# Structural Methodologies for Distributed Fault Detection and Isolation
## 2 Basic Definitions and Running Example  
This section introduces the basic concepts associated with the distributed diagnosis of dynamic systems. **Definition 1** (System model). : _A system model \(S\) is a four-tuple: (\(V\), \(M\), \(E\), \(F\)), where \(V\) is the set of variables, \(M\) is the set of measurements, \(E\) is the set of equations, and \(F\) is the set of system faults._  
It is assumed that the sets of \(V\) and \(E\) are sufficient to define the behavior of the system. The system \(S\) is partitioned into \(n\) subsystems, \(S_{1},S_{2},...S_{n}\), where each subsystem model is defined as:  
**Definition 2** (Subsystem model). : _A subsystem model \(S_{i}\) (\(1\leq i\leq k\)) associated with a system model, \(S\), is also a four-tuple: (\(V_{i}\), \(M_{i}\), \(E_{i}\), \(F_{i}\)), where \(V_{i}\subseteq V\), \(M_{i}\subseteq M\), \(E_{i}\subseteq E\), and \(F_{i}\subseteq F\). Furthermore, \(S_{1}\cup S_{2}\cup...S_{k}=S\)._  
We note that a variable can be shared between two or more subsystems describing the connection between the different subsystems. **Example 1**. : _A four-tank system is used as a running example in the paper; see Figure 1. The system is assumed to be divided into four non-overlapping subsystems, where each subsystem is constituted of one tank and the outlet pipe to its right. Two of the subsystems, 1 and 3, also have external inflows into their tanks. Associated with each subsystem is a set of measurements, \(\{y_{1},y_{2},...,y_{6}\}\), that are shown as encircled variables in the figure._  
_The first subsystem, \(S_{1}\), in the running example is described by the following set of equations:_  
\[ \begin{split} e_{1}:\dot{p}_{1}&=\frac{1}{C_{\text{T 1}}}(q_{in1}-q_{1}-f_{1})\\ e_{2}:q_{1}&=\frac{p_{1}-p_{2}}{R_{\text{P1}}+f_{2} }\end{split} \qquad\qquad \begin{split} e_{4}:q_{in1}& =u_{1}\\ e_{5}:p_{1}&=y_{1}\\ e_{6}:q_{1}&=y_{2}.\end{split}  \tag{1}\]  
\(E_{1}=\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}\) _defines the set of equations; \(V_{1}=\{\dot{p}_{1},p_{1},p_{2},q_{in1},q_{1}\}\) defines the set of subsystem unknown variables; \(M_{1}=\{u_{1},y_{1},y_{2}\}\) defines the set of subsystem known variables (measurements); and \(F_{1}=\{f_{1},f_{2}\}\) defines the set of faults associated with this subsystem model. It is assumed that the system parameters (\(C_{\text{T1}}\), and \(R_{\text{P1}}\) of the first subsystem) are known. This model representation is used to emphasize the model structure, which is useful, for example, when analyzing structural fault detectability and isolability properties [2]._  
_Similarly, the three other subsystem models are defined by the following equations:_  
\[ \begin{split} e_{7}:\dot{p}_{2}&=\frac{1}{C_{\text{T 2}}}(q_{1}-q_{2}-f_{3})\\ e_{8}:q_{2}&=\frac{p_{2}-p_{3}}{R_{\text{P2}}+f_{4} }\end{split} \qquad\begin{split} e_{12}:\dot{p}_{3}& =\frac{1}{C_{\text{T3}}}(q_{in2}+q_{2}-  
**Definition 3** (First order connected subsystems). : _Two subsystems, \(S_{i}\) and \(S_{j}\), are first order connected if and only if they have at least one shared variable._  
**Definition 4** (\(i^{\text{th}}\) order connected subsystems). : _Two subsystems, \(S_{k}\) and \(S_{j}\), are \(i^{\text{th}}\) order connected if there exists a subsystem model \(S_{m}\) that is \((i-1)^{\text{th}}\) order connected to \(S_{k}\) and is first-order connected to \(S_{j}\), or \(S_{m}\) is \((i-1)^{\text{th}}\) order connected to \(S_{j}\) and is first-order connected to \(S_{k}\)._  
**Example 2**. : _In the four-tank example, subsystems \(S_{1}\) and \(S_{2}\) are first order connected, and their shared variables are \(V_{1}\cap V_{2}=\{p_{2},q_{1}\}\). Similarly, \(S_{1}\) and \(S_{3}\) are second order connected because both of them are first order connected to \(S_{2}\)._  
In this paper, MSO sets are used as the primary approach for FDI and defined as follows [20]:  
**Definition 5**. : _(Structural over-determined set) Consider a set of equations and its associated variables, measurements, and faults: \((E,V,M,F)\).