# An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System
## 4 Simulation and Discussion
### Random Road Response  
The random road excitation was adopted as the first road disturbance input to validate the designed fault-tolerant controller, which was assumed as a vibration signal that is consistent and typically specified as a white noise process given by [34]:  
\[\dot{q}(t)=-2\pi f_{0}q(t)+2\pi n_{0}\omega(t)\sqrt{C_{q}(n_{0})v} \tag{25}\]  
where \(f_{0}\) is the lower cut-off frequency of road profile, \(n_{0}\) is the reference spatial frequency, \(G_{q}(n_{0})\) is the road roughness coefficient, \(\omega(t)\) is zero mean white Gaussian noise with identity power spectral density. In this case, we choose \(G_{q}(n_{0})=256\ \times 10^{-6}\ (\text{m}^{3})\) as C-class road, and \(v=20\ (\text{km}/\text{h})\). It is assumed that there was an actuator ramp fault at the front and rear suspension, i.e., the system faults occurred at \(t_{f}=1\ \text{s}\) with a slope of \(\beta_{f}=2\), and \(t_{f}=3\ \text{s}\) with a slope of \(\beta_{f}=-2\), successively. Figure 4a shows the time evolution of the fault with its estimate value, whereas Figure 4b reveals the fault error. It was observed that the amplitude of the fault error remained no more than \(\pm 0.2\). Additionally, the root-mean-square (RMS) values of the fault and fault estimation were \(1.4535\) and \(1.4756\), respectively, and the fault estimated error was only \(1.52\%\) for the fault signal. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline \(m_{\text{s}}\) & 600 & kg \\ \(m_{\text{uf}}\ (m_{\text{ur}})\) & 36 & kg \\ \(l_{y}\) & 3000 & kg\(\cdot\text{m}^{2}\) \\ \(k_{\text{f}}\ (k_{\text{r}})\) & 16,000 & N\(\cdot\text{m}^{-1}\) \\ \(k_{\text{ff}}\ (k_{\text{tr}})\) & 160,000 & N\(\cdot\text{m}^{-1 The comparisons of \(\ddot{z}_{s}\), \(\ddot{\phi}\), \(\Delta y_{t}\), \(\Delta y_{r}\), and \(F_{tt}\), \(F_{tt}\) for an active suspension system with different control schemes in a time-domain are presented in Figure 5, wherein the conventional \(H_{\infty}\) weighted output feedback controller was denoted as Controller I, the proposed active fault-tolerant controller was denoted as Controller II. It is seen from the black solid line in Figure 5 that the performance responses of Controller II and Controller I were almost the same under fault-free condition, which showed that the designed fault compensator controller was only activated when fault occurred. Besides, it can also be obtained from Figure 5 that with the proposed Controller II, the performance indicators of \(\ddot{z}_{s}\), \(\ddot{\phi}\), \(\Delta y_{t}\), \(\Delta y_{r}\), and \(F_{tt}\), \(F_{tt}\) for the active suspension system could stay stable within a certain range, which demonstrated the robustness of the proposed controller. By further observing Figure 5a,b, we can arrive at the conclusion that the vertical and pitch angular accelerations of the vehicle body with Controller I could obviously be deteriorated, however, the vertical accelerations of the active suspension system with Controller II could well suppress and eliminate the degradation phenomena caused by the faults, which improved the ride quality of the vehicle suspension system. Moreover, it was clear from Figure 5c,d that the dynamic suspension displacements of the front and rear suspension could be reduced using Controller II, thus preventing suspension breakdown when the suspension deflection was out of range, and further ensuring that vehicle suspension had a better ride comfort and handling stability. Simultaneously, from Figure 5e,f,  
Figure 5: Vehicle performance responses under random road (**a**) vertical acceleration (**b**) pitch angular acceleration (**c**) front suspension deflection (**d**) rear suspension deflection (**e**) front tire dynamic load (**f**) Rear tire dynamic load. the tire dynamic loads of the front and rear suspension yielded a smaller variation by using Controller II in the presence of the faults, which improved the tire lifespan and vehicle handling stability.