# A Review of Research on Diagnosability of Control Systems Based on Structural Analysis
## 5 Diagnosability Index System Based on Structural Analysis  
Assuming that the control system has \(N_{f}\) kinds of failure modes \(f_{i}\), \(i=0,1,\ldots,N_{f}\), \(f_{0}\) stands for a failure-free mode. The system has \(N_{t}\) kinds of tests \(\delta_{i}\), \(i=1,\ldots,N_{t}\) and \(N_{\delta}\) kinds of diagnosability influences \(\sigma_{i}\), \(i=1,\ldots,N_{\delta}\), such as noise, interference, uncertain inputs, selection of troubleshooting algorithms, and other diagnosability influences. The system has \(N_{\Delta}\) kinds of diagnosability influence modes \(\Delta_{i}\), \(i=0,1,\ldots,N_{\Delta}\). \(\Delta_{i}\) is the combination of influences \(\sigma_{i}\). \(\Delta_{0}\) stands for not considering the influences of diagnosability and only the influence of the structure and function on the diagnosability. \(\epsilon_{f_{i}}\big{(}\Delta_{j}\big{)}\) represents the detectability metric of \(f_{i}\) under \(\Delta_{j}\). \[\epsilon_{f_{i}}\big{(}\Delta_{j}\big{)}=FD(f_{i})=\min[K(p_{i}\parallel p_{ NF})] \tag{14}\]  
\(\epsilon_{f_{ij}}(\Delta_{k})\), \(i\
eq j\) represents an indicator of the isolability of \(f_{i}\) from \(f_{j}\) under \(\Delta_{k}\):  
\[\epsilon_{f_{ij}}(\Delta_{k})=FI\big{(}f_{i},f_{j}\big{)}=\min[K\big{(}p_{i} \parallel p_{j}\big{)}] \tag{15}\]  
\(\epsilon_{th,f_{i}}\big{(}\Delta_{j}\big{)}\) represents the threshold of detectability metrics for \(f_{i}\) under \(\Delta_{j}\). \(\epsilon_{th,f_{ij}}(\Delta_{k})i\
eq j\) represents the threshold of isolability metrics for \(f_{i}\) and \(f_{j}\) under \(\Delta_{k}\). \(w_{i}\) is the analytical weight of \(f_{i}\). Here, the diagnosability quantification adopts the diagnosability quantification method based on K-L dispersion, as referred to in [36]. (1) (Structural) Detectability rate  
The detectability rate \(a\) is used to describe the coverage of fault diagnosis, i.e., the ratio of fault modes to the total number of fault modes as determined by the structure and function:  
\[a=\sum_{i=1}^{N_{f}}a_{i} \tag{16}\]\(a\in[0,1]\) represents the diagnostic coverage. The larger \(a\) represents the higher structural detectability. \(a_{i}\) describes the diagnostic coefficient of \(f_{i}\), and the larger \(a_{i}\) represents the more significant contribution of \(f_{i}\) to the detectability. \[a_{i}=w_{i}v_{i} \tag{17}\]  
\[w_{i}=\frac{\lambda_{f_{i}}}{\sum\limits_{i=1}^{N_{f_{i}}}\lambda_{f_{i}}} \tag{18}\]  
\[v_{i}=\left\{ \begin{array}{l}1,e_{f_{i}}\in\left(E\backslash\left\{e_{f_{j}} \right\}\right)^{+}\\ 0,e_{f_{i}}\
otin\left(E\backslash\left\{e_{f_{j}}\right\}\right)^{+}\end{array} ,i=1,\ldots,N_{f}\right. \tag{19}\]  
The analysis weight of \(f_{i}\) is measured by the failure rate \(\lambda_{f_{i}}\). The higher the \(\lambda_{f_{i}}\), the higher the weight \(w_{i}\) of \(f_{i}\). \(w_{i}\) can be determined according to the diagnosability requirements. \(v_{i}\) is used to determine whether \(f_{i}\) is detected or not. (Structural) isolatability ratio  
The isolatable rate indicates the isolation of faults as determined by the system structure and function. It is described by the ratio of isolatable fault modes to total fault modes:  
\[b=\sum\limits_{i=1}^{N_{f}}b_{i} \tag{20}\]  
\[b_{i}=w_{i}n_{i} \tag{21}\]  
\[n_{i}=\left\{ \begin{array}{l}1,\forall f_{j}\in(F\backslash f_{i}),e_{f_{i} }\in\left(E\backslash\left\{e_{f_{j}}\right\}\right)^{+}\\ 0,otherwise\end{array} \right. \tag{22}\]  
where \(i,j=1,\ldots,N_{f},i\
eq j\); \(b\in[0,1]\) represents the isolation coverage. The larger \(b\) represents the higher structural isolability. \(b_{i}\) describes the isolability coefficient of \(f_{i}\), and the larger \(b_{i}\) indicates that \(f_{i}\) contributes more to the isolability.