# Structural Methodologies for Distributed Fault Detection and Isolation
## 4 Equation-Based Distributed Fault Detection and Isolation
### Equation-Based Fault Detection Approach
Subsystem \(S_{2}\) is just determined, but a subsystem can have an under-determined part as well. For example, consider subsystem \(S_{3}\) in Equation (2) where the DM decomposition is shown in Figure 8. Fault \(f_{5}\) is in the under-determined part of the structure. \(q_{in2}\) and \(q_{3}\) are in the just determined part of the system, and we can compute them using \(e_{15}\) and \(e_{16}\), respectively. However, to compute the other four variables in the subsystem, \(p_{3}\), \(q_{2}\), \(\dot{p}_{3}\), and \(p_{4}\), we only have three constraints, which makes complete matching between constraints and variables impossible. To make this part of the subsystem just determined, we need to augment a set of equations from the neighboring subsystems. Unlike previous work [18] where an algorithm was developed for subsystems with under-determined parts, Algorithm 3 automatically takes care of this. Using Algorithm 3 gives \(V_{m3}=\{q_{2},q_{3}\}\) as a minimal set of required shared variables to make \(f_{5}\) detectable. Having the set of required shared variables, Algorithm 4 gives \(A_{6}=\{e_{11},e_{17},e_{18},e_{19}\}\) as a minimal sets of equations from neighboring subsystems can be used to augment \(S_{3}\) to make \(f_{5}\) locally detectable. Figure 9 shows the DM decomposition of \((S_{3}|A_{6})\). Figure 8: DM decomposition of \(S_{3}\). Figure 7: DM decomposition of \((S_{2}|A_{2})\). Figure 9: DM decomposition of \((S_{3}|A_{6})\). In some cases, it is possible that an augmented minimal set, \(A_{i}\), also adds a set of faults \(F_{A_{i}}\) to the subsystem model \(S_{i}\). These faults can be sensor faults or faults in other equations. The following theorem states that these faults are locally detectable in subsystem model \(S_{i}\). Consider local subsystem model \(S_{i}=\{V_{i}\), \(M_{i}\), \(E_{i}\), \(F_{i}\}\) and \(E_{k}\) a set of minimal equations that makes set of faults \(F_{i}\) detectable in the augmented subsystem \((M_{i}|C_{augments})=\{V_{j},C_{j},F_{j}\}\), then the set of faults \(F_{j}\) in the augmented subsystem \((S_{i}|E_{k})\) is locally detectable. Proof. : The proof of this theorem is straight forward, since the minimal set makes a part of the system that includes the fault over-determined, and the set itself should be in the over-determined part as well. This means the associated faults in the set are detectable. For example, \(f_{6}\) is locally detectable in \((S_{3}|A_{6})\); see Figure 9. As long as fault detection is considered, the augmented faults do not cause any problem. The fault detection algorithm is summarized in Algorithm 5. ```
1:input: subsystem \(S_{i}\)
2:input: subsystem model neighbors \(\mathcal{N}_{Si}\)
3:\(\mathcal{M}=\{\}\)
4:\(V_{d}=\) set of determined variables in \(S_{i}\)
5:if\(\forall f\in F_{i}\) therefore \(e_{f}\in{S_{i}}^{+}\)then
6:return
7:\(U=\) Minimal-Shared-Variables(\(S_{i}\))
8:\(D=V_{d}\setminus U\)
9:\(E_{d}=\) Count-Matchings (\(\mathcal{M}\), \(D\), \(U\), \(\mathcal{N}_{Si}\))
```  
**Algorithm 5** Detectability.