## 3 Reliable Robust \(H_{\infty}\) Controller Design
\begin{array}{ll}\left[\begin{array}{cc}\Lambda_{11}&\mathbf{P}(\lambda) \mathbf{B}_{1}(\lambda)&\Lambda_{13}\\ *&-\gamma^{2}\mathbf{I}&0\\ *&*&-\mathbf{I}\end{array}\right]+\left[\begin{array}{cc}\mathbf{P}(\lambda )\mathbf{B}_{2}(\lambda)\\ 0\\ \mathbf{D}_{1}(\lambda)\end{array}\right]\mathbf{L}\Big{[}\begin{array}{cc} \xi_{0}\mathbf{k}(\lambda)&0&0\end{array}\Big{]}\\ &+\left(\left[\begin{array}{cc}\mathbf{P}(\lambda)\mathbf{B}_{2}(\lambda) \\ 0\\ \mathbf{D}_{1}(\lambda)\end{array}\right]\mathbf{L}\Big{[}\begin{array}{cc} \xi_{0}\mathbf{k}(\lambda)&0&0\end{array}\Big{]}\right)^{T}<0\end{array}  \tag{38}\]  
where  
\[\Lambda_{11}\!=\!\mathrm{sym}(\mathbf{P}(\lambda)\mathbf{A}(\lambda)+\mathbf{ P}(\lambda)\mathbf{B}_{2}(\lambda)\xi_{0}\mathbf{k}(\lambda)),\;\Lambda_{13}=[ \mathbf{C}_{1}(\lambda)\!+\!\mathbf{D}_{1}(\lambda)\xi_{0}\mathbf{k}(\lambda) ]^{T}.\]  
Based on Lemma 1, inequality (38) can be converted into inequality (39). \[\left[ \begin{array}{cc}\Psi_{11}&\mathbf{P}(\lambda)\mathbf{B}_{1}(\lambda )&\Psi_{13}\\ *&-\gamma^{2}\mathbf{I}&0\\ *&*&-\mathbf{I}+\eta\mathbf{D}_{1}(\lambda)\mathbf{J}\mathbf{D}_{1}^{T}( \lambda)\end{array} \right]<0 \tag{39}\]  
where  
\[\Psi_{11}=\Lambda_{11}+\eta\mathbf{P}(\lambda)\mathbf{B}_{2}(\lambda)\mathbf{ B}_{2}^{T}(\lambda)\mathbf{P}(\lambda)+\mathbf{k}^{T}(\lambda)\xi_{0}^{T} \eta^{-1}\mathbf{J}\xi_{0}\mathbf{k}(\lambda),\;\Psi_{13}=\Lambda_{13}+\eta \mathbf{P}(\lambda)\mathbf{B}_{2}\mathbf{J}\mathbf{D}_{1}^{T}(\lambda).\]  
By the Schur complement, we get  
\[\left[ \begin{array}{cccc}\Lambda_{11}&\mathbf{P}(\lambda)\mathbf{B}_{1}( \lambda)&\Lambda_{13}&\eta\mathbf{P}(\lambda)\mathbf{B}_{2}&\mathbf{k}^{T}( \lambda)\xi_{0}^{T}\\ *&-\gamma^{2}\mathbf{I}&0&0&0\\ *&*&-\mathbf{I}&\eta\mathbf{D}_{1}(\lambda)&0\\ *&*&*&-\eta\mathbf{J}^{-1}&0\\ *&*&*&*&-\eta\mathbf{J}^{-1}\end{array} \right]<0 \tag{40}\]  
Pre- and post-multiplying \(\mathrm{diag}\left\{\mathrm{P}(\lambda)^{-1}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I }\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\, \mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\, \mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I }\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{I}\,\mathrm{Iwhere \(\theta_{max}(\cdot)\) represents the maximal eigenvalue. Using the Schur complement and Lemma 1, inequality (41) can be written as  
\[\left[ \begin{array}{ccc}-\mathbf{I}+\varepsilon\mathbf{J}&\sqrt{\rho}\xi_{0} \mathbf{k}(\lambda)&0\\ *&-u_{\max}^{2}\mathbf{P}(\lambda)&\sqrt{\rho}\mathbf{k}^{T}(\lambda)\xi_{0}^{T }\\ *&*&\varepsilon\mathbf{J}^{-1}\end{array} \right]<0 \tag{43}\]  
Pre- and post-multiplying inequality (43) by \(\mathrm{diag}\left\{\mathrm{I}\;\mathrm{P}(\lambda)^{-1}\;\mathrm{I}\;\right\}\), defining \(\overline{\mathbf{P}}(\lambda)\mathbf{=P(\lambda)}^{-1}\) and \(\overline{\mathbf{k}}(\lambda)=\mathbf{k}(\lambda)\mathrm{P}(\lambda)^{-1}\), the inequality is equivalent to the second LMI in inequality (33). Furthermore, based on the Schur complement, the inequality matrix in inequality (42) can be change to  
\[\left[ \begin{array}{cc}-\mathbf{I}&\sqrt{\rho}\mathbf{C}_{2}(\lambda)\\ *&-2_{\max}^{2}\mathbf{P}(\lambda)\end{array} \right]<0 \tag{44}\]  
Pre- and post-multiplying inequality (44) by \(\mathrm{diag}\left\{\mathrm{I}\;\mathrm{P}(\lambda)^{-1}\right\}\), using \(\overline{\mathbf{P}}(\lambda)\) to replace \(\mathrm{P}(\lambda)^{-1}\), the LMI inequality (44) is equal to the third LMI in inequality (34). Thus, the proof is completed. According to the inner property of the polytopic tire stiffness uncertainties and sprung mass uncertainties, Equations (32)-(34) in Theorem 1 are equivalent to the inequalities as follows:  
\[\left[