## 4 Conclusions  
To address the problems of the weak adaptability of passive X-type interconnected hydropneumatic suspensions to different road surfaces and the poor effect of the traditional single-control mode, an active controller of hydropneumatic suspensions based on ESO and MPC was designed. On the basis of the analysis of X-type interconnected hydropneumatic suspensions and the seven-degree-of-freedom suspension model, a three-degree-of-freedom linearized hydropneumatic suspension model with disturbance was established. The disturbance was analyzed, the disturbed ESO extended state observer was established, and, utilizing the linearized state space model as its foundation, the controller for ESO-MPC was constructed from MPC, thereby notably enhancing the performance of the heavy-duty truck's suspension system.  
In the C-class random road simulation verification, compared to passive suspension, active suspension using the controller for ESO-MPC reduced the vertical acceleration, pitch acceleration, and roll acceleration by 18.7%, 24.7%, and 26.1%, respectively. After fuzzy PID control, the vertical acceleration, pitch acceleration, and roll acceleration were reduced by 5.59%, 7.99%, and 15.54%, respectively. Simulation verification on a convex road surface showed that active suspension using the controller for ESO-MPC reduced the vertical acceleration of the vehicle by 36.5%, the pitch acceleration by 21.2%, and the roll acceleration by 18.1% compared to passive suspension. After adopting fuzzy PID control, the vertical acceleration, pitch acceleration, and roll acceleration were reduced by 14.04%, 10.6%, and 7.92%, respectively. The results indicated that the control effect of ESO-MPC was superior to that of fuzzy PID.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Index** & **Passivity** & **Fuzzy PID** & **Reduction** & **ESO-MPC** & **Reduction** \\ \hline Vertical acceleration (m/s\({}^{2}\)) & 0.23744 & 0.20410 & 14.041\% & 0.15076 & 36.507\% \\ Pitch-angle acceleration (rad/s\({}^{2}\)) & 0.029651 & 0.026508 & 10.599\% & 0.023365 & 21.199\% \\ Roll-angle acceleration (rad/s\({}^{2}\)) & 0.0044827 & 0.0041275 & 7.923\% & 0.0036723 & 18.079\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Convex pavement index root mean square statistics.  
The ESO-MPC algorithm has better active control on hydropneumatic suspensions, especially on random road surfaces, where the control effect is more obvious than with fuzzy PID, achieving the precise control of X-type interconnected hydropneumatic suspension systems and improving the smoothness and stability of heavy-duty special vehicles.  
G.L.: supervision and writing--review and editing. Y.Y.: writing--original draft, software, and validation. Y.L.: conceptualization, project administration, and writing--review and editing. S.W.: supervision, data curation, validation, and funding acquisition. All authors have read and agreed to the published version of the manuscript.  
The research was supported by the Henan Province Science and Technology Research Project (242102220026).  
Not applicable.  
The detailed data are available upon request. The data are not publicly available due to privacy.  
The authors declare no conflicts of interest.