# Backstepping Adaptive Neural Network Control for Electric Braking Systems of Aircrafts
## 3 Control Strategy
### The RBF Neural Network  
The structure of an RBF neural network is shown in Figure 3.  
In Figure 3, the RBF neural network consists of the input layer, the basis function layer (hidden layer), and the output layer [35; 36]. The input layer is used to complete the information transfer, the basis function layer is the respective field, and the output layer is the outputs of the RBF neural network. The numbers of the input layer and the output layer are usually determined by the characteristics of the control system. In this paper, there are three neurons in the input layer and one neuron in the output layer. A k-means clustering algorithm is used on selecting the number of the hidden layer and the corresponding control parameters of the neural network [37], and the number of neurons in the hidden layer is eventually determined to be five. The Gaussian function is chosen as the radial basis function, as shown in Equation (38):  
\[h_{i}(x)=\exp\!\left(-\frac{\left\|x-c_{i}\right\|^{2}}{2d_{i}^{2}}\right)\!i =1,2,\cdots M, \tag{38}\]  
where \(c_{i}\) is the center of the basis function,\(d_{i}\) is the scaling factor of the basis function, and \(h_{i}(\cdot)\) is the base Gaussian function. From Equation (37), it can be found that there are three unknown functions \(f_{1}\), \(f_{2}\) and \(f_{3}\), so we use three RBFs to approximate them as follows (39):  
\[\left\{ \begin{array}{l}f_{1}=W_{1}^{T}H_{1}+\varepsilon_{1}\\ f_{2}=W_{2}^{T}H_{2}+\varepsilon_{2}\\ f_{3}=W_{3}^{T}H_{3}+\varepsilon_{3}\end{array} \right., \tag{39}\]  
where \(W_{i}\) is the ideal neural network weight vector and \(W_{i}=\left[w_{1i}\ w_{2i}\ w_{3i}\ w_{4i}\ w_{5i}\right]^{T}\!,w_{Mi}(M=1,2,3,4,5)\) is the weight value, \(H_{i}\) is the radial basis vector, and \(H_{i}=\left[h_{1i}\ h_{2i}\ h_{3i}\ h_{4i}\ h_{5i}\right]^{T}\!,h_{Mi}(M=1,2,3,4,5)\) is the Gaussian function, \(\varepsilon_{i}\) is the approximation error,\(i=1,2,3\), and \(\left\|\varepsilon\right\|=\left\|\left[\varepsilon_{1}\varepsilon_{2}\varepsilon_ {3}\right]^{T}\right\|\leq\varepsilon_{N}\), \(\left\|W_{i}\right\|_{F}\leq W_{M}\) (\(\left\|\varepsilon\right\|\) is the Euclidean norm of \(\varepsilon\), \(\left\|W_{i}\right\|_{F}\) is the Forbenius norm of \(W_{i}\)). Define \(\hat{f_{1}}\), \(\hat{f_{2}}\) and \(\hat{f_{3}}\) as follows (40):  
\[\left\{ \begin{array}{l}\hat{f_{1}}=\hat{W}_{1}^{T}H_{1}\\ \hat{f_{2}}=\hat{W}_{2}^{T}H_{2}\\ \hat{f_{3}}=\hat{W}_{3}^{T}H_{3}\end{array} \right., \tag{40}\]  
where \(\hat{W}_{i}\) is the weight vector estimation. Similarly, define \(Z\), \(\hat{Z}\) and \(\overline{Z}\) as follows (41):  
\[Z=\left[ \begin{array}{cccc}W_{1}&&&\\ &&W_{2}&&\\ &&&W_{3}\end{array} \right]\|Z\|_{F}\leq_{M},\hat{Z}=\left[ \begin{array}{cccc} \hat{W}_{1}&&&\\ &&\hat{W}_{2}&&\\ &&\hat{W}_{3}\end{array} \right]\overline{Z}=Z-\hat{Z}. \tag{41}\]  
Figure 3: The RBF neural network structure. Design the Lyapunov function as:  
\[V=\frac{1}{2}\xi^{T}\xi+\frac{1}{2}\text{tr}\Big{(}\overline{Z^{T}}\Gamma^{-1} \overline{Z}\Big{)}, \tag{42}\]  
where \(V_{3}=\frac{1}{2}\xi^{T}\xi\), \(\|Z\|_{F}\) is the Forbenius norm of \(Z\), \(\Gamma\) is a positive definite matrix with proper dimensions, \(\Gamma=\left[ \begin{array}{cc}\Gamma_{1}&\\ &\Gamma_{2}\\ &\Gamma_{3}\end{array} \right]\), \(\xi=[\varepsilon_{1}e_{2}e_{3}]^{T}\). Design the RBF neural network weights adaptive law as follows:  
For practical purposes, the EMA is installed in the aircraft wheel as shown in Figure 4a, and the HIL test bench is shown in Figure 4b. First, we need to design the control parameters of the neural networks, including the cluster centers of the basis function, the scaling factors of the basis function, and the weight value vectors of the output layer. As mentioned above, a k-means clustering algorithm is adopted to calculate the cluster centers and the scaling factors, and the weight value vectors can be obtained by a pseudo-inverse method.