## 2 Methods
### State Detection Based on FNN  
The maintenance activities performed on an HSV typically include corrective and preventive maintenance. Corrective maintenance operations are implemented after failure and usually have high costs associated with maintenance and replacement parts. Traditional preventive maintenance typically involves regular repair of the HSV according to specified standards and is unrelated to the condition of the valve. While these maintenance strategies usually improve the reliability of equipment, they are also significantly costly. Therefore, an approach to balancing the reliability and cost of HSVs is to monitor the condition and assess the health of HSV components, known as predictive maintenance (PdM). PdM is the maintenance policy that predicts the future state of equipment based on current data. The goal of PdM is to use the monitoring data to predict the RUL and potential faults of the equipment, avoid unexpected shutdown, and reduce the maintenance cost by maximizing the residual value of the equipment [31]. Therefore, the present study used neural network technology to detect the state of the HSV, whereas monitoring of the valve degradation performance was the basis for predicting its RUL.  
Computer fuzzy reasoning uses fuzzy rules to simulate human fuzzy logic. This method can be used to improve the accuracy of computer simulation. The FNN, formed by a combination of fuzzy reasoning and a neural network, increased the self-learning ability of the system [32; 33]. The FNN was based on a traditional neural network which fuzzifies the input and output signals. The topological structure of the FNN is shown in Figure 1c. The FNN is divided into five layers [34]:  
1. Input layer: The input layer node is connected with an input quantity \(X=(X_{1},X_{2},\ldots,X_{n})\), and \(n\) is the number of the input variables. The purpose of the input layer is to transfer the input directly to the next layer.
2. Fuzzification layer: This layer calculates the membership function of each input component. The Gaussian membership function is generally used, represented by Equation (3). \[\mu_{ij}=\exp\left[-\frac{\left(x_{i}-c_{ij}\right)^{2}}{\sigma_{ij}^{2}} \right],i=1,2,\ldots,n\ j=1,2,\ldots,m\] (3) where \(\mu_{ij}\), \(c_{ij}\) and \(\sigma_{ij}\) are membership function parameters.
3. Fuzzy inference layer: This layer calculates the applicability of each rule, i.e., activation strength, and can be represented by Equation (4). \[\alpha_{j}=\mu_{1j}(x_{1})\mu_{2j}(x_{2})\ldots\mu_{nj}(x_{n})\] (4)
4. De-fuzzification layer: This layer normalizes the input quantity, as shown in Equation (5). \[\overline{\alpha}_{j}=\alpha_{j}/\sum_{j=1}^{m}\alpha_{j}\] (5)(e) Output layer: The output of this layer is shown in Equation (6).  
\[y=\sum_{j=1}^{m}w_{j}\bar{\alpha}_{j} \tag{6}\]  
where \(w_{j}\) is the connection weight.  
The FNN learning process mainly includes structure adjustment and parameter learning. The FNN uses the gradient descent method to learn connection weight \(w_{j}\), the center \(c_{ij}\), and the membership function \(\sigma_{ij}\). This allows the FNN to obtain an FNN model with improved parameters.  
During this process, the objective function is defined as Equation (7).  
\[E=\frac{1}{2}(y_{\text{d}}-y)^{2} \tag{7}\]  
where \(y_{d}\) is the expected output of the system.  
Parameter learning is used to minimize expected objective function, and the parameter updating formula is shown in Equation (8).  
\[\left\{\begin{array}{l}w_{j}(t+1)=w_{j}(t)-\eta\partial E/\partial w_{j}\\ c_{ij}(t+1)=c_{ij}(t)-\eta\partial E/\partial c_{ij}\\ c_{ij}(t+1)=\sigma_{ij}(t)-\eta\partial E/\partial\sigma_{ij}\end{array}\right. \tag{8}\]  
where \(\eta\) is the learning rate.