# Structural Methodologies for Distributed Fault Detection and Isolation
## 5 Case Study
### Distributed Diagnoser Using the Equation-Based Method  
Instead of generating all the MSO sets and selecting a subset of MSO sets for each local diagnoser, Algorithm 7 is used.  
The results are shown in Table 5, summarizing the set of equations to augment each subsystem to achieve maximum possible detectability and isolability. In some cases, the first order neighboring subsystems were not enough to detect and isolate all the faults, and the algorithm had to extend to include higher order neighbors. For example, for Subsystem 2, the algorithm cannot find any solution when it considered the first order neighbors (Subsystem 1 and Subsystem 3). Therefore, it extended the search to a second order neighboring subsystem (Subsystem 4).  
Table 5 also represents the set of additional measurements that we need to transfer to each ADAPT subsystem. As mentioned earlier, the equation-based algorithm does not guarantee globally-minimum communication. For example, Subsystem 2 required three measurements from other subsystems (see  
\begin{table}
\begin{tabular}{c c} \hline
**Subsystem** & **Set of MSO Sets** \\ \hline
1 & \(MSO_{a12}=\{e_{a8},e_{a10},e_{a11},e_{a12},e_{a13},e_{a14},e_{a16},e_{a17},e_{ a18},e_{a20},e_{a21},e_{a22},e_{a24}\}\) \\  & \(MSO_{a13}=\{e_{a27},e_{a8},e_{a9},e_{a11},e_{a12},e_{a13},e_{a14},e_{a16},...\) \\  & \(e_{a17},e_{a18},e_{a20},e_{a21},e_{a22},e_{a24}\}\) \\ \hline
2 & \(MSO_{a21}=\{e_{a1},e_{a22},e_{a3},e_{a4},e_{a5},e_{a6},e_{a7},e_{a8},e_{a12},e_{ a13},e_{a14},e_{a15},e_{a16},...\) \\  & \(e_{a18},e_{a20},e_{a21},e_{a22},e_{a23},e_{a24}\}\) \\  & \(MSO_{a22}=\{e_{a1},e_{a22},e_{a3},e_{a4},e_{a5},e_{a6},e_{a7},e_{a8},e_{a12},e_{ a13},e_{a14},e_{a15},e_{a16},...\) \\  & \(e_{a17},e_{a18},e_{a23}\}\) \\ \hline
3 & \(MSO_{a31}=\{e_{a15},e_{a16},e_{a17},e_{a19}\}\) \\ \hline
4 & \(MSO_{a41}=\{e_{a21},e_{a22},e_{a23},e_{a24},e_{a25},e_{a26},e_{a27}\}\) \\  & \(MSO_{a42}=\{e_{a1},e_{a2},e_{a3},e_{a4},e_{a5},e_{a6},e_{a7},e_{a8},e_{a12},e_{ a13},e_{a14},e_{a16},...\) \\  & \(e_{a17},e_{a18},e_{a20},e_{a21},e_{a22},e_{a23},e_{a25},e_{a26},e_{a27}\}\) \\ \hline
5 & \(MSO_{a51}=\{e_{a19},e_{a20},e_{a24},e_{a25},e_{a26},e_{a27}\}\) \\ \hline \end{tabular}
\end{table}
Table 4: Set of Minimal Structurally-Over-determined (MSO) sets for each local diagnoser using the MSO-based method.  
\begin{table}
\begin{tabular}{l c} \hline
**Subsystem** & **Set of Augmented Measurements** \\ \hline Subsystem 1 & - \\ Subsystem 2 & \(y17267\) \\ Subsystem 3 & \(y_{E281}\) \\ Subsystem 4 & \(y_{ST516}\) \\ Subsystem 5 & \(y_{E265}\) \\ \hline \end{tabular}
\end{table}
Table 3: Set of augmented measurements for each ADAPT subsystem using the global method.  
Table 5). However, Table 3 shows that complete diagnosability was achievable by adding only one additional measurement. To detect and isolate faults in each subsystem, the augmented subsystem equations were used to generate MOS sets. Table 6 shows the set of MSO sets for each local diagnoser using the equation-based method. The experiment was run on the same desktop where total execution time was 0.32s. This demonstrates the computational advantage of this method.