# A Review of Research on Diagnosability of Control Systems Based on Structural Analysis
## 5 Diagnosability Index System Based on Structural Analysis
(6) Real-Time Isolation Rate  
The real-time isolation rate \(g\big{(}\Delta_{j}\big{)}\) characterizes the speed of fault isolation under \(\Delta_{j}\):  
\[g\big{(}\Delta_{j}\big{)}=\sum_{i=1}^{N_{f}}w_{i}g_{i}(\Delta_{j}) \tag{34}\]  
\[g_{i}\big{(}\Delta_{j}\big{)}=\frac{t_{th,i}\big{(}\Delta_{j}\big{)}-t_{i}^{I} \big{(}\Delta_{j}\big{)}}{t_{th,i}\big{(}\Delta_{j}\big{)}} \tag{35}\]  
\[t_{i}^{I}\big{(}\Delta_{j}\big{)}=\left\{ \begin{array}{l}t_{tm,i}\big{(} \Delta_{j}\big{)},t_{tm,j}\big{(}\Delta_{j}\big{)}<t_{th,i}\big{(}\Delta_{j} \big{)}\\ t_{th,i}\big{(}\Delta_{j}\big{)},t_{tm,j}\big{(}\Delta_{j}\big{)}\geq t_{th,i} \big{(}\Delta_{j}\big{)}\end{array} \right. \tag{36}\]\(g_{i}\big{(}\Delta_{j}\big{)}\) in the above equation represents the isolation speed of \(f_{i}\) under \(\Delta_{j}\). \(t_{1m,i}\big{(}\Delta_{j}\big{)}\) is the average isolation time of \(f_{i}\) in \(\Delta_{j}\). \(t_{1 th,i}\big{(}\Delta_{j}\big{)}\) is the shortest isolation time of \(f_{i}\) in \(\Delta_{j}\), which characterizes the limit value of the isolation time. (7) Resource Consumption Rate  
The resource consumption rate \(h\big{(}\Delta_{j}\big{)}\) characterizes the level of resource consumption under \(\Delta_{j}\):  
\[h\big{(}\Delta_{j}\big{)}=\sum\limits_{i=1}^{N_{t}}w_{t_{i}}h_{i}\big{(} \Delta_{j}\big{)} \tag{37}\]  
\[h_{i}\big{(}\Delta_{j}\big{)}=\frac{\cos\mathfrak{t}_{i}-\cos\mathfrak{t}_{ \min,i}}{\cos\mathfrak{t}_{\max,i}-\cos\mathfrak{t}_{\min,i}} \tag{38}\]  
where \(h_{i}\big{(}\Delta_{j}\big{)}\) characterizes the resource consumption level of test \(\delta_{i}\) under \(\Delta_{j}\). The larger \(h_{i}\big{(}\Delta_{j}\big{)}\) is, the higher the resource consumption level of \(\delta_{i}\). \(\cos\mathfrak{t}_{i}\) denotes the comprehensive cost of test resources selected by \(\delta_{i}\). \(\cos\mathfrak{t}_{\min,i}\) is the smallest value of the total cost among the alternative test resources of test \(\delta_{i}\). \(\cos\mathfrak{t}_{\max,i}\) is the largest value of the complete cost among the alternative test resources of test \(\delta_{i}\). \(w_{t_{i}}\) is the weight of \(\delta_{i}\), which can be obtained by analyzing the system requirements. (8) Comprehensive Evaluation Indicators of Diagnosability  
The detectability and isolability rates are diagnosable characteristics determined by the structure and function of the system. Therefore, the detectability and isolability rates are mainly utilized in the diagnosability structure and function design stage to evaluate the diagnosability. If the detectability and isolation rates fail to meet the diagnosability requirements in the structural and functional design phase, structural and functional optimization design is required. Otherwise, the detection and isolation of undetected or isolated faults cannot be guaranteed through the design of diagnostic strategies and methods. Detection and isolation accuracy are evaluated from the perspective of actual operation by considering various influencing factors and assessing the diagnostic stratagem. To a certain extent, it reflects the difficulty of fault diagnosis. The real-time detection rate and the real-time isolation rate reflect the real-time index of fault diagnosis, while the resource consumption rate reflects the level of the diagnosis of resource consumption. Combining the above seven diagnosability assessment indicators, the comprehensive diagnosability evaluation indicators are obtained:  
\[\mathcal{M}=\frac{\sum\limits_{i=1}^{N_{f}}a_{i}\cdot I}{\beta h\big{(}\Delta _{j}\big{)}} \tag{39}\]  
\[I=\alpha_{i}^{1}b_{i}+\alpha_{i}^{2}c_{i}\big{(}\Delta_{j}\big{)}+\alpha_{i}^ {3}d_{i}\big{(}\Delta_{j}\big{)}+\alpha_{i}^{4}e_{i}\big{(}\Delta_{j}\big{)}+ \alpha_{i}^{5}g_{i}\big{(}\Delta_{j}\big{)} \tag{40}\]  
In the above equation \(\sum\limits_{j=1}^{5}\alpha_{i}^{j}=1\). \(\alpha_{i}^{j}\) represents the analytical weight of the indicator for \(f_{i}\) under \(\Delta_{j}\). \(\beta\) is the analytical weight of the resource consumption rate indicator.