## 3 Design of Vehicle Stability Controller
### Design of RBFNN-SMC Controller
The structure of the FRBFNN is 2-25-25-1 (input layer- fuzzification layer- fuzzy reasoning layer). Let the input be \(x=\left[s,s\right]\). Let \(\overset{\wedge}{d}(t)=h_{4}(l)\), then \(\overset{\wedge}{d}(t)\) and \(d(\omega_{r},\beta,t)\) are the network output value and the actual value respectively, so the approximation error of the network is:  
\[\varepsilon_{1}(t)=d(\omega_{r},\beta,t)-\overset{\wedge}{d}(t). \tag{43}\]  
The adjustable parameters are calibrated using the gradient descent method. The target function is defined as:  
\[E=\frac{1}{2}\varepsilon(t)^{2}. \tag{44}\]  
The learning algorithm of the FRBFNN is:  
Adjust the weights of the output layer:  
\[\Delta w(t)=-\phi\frac{\partial E}{\partial w}=-\phi\frac{\partial E}{\partial w }\overset{\wedge}{\partial\overset{\wedge}{d}}\frac{\partial\overset{\wedge}{ \partial}}{\partial w}. \tag{45}\]  
The learning algorithm of the output layer is:  
\[ \begin{array}{ll}w(t)&=w(t-1)+\Delta w(t)\\ &+a[w(t-1)-w(t-2)]\end{array}  \tag{46}\]  
where \(\phi\) is the learning rate, \(\phi\in[0,1]\), and \(\alpha\) is the momentum factor, \(\alpha\in[0,1]\). The membership function parameters of the input layer are adjusted as follows:  
\[\left\{ \begin{array}{ll}\Delta c_{ij}=-\phi\frac{\partial E}{\partial c_{ij }}=-\phi\frac{\partial E}{\partial int_{j}^{2}}\frac{\partial int_{j}^{2}}{ \partial c_{ij}}=-\phi\delta_{j}^{2}\frac{2\left(x_{j}-c_{ij}\right)}{b_{j}^{2} }\\ \Delta b_{j}=-\phi\frac{\partial E}{\partial b_{j}}=-\phi\frac{\partial E}{ \partial int_{j}^{2}}\frac{\partial int_{j}^{2}}{\partial b_{j}}=-\phi\delta_ {j}^{2}\frac{2\left(x_{j}-c_{ij}\right)}{b_{j}^{2}}\end{array} \right., \tag{47}\]where:  
\[\delta_{j}^{2}=\frac{\partial E}{\partial net_{j}^{2}}=-\varepsilon(t)\frac{ \overset{\wedge}{\partial d}(t)}{\partial net_{j}^{2}}. \tag{48}\]  
The learning process of membership function parameters is:  
\[\left\{ \begin{array}{rl}c_{ij}(t)&=c_{ij}(t-1)+\Delta c_{ij}(t)\\ &+a\left[c_{ij}(t-1)-c_{ij}(t-2)\right]\\ b_{j}(t)&=b_{j}(t-1)+\Delta b_{j}(t)\\ &+a\left[b_{j}(t-1)-b_{j}(t-2)\right]\end{array} \right.. \tag{49}\]  
From the above equation, the output of the FRBFNN system is:  
\[\overset{\wedge}{d}(t)=\sum_{j=1}^{N}w(l,j)\bullet h_{3}(j)=W^{*T}h(x), \tag{50}\]  
where \(W^{*}\) is the ideal network weight. Then:  
\[\left\{ \begin{array}{rl}d(\omega_{r},\beta,t)=\overset{\wedge}{d}(t)+ \varepsilon(t)=W^{*T}h(x)+\varepsilon(t)\\ d(\omega_{r},\beta,t)-\overset{\wedge}{d}(t)&=W^{T}h(x)-W^{*T}h(x)+ \varepsilon(t)\\ &=W^{T}h(x)+\varepsilon(t)\end{array} \right., \tag{51}\]  
where \(\overset{\sim}{W}=W-W^{*}\). In all, the additional yaw moment sliding mode control rate based on the FRBFNN is:  
\[u=\frac{-1}{g(\omega_{r},\beta,t)}\bigg{[}f(\omega_{r},\beta,t)+u_{\text{eq}} +u_{s}+\overset{\wedge}{d}(t)\bigg{]}. \tag{52}\]  
Substituting the above equation into Equation (34), we have:  
\[\overset{\bullet}{s}=-k_{1}s-\eta sgn(s)+d(\omega_{r},\beta,t)-\overset{ \wedge}{d}(t). \tag{53}\]  
With the above controller, the Lyapunov stability of the system is proved, and then a  
In all, when the system is under the AFRBF-SMC, the condition that the system meets Lyapunov stability is \(\varepsilon<\eta\). The approximation error \(\varepsilon\) of the FRBFNN nonlinear estimator is far smaller than the upper boundary \(d\) of the disturbance term. Compared with SMC, the controller with additional yaw moment and based on AFRBF-SMC only needs a very small positive number \(\varepsilon\). While the system meets convergence, it decreases slip mode vibration.