## 3 Main Results
### Stability of Auxiliary State Dynamics System  
In this subsection, we establish the stability of the auxiliary state dynamics system. **Theorem 3**. : _For given positive scalars \(e_{i},i=1,2,3,...,6\), \(\rho\), \(0\leq\tau_{m}<\tau_{M}\), the auxiliary system (10) under controller (2) is asymptotically stable and satisfies the \(H_{\infty}\) tracking performance level \(\gamma\), if there exist symmetric positive definite matrices \(\hat{P}\), \(\hat{P}\), \(\hat{P}\), \(\hat{T}_{1}\), \(\hat{T}_{2}\), \(\hat{T}_{3}\), \(\hat{T}_{4}\), \(\hat{T}_{5}\), \(\hat{T}_{r}\), and for any matrix \(\hat{S}\), \(\hat{K}\) with compatible dimensions, one satisfies \( \begin{bmatrix}\hat{T}_{5}&\hat{S}\\ \hat{S}^{T}&\hat{\tau}_{5}\end{bmatrix} \geq 0\), such that the following LMI holds:_  
\[ \begin{array}{rcl}\dot{V}_{3}(t,z(t))&=&\tau_{m}\int_{-\tau_{m}}^{0}\dot{z}^{T}(t )T_{4}\dot{z}(t)\mathrm{d}\theta-\tau_{m}\int_{-\tau_{m}}^{0}\dot{z}^{T}(t+ \theta)T_{4}\dot{z}(t+\theta)\mathrm{d}\theta\\ &&+\delta\int_{-\tau_{M}}^{-\tau_{m}}\dot{z}^{T}(t)T_{5}\dot{z}(t)\mathrm{d} \theta-\delta\int_{-\tau_{M}}^{-\tau_{m}}\dot{z}^{T}(t+\theta)T_{5}\dot{z}(t+ \theta)\mathrm{d}\theta\\ &\leq&\dot{z}^{T}(t)(\tau_{m}^{2}\mathcal{T}_{4}+\delta^{2}\mathcal{T}_{5}) \dot{z}(t)-\tau_{m}\int_{t-\tau_{m}}^{t}\dot{z}^{T}(s)T_{4}\dot{z}(s)\mathrm{d }s\\ &&-\delta\int_{t-\tau_{M}}^{t-\tau_{m}}\dot{z}^{T}(s)T_{5}\dot{z}(s)\mathrm{d}s.\end{array}  \tag{31}\]  
It should be noted that the positive definite matrices \(\mathcal{P}=\mathrm{diag}\left\{P,\,P_{r}\right\}\), \(\mathcal{T}_{4}=\mathrm{diag}\left\{T_{4},\,T_{r}\right\}\), and \(\mathcal{T}_{5}=\mathrm{diag}\left\{T_{5},\,T_{r}\right\}\). From Equation (29)-(31), the solution is similar to that of Equations (17)-(21). Now, consider the \(H_{\infty}\) condition following index:  
\[\mathcal{J}=\int_{0}^{\infty}[\ e^{T}(t)e(t)-\gamma^{2}w^{T}(t)w(t)+\dot{V}(t,z(t))]\ dt+V(t,z(t))|_{t=0}-V(t,z(t))|_{t=\infty}.\]  
Under zero-initial conditions, we have \(V(t,z(t))|_{t=0}=0\) and \(V(t,z(t))|_{t=\infty}\geq 0\), which leads to the following result:  
\[\mathcal{J}\leq\int_{0}^{\infty}[\ e^{T}(t)e(t)-\gamma^{2}w^{T}(t)w(t)+\dot{V} (t,z(t))]\ dt. \tag{32}\]  
Then, combining (29)-(32), we have the expression:  
\[e^{T}(t)e(t)-\gamma^{2}w^{T}(t)w(t)+\dot{V}(t,z(t))\leq\xi^{T}(t)[\bar{\Pi}] \ \zeta(t) \tag{33}\]  
where \(\xi^{T}(t)=[z^{T}(t)\ z^{T}(t-\tau(t))\ z^{T}(t-\tau_{m})\ z^{T}(t-\tau_{M})\ w^{T}(t)\ e^{T}(t)\ z^{T}(t)\ z^{T}(t)]\) and \(\bar{\Pi}\) is defined as later. Using the [PERSON] complement lemma, the right-hand side of inequality (33) is equivalent to the matrix \(\Pi_{1}+\Pi_{2}+\Pi_{3}\), resulting in:  
\[[\Pi]_{12\times 12}= \begin{bmatrix}\Pi_{1,1}&0&\Pi_{1,3}&\Pi_{1,4}&0&\Pi_{1,6 }&\Pi_{1,7}&0&\Pi_{1,9}&0\\ *&\Pi_{2,2}&0&0&0&0&\Pi_{2,8}&0&\Pi_{2,10}\\ *&*&\Pi_{3,3}&\Pi_{3,4}&\Pi_{3,5}&0&\Pi_{3,7}&0&\Pi_{3,9}&0\\ *&*&*&\Pi_{4,4}&S&0&0&0&0&0\\ *&*&*&*&\Pi_{5,5}&0&0&0&0&0\\ *&*&*&*&*&-\gamma^{2}&\tau_{m}B_{w}^{T}T_{4}&0&\delta B_{w}^{T}T_{5}&0\\ *&*&*&*&*&*&\Pi_{7,7}&0&0&0\\ *&*&*&*&*&*&\Pi_{8,8}&0&0\\ *&*&*&*&*&*&*&\Pi_{9,9}&0\\ *&*&*&*&*&*&*&*&\Pi_{10,10}\end{bmatrix} \]  
where \(\Pi_{1,1}=P\bar{A}+\bar{A}^{T}P+PBK+K^{T}B^{T}P+T_{1}+T_{2}+T_{3}-T_{4}\), \(\Pi_{1,3}=P\bar{A}_{d}\), \(\Pi_{1,4}=T_{4}\), \(\Pi_{1,6}=PB_{w}\), \(\Pi_{1,7}=\tau_{m}(\bar{A}^{T}+K^{T}B^{T})T_{4}\), \(\Pi_{1,9}=\delta(\bar{A}^{T}+K^{T}B^{T})T_{5}\), \(\Pi_{1,11}=C^{T}\), \(\Pi_{2,2}=-P_{r}A_{r}-A_{r}^{T}P_{r}\), \(\Pi_{2,8}=-\tau_{m}A_{r}^{T}T_{r}\), \(\Pi_{2,10}=-\delta A_{r}^{T}T_{r}\), \(\Pi_{2,12}=-C_{r}^{T}\), \(\Pi_{3,3}=-(1-\mu)T_{3}-2T_{5}+S+S^{T}\), \(\Pi_{3,4}=T_{5}-S^{T}\), \(\Pi_{3,5}=T_{5}-S\), \(\Pi_{3,7}=\tau_{m}\bar{A}_{d}^{T}T_{4}\), \(\Pi_{3,9}=\delta\bar{A}_{d}^{T}T_{5}\), \(\Pi_{4,4}=-T_{1}-T_{4}-T_{5}\), \(\Pi_{5,5}=-T_{2}-T_{5}\), \(\Pi_{7,7}=-T_{4}^{T}\), \(\Pi_{8,8}=-T_{r}^{T}\), \(\Pi_{9,9}=-T_{5}^{-1}\), \(\Pi_{10,10}=-T_{r}^{-1}\).