# A Review of Research on Diagnosability of Control Systems Based on Structural Analysis
## 5 Diagnosability Index System Based on Structural Analysis
(3) Detection Accuracy  
Detection accuracy \(c\left(\Delta_{j}\right)\) measures the accuracy of system fault diagnosis under the influence mode \(\Delta_{j}\):  
\[c\left(\Delta_{j}\right)=\sum\limits_{i=1}^{N_{f}}w_{i}c_{i}\left(\Delta_{j}\right) \tag{23}\]  
\[c_{i}\left(\Delta_{j}\right)=w_{i}^{fa}r_{i}^{fa}\left(\Delta_{j}\right)+w_{i} ^{ta}r_{i}^{ta}\left(\Delta_{j}\right) \tag{24}\]  
\[r_{i}^{fa}\left(\Delta_{j}\right)=P\left(\varepsilon_{th,f_{i}}\left(\Delta_{ j}\right)\geq\varepsilon_{f_{0}}\left(\Delta_{j}\right)\left|f=f_{0}\right) \tag{25}\]  
\[r_{i}^{ta}\left(\Delta_{j}\right)=P\left(\varepsilon_{f_{i}}\left(\Delta_{j} \right)\geq\varepsilon_{th,f_{i}}\left(\Delta_{j}\right)\left|f=f_{i}\right) \tag{26}\]  
The larger \(c\left(\Delta_{j}\right)\) is, the higher the fault detection rate under \(\Delta_{j}\). In the above equation, \(c_{i}\left(\Delta_{j}\right)\) denotes the detection accuracy of \(f_{i}\) under \(\Delta_{j}\). \(r_{i}^{fa}\left(\Delta_{j}\right)\) indicates the conditional probability that no false alarm will occur under \(\Delta_{j}\) when no fault occurs, which is another sense of the \"correct\" diagnosis probability. \(r_{i}^{ta}\left(\Delta_{j}\right)\) denotes the conditional probability that \(f_{i}\) is correctly detected under \(\Delta_{j}\) when \(f_{i}\) occurs. \(w_{i}^{fa}\) and \(w_{i}^{ta}\) are the weights of the indicators, which can be adjusted according to the needs of the fault diagnosis, and \(w_{i}^{fa}+w_{i}^{ta}=1\). (4) Isolation Accuracy  
The isolation accuracy rate \(d(\Delta_{k})\) measures the accuracy of the system fault isolation under \(\Delta_{k}\):  
\[d(\Delta_{k})=\sum_{i=1}^{N_{f}}w_{i}d_{i}(\Delta_{k}) \tag{27}\]  
\[d_{i}(\Delta_{k})=w_{i}^{fb}r_{i}^{fb}(\Delta_{k})+w_{i}^{tb}r_{i}^{tb}(\Delta _{k}) \tag{28}\]  
\[r_{i}^{fb}(\Delta_{k})=P\Big{(}\varepsilon_{th,f_{ij}}(\Delta_{k})\geq \varepsilon_{f_{0}}(\Delta_{k})\Big{|}\,f=f_{0},\varepsilon_{th,f_{i}}(\Delta _{k})\geq\varepsilon_{f_{0}}(\Delta_{k})\Big{)} \tag{29}\]  
\[r_{i}^{tb}(\Delta_{k})=P\Big{(}\varepsilon_{f_{ij}}(\Delta_{k})\geq \varepsilon_{th,f_{ij}}(\Delta_{k})\Big{|}\,f=f_{i},\varepsilon_{f_{i}}( \Delta_{k})\geq\varepsilon_{th,f_{i}}(\Delta_{k})\Big{)} \tag{30}\]  
The larger \(d(\Delta_{k})\) is, the higher the isolation rate under the influence of \(\Delta_{k}\). In the above equation, \(d_{i}(\Delta_{k})\) denotes the isolation accuracy rate of \(f_{i}\) under \(\Delta_{k}\). \(r_{i}^{fb}(\Delta_{k})\) indicates the conditional probability that no false alarm occurs under \(\Delta_{k}\) when no fault occurs, which is another sense of the likelihood of \"correct\" isolation. \(r_{i}^{tb}(\Delta_{k})\) denotes the conditional probability that \(f_{i}\) is correctly isolated under \(\Delta_{k}\) when \(f_{i}\) occurs. \(w_{i}^{fb}\) and \(w_{i}^{tb}\) are the weights of the indicators, which may be altered to meet the requirements of the fault diagnostics, and \(w_{i}^{fb}+w_{i}^{tb}=1\). (5) Real-Time Detection Rate  
The real-time detection rate \(e\big{(}\Delta_{j}\big{)}\) characterizes the speed of fault detection under \(\Delta_{j}\):  
\[e\big{(}\Delta_{j}\big{)}=\sum_{i=1}^{N_{f}}w_{i}e_{i}\big{(}\Delta_{j}\big{)} \tag{31}\]  
\[e_{i}\big{(}\Delta_{j}\big{)}=\frac{t_{th,i}\big{(}\Delta_{j}\big{)}-t_{i} \big{(}\Delta_{j}\big{)}}{t_{th,i}\big{(}\Delta_{j}\big{)}} \tag{32}\]  
\[t_{i}\big{(}\Delta_{j}\big{)}=\left\{ \begin{array}{l}t_{dm,i}\big{(}\Delta_ {j}\big{)},t_{dm,i}\big{(}\Delta_{j}\big{)}<t_{th,i}\big{(}\Delta_{j}\big{)} \\ t_{th,i}\big{(}\Delta_{j}\big{)},t_{dm,i}\big{(}\Delta_{j}\big{)}\geq t_{th,i} \big{(}\Delta_{j}\big{)}\end{array} \right. \tag{33}\]  
The above equation \(e_{i}\big{(}\Delta_{j}\big{)}\) denotes the detection speed of \(f_{i}\) under \(\Delta_{j}\). \(t_{dm,i}\big{(}\Delta_{j}\big{)}\) is the average detection time of \(f_{i}\) in \(\Delta_{j}\). \(t_{th,i}\big{(}\Delta_{j}\big{)}\) is the shortest detection time of \(f_{i}\) in \(\Delta_{j}\), which characterizes the limit value of the diagnosis time.