# Sequential-Fault Diagnosis Strategy for High-Speed Train Traction Systems Based on Unreliable Tests
## 3 Methodology
### Fault Diagnosis Reasoning Based on Test Reliability Dependency Matrix  
(1) Reasoning theory of fault diagnosis when testing reliability  
When the test is reliable, which refers to when the dependency matrix is determined, there are two kinds of test output. Assuming that the current fault set is \(X\), the fault set is divided into two fault subsets \(X_{p}\) and \(X_{r}\), which refer to the test result is binary output. If the test \(t_{j}\) passes, there are:  
\[X_{pj}=\left\{f_{i}|ft_{ij}=0,f_{i}\in X\right\} \tag{19}\]  
If the test fails, there are:  
\[X_{rj}=\left\{f_{i}|ft_{ij}=1,f_{i}\in X\right\} \tag{20}\]  
The value of \(f_{i}\) in the fault-test dependency matrix is divided into corresponding subsets. \(f_{i}\) must belong to one of the two subsets, which means that \(f_{i}\) cannot belong to both \(X_{pj}\) and \(X_{rj}\). (2) Reasoning theory for fault diagnosis when a test is unreliable  
When a test is unreliable, false alarms may occur if the test passes, and missed detection may occur if the test fails. The test output still divides the fault set into two fault subsets \(X_{p}\) and \(X_{r}\), but there are two other cases, namely:When the test \(t_{j}\) passes, there may be the following in set \(X_{p}\):  
\[X_{pj}=\left\{f_{i}|ft_{ij}=1,f_{i}\in X\right\} \tag{21}\]  
When the test \(t_{j}\) fails, there may be the following in set \(X_{r}\):  
\[X_{rj}=\left\{f_{i}|ft_{ij}=0,f_{i}\in X\right\} \tag{22}\]
3. Construction method of fault diagnosis strategy based on test information The heuristic search method is the most commonly used method and has high practicability. The heuristic search method is relatively simple to some extent and can reduce the steps in a traversal search and achieve rapid fault isolation. (1) Criteria for test selection In the process of fault diagnosis reasoning, how to determine the next test is the key problem to be solved. This refers to determining the reason for the criterion of fault diagnosis, which is also known as the evaluation function. According to information theory, when the test \(t_{j}\) is unreliable, we can conclude that, the sum of the information gain for each part of the system after the test is:  
\[ \begin{array}{l}I(F,t_{j})=-\left\{\frac{p_{j}\partial\left(X_{pj}\right)}{ \partial\left(X\right)}\log_{2}\frac{p_{j}\partial\left(X_{pj}\right)}{ \partial\left(X\right)}+\frac{p_{j}\partial\left(X_{jj}\right)}{\partial \left(X\right)}\log_{2}\frac{p_{j}\partial\left(X_{jj}\right)}{\partial\left(X \right)}\right\}-\\ \left\{\frac{\left(1-p_{j}\right)\partial\left(X_{pj}\right)}{\partial\left(X \right)}\log_{2}\frac{\left(1-p_{j}\right)\partial\left(X_{pj}\right)}{ \partial\left(X\right)}+\frac{\left(1-p_{j}\right)\partial\left(X_{jj}\right) }{\partial\left(X\right)}\log_{2}\frac{\left(1-p_{j}\right)\partial\left(X_{j j}\right)}{\partial\left(X\right)}\right\}\end{array}  \tag{23}\]  
The proportion of failures that pass the test is:  
\[\frac{p_{j}p\left(X_{pj}\right)}{p\left(X\right)} \tag{24}\]  
The proportion of failures that fail the test is:  
\[\frac{\left(1-p_{j}\right)p\left(X_{fj}\right)}{p\left(X\right)} \tag{25}\]  
where \(p_{j}\) represents the reliability of test J, \(\partial\left(X\right)\) represents the sum of the detection probabilities for fault sets before the test, \(\partial\left(X_{pj}\right)\) represents the sum of detection probabilities for fault sets passing the test, and \(\partial\left(X_{fj}\right)\) represents the sum of detection probabilities for fault sets that fail the test. The evaluation function \(k^{*}\) can be designed. Considering the maintenance cost, we specify \(I(F,t_{j})\) as the evaluation function of fault diagnosis reasoning. When \(I(F,t_{j})\) is larger, the greater the amount of information obtained by the test. Therefore, we select the test that can maximize \(I(F,t_{j})\) as the next test, and make the test number \(k^{*}\).