# Structural Methodologies for Distributed Fault Detection and Isolation
## 4 Equation-Based Distributed Fault Detection and Isolation  
To avoid the computational complexity of the MSO-based algorithm, in the previous section, a distributed diagnosis method that works directly with the system of equations is proposed. To recap from earlier work [27], a structural model representation is used. A structural model describes which variables are included in which model equations. A useful tool is the Dulmage-Mendelsohn (DM) decomposition that decomposes a system model into three parts: (1) under-determined, (2) exactly determined, and (3) over-determined. The over-determined part introduces redundancy in the system description and forms the basis for fault detection and isolation [2]. Figure 2 shows the DM decomposition of subsystem \(S_{1}\) in the running example. The figure represents the set of equations in the just determined part, \(S_{1}{}^{0}\), the set of equations in the over-determined part, \(S_{1}{}^{+}\), and the set of unknown variables in each equation. The shared variables are shown as encircled variables in the figure. Without loss of generality, it is assumed that every fault parameter is included in exactly one equation, i.e., each fault \(f\) appears in one equation \(e_{f}\). This is not a restricting assumption because if a fault is included in more than one equation, we can replace the fault signal by a new variable and add a new equation where the new variable is equal to the fault. Similar to the definitions of detectability and isolability for a structural model in, e.g., [20], local detectability and isolability can be defined as:  
**Definition 11**. : _(Locally detectable) A fault \(f\in F_{i}\) is locally detectable in subsystem \(S_{i}\) if \(e_{f}\in{S_{i}}^{+}\), where \({S_{i}}^{+}\) is the over-determined part of subsystem \(S_{i}\)._  
**Definition 12**. : _(Locally isolable) A fault \(f_{i}\in F_{i}\) is locally isolable from fault \(f_{j}\in F\) if \(e_{f_{i}}\in(S_{i}\backslash e_{f_{j}})^{+}\), where \((S_{i}\backslash e_{f_{j}})^{+}\) is the over-determined part of subsystem \(S_{i}\) without equation \(e_{f_{j}}\)._  
Figure 2: Dulmage–Mendelsohn (DM) decomposition of the first subsystem model. Note that these definitions are equivalent to Definitions 8 and 10 since an MSO set is an over-determined equation set. Consider Definition 11 and Figure 2. Fault \(f_{1}\) is locally detectable because \(e_{1}\in{S_{1}}^{+}\), but \(f_{2}\) is not locally detectable since \(e_{2}\
otin{S_{1}}^{+}\). To expand the over-determined part and make \(f_{2}\) detectable, the diagnosis subsystem needs to include at least one additional equation. The extension to the original subsystem is defined as:  
**Definition 13**. : _(Augmented subsystem) Given subsystem \(S_{i}\) and a set of equations, \(E_{k}\), the augmented subsystem model \(S_{iE_{k}}=(S_{i}|E_{k})\) is (\(V_{iE_{k}}\), \(M_{iE_{k}}\), \(E_{iE_{k}}\), \(F_{iE_{k}}\)), where \(V_{iE_{k}}\) is the union of \(V_{i}\) and the unknown variables that appear in \(E_{k}\), \(M_{iE_{k}}\) is the union of \(M_{i}\) and the known variables that appear in \(E_{k}\), \(E_{iE_{k}}\) is the union of \(E_{i}\) and \(E_{k}\), and \(F_{iE_{k}}\) is the union of \(F_{i}\) and the possible faults associated with \(E_{k}\)._  
**Example 5**. : _Consider the running example. \(S_{1\epsilon 10}=(S_{1}|e_{10})=(V_{1\epsilon 10}\), \(M_{1\epsilon 10}\), \(E_{1\epsilon 10}\), \(F_{1\epsilon 10}\)), where \(V_{1\epsilon 10}=\{\hat{p}_{1},p_{1},p_{2},q_{in1},q_{1}\}\), \(E_{1\epsilon 10}=\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{10}\}\), \(M_{1\epsilon 10}=\{u_{1},y_{1},y_{2},y_{3}\}\), and \(F_{1\epsilon 10}=\{f_{1},f_{2}\}\). Note that \(e_{10}\) did not add any new unknown variables or faults to the subsystem model. Figure 3 represents the DM decomposition of the augmented subsystem, \(S_{1\epsilon 10}\). This figure shows that \(e_{2}\in S_{1\epsilon 10}^{+}\), and therefore, \(f_{2}\) is locally detectable for the augmented subsystem \(S_{1\epsilon 10}\)._  
Figure 4 shows DM decomposition of the \(S_{1\epsilon 10}\setminus e_{1}\).