## 3 Reliable Robust \(H_{\infty}\) Controller Design
\begin{array}{cccc}\overline{\Lambda}_{11}&\mathbf{B}_{1}(\mathbf{\lambda })&\overline{\Lambda}_{13}&\eta\mathbf{B}_{2}&\overline{\mathrm{k}}^{T}(\mathbf{ \lambda})\xi_{0}^{T}\\ *&-\gamma^{2}\mathbf{I}&0&0&0\\ *&*&-\mathbf{I}&\eta\mathbf{D}_{1}(\mathbf{\lambda})&0\\ *&*&*&-\eta\mathbf{I}^{-1}&0\\ *&*&*&*&-\eta\mathbf{J}^{-1}\end{array} \right]<0 \tag{32}\]  
\[\left[ \begin{array}{cccc}-\mathbf{I}+\varepsilon\mathbf{I}&\sqrt{\rho} \xi_{0}\overline{\mathrm{k}}(\mathbf{\lambda})&0\\ *&-u_{\max}^{2}\overline{\mathrm{P}}(\mathbf{\lambda})&\sqrt{\rho}\overline{ \mathrm{k}}^{T}(\mathbf{\lambda})\xi_{0}^{T}\\ *&*&\varepsilon\mathbf{J}^{-1}\end{array} \right]<0 \tag{33}\]  
\[\left[ \begin{array}{cccc}-\mathbf{I}&\sqrt{\rho}\mathbf{C}_{2}(\mathbf{\lambda })\overline{\mathrm{P}}(\mathbf{\lambda})\\ *&-2_{\max}^{2}\overline{\mathrm{P}}(\mathbf{\lambda})\end{array} \right]<0 \tag{34}\]  
with  
\[\overline{\Lambda}_{11}=\text{sym}(\mathbf{A}(\mathbf{\lambda})\overline{\mathrm{ P}}(\mathbf{\lambda})+\mathbf{B}_{2}(\mathbf{\lambda})\xi_{0}\overline{\mathrm{k}}(\mathbf{ \lambda})),\ \overline{\Lambda}_{13}=\overline{\mathrm{P}}(\mathbf{\lambda})\mathbf{C}_{1}^{T}( \mathbf{\lambda})+\overline{\mathrm{k}}^{T}(\mathbf{\lambda})\xi_{0}^{T}\mathbf{D}_{1}^ {T}(\mathbf{\lambda}).\]  
**Proof. ** Choose the Lyapunov-Krasovskii functional candidate as follows:  
\[\
u(t,\mathbf{\lambda})=\mathbf{\mathrm{x}}^{T}(t)\mathbf{P}(\mathbf{\lambda})\mathbf{ \mathrm{x}}(t) \tag{35}\]  
To establish a system in Equation (29) that is robust asymptotically stable with disturbance attenuation \(\gamma>0\), the following Hamiltonian should be less than zero  
\[\mathrm{H}(\mathbf{\mathrm{x}},\mathbf{\lambda},\mathbf{\mathrm{w}},t)=\mathbf{\mathrm{z}}_{1} ^{T}(t)\mathbf{z}_{1}(t)-\gamma^{2}\mathbf{w}^{T}(t)\mathbf{w}(t)+\hat{v}(\mathbf{ \lambda},t)<0 \tag{36}\]  
\[\begin{array}{ll}\mathrm{H}(\mathbf{\mathrm{x}},\mathbf{\lambda},\mathbf{\mathrm{w}},t) =&\mathbf{\mathrm{x}}^{T}(t)\big{[}[\mathbf{A}(\mathbf{\lambda})+\mathbf{B}_{2}(\mathbf{ \lambda})\xi_{0}(\mathbf{I}+\mathbf{L})\mathbf{k}(\mathbf{\lambda})]^{T}\mathbf{P}( \mathbf{\lambda})+\mathbf{P}(\mathbf{\lambda})[\mathbf{A}(\mathbf{\lambda})+\mathbf{B}_{2}( \mathbf{\lambda})\xi_{0}(\mathbf{I}+\mathbf{L})\mathbf{k}(\mathbf{\lambda})]\big{]} \mathbf{x}(t)+\\ &\mathbf{x}^{T}(t)\mathbf{P}(\mathbf{\lambda})\mathbf{B}_{1}(\mathbf{\lambda})\mathbf{w }(t)+\mathbf{w}^{T}\mathbf{B}_{1}^{T}(\mathbf{\lambda})\mathbf{P}(\mathbf{\lambda}) \mathbf{x}(t)-\gamma^{2}\mathbf{w}(t)^{T}\mathbf{w}(t)+\mathbf{x}^{T}(t)[ \mathbf{C}_{1}(\mathbf{\lambda})+\\ &\mathbf{D}_{1}(\mathbf{\lambda})\xi_{0}(\mathbf{I}+\mathbf{L})\mathbf{k}(\mathbf{ \lambda})]^{T}[\mathbf{C}_{1}(\mathbf{\lambda})+\mathbf{D}_{1}(\mathbf{\lambda})\xi_{0 }(\mathbf{I}+\mathbf{L})\mathbf{k}(\mathbf{\lambda})]\mathbf{x}(t)\\ =&\mathbf{\mathrm{r}}^{T}(t) \begin{bmatrix}\Pi_{1}&\mathbf{P}(\mathbf{\lambda})\mathbf{B }_{1}(\mathbf{\lambda})\\ \mathbf{B}_{1}^{T}(\mathbf{\lambda})\mathbf{P}(\mathbf{\lambda})&-\gamma^{2}I\end{bmatrix}  \mathbf{\mathrm{r}}^{T}(t)\end{bmatrix}\]where  
\[ \begin{array}{ll}\Gamma^{T}(t)\!=[\mathbf{x}^{T}(t),\mathbf{w}(t)^{T}]^{T}&\Pi_{ 1}=[\mathbf{A}(\lambda)+\mathbf{B}_{2}(\lambda)\xi_{0}(\mathbf{I}+\mathbf{L}) \mathbf{k}(\lambda)]^{T}\mathbf{P}(\lambda)+\mathbf{P}(\lambda)[\mathbf{A}( \lambda)+\mathbf{B}_{2}(\lambda)\xi_{0}(\mathbf{I}+\mathbf{L})\mathbf{k}( \lambda)]+\\ &[\mathbf{C}_{1}(\lambda)\mathbf{x}(t)\!+\!\mathbf{D}_{1}(\lambda)\xi_{0}( \mathbf{I}+\mathbf{L})\mathbf{k}(\lambda)]^{T}[\mathbf{C}_{1}(\lambda)\mathbf{ x}(t)\!+\!\mathbf{D}_{1}(\lambda)\xi_{0}(\mathbf{I}+\mathbf{L})\mathbf{k}( \lambda)]\end{array} \]  
To achieve \(\mathrm{H}(\mathbf{x},\lambda,w,t)<0\) for all \(\Gamma(t)\
eq 0\), the following inequality should hold  
\[\left[ \begin{array}{cc}\Pi_{1}&\mathbf{P}(\lambda)\mathbf{B}_{1}(\lambda)\\ \mathbf{B}_{1}{}^{T}(\lambda)\mathbf{P}(\lambda)&-\gamma^{2}\mathbf{I}\end{array}  \right]<0 \tag{37}\]  
By using the Schur complement, inequality (37) can be converted into inequality (38). \[