# 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
The ARR is a constraint relationship of the known variables. It equals zero when no fault occurs and non-zero when a fault occurs. The signal that describes the ARR is also termed the residual. Therefore, solving for the ARR starts with determining the redundancy equation, also known as the non-matched equation. If \(|E|>|X|\), then there is redundancy. If there is column matching, the unknown variables can be solved according to the matching equations. The remaining non-matched equations, which are the redundant equations, are used to construct the ARR. **Definition 14**. : _(Structural Redundancy) Given a system model \(M=(E,V)\), whose structural model is \(G\big{(}E,V,\overline{A}\big{)}\), let \(\varphi:2^{M}\rightarrow\mathbb{Z}\) be a mapping, then:_  
\[\varphi(M)=v\big{(}G\big{(}E_{X,\overline{A}_{X}},\mathbf{X}\cup\mathbf{Z},\overline{ A}_{X,\overline{A}_{X}}\big{)}\big{)}-v\big{(}G\big{(}E_{X},\mathbf{X},\overline{A}_{X} \big{)}\big{)} \tag{7}\]  
_is the structural redundancy._  
The basic principles of matching-based diagnostic system design:  
First, construct the structural model \(G\big{(}E_{X},X,\overline{A}_{X}\big{)}\);  
Second, determine the column complete matching;  
Third, solve for the unknown variables based on the matching constraints;  
Fourth, build the ARRs based on the non-matched constraints;  
Fifth, based on the sensitivity relationship between the residuals and the faults, construct the residual characteristic matrix and analyze fault detectability. In performing the theoretical analysis, it is not necessary to actually compute the residuals but only to discover the equations required to generate them. Find the matching constraints corresponding to the matching variables for each non-matching constraint. Thus,a non-matching constraint plus its matching constraints determines the set of equations (MSOs) required to create the residuals. 2. Structural decomposition:  
Based on graph theory, a given structural model can be decomposed into three structural subgraphs: structurally overdetermined, structurally just-determined, and structurally underdetermined subsystems. **Definition 15**. : _(Structurally overdetermined, structurally just-determined, and structurally underdetermined diagrams) \(G\big{(}E,V,\overline{A}\big{)}\) is said to be:_  
* _Structurally overdetermined if and only if there is column complete matching and no row complete matching;_
* _Structurally just-determined if and only if there is perfect matching;_
* _Structurally underdetermined if and only if row and column complete matching exists._  
For any given system model \(M=(E,V)\), its structural model may not contain the three properties defined above. Therefore, \(M\) can be decomposed into three subsystems:  
\[M^{+}=\big{(}E^{+},V^{+}\big{)} \tag{8}\]  
\[M^{0}=\Big{(}E^{0},V^{0}\cup V^{+}\Big{)} \tag{9}\]  
\[M^{-}=\Big{(}E^{-},V^{-}\cup V^{0}\cup V^{+}\Big{)} \tag{10}\]  
The set of equations is decomposed as \(E=\big{\{}E^{+},E^{0},E^{-}\big{\}}\), and the set of variables is decomposed as \(V=\big{\{}V^{+},V^{0},V^{-}\big{\}}\). \((E^{+},V^{+})\) is structurally overdetermined. \(\big{(}E^{0},V^{0}\big{)}\) is structurally just-determined. \((E^{-},V^{-})\) is structurally underdetermined. \(M^{+}\) is also said to be a structurally overdetermined subsystem and \(|E^{+}|>|V^{+}|\). \(M^{-}\) is structurally just-determined and \(|E^{0}|=|V^{0}|\). \(M^{-}\) is structurally underdetermined and \(|E^{-}|<|V^{-}|\). According to Definition 13, \(M^{-}\) satisfies \(\varphi(M^{-})<0\); \(M^{0}\) satisfies \(\varphi\big{(}M^{0}\big{)}=0\); and \(M^{+}\) satisfies \(\varphi(M^{+})\geq 1\). If \(\varphi(M^{+})=1\), it is MSO. According to the consistent diagnosis principle, the structurally overdetermined section implies the ARR, which can be used for diagnosability analysis. The critical tool for structural analysis is DM decomposition, as shown in Figure 1. DM decomposition transforms the incidence matrix into an upper triangular matrix by appropriately ordering the constraint equations and variables.