## 2 Nonlinear Quarter-Vehicle Model  
The schematic in Figure 1 illustrates the quarter-vehicle dynamics, featuring a nonlinear suspension spring, where \(z_{2}\) and \(z_{1}\) represent the positions of the unsprung and sprung masses, respectively. The selection of the quarter-car model within this study is motivated by the clear focus on analyzing heave motion of the sprung mass. This model comprises four main components: a quarter-car body, a wheel, a suspension spring, and a damper connecting the body to the wheel. [PERSON] et al. (2018) conducted a comprehensive investigation into different types of nonlinearities present in suspension systems. Through their experiments, they illustrated that the cubic nonlinearity in the suspension spring is an essential characteristic for the accurate modeling of suspension systems. Following ([PERSON], 2018; [PERSON], 2018), the suspension spring used in this work has cubic nonlinear characteristics, where the restoring force is composed of two distinctive components: a linear component denoted as \(k_{1}^{L}(z_{1}-z_{2})\) and a nonlinear one described by \(k_{1}^{NL}(z_{1}-z_{2})^{3}\).  
\[m_{1}\ddot{z}_{1}+c(\dot{z}_{1}-\dot{z}_{2})+k_{1}^{L}(z_{1}-z_{2})+k_{1}^{NL}(z _{1}-z_{2})^{3}=u \tag{1}\]  
\[m_{2}\ddot{z}_{2}+c(\dot{z}_{2}-\dot{z}_{1})+k_{1}^{L}(z_{2}-z_{1})-k_{1}^{NL}( z_{1}-z_{2})^{3}+k_{2}(z_{2}-r_{d})=-u \tag{2}\]  
Here, \(m_{1}\) and \(m_{2}\) represent the unsprung and sprung masses, respectively, while \(k_{1}^{L}\) and \(k_{1}^{NL}\) correspond to the linear and nonlinear spring stiffnesses. Additionally, \(c\) stands for the damping coefficient of the viscous damper installed between the sprung and unsprung masses. In the context of this study, a linear spring emulates the tire dynamics with a stiffness coefficient denoted as \(k_{2}\). The term \(u(t)\) signifies the dynamic force input applied to the suspension system by the actuator, serving the purpose of compensating for the road disturbance (\(r_{d}\)). The nonlinear quarter car model in Equations (1) and (2) can be written as follows:  
\[\dot{x}=Ax+N(x)+Bu+Gr_{d} \tag{3}\]  
where \(x\) is the state vector, which is defined as \(x=[z_{1}\quad z_{1}\quad z_{2}\quad z_{2}]^{T}\). The terms \(Ax\) and \(N(x)\) are the linear and nonlinear dynamics, respectively. The linear state space matrices are defined as follows:  
Figure 1: Nonlinear quarter-car model.  
\[A=\left[\begin{array}{cccc}0&1&0&0\\ \frac{-k_{1}^{L}}{m_{1}}&\frac{-c}{m_{1}}&\frac{k_{1}^{L}}{m_{1}}&\frac{c}{m_{1}} \\ 0&0&0&1\\ \frac{k_{1}^{L}}{m_{2}}&\frac{c}{m_{2}}&-\frac{k_{1}^{L}-k_{2}}{m_{2}}&\frac{-c }{m_{2}}\end{array}\right],B=\left[\begin{array}{c}0\\ \frac{1}{m_{1}}\\ 0\\ \frac{-1}{m_{2}}\end{array}\right],G=\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{k_{2}}{m_{2}}\end{array}\right]\]  
whereas the nonlinear part is defined as:  
\[N(x)=\left[\begin{array}{c}0\\ \frac{-k_{1}^{NL}(z_{1}-z_{2})^{3}}{m_{1}}\\ 0\\ \frac{k_{1}^{NL}(z_{1}-z_{2})^{3}}{m_{2}}\end{array}\right]\]  
The formulation in Equation (3) offers notable benefits for our subsequent analysis. Specifically, it lends itself readily to linearization by simply setting \(N(x)=0\). Additionally, setting \(r_{d}=0\) transforms the original system into an undisturbed system. These dynamic systems, as described, will be instrumental in the forthcoming controller design section.