## 2 Problem Formulation
### Semi-Active Suspension Model  
A quarter-car suspension model, as depicted in Figure 3, is considered for this work, wherein the vehicle body is assumed to be rigid and to possess freedom of movement in the vertical direction. Figure 2: A typical curve of the proposed model (**a**) Suspension deflection vs. Damper force, (**b**) Suspension deflection rate vs. Damper force. The governing equations of motion for the quarter-car model, which account for the semi-active suspension behavior resulting from the MR damper (1), are as follows:  
\[ \begin{split}& m_{s}z_{s}(t)+k_{s}\big{(}z_{s}(t)-z_{u}(t)\big{)}+F_{ md}(t)=0\\ & m_{u}z_{u}(t)+k_{s}\big{(}z_{u}(t)-z_{s}(t)\big{)}-F_{md}(t)\\ &+k_{t}\big{(}z_{u}(t)-z_{r}(t)\big{)}+c_{t}\big{(}\dot{z}_{u}(t)- \dot{z}_{r}(t)\big{)}=0\end{split}  \tag{3}\]  
where \(z_{s}\), \(z_{u}\), and \(z_{r}\) denote the displacements of the sprung mass, unsprung mass, and road profile, respectively. The quarter-car model parameters are presented in Table 1.  
Furthermore, the following performance aspects must be taken into account with regard to the vehicle suspension [40]:  
1. In order to enhance ride comfort, the vertical acceleration \(\dot{z}_{s}\) has to be minimized \[\min\ddot{z}_{s}(t)=-\frac{k_{s}\big{(}z_{s}(t)-z_{u}(t)\big{)}+F_{md}(t)}{m_ {s}}\] (4)
2. The suspension deflection is limited by the mechanical structure and cannot exceed a maximum value \[\big{|}z_{s}(t)-z_{u}(t)\big{|}\leq z_{max}\] (5)
3. In order to guarantee a firm and uninterrupted contact of the wheels on the road, it is necessary that the dynamic tye load be less than the static tye load \[k_{t}\big{(}z_{u}(t)-z_{r}(t)\big{)}<(m_{s}+m_{u})g\] (6) \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Name** & **Value** \\ \hline \(m_{s}\) & Vehicle sprung mass & 243.2 kg \\ \(m_{u}\) & Vehicle unsprung mass & 28.5 kg \\ \(k_{s}\) & Spring stiffness & 10,680 N/m \\ \(k_{t}\) & Tire stiffness & 25,278 N/m \\ \(c_{t}\) & Tire damping & 3.65 Ns/m \\ \(z_{max}\) & Maximum suspension deflection & 0.05 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quarter-car model parameters. Figure 3: Semi-active quarter-car suspension model. Following (1), (2) and (3)-(6), the state-space representation of the quarter-car semi-active suspension model is written as  
\[ \begin{split}\dot{x}(t)&=\mathcal{A}x(t)+\mathcal{B}u (t)+\mathcal{B}_{\omega}\omega(t)\\ y(t)&=C_{y}x(t)\\ z(t)&=C_{z}x(t)+D_{z}u(t)\end{split}  \tag{7}\]  
where \(x\) is the state vector, \(u\) is the control input, and \(\omega\) is the road disturbance. \(y\) represents the measured output from the suspension system; in this case, it is assumed that the suspension deflection and its rate can be measured by a linear variable differential transformer (LVDT) [41]. The LVDT serves to convert a linear displacement into a proportional electrical signal. LVDT sensors can be situated in close proximity to the suspension shock-absorbers, on the same screws as the dampers. They are capable of measuring the relative displacement of the suspension system. Due to the high precision of the LVDT signals, they can be differentiated for the estimation of the relative velocity of the suspension elements. The performance vector \(z\) is a quantitative indicator utilized to assess whether the suspension deflection, vertical acceleration, and dynamic tire load remain within an acceptable range. \[ \begin{split} x(t)&=\begin{bmatrix}z_{s}(t)-z_{u}(t )\\ z_{u}(t)-z_{r}(t)\\ z_{s}(t)\\ z_{u}(t)\end{bmatrix},\;y(t)=\begin{bmatrix}z_{s}(t)-z_{u}(t)\\ z_{s}(t)-z_{u}(t)\end{bmatrix},\;\omega(t)=\dot{z}_{r}(t),\\ u(t)&=(a_{0}+a_{1}i(t))\text{tanh}\bigg{(}a_{2}\bigg{(}\dot{z}_{ def}(t)+\frac{V_{0}}{X_{0}}z_{def}(t)\bigg{)}\bigg{)},\\ z_{def}(t)&=z_{s}(t)-z_{u}(t)\end{split}  \tag{8}\]  
The state space matrices of system (7) are  
\[