# Structural Methodologies for Distributed Fault Detection and Isolation
## 2 Basic Definitions and Running Example
This set of equations is structurally over-determined (SO) if the cardinality of the set \(\{E\}\) is greater than the cardinality of set \(\{V\}\), i.e., \(|E|>|V|\)._  
**Definition 6**. : _(Minimal Structurally-Over-determined (MSO) set) A set of over-determined equations is minimal structurally-over-determined if it has no subset of structurally-over-determined equations._  
The MSO sets are minimal sets of equations that can be used to generate residuals, for example by using the Fault diagnosis toolbox developed by [PERSON] and [PERSON] [22]. MSO sets represent redundant equation sets that capture the redundancies in the system: \(MSO_{l}=(E_{l},V_{l},M_{l},F_{l})\). For example, \(MSO_{11}=(E_{11},V_{11},M_{11},F_{11})\), where \(E_{11}=\{e_{1},e_{3},e_{4},e_{5},e_{6}\}\), \(V_{11}=\{p_{1},p_{1},q_{in1},q_{1}\}\), \(M_{11}=\{u_{1},y_{1},y_{2}\}\), and \(F_{11}=\{f_{1}\}\) represent an MSO set in subsystem \(S_{1}\) (1) of our running example. For brevity and simplification, we simply say a specific equation, variable, measurement, or fault is a member of an MSO in the rest of the paper, e.g., \(f\in MSO_{l}\). Each MSO set represents a part of the system model that can be used to design a residual that is only sensitive to certain faults. A set of MSO sets can be used to generate residuals that together can isolate a set of faults. To discuss the fault detectability and isolability properties of the global system and its subsystems, we define global and local fault detectability and isolability as follows. **Definition 7**. : _(Globally-detectable fault) A fault \(f\in F\) is globally detectable in system \(S\) if there is a minimal structurally over-determined set \(MSO_{l}\) in the system, such that \(f\in MSO_{l}\)._  
**Definition 8**. : _(Locally-detectable fault) A fault \(f\in F_{i}\) is locally detectable in subsystem \(S_{i}\) if there is a minimal structurally over-determined set \(MSO_{l}\) in the subsystem such that \(f\in MSO_{l}\)._  
Figure 1: Running example: four-tank system. **Example 3**. : _Fault \(f_{1}\) in (1) is locally detectable because \(f_{1}\in MSO_{11}\). However, \(f_{2}\) is not locally detectable because there is no MSO set in this subsystem that includes \(f_{2}\). To detect \(f_{2}\) locally, the diagnosis subsystem requires additional measurements._  
**Definition 9**. : _(Globally-isolable fault) A fault \(f_{i}\in F\) is globally isolable from fault \(f_{j}\in F\) if there exists a minimal structurally-over-determined set \(MSO_{i}\) in the system \(S\), such that \(f_{i}\in MSO_{i}\) and \(f_{j}\
ot\in MSO_{i}\)._  
**Definition 10**. : _(Locally-isolable fault) A fault \(f_{i}\in F_{i}\) is locally isolable from fault \(f_{j}\in F\) if there exists a minimal structurally-over-determined set \(MSO_{i}\) in subsystem \(S_{i}\), such that \(f_{i}\in MSO_{i}\) and \(f_{j}\
ot\in MSO_{i}\)._  
Note that if a fault \(f_{j}\) is locally detectable in a subsystem \(S_{i}\), it is globally detectable as well, and if a fault \(f_{j}\) is locally isolable from a fault \(f_{k}\), it is globally isolable from \(f_{k}\), as well.