## 2 Problem Formulation
### MR Damper Current Control  
This section presents a static output feedback controller designed to vary the current in the MR damper, enabling the vehicle to adapt to varying road conditions. This work emphasizes the constraints associated with the utilization of sensors in mass-produced vehicles. Given that not all vehicle states can be directly measured, the control signal must be generated in accordance with the available information. Furthermore, given that external disturbances (in this instance, the unknown road profile variation) affect the vehicle, it is essential that the designed controller is robust against such influences, thereby ensuring an adequate response in any scenario [42]. First, in order to facilitate the real-time implementation, the system (7) is transformed into its discrete counterpart by applying [PERSON]'s discretization [43]  
\[\dot{x}(t)\simeq\frac{x(k+1)-x(k)}{T_{d}},\;\forall k\in\mathbb{Z}_{+} \tag{9}\]leading to the discrete state-space model  
\[ \begin{split} x(k+1)&=Ax(k)+Bu(k)+B_{\omega}\omega(k)\\ y(k)&=C_{y}x(k)\\ z(k)&=C_{z}x(k)+D_{z}u(k)\end{split}  \tag{10}\]  
with the matrices  
\[A=I+T_{d}\mathcal{A},\;B=T_{d}\mathcal{B},\;B_{\omega}=T_{d}\mathcal{B}_{\omega} \tag{11}\]  
The control signal \(u\) is generated by  
\[u(k)=FG^{-1}y(k) \tag{12}\]  
where \(F\in\mathbb{R}^{1\times 2}\) and \(G\in\mathbb{R}^{2\times 2}\) are matrices to be designed. At a given instant, the MR damper current \(i\) is obtained by solving (8), for known \(z_{def}\), \(\dot{z}_{def}\) and control signal \(u\)  
\[i(k)=\frac{u(k)}{a_{1}\text{tanh}\left(a_{2}\left(\dot{z}_{def}(k)+\frac{V_{0}} {X_{0}}z_{def}(k)\right)\right)}-\frac{a_{0}}{a_{1}} \tag{13}\]  
The following theorem presents a set of sufficient conditions for the design of an MR damper current controller for the semi-active suspension (3), taking into account the available sensors on the suspension system. Theorem 1: _The semi-active suspension system (10) is stable and robust with an \(\mathcal{H}_{\infty}\) performance index \(\gamma>0\) if there exists a symmetric matrix \(Q>0\), matrices \(F\) and \(G\) and scalars \(\varepsilon_{1}\) and \(\varepsilon_{2}\), if the following linear matrix inequality (LMI) convex optimization problem returns a feasible solution:_  
\[ \begin{split}\text{min}&\gamma\quad\text{s.t.}\quad( \ref{eq:LMI})\end{split} \]  
_where_  
\[ \begin{split}\left[\begin{array}{ccccc}-Q&\star&\star& \star&\star&\star\\ 0&-\gamma^{2}I&\star&\star&\star&\star\\ AQ+BFC_{y}&B_{\omega}&-Q&\star&\star&\star\\ C_{y}Q-GC_{y}&0&\varepsilon_{1}F^{\top}B^{\top}&-\varepsilon_{1}\text{He}(G )&\star&\star\\ C_{z}Q+D_{z}FC_{y}&0&0&0&-I&\star\\ C_{y}Q-GC_{y}&0&0&0&\varepsilon_{2}F^{\top}D_{z}^{\top}&-\varepsilon_{2} \text{He}(G)\end{array}\right]<0\end{split}  \tag{14}\]  
Proof: First define  
\[ \begin{bmatrix}I&0&0&0&0&0\\ 0&I&0&0&0&0\\ 0&0&I&0&0&0\\ 0&0&0&I&0&0\\ 0&0&0&0&I&D_{z}FG^{-1}\end{bmatrix}  \tag{15}\]  
now perform a congruent transformation by pre and post multiplying the LMI in (14) by (15) and its transpose to return  
\[ \begin{bmatrix}-Q&\star&\star&\star&\star\\ 0&-\gamma^{2}I&\star&\star&\star\\ AQ+BFC_{y}&B_{\omega}&-Q&\star&\star&\star\\ C_{y}Q-GC_{y}&0&\varepsilon_{1}F^{\top}B^{\top}&-\varepsilon_{1}\text{He}(G)& \star\\ C_{z}Q+D_{z}FG^{-1}C_{y}Q&0&0&0&-I\end{bmatrix} <0 \tag{16}\]now perform the Schur complement to (16) to obtain  
\[ \begin{bmatrix}-Q+Q(C_{z}+D_{z}FG^{-1}C_{y})^{\top}(C_{z}+D_{z}FG^{-1}C_{y})Q& \star&\star&\star\\ 0&-\gamma^{2}I&\star&\star\\ AQ+BFC_{y}&&B_{\omega}&-Q&\star\\ C_{y}Q-GC_{y}&&0&\varepsilon_{1}F^{\top}B^{\top}&-\varepsilon_{1}\text{He}(G) \end{bmatrix} <0 \tag{17}\]  
Now define  
\[ \begin{bmatrix}I&0&0&0\\ 0&I&0&0\\ 0&0&I&BFG^{-1}\end{bmatrix}  \tag{18}\]  
perform a congruent transformation by pre and post multiplying the LMI (17) by (18) and its transpose to obtain  
\[ \begin{bmatrix}-Q+Q(C_{z}+D_{z}FG^{-1}C_{y})^{\top}(C_{z}+D_{z}FG^{-1}C_{y})Q& \star&\star\\ 0&-\gamma^{2}I&\star\\ AQ+BFG^{-1}C_{y}Q&B_{\omega}&-Q\end{bmatrix}