## 2 General Structure of the Methodology
### Fuzzy Fusion Estimation Module
#### 2.3.2 Fuzzy Controller Design Ideas for Integrated Rail Surface Conditions  
The rail surface condition (dry, wet, oil contaminated), the ratio of the similarity between the current rail surface and standard rail surface 1 to the sum of all similarities (\(u_{1}\)), and the ratio of the similarity between the current rail surface and standard rail surface 4 to the sum of all similarities (\(u_{4}\)) are selected as the three input variables. The four outputs are the correction coefficients (\(k_{1}\), \(k_{2}\), \(k_{3}\), \(k_{4}\)) for the similarity values, adjusted based on the identified rail surface condition. The values of \(u_{1}\) and \(u_{4}\) are used to determine which standard rail surface the current rail surface is more closely associated with. If the identified bias differs across standard rail surfaces, the corresponding correction coefficients will also vary accordingly, enhancing the robustness of the system. Among these inputs, the rail surface condition is a categorical state variable, while \(u_{1}\) and \(u_{4}\) are continuous numerical values. To align with fuzzy logic processing, \(u_{1}\) and \(u_{4}\) must be fuzzified before logical inference. The rail surface condition is classified into three discrete states: dry, wet, and oil contaminated. The values of \(u_{1}\) and \(u_{4}\) are divided into five fuzzy levels: very large, large, medium, small, and very small. The correction coefficients for rail surface conditions are categorized into four fuzzy subsets: decreased, unchanged, increased, and significantly increased. The inclusion of the \"significantly increased\" subset accounts for the fact that oil contamination has a more severe impact on adhesion compared to wet conditions. Consequently, when the rail surface is identified as oil contaminated, a larger correction factor is applied. The fuzzy control rules governing these corrections are presented in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Input Quantity**} & \multicolumn{4}{c}{**Output Quantity**} \\ \hline
**Creep and** & **Adhesion** & \(x_{1}\) & \(x_{2}\) & \(x_{3}\) & \(x_{4}\) \\
**Slide Speed** & **Coefficient** & & & & \\ \hline Small & 1 & Very similar & Similar & Not similar & Not similar \\ Small & 2 & Similar & Very similar & Generally similar & Not similar \\ Small & 3 & Not similar & Generally similar & Very similar & Similar \\ Small & 4 & Not similar & Not similar & Similar & Very similar \\ Medium & 1 & Very similar & Generally similar & Not similar & Not similar \\ Medium & 2 & Similar & Very similar & Generally similar & Generally similar \\ Medium & 3 & Not similar & Similar & Very similar & Similar \\ Medium & 4 & Not similar & Generally similar & Similar & Very similar \\ Large & 1 & Very similar & Generally similar & Not similar & Not similar \\ Large & 2 & Generally similar & Very similar & Generally similar & Not similar \\ Large & 3 & Not similar & Generally similar & Very similar & Similar \\ Large & 4 & Not similar & Not similar & Generally similar & Very similar \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of logical inference rules for unintegrated rail surface states.