## 3 Reliable Robust \(H_{\infty}\) Controller Design
\right]\!\!,\ \ \textbf{D}_{2}=[00]^{T}\!,\]  
where \(\gamma\) and \(\beta\) are weight coefficients. To obtain a better control performance of the suspension, a state-feedback controller as \(u_{d}=k(\lambda)x(t)\) is designed. Then, by considering the actuator fault, a state-feedback controller can be modelled as  
\[u_{actual}(t)=\xi u_{d}=\xi\mathbf{k}(\mathbf{\lambda})\mathbf{x}(t) \tag{27}\]  
where \(\mathbf{k}(\mathbf{\lambda})\) is a gain matrix of the feedback controller that needs to be determined, \(\xi\) represents the possible actuator fault, \(u_{d}\) is the desired force calculated by the controller, and \(u_{actual}\) is the actual force generated by the actuator. Assuming that \(\xi\) is constrained by its maximum \(\xi_{max}\) value and minimum value \(\xi_{min}\), the control law considering actuator faults and parameter uncertainties can be denoted as  
\[u_{actual}(t)=\xi\mathbf{k}(\mathbf{\lambda})\mathbf{x}(t)=\xi_{0}(\mathbf{I} +\mathbf{L})\mathbf{k}(\mathbf{\lambda})\mathbf{x}(t) \tag{28}\]  
where \(\xi_{0}=(\xi_{max}+\xi_{min})/2\) and \(\mathbf{L}=(\xi-\xi_{0})/\xi_{0}\). Defining \(\mathbf{J}=(\xi_{max}-\xi_{min})/(\xi_{max}+\xi_{min})\), then, we have \(\mathbf{L}^{T}\mathbf{L}\leq\mathbf{J}^{T}\mathbf{J}\leq\mathbf{I}\). Therefore, the closed-loop system with state-feedback controller is rewritten as:  
\[\begin{array}{l}\dot{\mathbf{x}}(t)\textbf{=}(\mathbf{A}(\mathbf{\lambda}) +\xi_{0}(\mathbf{I}+\mathbf{L})\textbf{B}_{2}(\mathbf{\lambda})\mathbf{k}( \mathbf{\lambda}))\mathbf{x}(t)\textbf{+}\textbf{B}_{1}(\mathbf{\lambda}) \mathbf{w}(t)\\ \textbf{z}_{1}\textbf{=}\textbf{C}_{1}(\mathbf{\lambda})\mathbf{x}(t)\textbf{ +}\textbf{D}_{1}(\mathbf  
Additionally, the relationship between the uncertain masses (\(m_{b}\) and \(k_{t}\)) and the vector \(a(\mathbf{\lambda})=(a_{1}(\mathbf{\lambda}),a_{2}(\mathbf{\lambda}),a_{3}(\mathbf{\lambda}),a_{4 }(\mathbf{\lambda}))\) is expressed by  
\[ \begin{array}{l}a_{1}(\mathbf{\lambda})=M_{1}(\mathbf{\lambda})\times N_{1}(\mathbf{ \lambda})a_{2}(\mathbf{\lambda})=M_{1}(\mathbf{\lambda})\times N_{2}(\mathbf{\lambda})\\ a_{3}(\mathbf{\lambda})=M_{2}(\mathbf{\lambda})\times N_{1}(\mathbf{\lambda})a_{4}(\mathbf{ \lambda})=M_{2}(\mathbf{\lambda})\times N_{2}(\mathbf{\lambda})\end{array}  \tag{31}\]  
where  
\[ \begin{array}{l}M_{1}(\mathbf{\lambda})=\Big{(}m_{b}^{-1}-m_{bu}^{-1}\Big{)}/ \Big{(}m_{bl}^{-1}-m_{bu}^{-1}\Big{)}M_{2}(\mathbf{\lambda})=\Big{(}m_{bl}^{-1}-m_ {b}^{-1}\Big{)}/\Big{(}m_{bl}^{-1}-m_{bu}^{-1}\Big{)}\\ N_{1}(\mathbf{\lambda})=\Big{(}k_{t}^{-1}-k_{tu}^{-1}\Big{)}/\Big{(}k_{tl}^{-1}-k_{ tu}^{-1}\Big{)}N_{2}(\mathbf{\lambda})=\Big{(}k_{rl}^{-1}-k_{r}^{-1}\Big{)}/ \Big{(}k_{rl}^{-1}-k_{ru}^{-1}\Big{)}\end{array} \]  
**Lemma 1**. : [18] _For matrices \(R\) and \(S\), the inequality \(\mathrm{R}\Sigma S+S^{T}\Sigma^{T}R^{T}\leq\varepsilon\mathrm{R}\mathrm{V}R^{ T}+\varepsilon^{-1}S^{T}\mathrm{V}S\) holds, if \(\varepsilon>0\), \(\Sigma(t)=\text{diag}\Big{(}\sigma(t)_{1},\sigma(t)_{2},\ldots,\sigma(t)_{p} \Big{)}\) is a time varying diagonal matrix, and satisfies \(|\Sigma|\leq\mathrm{V}\)._  
**Theorem 1**. : _For given positive constant \(\rho\), a robust reliable \(H_{\infty}\) controller in the form of Equation (28) exists, such that the closed-loop system in Equation (29) is asymptotically stable and satisfies \(\|\mathbf{z}_{1}(t)\|_{2}<\gamma\|\mathrm{w}(t)\|_{2}\) for all \(w\) satisfying \(\|\mathrm{w}(t)\|_{2}^{2}\leq\|\mathrm{w}(t)\|_{max}=\rho/\gamma^{2}\), while the constrains in inequalities (32-34) are guaranteed, if there exist symmetric matrices \(\overline{\mathrm{P}}(\mathbf{\lambda})\), general matrix \(\overline{\mathrm{k}}(\mathbf{\lambda})\) and any scalar \(\eta>0\), \(\varepsilon>0\) satisfying_  
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