## 3 Design of Vehicle Stability Controller
### Design of RBFNN-SMC Controller  
As for synchronous control of yaw rate and sideslip angle, we design an SMC:  
\[s=(\omega_{r}-\omega_{rd})+\lambda(\beta-\beta_{d}), \tag{31}\]  
where \(\lambda\) is the weight coefficient of joint control of yaw rate and sideslip angle. During the convergence of the control system, the sliding mode function should meet: when \(s\to 0\), \(\overset{\bullet}{s}\to 0\), namely: \(\omega_{r}-\omega_{rd}\to 0,\beta-\beta_{d}\to 0,\,\overset{\bullet}{ \omega_{r}}-\omega_{rd}^{\bullet}\to 0,\overset{\bullet}{\beta}-\overset{ \bullet}{\beta_{d}}\to 0\), which ensures the convergence of the system. Derivation of Equation (31) yields:  
\[\overset{\bullet}{s}=(\overset{\bullet}{\omega_{r}}-\overset{\bullet}{ \omega_{rd}})+\lambda(\overset{\bullet}{\beta}-\overset{\bullet}{\beta_{d}}). \tag{32}\]  
Substituting it into Equation (26), we have:  
\[\overset{\bullet}{s}= \begin{array}{l}\left[\frac{a^{2}C_{f}+b^{2  
The SMC law of additional yaw moment is designed:  
\[u=\frac{-1}{g(\omega_{r},\beta,t)}\left[f(\omega_{r},\beta,t)+u_{\text{eq}}+u_{s} \right], \tag{36}\]  
where \(u_{\text{eq}}=k_{1}s\), \(u_{\text{eq}}\) is the equivalent term; \(u_{s}=\eta\text{sgn}(s)\), \(u_{s}\) is the robust term, \(\eta>d\). By substituting Equation (36) to Equation (34), then:  
\[\overset{\bullet}{s}=d(\omega_{r},\beta,t)-\eta\text{sgn}(s)-k_{1}s. \tag{37}\]  
Namely:  
\[\overset{\bullet}{s}\quad=d(\omega_{r},\beta,t)s-\eta|s|-k_{1}|s| \tag{38}\] \[\leq d(\omega_{r},\beta,t)s-\eta|s|<0\]  
Since \(d(\omega_{r},\beta,t)\) is the uncertainty disturbance, its value is affected by multiple factors. When \(d(\omega_{r},\beta,t)\) is large, the robustness coefficient \(\eta\) of the controller is also large, which will lead to severe vibration. Hence, to ensure the system vibration is small, we adopt an approximation function to approximate the nonlinear disturbance \(d(\omega_{r},\beta,t)\). As for a nonlinear approximation problem, we combine the merits of RBFNN and fuzzy systems, and integrate them to approximate the nonlinear uncertainty \(d(\omega_{r},\beta,t)\). The structure of a FRBFNN is shown in Figure 4, which consists of an input layer, a fuzzification layer, a fuzzy reasoning layer, and an output layer. After the input components are entered into the input layer, they are transferred to the fuzzification layer, and after connection with the fuzzy reasoning layer, the fuzzy rule matching is finished before fuzzy computation. Finally, the nodes in the output layer are weighed and outputted. The first layer: the input layer. The input and output of each node \(i\) in this layer can be expressed as:  
\[h_{1}(i)=x_{i}. \tag{39}\]  
The second layer: the fuzzification layer. Each node of this layer has a membership function, which is a Gaussian basis function. For the \(j\)-th node:  
\[\left\{ \begin{array}{l}h_{2}(i,j)=\exp\Bigl{(}net_{j}^{2}\Bigr{)}\\ net_{j}^{2}=\frac{\|x-c_{j}\|^{2}}{b_{j}^{2}}\end{array} \right., \tag{40}\]  
Figure 4: Fuzzy radial basis function neural network structure diagram. where \(c_{ij}\) is the central vector value of the \(j\)-th neuron in the hidden layer; \(b_{j}>0\) is the Gaussian basis function width of neuron \(j\) in the hidden layer. The third layer: the fuzzy reasoning layer. This layer is connected to the fuzzification layer, so the fuzzy-rule matching and between-node fuzzy operation are realized. The output from each node \(j\) is the product of the inputted signals of this node:  
\[h_{3}(j)=\prod\limits_{j=1}^{N}h_{2}(i,j), \tag{41}\]  
where \(N=\prod\limits_{i=1}^{n}N_{i}\), \(N_{i}\) is the number of \(i\)-th membership functions in the input layer, and the number of nodes in the fuzzification layer. The fourth layer: the output layer. The output in each node of this layer is the weighted sum of the inputted signals:  
\[h_{4}(l)=w\bullet h_{3}=\sum\limits_{j=1}^{N}w(l,j)\bullet h_{3}(j), \tag{42}\]  
where \(l\) is the number of nodes in the output layer; \(w\) is the connection matrix between the output nodes and the third-layer nodes.