## 5 Results and Discussion  
In this section, the proposed methodology is considered for a quarter-vehicle active suspension system and illustrated in Figure 1[31,32]. The quarter-vehicle suspension model parameters used in this study are displayed in Table 1.  
The mathematical model of the quarter-vehicle active suspension system is a nonlinear system described by the following nonlinear differential equations [27]:  
\[ \begin{split}& m_{s}\ddot{z}_{s}(t)+c_{s}\big{(}\dot{z}_{s}(t)- \dot{z}_{us}(t)\big{)}+k_{s}(z_{s}(t)-z_{us}(t))+N_{s}(z_{s}(t)-z_{us}(t))^{3}=-u (t)\\ &\quad m_{us}\ddot{z}_{us}(t)+c_{s}\big{(}\dot{z}_{us}(t)-\dot{z}_ {s}(t)\big{)}+k_{s}(z_{us}(t)-z_{s}(t))+N_{s}(z_{us}(t)-z_{s}(t))^{3}\\ &\quad\quad\quad\quad\quad\quad\quad+c_{us}\big{(}\dot{z}_{us}(t )-\dot{z}_{0}(t)\big{)}+k_{us}(z_{us}(t)-z_{0}(t))=u(t)\end{split}  \tag{37}\]  
where \(z_{us}(t)\) and \(z_{s}(t)\) represent the vertical displacements of the unsprung mass and the vertical displacements of the sprung mass, respectively. The vector \(z_{o}(t)\) represents the road displacements. The active force (control signal) of the suspension system is given by \(u(t)\). The sprung mass, which is also known as the car chassis, is given by \(m_{s}\). The mass of the wheel assembly, also known as the unsprung mass, is given by \(m_{us}\). The parameters \(k_{s}\) and \(c_{s}\) are the spring coefficient and the damping coefficient of the suspension system, respectively. In addition, the parameters \(k_{us}\) and \(c_{us}\) are the compressibility and the damping of the pneumatic tire, respectively. Finally, \(N_{s}\) is the nonlinear stiffness property of the primary suspension. In this study, more attention was paid to the vertical behavior of the tire, which is useful in the design and the analysis of the comfort of the passengers and the road handling of the vehicle. The tire can be modeled by a simple spring \(k_{us}\) that is in parallel with a damper \(c_{us}\). In addition, the stiffness of the spring is considered constant while its damping ratio is negligible. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Description** & **Numerical Value** \\ \hline \(m_{s}\) & The mass of the quarter body & 972 (g) \\ \(m_{us}\) & Unsprung mass & 113.6 (kg) \\ \(k_{s}\) & Suspension stiffness coefficient & 42,719.6 (N/m) \\ \(k_{us}\) & Type stiffness coefficient & 101,115 (N/m) \\ \(c_{s}\) & The damping ratio of the damper & 1095 (N*s/m) \\ \(c_{us}\) & The damping ratio of the pneumatic tire & 14.6 (N*s/m) \\ \(N_{s}\) & The passive suspension stiffness & 4271.96 (10\% of \(k_{s}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the quarter-vehicle suspension system. Figure 1: Nonlinear active quarter-car suspension model. The nonlinear system described in Equation (32) can be modified to have the following state-space representation:  
\[\left\{ \begin{array}{c}x(k+1)=A(\theta_{k})x(k)+B(\theta_{k})u(k)+D(\theta_{k}) \omega(k)\\ y_{f}(k)=C(\theta_{k})x(k)+f_{s}(k)\end{array} \right. \tag{38}\]  
Note that the tire deflection is the state variable \(x_{1}(t)=z_{us}(t)-z_{0}(t)\). The rest of the state variables are the unsprung mass velocity \(x_{2}(t)=\dot{z}_{us}(t)\), the suspension stroke \(x_{3}(t)=z_{s}(t)-z_{us}(t)\), and the sprung mass velocity \(x_{4}(t)=\dot{z}_{s}(t)\). The disturbance input that affects the system is represented as \(\omega(t)=z_{0}(t)\). Note that the system matrix A can be expressed in polytopic form as:  
\[A(\theta_{k})=\sum\
olimits_{r=1}^{2}\rho_{r}(\theta_{k})A_{r} \tag{39}\]  
where \(\rho_{r}(\theta_{k})\) are the vertex coefficients with a square signal \(\alpha\in[\alpha_{1}\ \alpha_{2}]\) and \(\rho_{1}=(\alpha_{2}-\alpha)/(\alpha_{2}-\alpha_{1})\), \(\rho_{2}=(\alpha-\alpha_{1})/(\alpha_{2}-\alpha_{1})\). The values of the vertex matrices \(A(\theta_{k})\) are \(A_{1}=\left.A\right|_{\alpha=\alpha_{1}}\) and \(A_{2}=\left.A\right|_{\alpha=\alpha_{2}}\).