## 3 Proposed Control Methods
### Snn  
SNNs are an important algorithm in the field of machine learning, which uses mathematical modeling techniques to build versatile approximators. These networks effectively model and predict intricate data relationships while approximating complex functional relationships in a highly non-linear manner. SNNs primarily consist of a single hidden layer architecture with an input layer and an output layer. According to the method of learning connection weights, SNNs follow the universal learning process of neural networks and can be categorized into two types: those utilizing backward learning (BL) approaches and those implementing forward learning (FL) methodologies. As previously mentioned, SNNs have fewer parameters, simpler structure, fewer hidden layers, and lower overfitting risk than DNNs. The main characteristic that distinguishes SNNs from DNNs is their fewer number of hidden layers and relatively simpler structure. However, SNNs still follow the backpropagation (BP) training mechanism, maintaining consistency with DNNs and traditional BP neural networks in terms of basic structure and training principles. The basic structure and theoretical expressions of SNNs used in this paper are described below. In Figure 3, each of the three layers (input, hidden, and output) contains \(n\), \(m\), and \(l\) neurons respectively. The network's output vectors comprise \(x_{i}\) (\(i\) = 1, 2, \(\ldots\), \(n\)) in the input layer, \(y_{j}\) (\(j\) = 1, 2, \(\ldots\), \(m\)) in the hidden layer, and \(\omega_{k}\) (\(k\) = 1, 2, \(\ldots\), \(l\)) in the output layer, with \(d_{k}\) denoting the target output. The interconnections between layers are characterized by neuron weights \(v_{ij}\) and \(w_{jk}\), while the output and hidden layers utilize thresholds \(\alpha_{k}\) and \(\theta_{j}\), respectively. The following equation applies to the hidden layer:  
\[y_{j}=f(net_{j})=f(\sum_{i=1}^{n}v_{ij}x_{i}) \tag{10}\]  
For the output layer, the equation is as follows:  
\[o_{k}=f(net_{k})=f(\sum_{j=1}^{m}w_{jk}y_{j}) \tag{11}\]  
Within these equations, the unipolar Sigmoid function is employed as the transformation function \(f(x)\):  
\[f(x)=1/(1+e^{-x}) \tag{12}\]  
When the network output \(O\) differs from the expected output \(D\), an output error \(E\) is defined as follows:  
\[E=\frac{1}{2}(\mathbf{D}-\mathbf{O})^{2}=\frac{1}{2}\sum_{k=1}^{l}\left(d_{k}-o_{k} \right)^{2}=\frac{1}{2}\sum_{k=1}^{l}\left[d_{k}-f(\sum_{j=1}^{m}w_{jk}y_{j}) \right]^{2} \tag{13}\]  
Figure 3: SNN topology. The output error \(\mathbf{E}\) can also be expanded further into the hidden layer:  
\[\mathbf{E}=\frac{1}{2}\sum_{k=1}^{l}\left\{d_{k}-f[\sum_{j=1}^{m}w_{jk}f(\sum_{i=1}^{ n}v_{ij}x_{i})]\right\}^{2} \tag{14}\]  
Since SNNs use the standard BP training mechanism to adjust their learning rate, weights, and thresholds, this part will not be elaborated further. Neural network training follows a systematic process that begins with data preparation and network design. After parameter initialization, the network uses forward propagation to generate predictions. A loss function measures the difference between predictions and actual values, while the backpropagation algorithm updates network parameters. This process repeats until performance reaches a satisfactory level, with an independent test set verifying the model's ability to generalize. Based on validation results, the network structure and hyperparameters are then fine-tuned to enhance overall performance. The flowchart for SNN fitting is shown in Figure 4.  
Here, data fitting with SNNs is a critical core step, but a challenge remains. That challenge is the time series nature of the data, specifically, whether a one-to-one correspondence exists between data points for control systems. The requirements for this real-time nature  
Figure 4: Flowchart for SNN fitting. and the degree of matching between them are relatively high. Next, further data processing will be conducted to achieve high-precision predictive control.