## 3 Resistance Analysis Based on RBFNN  
When the train operates under the actual route conditions with large slopes and small curves, the additional resistance and unknown disturbances caused by the complex route may be large, leading to a low accuracy of train control and affecting the safety of train operation. Therefore, RBFNN is used to learn and estimate the additional resistance and unknown disturbance in the train dynamics model, which mainly solves the problem of the large position and speed tracking error caused by factors such as dynamic model simplification, model parameter uncertainty, and external environment change in order to enhance the adaptability and anti-interference performance of the FAO. RBFNN is a locally convergent forward network, which has simple structure, fast learning speed, and good generalization ability. Theoretically, it can approximate any smooth function with any precision, which can be regarded as a universal approximator [18; 19]. There are many research results in the adaptive control of nonlinear systems [20; 21; 22]. The structure of RBFNN is shown in Figure 1, which is composed of three layers of a forward network. The input layer is composed of signal source nodes. The number of neurons in the hidden layer depends on the exact requirements of the approximation function. The output layer outputs the response to the input layer. The mapping from input to output is nonlinear, while the mapping from hidden layer to output space is linear, which speeds up learning and effectively avoids the local minimum problem [23; 24; 25; 26]. The output of the output layer is the linear weighted sum of the output signals of the hidden layer, and the 'weight' represents the adjustable parameters of the network structure. By adjusting the appropriate output weight, the RBFNN can realize the approximation of the unknown nonlinear additional resistance and disturbance function. In Figure 1, \(\mathbf{x}=[x_{1},x_{2},\cdots,x_{n}]^{\mathrm{T}}\subset R^{n}\) is the network input, \(n\) is the dimension of the input vector, \(\mathbf{W}=[w_{1},w_{2},\cdots,w_{m}]^{\mathrm{T}}\) is an adjustable weight parameter vector, \(m\) is the number of neurons in the hidden layer, \(\mathbf{h}(\mathbf{x})=[h_{1}(x),h_{2}(x),\cdots,h_{m}(x)]^{\mathrm{T}}\) is the output vector of Gaussian function for the hidden layer, and \(h_{j}(\mathbf{x})\) is the output of the \(j\)-th neuron in the hidden layer. \(h_{j}(\mathbf{x})\) can be expressed as follows:  
\[h_{j}(\mathbf{x})=\exp\left(-\frac{\left\|\mathbf{x}-c_{j}\right\|^{2}}{2b_{j}^{2}} \right)j=1,2,\cdots,m, \tag{5}\]  
where \(\|\cdot\|\) is a Euclidean norm, and \(c_{j}\) and \(b_{j}\) are the center parameter and width parameter of the Gaussian function of the j-th neuron in the hidden layer. The width vector of the  
Figure 1: Structure of RBFNN. hidden layer Gaussian function is \(\mathbf{b}=[b_{1},b_{2},\cdots,b_{m}]^{\mathsf{T}}\), and \(b_{j}>0\). The coordinate vector of the center point of the hidden layer Gaussian function is as follows:  
\[\mathbf{c}= \begin{bmatrix}c_{j1},c_{j2},\cdots,c_{ji},\cdots,c_{jn}\end{bmatrix} ^{ \mathsf{T}}\quad i=1,2,\cdots,n. \tag{6}\]  
RBFNN can approximate the unknown nonlinear additional resistance and interference function with any precision in the compact working domain \(\Theta_{\mathbf{x}}\subset R^{n}\). \(f(\mathbf{x})\) is as follows:  
\[f(\mathbf{x})=\mathbf{W}^{*\mathsf{T}}\mathbf{h}(\mathbf{x})+\varepsilon(\mathbf{x}), \tag{7}\]  
where \(\mathbf{W}^{*}\) is the weight vector that minimizes \(\varepsilon(\mathbf{x})\) among all the estimated weights, in which  
\[\mathbf{W}^{*}=\arg\min_{W\in R^{m}}\left\{\sup_{\mathbf{x}\in\Theta_{\mathbf{x}}}\left|f( \mathbf{x})-\mathbf{W}^{\mathsf{T}}\mathbf{h}(\mathbf{x})\right|\right\}, \tag{8}\]  
where \(\varepsilon(\mathbf{x})\) is a bounded approximation error; that is, there is a small normal \(\varepsilon^{*}\) number satisfying \(|\varepsilon(\mathbf{x})|\leq\varepsilon^{*}\).