# Motor On-Line Fault Diagnosis Method Research Based on 1D-CNN and Multi-Sensor Information
## 3 Multi-Source Homogeneous Sensor Information Data Layer Fusion  
To fully use the subtle information and features of the original sensor data, this section combined correlation and adaptive weighting methods to perform the data layer fusion of the multi-source homogeneous sensor information. Firstly, the correlation signal energy function was constructed and normalized. Secondly, according to the conditions of the adaptive weighting method (i.e., the total mean square error was the minimum), the optimal weighting factor among sensors was adjusted adaptively [16], the fused signal reached the optimum, and the dynamic fusion of multi-source homogeneous sensor information is realized.  
The flow chart of data layer fusion, multi-source homogeneous sensor information based on the correlation adaptive weighting method, is shown in Figure 3.  
Figure 2: Multi-sensor fusion framework of the motor.  
Let \(X(n)\) and \(Y(n)\) to be deterministic signals collected by sensors \(X\) and \(Y\), then, the correlation function between signal \(X(n)\) and \(Y(n)\) could be acquired as shown in Equation (1).  
\[R_{XY}=\sum_{n=0}^{\infty}X(n)\cdot Y(n) \tag{1}\]  
Suppose there were m homogeneous sensor signals \(Z_{1}(n),Z_{2}(n),\ldots,Z_{m}(n)\), the cross-correlation function between any two signals was shown as Equation (2).  
\[R_{Z_{i}Z_{i}}(t)=\frac{1}{n-t}\sum_{t_{0}=1}^{n-1}Z_{i}(t_{0})\cdot Z_{j}(t_{ 0}+t) \tag{2}\]  
where \(n\) was the number of data points of each signal, and \(t=0,1,2,\ldots,n-1\) was the time series of the signal. The correlation energy of the discrete signals between two sensors is shown in Equation (3).  
\[E_{ij}=\sum_{t=0}^{n-1}[R_{ij}(t)]^{2} \tag{3}\]  
The correlation energy matrix of discrete signals between two sensors is shown in Equation (4).  
\[E=\begin{bmatrix}1&E_{12}&\cdots&E_{1m}\\ E_{21}&1&\cdots&E_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ E_{m1}&E_{m2}&\cdots&1\end{bmatrix} \tag{4}\]  
The correlation signal energy between the \(i\)th sensor and all homogeneous sensors is shown in Equation (5).  
\[E_{i}=\sum_{j=1,j\
eq i}^{m}E_{ij} \tag{5}\]  
After normalizing \(E_{i}\), the normalized signal of correlation energy could be acquired, as shown in Equation (6).  
\[R_{i}(n)=\frac{Z_{i}(n)}{\sqrt{E_{i}}} \tag{6}\]  
where \(Z_{i}(n)\) was the data signal sequence collected by the \(i\)th sensor within time \(T\), and \(R_{i}(1),R_{i}(2),\ldots,R_{i}(n)\) were the correlation energy normalized discrete signals.  
Let \(\sigma_{1},\sigma_{2},\ldots,\sigma_{m}\) to be the mean square error of \(m\) sensor signals, respectively, after correlation energy normalization. Let \(T\) to be the true value after multi-sensor data fusion. Let \(W_{1},W_{2},\ldots,W_{m}\) to be the weighting factor of weighted fusion of each sensor, respectively. Then, the gross mean square error of \(R_{i}(n)(i=1,2,\ldots,m)\) was shown as Equation (7).  
\[\sigma^{2}=E[\sum_{i=1}^{m}W_{i}^{2}(R-R_{i})^{2}]=\sum_{i=1}^{m}W_{i}^{2} \sigma_{i}^{2} \tag{7}\]  
Figure 3: Data layer fusion flow chart of multi-source homogeneous information.  
The weighting factor of each sensor satisfied Equation (8).  
\[\sum\limits_{i=1}^{m}W_{i}=1 \tag{8}\]  
To make the weighting factor reach the optimal state and the total mean square error reach the minimum value, the minimum total mean square error was shown as Equation (9).  
\[\sigma_{\min}^{2}=\sum\limits_{i=1}^{m}\frac{1}{\sigma_{i}^{2}} \tag{9}\]  
The optimal weighting factor could be obtained by combining Equations (7)-(9).  
According to Equation (10), the multiple sensors signal fused by correlation weighted average could be acquired, as shown in Equation (11).  
\[{W_{i}}^{*}=\frac{1}{\sigma_{i}^{2}\sum\limits_{i=1}^{m}\sigma_{i}^{2}} \tag{10}\]  
\[T(n)=\sum\limits_{i=1}^{m}{W_{i}}^{*}R_{i}(n) \tag{11}\]