# Structural Methodologies for Distributed Fault Detection and Isolation
## 5 Case Study
### Designing the Diagnosers  
After the augmented subsystem models have been selected, computational tools, for example the fault diagnosis toolbox [22], can be used to generate the set of residuals to be used for each subsystem. For example, the fault diagnosis toolbox generates the following residual from \(MSO_{a11}\).  
\[r_{a11}=y_{E240}-y_{E242}. \tag{18}\]  
Each residual is sensitive to a set of faults, and the set of residuals for each subsystem can detect all the globally-detectable faults and isolate all the globally-isolable faults in the subsystem. For example, \(r_{a11}\) in (18) is sensitive to \(f_{E240}\) and \(f_{E242}\) and, therefore, can be used to detect these faults. In realistic situations, sensor noise and model uncertainties can have a negative impact on each diagnoser's performance. For example, consider the case where each sensor in the ADAPT-Lite system has an additive noise with a normal distribution, \(N(0.0.1)\). Figure 12 shows that because of noise, residual \(r_{a11}\) is not zero even when there is no fault in the system.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Subsystem** & **Set of MSO Sets** \\ \hline
1 & \(MSO_{011}=\{e_{47},e_{49},e_{410}\}\) \\  & \(MSO_{012}=\{e_{41},e_{42},e_{43},e_{44},e_{45},e_{46},e_{49},e_{411}\}\) \\  & \(MSO_{013}=\{e_{41},e_{42},e_{43},e_{44},e_{45},e_{46},e_{47},e_{410},e_{411}\}\) \\
2 & \(MSO_{021}=\{e_{411},e_{413},e_{414},e_{415},e_{416},e_{418},e_{419},e_{423}\}\) \\
3 & \(MSO_{031}=\{e_{415},e_{416},e_{417},e_{418},e_{419}\}\) \\
4 & \(MSO_{041}=\{e_{419},e_{420},e_{421},e_{422},e_{423},e_{425},e_{426},e_{427}\}\) \\
5 & \(MSO_{051}=\{e_{419},e_{420},e_{420},e_{424},e_{425},e_{426},e_{427}\}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Set of MSO sets for each local diagnoser using the equation-based approach.  
Figure 12: Residual \(r_{a11}\) for 24 h when each sensor in the ADAPT-Lite system has an additive noise with a normal distribution, \(N(0.0.1)\), and the system is in normal operation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Subsystem** & **Augmented Equations** & **Augmented Measurements** \\ \hline Subsystem 1 & - & - \\ Subsystem 2 & \(e_{a11}\), \(e_{a16}\), \(e_{a18}\), \(e_{a19}\), \(e_{a23}\) & \(\mathit{y17240y,y265y,y17267}\) \\ Subsystem 3 & \(e_{a15}\) & \(y_{E281}\) \\ Subsystem 4 & \(e_{a19}\), \(e_{a25}\), \(e_{a26}\), \(e_{a27}\) & \(\mathit{y2E265y,y17516}\) \\ Subsystem 5 & \(e_{a19}\), \(e_{a20}\) & \(\mathit{yE265}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Set of augmented equations and measurements for each subsystem model using the equation-based approach.  
This can impact fault diagnosis performance negatively by increasing false positive rates. Moreover, noise can hide the effect of faults on the residuals and lead to high false negative rates. Figure 13 shows residual \(r_{a11}\) when an additive fault \(f_{E240}=0.25\) occurs at \(t=20\) h. Sensor noise conceals the fault signal and makes fault detection and isolation more challenging.  
To achieve acceptable performance in practice, it is necessary to design a set of hypothesis tests, such as the Z-test [31], to distinguish faults from noise and uncertainty and determine what residual outputs are significant enough to reject the normal operation assumption and trigger the alarms. Figure 14 shows that a simple hypothesis test can achieve a zero false positive rate and detect \(f_{E240}\) in less than 5 min using residual \(r_{a11}\).