## 3 Multi-Motor Power Distribution Control Strategy
### Neural Network Regulator Design  
With the rapid advancement of modern industrial automation, energy-saving control methods for belt conveyor systems have become a focal point of research. The Radial Basis Function (RBF) neural network, known for its superior approximation capabilities and rapid learning speed, has been widely applied in energy consumption prediction and control [20]. However, despite the good performance of RBF-based energy consumption prediction methods in belt conveyor systems, there remain some limitations and challenges in practical applications [21]. This study optimizes the RBF neural network model, successfully reducing its complexity and computational burden. Compared to the standard RBF neural network presented in [22], the proposed model reduces the number of parameters by 44.4% (from 50 to 28), while maintaining a high level of predictive accuracy (R\({}^{2}\) increased from 0.9300 to 0.9432). Additionally, through algorithmic optimization, the computational complexity of the model has been reduced from O(n\({}^{3}\)) to O(n\({}^{2}\)), significantly enhancing the efficiency of large-scale data processing. The training time was reduced from 150 s to 120 s, and the prediction time was shortened by 20%. To further validate the superiority of the proposed model, it was applied to motor power prediction in belt conveyor systems and compared with the method in [22; 23]. The results demonstrated that the proposed model not only improved predictive accuracy but also performed excellently in capturing dynamic changes in motor load. The power prediction results presented in Figure 3 clearly illustrate the improvements in the dynamic response and energy-saving efficiency of the proposed method compared to the method in [22]. Figure 4 further presents a bar chart comparison between the proposed model and the method in [23], covering multiple metrics including training time, model complexity, predictive accuracy, and prediction time. This paper designs an RBF neural network as multi-input and multi-output. \(x=\left[x_{1},x_{2},\ldots,x_{n},x_{n+1}\right]^{T}\) is the network input and \(x_{i}\) is the load power value of each motor (\(i=1,2,\ldots,n\)). \(x_{n+1}\) is the optimal number of motors' output by the fuzzy controller. The RBF neural network structure is shown in Figure 5.  
Figure 4: Comparative analysis of training time, complexity, accuracy, and prediction time [23]. Figure 3: Comparison of the predicted power output of each motor for belt conveyors [22]. The output of the neuron in the hidden layer is as follows:  
\[h_{j}=\exp(-\frac{||x-c_{i}||^{2}}{2b_{j}^{2}}),j=1,2,\cdots m \tag{9}\]  
where \(c_{j}=[c_{j1},c_{j2},\ldots,c_{jn+1}]\) is the neutral point vector map of the \(j\) hidden layer neuron, and \(b_{j}>0\) is the Gaussian function width of the hidden layer neuron \(j\). The actual output of the RBF network shows the probability that each motor needs to keep running, which is the following:  
\[y_{i}=\sum_{j=1}^{n}p_{ij}h_{j},\ i=1,2,\cdots n \tag{10}\]  
where \(p_{i}=[p_{i1},p_{i2},\ldots,p_{im}]^{T}\) represents the network weight associated with the \(i\) output. The error function is defined as the mean square error:  
\[E=\frac{1}{2}\sum_{i=1}^{n}\left(Y_{i}-y_{i}\right)^{2} \tag{11}\]  
where \(Y_{i}\) is the expected value. According to the back-propagation error of the BP network algorithm, the gradient descent method and the momentum method are used, and the moving amplitude of the independent variable in each direction depends not only on the current gradient but also on the consistency of each gradient in each direction in the past. The learning correction formula of the relevant parameters of the RBF neural network is as follows:  
\[\left\{ \begin{array}{l}\triangle p_{ij}=\frac{\partial E}{\partial p_{ij}}=( Y_{i}-y_{i})h_{j}\\ p_{ij}(k)=p_{ij}(k-1)+\eta\Delta p_{ij}(k)+\alpha\eta\Delta p_{ij}(k-1)\\ \end{array} \right. \tag{12}\]  
\[\left\{