# Sensor Fusion Architecture for Fault Diagnosis with a Predefined-Time Observer
## 4 Results
### Experimental Results  
The architecture and the diagnosers are executed in Simulink, with the real data obtained from the prototype shown in Figure 3. The sampling time in Simulink is 0.001 s and the selected solver is ode3 (Bogacki-Shampine). The HOSM parameters are chosen as \(\lambda=[3,1.5,1.1,3,1.5,1.1,3.1,5.1,1]^{T}\) and \(L=[\)50,000, 50,000, 50,000], according to [15]. The chosen parameters for the proposed predefined-time diagnoser are reported in Table 3, while the selected parameters for the sensor fusion architecture are reported in Section 3. When the proposed sensor fusion architecture is not applied, the GPS signals are used by the diagnosers and the differentiator \(\dot{z}=-\tau(z-y)\) is computed for the predefined-time diagnoser, where \(\dot{z}\) is the estimated derivative of the signal \(y\), and \(\tau\) is a designing parameter which is chosen as \(\tau=5\).  
The considered fault scenario consists of abrupt faults occurring either simultaneously or non-simultaneously at the actuators of the mobile robot. The detection fault threshold is \(\pm 0.035\) and it is represented as a red line in the graphics of the experiments. The time intervals and magnitudes of fault occurrences are:  
* Faults at the first actuator: \(m_{1}=0.12\) during the interval \([5\,\mathrm{s},8\,\mathrm{s}]\) and \(m_{1}=0.7\) during the interval \([22\,\mathrm{s},22.5\,\mathrm{s}]\).
* Faults at the second actuator: \(m_{2}=0.8\) during the interval \([10\,\mathrm{s},10.8\,\mathrm{s}]\) and \(m_{2}=0.5\) during the interval \([22\,\mathrm{s},23\,\mathrm{s}]\).
* Faults at the third actuator: \(m_{3}=-0.5\) during the interval \([10\,\mathrm{s},12\,\mathrm{s}]\), \(m_{3}=-0.1\) during the interval \([18\,\mathrm{s},18.5\,\mathrm{s}]\) and \(m_{3}=-0.12\) during the interval \([22.5\,\mathrm{s},24\,\mathrm{s}]\).  
Notice that the faults \(m_{1}\) and \(m_{2}\) are concurrent during the interval \([22\,\mathrm{s},23\,\mathrm{s}]\), also \(m_{2}\) and \(m_{3}\) occur simultaneously during the interval \([10\,\mathrm{s},10.8\,\mathrm{s}]\). At the considered fault scenario, the faults correspond to an actuator loss of effectiveness of around 20%, 50%, and 70%. Also, the faults occur in small intervals of time such as 0.5 s. Although the system is affected by the faults during small intervals of time, Figure 4 shows that the faults cause a significant deviation from the fault-free trajectory.  
Figures 5 and 6 present the fault detection using the proposed predefined-time diagnoser and the HOSM diagnoser reported in [26], respectively. In both figures, the simulations on the left correspond to the case where the proposed sensor fusion architecture is used, and on the right when it is not used. We highlight that the use of the proposed sensor fusion architecture is of major importance, in fact when this architecture is omitted, the graphics in the right columns of Figures 5 and 6 show a poor performance in fault detection, exacerbating the number of false fault detections. In addition, the predefined-time diagnoser estimates faults faster than the HOSM, it also reduces false fault detections. For example, HOSM (Figure 6) detects a false fault detection in actuator 3, when the estimated fault reaches the threshold of 0.035 at 6.45 s.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(\gamma\) & 4.9968 & \(q\) & 3 \\ \(T_{c}\) & 0.15 & \(k\) & 0.5 \\ \(\alpha\) & 1 & \(\xi_{1}\) & 100000 \\ \(\beta\) & 2 & \(\xi_{2}\) & 100000 \\ \(p\) & 1.5 & \(\xi_{3}\) & 300000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Selected parameters for the diagnoser.  
Figure 4: Mobile robot trajectory comparison.  
Figure 5: The proposed diagnoser based on a predefined-time observer is used, with (**left**) and without (**right**) the proposed sensor fusion architecture.