## 3 The Establishment of Self-Architected Fuzzy Calibration Network
### Construction of the Calibration Network  
In reality, there are errors in the sensor installation and data collection, which lead to subsequent displacement reconstruction errors. In this paper, the self-architected fuzzy network (SAFN) algorithm is used to calibrate the reconstruction errors. The algorithm uses the triangular membership functions (MF), which independently increases the membership functions and fuzzy rules together with adjusting their distributions, thereby improving the structure of the fuzzy system. Based on the cubic B-splines, large-scale data is obtained, and a further fuzzy network is derived to approximate the measured strain and reconstruction error.  
The generation of self-architected fuzzy networks can be divided into three parts: (1) adding MF and generating the rules, (2) establishing self-adapting fuzzy rules and consequent parts, and (3) preserving fuzzy networks [24]. The algorithm flowchart is shown in Figure 6.  
1. Adding MF and generation rules (1) Error criterion  
The root-mean-square error is used to describe the system error, and its calculation is given by Equation (16). In the equation, \(y(k)\) represents the output value from the fuzzy network, and \(y_{d}(k)\) signifies the reconstruction deformation error. Hence, the system error \(RMSE_{2}\) can be calculated. The symbol \(E_{r}\) represents the error threshold in the training phase. If \(RMSE_{2}>E_{r}\), the membership function needs to be increased at the input.  
\[RMSE=\sqrt{\frac{\sum_{k=1}^{N}\left(y(k)-y_{d}(k)\right)^{2}}{N}} \tag{16}\]  
(2) Completeness criteria  
For any input variable \(x_{j}\) in the working range, the current value \(x_{j}(k)\) can help activate an MF, and the maximum value for \(\mu_{m}\left(x_{j}(k)\right)\) of the obtained membership degree is either  
Figure 6: Flowchart of the self-structuring fuzzy network algorithm.  
greater than or equal to the preset \(\beta\) value. If \(\mu_{m}\left(x_{j}(k)\right)<\beta\), MF needs to be increased or is otherwise unchanged.
2. The consequence of self-adaptive rules  
Based on the current estimated output value and the expected output value from the fuzzy network, the rule consequences are adjusted. At \(k\) moment, the expression for adjusting the consequence of the \(j\)-th rule \(\alpha_{j}(k)\) is as follows:  
\[\Delta\alpha_{j}(k)=\gamma\cdot\mu_{j}(k-1)\cdot(r(k-1)-y(k)), \tag{17}\]  
where \(\mu_{j}(k-1)\) represents the activation of the \(j\)-th rule at time \(k-1\), \(r(k-1)\) represents the estimated displacement that is input to the network at the previous time, and \(y(k)\) is the calibrated displacement error output from the fuzzy network at the current time. The value of \(\gamma\) is artificially adjusted to change the speed of the self-adaptive process of the rule consequence.
3. Fixing the rules and obtaining the fuzzy network  
After adding the data, the estimated output error from the network for all training input values becomes very small. For \(E_{FN}<E_{r}\), it means that the autonomous architecture stage of the network is completed. The network training is then stopped, and the fuzzy rules in the current network are saved.  
After expanding the samples of strain and reconstruction error through the cubic B-spline theory, the expanded samples are used to train the fuzzy network and generate the fuzzy calibration network. In engineering practice, when the real-time measured strain \(\varepsilon(x_{i},\theta_{i},\beta_{i})\) is input into the fuzzy calibration network, the calibrated displacement error value can be presented in real-time. After that, the accurately reconstructed displacements will be obtained.