## 3 Results
### Simulation Verification
In Figure 7b, the value of fault detection function \(\gamma\) for GPS signal channel by using residuals chi-square detector remains in the order of magnitude \(10^{-7}\), and this value is proportional to the absolute value of the yaw angle error within the time intervals of 3-5 and 8-10 s. However, the order of magnitude of the fault detection function \(\gamma\) for vision and LIDAR sensing systems is \(10^{-8}\). Through the above analysis, we know that, by setting the appropriate threshold, the fault signals can be easily detected when sensor failure occurs. Figure 8a shows the yaw angle changes detected by vision, GPS, and LIDAR sensing systems and Figure 8b shows the errors between the reference yaw angel and the yaw angles obtained vision, GPS, and LIDAR sensing systems. In Figure 8, we can see that the lateral position detected by GPS sensing system has obvious fault due to the sensor failure within the time intervals of 3-5 and 8-10 s. The peak lateral position error of GPS sensing system reaches 3.5 m, which is extremely large considering the maximum lateral position of vehicle in simulation is about 4 m and width of road lane is 4 m. Therefore, the sensor fault of GPS channel cannot be ignored and needs to be detected and isolated. The fault detection results for lateral position in simulation are shown in Figure 9.  
Figure 8: The lateral position changes in simulation: (**a**) lateral position detected by using different sensors; and (**b**) the lateral position error. Figure 9: The changes of fault detection function for lateral positions based on: (**a**) double state chi-square detector; and (**b**) residuals chi-square detector. In Figure 9, we can see that the detection results of the fault signal are obviously different compared to the normal signals. In Figure 9a, the peak value of fault detection function \(\lambda\) for GPS signal channel by using double state chi-square detector reaches \(2.5\times 10^{-3}\), \(1.1\times 10^{-2}\), \(1.2\times 10^{-2}\), and \(4.5\times 10^{-3}\) at 3, 6, 8, and 10 s, which correspond to the moments when the fault signal of GPS sensing system appears and disappears. However, the peak values of fault detection function \(\lambda\) for vision and LIDAR sensing system reach no more than \(2\times 10^{-4}\) and \(5\times 10^{-4}\), respectively. In Figure 9b, the value of fault detection function \(\gamma\) for GPS signal channel by using residuals chi-square detector remains in the order of magnitude of \(10^{-3}\) when sensor failure occurs, and this value is proportional to the absolute value of the lateral position error within the time intervals of 3-5 and 8-10 s. However, the order of magnitude of the fault detection functions \(\gamma\) for vision and LIDAR sensing systems are \(10^{-6}\). Hence, the fault lateral position signal can be easily detected in simulation process by setting appropriate thresholds for the fault detection functions. To confirm the effect of sensor fault to the path tracking control system of autonomous vehicle, the simulation results are compared by using MPC, adaptive MPC (AMPC), fault-tolerant MPC (FTMPC), and fault-tolerant Adaptive MPC FTAMPC algorithms, as shown in Figure 10.  
In Figure 10a,b, we can see that the curves of lateral position and lateral position error are divergent, which means that the vehicle is gradually moving away from the reference path, and the accurately path tracking of the autonomous vehicle cannot be realized using the fault sensor information. In Figure 10c,d, we can see that the curves of yaw angle and yaw angle error are irregular and the changes are very dramatic, which also means that the autonomous vehicle cannot track the reference path, and the risk of overturning exists without fault detection and isolation. In addition, because the longitudinal velocity is a constant, FTMPC and FTAMPC reflect almost the same path tracking control performance.