# A Review of Research on Diagnosability of Control Systems Based on Structural Analysis
## 2 Diagnosability Architecture for Control Systems Based on Structural Analysis
### Fault Diagnosis Based on Structural Analysis
#### 2.2.2 Principles of Fault Diagnosis Based on Structural Analysis  
Consistency relations are represented as redundancy relations. The redundancy relations are derived by removing the unknown variables and creating the constraints between the known variables by algebraic elimination between the equations [28]. The concept of the redundancy relationship is that the number of equations is more significant than the number of unknown variables. In structural analysis, the redundancy connection is reflected as structural redundancy, which commonly appears in the overdetermined section. In this scenario, the solution of the redundancy relation is no longer determined by algebraic elimination. However, it is further characterized by the matching connection between the equations and the unknown variables. It can solve for the unknown variables by matched constraints and introduce the analytical expressions of the unknown variables into the non-matched constraints to produce the ARR. Therefore, the consistency-based fault diagnosis problem can be transformed into a structural overdetermination problem. The sensitive relationship between structural overdetermination and the faults is utilized to qualitatively determine structural detectability and isolability. 1. Based on matching relationships:  
Given a structural model, \(G(E,V,\overline{A})\), \(a\in\overline{A}\), and \(a=(e,v)\) are the edges between the equation \(e\) and the variable \(v\). Define the projections \(p_{E}\), \(p_{Z}\):  
\[ \begin{array}{c}p_{V}:\overline{A}\to V\\ a\mapsto p_{V}(a)=v\end{array}  \tag{4}\]  
\[ \begin{array}{c}p_{E}:\overline{A}\to E\\ a\mapsto p_{E}(a)=e\end{array}  \tag{5}\]  
The equation projection \(p_{E}\) represents the projection \(p_{E}(a)=e\) of the edges to the vertices of the equation. The variable projection \(p_{V}\) represents the projection \(p_{V}(a)=v\) of the edges to the vertices of the variables. **Definition 8**. : _(Matching) Given a system structure graph \(G(E,V,\overline{A})\), matching \(\mathcal{N}\) is a subset \(\mathcal{N}\subseteq\overline{A}\) of the edge set \(\overline{A}\), and:_  
\[\forall a_{1},a_{2}\in\mathcal{N},\,\text{s.t.}\,\,a_{1}\
eq a_{2}\Rightarrow p _{E}(a_{1})\
eq p_{E}(a_{2})\wedge p_{V}(a_{1})\
eq p_{V}(a_{2}) \tag{6}\]  
**Definition 9**. : _(Maximum Matching)\(\mathcal{N}\) is a maximum matching if and only if \(\forall\mathcal{N}_{i}\in 2^{\overline{A}}\), \(\mathcal{N}\subset\mathcal{N}_{i}\), and \(\mathcal{N}_{i}\) is not a matching._  
**Definition 10**. : _(Complete Matching) A match is a complete matching when and only when \(|\mathcal{N}|=|E|\) or \(|\mathcal{N}|=|V|\)._  
**Definition 11**. : _(Perfect Matching) A matching is a perfect matching when and only when \(|\mathcal{N}|=|E|=|V|\)._  
According to the incidence matrix, the complete matching at \(|\mathcal{N}|=|E|\) is row complete matching, and the complete matching at \(|\mathcal{N}|=|V|\) is column complete matching. The known variables do not need to be defined by particular constraint equations. The matching is more significant for the solution of the unknown variables. Consequently, the matching connection of the structural subgraphs of the unknown variables is principally addressed. **Definition 12**. : _(ARR) Given a system model \(M=(E,V)\), there exists an equation \(e\in E\), where \(Q(e)=\{\mathbf{z},\mathbf{x}\}_{\mathbf{x}}\subseteq X\) and \(\mathbf{z}\subseteq Z\). \(e\) is said to be the ARR if \(\mathbf{x}\) can be determined by an equation in \(E\backslash e\), i.e., \(x=h(z)\)._  
**Definition 13**. : _(Redundant Equation) Given a bipartite graph \(G\big{(}E_{X},X,\overline{A}_{X}\big{)}\), \(\mathrm{N}\subseteq\overline{A}_{X}\) is column complete matching. If \(e_{i}\in\overline{A}_{X}\backslash\mathrm{N}\), then \(e_{i}\) is said to be a redundant equation._  
By definition, it is recognized that the redundant equations exist only in the structurally overdetermined section. Each redundant equation may generate an ARR for producing system residuals for fault diagnostics.