# An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System
## 5 Conclusions  
In this paper, an observer-based active fault-tolerant controller was proposed for a half-vehicle active suspension system subjected to some generalized stochastic faults. By constructing the augmented fault suspension model, a robust \(H_{\infty}\) weighted output feedback controller was developed to improve the vehicle dynamics performance under a fault-free condition. Then, an \(H_{\infty}\) observer was designed to accurately estimate the fault information with an auxiliary diagnosis system and further to design an \(H_{\infty}\) fault compensation controller to eliminate the effect of the faults on vehicle system performances. Finally, a numerical simulation investigation was provided to verify the effectiveness of the proposed controller under random and bump road surface compared with the classical \(H_{\infty}\) output feedback controller. The results showed that under a fault-free condition, both of the proposed controller and \(H_{\infty}\) output feedback controller could stabilize a vehicle active suspension system with better dynamic performances within a finite time; while under fault condition, compared with the \(H_{\infty}\) output feedback controller, our designed active fault-tolerant controller could accurately estimate the fault information and compensate for the system performance penalties to make the fault suspension system approximate to the performance indicators of the active suspension without faults, which could \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **Controller I/II without Fault** & **Controller I with Fault** & **Controller II with Fault** \\ \hline \(\tilde{z}_{\text{s}}/(\text{m}/\text{s}^{2})\) & 0.8770 & 1.5053 & 0.8772 \\ \(\tilde{\varphi}/(\text{m}^{2}/\text{s}^{3})\) & 0.4141 & 0.4290 & 0.4145 \\ \(\Delta y_{\text{tf}}/\text{m}\) & 0.0196 & 0.0277 & 0.0196 \\ \(\Delta y_{\text{r}}/\text{m}\) & 0.0190 & 0.0276 & 0.0191 \\ \(F_{\text{tf}}/\text{N}\) & 411.8726 & 470.1569 & 413.0139 \\ \(F_{\text{tr}}/\text{N}\) & 439.6549 & 486.6650 & 439.0177 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparisons of root mean square (RMS) values for vehicle suspension performance under bump road. Figure 9: The control forces of the proposed controller at (**a**) the front wheel and (**b**) the rear wheel. ensure the reliability of the vehicle active suspension system with the stochastic faults. Future work will focus on the robustness of the control strategy and the experiment for the auxiliary system. [PERSON], [PERSON] conceived and designed the controller. [PERSON] performed the simulation. [PERSON], [PERSON], and [PERSON] wrote this paper. This research was supported by National Natural Science Foundation of China (Grant No. 51675423), National Natural Science Foundation of China (Grant No. 51305342), and Primary Research and Development Plan of Shaanxi Province (Grant No. 2017 GY-029). The authors declare no conflicts of interest. The coefficient matrices of \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{B}_{d}\), \(\mathbf{C}_{1}\), \(\mathbf{C}_{2}\), and \(\mathbf{D}_{2}\) for Equation (3) are expressed as follows:  
\[\mathbf{A}=\left[