## 3 Main Results
### Stability of Auxiliary State Dynamics System
Pre- and post-multiplying both sides of \(\Pi_{1}+\Pi_{2}+\Pi_{3}\) by \(\mathrm{diag}\left\{P^{-1},\,P^{-1},\,P^{-1},\,P^{-1},\,P^{-1},\,P^{-1},\,P^{-1},\,P ^{-1},\,I,\,P^{-1},\,P^{-1},\,P^{-1},\,P^{-1},\,I,\,I\right\}\) and using similar arguments from Theorem 1 with \(\hat{P}\hat{T}_{i}^{-1}\hat{P}\leq-2\rho\hat{P}+\rho^{2}\hat{T}_{r}^{T}\), we obtain the following result. Separating the uncertainties from \(\bar{A}\) and \(\bar{A}_{d}\) in \(\Pi_{1}\), we have:  
\[[\Pi]_{24\times 24}=\hat{\Pi}_{1}+\hat{\Pi}_{2}+\hat{\Pi}_{3}+\sum_{i=1}^{6}( \mathcal{M}_{i}\Delta(t)\mathcal{N}_{i}+\mathcal{N}_{i}^{T}\Delta(t)^{T} \mathcal{M}_{i}^{T}).\]Using Lemma 2 of [20], we have:  
\[[\Pi]_{24\times 24}=\hat{\Pi}_{1}+\hat{\Pi}_{2}+\hat{\Pi}_{3}+\sum_{i=1}^{6}( \epsilon_{i}^{-1}\mathcal{M}_{i}\mathcal{M}_{i}^{T}+\epsilon_{i}\mathcal{N}_{i} \mathcal{N}_{i}^{T}).\]  
By applying Schur complement Lemma 1 in \(\Pi\), we can guarantee that LMI (27) holds, that is, \(\hat{\Pi}_{s}<0\). Thus, we concludes that the auxiliary system (10) is asymptotically stable and satisfies the \(H_{\infty}\) performance level \(\gamma\). This completes the proof. Theorem 4: _For given positive scalars \(\epsilon_{i},i=1,2,3,...,6\), \(\rho\), \(0\leq\tau_{m}<\tau_{M}\), the auxiliary system (11) under controller (2) is asymptotically stable and satisfies the \(H_{\infty}\) tracking performance level \(\gamma\), if there exist symmetric positive definite matrices \(\hat{P}\), \(\hat{P}_{r}\), \(\hat{\Gamma}_{1}\), \(\hat{\tau}_{2}\), \(\hat{\tau}_{3}\), \(\hat{\tau}_{4}\), \(\hat{\tau}_{5}\), \(\hat{\tau}_{r}\), and for any matrix \(\hat{S}\), \(\hat{K}\) with compatible dimensions, one satisfies \( \begin{bmatrix}\hat{\Gamma}_{5}&\hat{S}\\ \hat{S}^{T}&\hat{\Gamma}_{5}\end{bmatrix} \geq 0\), such that the following LMI holds:_  
\[[\hat{\Gamma}_{s}]_{24\times 24}= \begin{bmatrix}[\hat{\Gamma}_{1}]_{10\times 10}&[ \hat{\Gamma}_{2}]_{10\times 1}&[\hat{\Gamma}_{3}]_{10\times 1}&[\hat{\Gamma}_{4}]_{ 10\times 4}&[\hat{\Gamma}_{5}]_{10\times 8}\\ *&-I&0&0&0\\ *&*&-I&0&0\\ *&*&*&[\hat{\Omega}_{4}]_{4\times 4}&0\\ *&*&*&*&[\hat{\Omega}_{5}]_{8\times 8}\end{bmatrix} <0,\]  
_where_  
\[[\hat{\Gamma}_{1}]_{10\times 10}= \begin{bmatrix}\hat{\Gamma}_{1,1}&0&\hat{ \Gamma}_{1,3}&\hat{\Gamma}_{1,4}&0&\hat{\Gamma}_{1,6}&\hat{\Gamma}_{1,7}&0& \hat{\Gamma}_{1,9}&0\\ *&\hat{\Gamma}_{2,2}&0&0&0&0&\hat{\Gamma}_{2,8}&0&\hat{\Gamma}_{2,10}\\ *&*&\hat{\Gamma}_{3,3}&\hat{\Gamma}_{3,4}&\hat{\Gamma}_{3,5}&0&\hat{\Gamma}_ {3,7}&0&\hat{\Gamma}_{3,9}&0\\ *&*&*&\hat{\Gamma}_{4,4}&\hat{S}&0&0&0&0&0\\ *&*&*&*&\hat{\Gamma}_{5,5}&0&0&0&0&0\\ *&*&*&*&*&-\gamma^{2}&\tau_{m}B_{w}^{T}&0&\delta B_{w}^{T}&0\\ *&*&*&*&*&\hat{\Gamma}_{17,7}&0&0&0\\ *&*&*&*&*&*&\hat{\Gamma}_{8,8}&0&0\\ *&*&*&*&*&*&*&\hat{\Gamma}_{9,9}&0\\ *&*&*&*&*&*&*&\hat{\Gamma}_{10,10}\end{bmatrix} ,\]  
\(\hat{\Gamma}_{1,3}=A_{d}\hat{P}+B_{d}\hat{K}\), \(\hat{\Gamma}_{3,7}=\tau_{m}(\hat{P}A_{d}^{T}+\hat{K}^{T}B_{d}^{T})\), \(\hat{\Gamma}_{3,9}=\delta(\hat{P}A_{d}^{T}+\hat{K}^{T}B_{d}^{T})\), and all other parameters are defined in Theorem 3. Furthermore, the controller gain matrix is given by \(K=\hat{K}\hat{P}^{-1}\). Proof: Let us consider the Lyapunov-Krasovskii functional (28) as in Theorem 3 and take its derivative along the trajectories of system (11), which yields:  
\[\dot{V}_{1}(t,z(t)) = 2z^{T}(t)\mathcal{P}\dot{z}(t)\] \[= 2z^{T}(t)\mathcal{P}[(A+\Delta A)z(t)+(A_{d}+\Delta A_{d})z(t-\tau (t))+Bu(t)\] \[+B_{d}u(t-\tau(t))+B_{w}w(t)+\Delta AG_{r}(t)+(A_{d}+\Delta A_{d}) Gx_{r}(t-\tau(t))].\]  
By utilizing Equations (31)-(32) and Definition 1, the rest of the proof is similar to that in Theorem 3. Then, we can easily obtain the LMI \([\hat{\Gamma}_{s}]_{24\times 24}\). Hence, the auxiliary system (11) is asymptotically stable and satisfies the \(H_{\infty}\) tracking performance level \(\gamma\). This completes the proof.