## 2 Material and Methods
### Dynamic Model of Hydroneumatic Suspension
#### 2.1.3 Linearization of a Seven-DOF Model  
The controller design of this type of system is complicated and usually requires many calculations, so the seven-degree-of-freedom model should be decoupled to a single-input single-output system first. Simplifying the seven-degree-of-freedom model to a three-degree-of-freedom model results in low accuracy loss, as ESO-MPC combines the advantages of ESO and MPC. ESO can estimate and compensate for system uncertainty and external disturbances, whereas MPC improves performance by optimizing future control inputs. This combination has unique advantages in addressing system nonlinearity and uncertainty, especially in complex dynamic systems. The traditional seven-DOF equation is transformed into a three-DOF linear system of the car body. The mathematical description of this type of system is as follows:  
\[\left\{ \begin{aligned} \dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f(x,t)+\Delta f(x,t)+bu+d(t)\end{aligned} \right. \tag{8}\]\(x_{1}\) and \(x_{2}\) are based on the system state variables, \(\Delta f(x,t)\) represents the nonlinear part of the system, \(|\Delta f(x,t)|\leq F(x,t)\), \(d\left(t\right)\) represents the disturbance, and \(|d(t)|\leq D\). Considering that the optimization objective of the controller is the movement of the sprung mass, namely, the vertical \(z_{s}\), roll \(\varphi\), and pitch \(\theta\) movements of the body, the vehicle model with seven DOFs is simplified into a vehicle model with three DOFs for the purpose of research [17]. The three degrees of freedom of the body are vertical uncertain dynamics, pitch uncertain dynamics, and roll uncertain dynamics. The vibration equation is composed of a linear part, a nonlinear part, and an uncertain part. The differential equation is expressed in detail as follows:  
\[\left\{ \begin{aligned} \ddot{z}_{s}&=-\frac{2}{m_{se}} \Big{(}k_{sf}z_{s}+c_{sf}\dot{e}\dot{z}_{s}\Big{)}-\frac{2}{m_{se}}(k_{sre}z_{ s}+c_{sr}\dot{z}_{s})+\frac{1}{m_{se}}\sum\limits_{i=1}^{4}f_{i}+\frac{1}{m_{se}} \delta_{z}\\ \ddot{\theta}&=-\frac{2a^{2}}{J_{xe}}\Big{(}k_{sf} \dot{e}\theta+c_{sf}\dot{e}\dot{\theta}\Big{)}-\frac{2b^{2}}{J_{xe}}\Big{(}k_ {sre}\theta+c_{sr}\dot{\theta}\Big{)}-\frac{a}{J_{xe}}(f_{1}+f_{2})+\frac{b}{ J_{xe}}(f_{3}+f_{4})+\frac{1}{J_{xe}}\delta_{\theta}\\ \ddot{\varphi}&=-\frac{2t^{f}}{J_{ye}}\Big{(}k_{sf} \dot{e}\varphi+c_{sf}\dot{e}\dot{\varphi}\Big{)}-\frac{2t^{\prime}}{J_{ye}}(k _{sre}\varphi+c_{sr}\dot{\varphi})+\frac{t_{f}}{J_{ye}}(f_{1}+f_{3})-\frac{t_{ r}}{J_{ye}}(f_{2}+f_{4})+\frac{1}{J_{ye}}\delta_{\varphi}\\ \end{aligned} \right. \tag{9}\]  
where \(m_{se}\), \(J_{xe}\), \(J_{ye}\), \(k_{sf}\), \(c_{sf}\), \(k_{sre}\), and \(c_{sr}\) are the modified parameters of the disturbed part after the transformation of the seven-DOF model. To satisfy \(m_{se}=m_{s}+\Delta m_{s}\), \(J_{xe}=J_{x}+\Delta J_{x}\), \(J_{ye}=J_{y}+\Delta J_{y}\), \(k_{sf}=k_{sf}+\Delta k_{sf}\), \(c_{sf}=c_{sf}+\Delta c_{sf}\), \(k_{sre}=k_{sr}+\Delta k_{sr}\), and \(c_{sre}=c_{sr}+\Delta c_{sr}\). Here, \(g_{z}\), \(g_{\theta}\), and \(g_{\varphi}\) are the nonlinear parts provided by the unsprung mass. The detailed expression of the nonlinear part is shown as follows:  
\[\left\{ \begin{aligned} g_{z}&=\sum\limits_{i=1}^{4}(k _{si}z_{ui}+c_{si}\dot{z}_{ui})+2a\Big{(}k_{sf}\dot{e}\theta+c_{sf}\dot{e} \dot{\theta}\Big{)}-2b\Big{(}k_{sre}\theta+c_{sr}\dot{\theta}\Big{)}\\ g_{\theta}&=-a\sum\limits_{i=1}^{2}(k_{si}z_{ui}+c_{si} \dot{z}_{ui})+b\sum\limits_{i=3}^{2}(k_{si}z_{ui}+c_{si}\dot{z}_{ui})+2a\Big{(} k_{sf}z_{s}+c_{sf}\dot{z}_{s}\Big{)}-2b(k_{sre}z_{s}+c_{sr}\dot{z}_{s})\\ g_{\varphi}&=t_{f}\sum\limits_{i=1,3}(k_{si}z_{ui}+c_{si} \dot{z}_{ui})-t_{r}\sum\limits_{i=2,4}(k_{si}z_{ui}+c_{si}\dot{z}_{ui})\\ \end{aligned} \right. \tag{10}\]  
Furthermore, \(d_{z}\), \(d_{\theta}\), and \(d_{\varphi}\) are defined as the comprehensive perturbations under the three modes.