# An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System
## 4 Simulation and Discussion
### Bump Road Response  
A pronounced bump on a smooth road surface was also adopted as the second road disturbance input to further demonstrate the effectiveness of the proposed controller, which is expressed by:  
\[x_{r}(t)=\left\{ \begin{array}{ll}\frac{A_{m}}{2}(1-\cos\frac{2\pi u}{L}t),& 0\leq t\leq\frac{L}{u}\\ 0,&t>\frac{L}{u}\end{array} \right. \tag{26}\]  
where \(A_{m}\), \(L\) and \(u\) denote the height and length of bump road, and vehicle forward speed, respectively, and their corresponding values were extracted from literature [27] as \(A_{m}=100\) (mm), \(L=5\) m and \(u=45\) (km/h). Meanwhile, in order to verify the effectiveness and applicability of the fault compensation controller, the random fault model of (4) was reconfigured as that, the system faults occurred at \(t_{f}=0.3\) s with a slope of \(\beta_{f}=5\), and \(t_{f}=3\) s with a slope of \(\beta_{f}=-5\), continuously. The fault input signal and its estimation are displayed in Figure 7a, and Figure 7b shows the corresponding fault error with a peak error value of \(\pm 0.6\). Moreover, the RMS values of the fault and fault estimation were 5.7276 and 5.7829, the fault estimated error was only 0.97% of the fault signal, all of which implied that the designed compensation controller could accurately estimate the fault. Figure 8 reveals the comparisons of time-domain responses of \(\tilde{z}_{s}\), \(\tilde{\phi}\), \(\Delta y_{t}\), \(\Delta y_{r}\), and \(F_{\text{tf}}\), \(F_{\text{tr}}\) for the active suspension system with different control schemes. It was concluded from Figures 7 and 8  
Figure 6: The control forces of the proposed controller: (**a**) Front wheel control force and (**b**) rear wheel control force. Figure 7: Fault estimation and fault estimation error: (**a**) Fault estimation and (**b**) fault error. that under the fault-free condition, the control performances of the active suspension system with Controller II and Controller I were identical since the fault compensation controller did not work when there was no fault in the system (15). Moreover, it could also be obtained that Controller II could ensure better dynamic responses as opposed to Controller I in the presence of system stochastic faults. Besides, with the Controller II, the performance responses of the active suspension system could reach an asymptotic stability within a short time (about 3 to 5 s), which showed that Controller II had a good robustness to the actuator faults and external disturbances. Additionally, the variation of control forces for the designed active fault-tolerant controller is shown in Figure 9, and it is seen that the control force will be tuned to compensate for the performance penalties of the active suspension system imposed by the actuator random ramp fault. Figure 8: 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. It is well known that the RMS value of the acceleration is strictly related to the ride comfort of passengers and it is often used to quantify the amount of the acceleration transmitted to the vehicle body [35]. Additionally, the RMS value of suspension deflection and tire dynamic loads are usually used to verify the performance of the controller [36]. The following equation is used to calculate the RMS value of an \(n\)-dimensional vector \(x\) as \(x_{\text{RMS}}=\dfrac{||x||}{\sqrt{n}}=\sqrt{\dfrac{1}{n}\sum\limits_{i=1}^{n}x _{i}^{2}}\ (i=1,2\cdots n)\). Table 2 shows the RMS values of \(z_{\text{s}}\), \(\varphi\), \(\Delta y_{\text{r}}\), \(\Delta y_{\text{r}}\), and \(F_{\text{tf}}\), \(F_{\text{tr}}\) for the active suspension system with the different fault-tolerant control scheme.