# Structural Methodologies for Distributed Fault Detection and Isolation
## 3 MSO-Based Distributed Fault Detection and Isolation
### MSO Set Selection for Distributed Fault Detection
Setting \(b(j)=-1\) for \(g<j\leq g*h\), where \(h\) is the number of faults in the system, \(h=|F|\), will make sure that there is at least one MSO set to isolate each of the subsystem faults from the other faults in the system. Using an MSO set is equivalent to using the measurements that are included in the MSO set. For example, using \(MSO_{11}\) from the water tank example in a subsystem diagnoser requires three measurements \(M_{11}=\{u_{1},y_{1},y_{2}\}\) transmitted to that subsystem. Therefore, a set of constraints is included that capture the relationship between the measurements and MSO sets in the distributed diagnosis system. Equation (10) represents this set of constraints in the \(A\) matrix. \[A(j+g*h,k)=\left\{ \begin{array}{ccc}-1&\text{if}&m_{k}\in MSO_{j}\\ |M_{j}|&\text{if}&k=j+|M|\\ 0&\text{otherwise,}\end{array} \right. \tag{10}\]  
where \(|M_{j}|\) is the cardinality number of the set of measurements in \(MSO_{j}\) and \(|M|\) is the number of measurements in the system. Furthermore, \(b(j)=0\) for \(g*h<j\leq g*h+r\), where \(r\) is the number of MSO sets in the system. **Example 4**. : _The water tank system has 165 MSO sets. The entire system includes eight measurements, where subsystem \(S_{1}\) includes three measurements. \(S_{1}\) has two faults of interest, and the goal is to be able to isolate them from the other six faults in the complete system. For the optimization problem (4) for \(S_{1}\), matrix \(A\in\mathbb{R}^{177\times 165}\), i.e., two local detectability constraints, 10 local isolability constraints, and 165 constraints to capture the relationship between the MSO sets and the measurements and 173 columns corresponding to the eight measurements and 165 MSO sets._  
_Table 1 shows the set of measurements to add for each of the subsystem diagnosers to achieve maximum possible detectability and isolability. To find the optimum measurements, the optimization problem (4) has been solved for each subsystem._  
_Subsystem \(S_{1}\) initially has three measurements \(M_{1}=\{u_{1},y_{1},y_{2}\}\). To achieve global diagnosability for its faults, \(y_{3}\) must be shared with its diagnoser from subsystem \(S_{2}\). Subsystem \(S_{2}\) is the only subsystem that shares a variable with a second order connected subsystem. All the other subsystems only need to communicate with their first order connected subsystems._ \begin{table}
\begin{tabular}{c c} \hline \hline
**Subsystem** & **Set of Augmented Measurements** \\ \hline \(S_{1}\) & \(y_{3}\) \\ \(S_{2}\) & \(u_{2},y_{2},y_{6}\) \\ \(S_{3}\) & \(y_{4},y_{6}\) \\ \(S_{4}\) & \(y_{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Set of augmented measurements for each subsystem model. In general, the worst case scenario for a system with connected subsystems will typically require a large number of measurements from other subsystems to be communicated to each subsystem diagnoser. In those situations, distributed subsystem diagnosers help overcome the single point of failure problem, but each subsystem diagnoser may require a large number of measurements to be communicated to it from all of the other subsystems. Even though there are efficient algorithms to solve BILP problems, this approach will not scale up for larger systems, since the search space is exponential in the number of MSO sets \(2^{r}\), even if the subsystem diagnoser design is performed off-line. In addition to the computational complexity, the availability of global models for large, complex systems is unlikely because of the issues discussed in Section 1. To overcome this problem, a heuristic search strategy is proposed based on an incremental search algorithm that works with the original system equations instead of MSO sets.