# 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
#### 2.2.1 Principles of Consistency-Based Fault Diagnosis  
Consistency-based fault diagnosis is based on a diagnosable model and the value of \(\mathbf{Z}\). The known variable \(\mathbf{z}\in\mathbb{Z}\) is typically the observed quantity \(z_{0}\) obtained by sampling the signals of the sensors and actuators.  
If \(\exists\mathbf{x}\in\mathbb{X}:\wedge_{e\in E}e(\mathbf{x},\mathbf{z})\), \(M\subseteq M\) is said to be consistent with \(\mathbf{z}\in\mathbb{Z}\). Let \(\wedge_{e\in E}e(\mathbf{x},\mathbf{z})=M(\mathbf{x},\mathbf{z})\). Given an observation, \(M\subseteq M\) is said to be inconsistent if the observation is inconsistent with \(M\subseteq M\). The set of observations with a consistent relationship with \(M\) is: \(\mathcal{O}(M)=\{\mathbf{z}\in\mathbb{Z}|\exists\mathbf{x}\in\mathbb{X}:M( \mathbf{x},\mathbf{z})\}\).  
Verifying that \(M\subseteq M\) is consistent with \(\mathbf{z}\in\mathbb{Z}\) verifies that \(M\subseteq M\) can satisfy a given observation. If \(\mathbf{z}_{0}\) is a given observation, \(x\) is an unknown variable, and \(f(\mathbf{z},\mathbf{x})\) is a differential-algebraic model. \(f(\mathbf{z},\mathbf{x})\) is said to be consistent with \(\mathbf{z}_{0}\) if \(\exists x\), \(s.t.\)\(f(z_{0},x)=0\). This problem is generally not easy to solve. Moreover, the time for obtaining observations and consistency checking must be as short as possible to ensure efficiency. Therefore, the consistency test's computational complexity must be reduced in order to ensure efficiency.  
(Structural Redundancy) Given a system model \(M\), let \(\varphi_{s}:2^{M}\rightarrow\mathbb{Z}\) be a mapping of the submodel \(M\subseteq M\), then:  
\[\varphi_{s}(M)=|E|-|X| \tag{3}\]  
is the structural redundancy.  
**Definition 4**.: _(Structurally Overdetermined) A system model \(M\subseteq\textbf{M}\) is said to be Structurally Overdetermined (SO) if \(\varphi_{s}(M)>0\)._  
**Definition 5**.: _(Proper Structurally Overdetermined) A structurally overdetermined model \(M\) is said to be Proper Structurally Overdetermined (PSO) if \(\forall M^{\prime}\subseteq M\) s.t.\(\varphi_{s}(M^{\prime})<\varphi_{s}(M)\)._  
**Definition 6**.: _(Minimal Structurally Overdetermined) An overdetermined model \(M\) is said to be Minimal Structurally Overdetermined (MSO) if \(\
eg\exists M^{\prime}\subseteq M\) s.t.\(\varphi_{s}(M^{\prime})<\varphi_{s}(M)\)._  
**Definition 7**.: _(Consistency Diagnosis) Given an observation \(\textbf{z}\in\mathbb{Z}\), a system's behavioral model \(b\in\textbf{B}\) is considered diagnostic if and only if \(M_{b}\) is consistent with \(\textbf{z}\)._  
The consistency relation is also known as the ARR, and the ARR exists in the structurally overdetermined sections. Therefore, the fault diagnosis is primarily diagnosed by finding the faults in the structurally overdetermined sections. In contrast, the faults in the other system sections cannot be diagnosed because of the absence of the ARR [27].  
The consistency-based diagnostic approach uses the consistency relationship to make the diagnosis. There is a fault if the observed model is inconsistent with the planned model based on the known variables. Based on the consistency relations, the fault diagnosis problem is further turned into the issue of splitting the diagnostic model based on the consistency relations. Identifying whether the fault is in the overdetermined section can simplify the consistency test's complexity.