ N}(N_{tao})\\ \\ X^{k}_{1}(1)&X^{k}_{1}(2)&\cdots&X^{k}_{1}(N_{tao})&\cdots&X^{k}_{N}(1)&X^{k}_ {N}(2)&\cdots&X^{k}_{N}(N_{tao})\\ \end{array}\right],\end{array} \]  
where \(Z^{k}\) is the system states, \(X^{k}\) is the performance output, and \(\Upsilon^{k}\) is the controlled output, and  
\[\begin{array}{l}A_{1}=\left[ \begin{array}{cc}(A_{d}+\Delta A_{d})&0\\ 0&0\\ \end{array} \right]\begin{array}{l}B_{1}=\left[ \begin{array}{cc}\psi(B_{d1} +\Delta B_{d1})&0\\ 0&0\\ \end{array} \right]\begin{array}{l}B_{2}=\left[ \begin{array}{cc}(B_{d2}+ \Delta B_{d2})&0\\ 0&0\\ \end{array} \right] \begin{array}{l}C_{d1}=\left[\begin{array}{cc}0&I\\ 0&0\\ \end{array}\right]\begin{array}{l}C_{d2}=H+\Delta H\\ \\ D_{11}=0,\,D_{12}=\left[\begin{array}{cc}0&I\\ 0&0\\ \end{array}\right]\begin{array}{l}D_{21}=\left[\begin{array}{cc}0&\psi(B_{d1 }+\Delta B_{d1})\\ 0&0\\ \end{array}\right]\end{array}\end{array}\end{array} \]  
**Definition 1**[44]. From Equation (34) after substituting \(\varphi=1\), the dynamics Equations (37)-(39) (with \(U^{k}=0\)) are said to be quadratic stabilized with \(H_{\infty}\) norm bound \(\varphi\), a positive definite matrix \(M\) exists such that for any allowable uncertainties \(\Delta A_{d1}\):  
\[\overline{A}^{T}{}_{d}M\overline{A}_{d}+\overline{A}^{T}{}_{d}MB_{1}(I- \varphi^{-2}B_{1}^{T}MB_{1})^{-1}B_{1}^{T}M\overline{A}_{d}-M+C_{d2}^{T}C_{d2}<0 \tag{43}\]with  
\[I-\varphi^{-2}B_{1}^{T}MB_{1}>0. \tag{44}\]  
Or equivalently, a positive definite matrix \(P>0\) exists such that for any allowable uncertainties \(\Delta A_{d1}\):  
\[\widetilde{A}^{T}{}_{d}P\widetilde{A}_{d}+\widetilde{A}^{T}{}_{d}PB_{1}(I- \varphi^{-2}BPB_{1}^{T})^{-1}B_{1}P\widetilde{A}_{d}{}^{T}-P+C_{d2}C_{d2}^{T}<0 \tag{45}\]  
with  
\[I-\varphi^{-2}B_{1}PB_{1}^{T}>0 \tag{46}\]  
where  
\[\widetilde{A}_{d}=A_{d}+\Delta A_{d}. \tag{47}\]  
**Definition 2**[44].