# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 4 ASMO and KF
\begin{array}{ll}\dot{V}_{1}&=e_{1}\dot{e}_{1}+\frac{1}{\rho_{1}}\widetilde{ L}_{1}\dot{\hat{L}}_{1}\\ &=e_{1}[-A_{11}e_{1}-A_{12}e_{2}-\hat{L}_{1}\text{sign}(e_{1})]+\frac{1}{\rho_ {1}}\widetilde{L}_{1}\dot{\hat{L}}_{1}\\ &\leq e_{1}[-A_{11}e_{1}-A_{12}e_{2}]-\hat{L}_{1}|e_{1}|+(\hat{L}_{1}-L_{1})| e_{1}|\\ &\leq e_{1}[-A_{11}e_{1}-A_{12}e_{2}]-L_{1}|e_{1}|.\end{array}  \tag{33}\]  
Thus, by considering Equation (27):  
\[\dot{V}_{1}\leq 0. \tag{34}\]  
Similarly, the Lyapunov function \(V_{2}\) for yaw rate convergence error \(e_{2}\) and adaptive gain convergence error \(\widetilde{L}_{2}=\hat{L}_{2}-L_{2}\) can be written as:  
\[V_{2}=\frac{1}{2}\varepsilon_{2}^{2}+\frac{1}{2\rho_{2}}\widetilde{L}_{2}. \tag{35}\]  
The time derivative of Equation (35) will asymptotically converge to zero, \(\dot{V}_{2}\leq 0\), by considering \(\dot{L}_{2}=0\) and \(L_{2}>\max(|a_{12}e_{1}|+|a_{22}e_{2}|)\). **Remark 1**. : _In the practical implementation, the direct strapdown of lateral acceleration may incorporate the small continuous noise to the lateral velocity that can diverge the ASMO estimation over time. Therefore, to deal with the issue, a lateral velocity-based damping term [40] is added to cancel out the incremental noise. Now Equation (24) can be written as:_  
\[\dot{y}(t)=\dot{y}(t-1)(1-\sigma)+\int(a_{y,sensor}-V_{x}\dot{\hat{y}})dt, \tag{36}\]_where \(\sigma>0\) is an adjustable small damping parameter._  
**Remark 2**. : _The designed ASMO may encounter high-frequency chattering due to the discontinuous signum function \(\mathrm{sign}()\); therefore, it is replaced by the continuous function \(e_{i}/\left(|e_{i}|+\varepsilon_{i}\right)\), such that Equations (29) and (30) are rewritten as:_  
\[\ddot{\bar{y}} =-A_{11}\dot{\bar{y}}-A_{12}\dot{\bar{y}}+B_{1}\delta_{fw}+\hat{L}_ {1}(t)\frac{e_{1}}{|e_{1}|+\varepsilon_{1}} \tag{37}\] \[\ddot{\bar{y}} =-A_{21}\dot{\bar{y}}-A_{22}\dot{\bar{y}}+B_{2}\delta_{fw}+\hat{L}_ {2}(t)\frac{e_{2}}{|e_{2}|+\varepsilon_{2}}. \tag{38}\]  
The estimation performance of the ASMO for lateral velocity and yaw rate primarily depends upon the knowledge of tire cornering stiffness coefficients \(C_{f}\) and \(C_{r}\), which are unknown in practice and cannot be measured directly from the onboard vehicle sensors. Therefore, a Kalman filter (KF) [41] is proposed in cooperation with ASMO to estimate these stiffness coefficients under different tire-road conditions. Once the KF estimates a sufficient set of tire cornering stiffness coefficients, the parameter estimation can be switched off. The KF algorithm [41] for tire cornering stiffness estimation is given in Table 1.  
\(P\) denotes the estimate error covariance, \(Q\) is the process noise covariance, and \(R=r_{s}^{2}\) is the measurement noise covariance, whereas \(r_{s}\) represents the sensor's zero-mean white noise. The tire cornering coefficients vector \(w\) and the measurement \(z\), consisting of the lateral acceleration \(a_{y}\), are defined as:  
\[w=\left[C_{f}\ C_{r}\right]^{T}\!,z=Hw, \tag{39}\]  
where  
\[z=a_{y} \tag{40}\] \[H=\left[-\frac{2}{m_{0}}\left(\frac{\dot{\bar{y}}+l_{f}\dot{\bar{y}}} {V_{x}}-\delta_{fw}\right)-\frac{2}{m_{0}}\left(\frac{\dot{\bar{y}}-l_{f}\dot {\bar{y}}}{V_{x}}\right)\right].\]  
It is to be noted that the tire cornering stiffness coefficient's vector \(w\) is considered as constant, therefore, the time derivative of \(w\) is zero, (\(\dot{w}=0\)). Then, \(w\) and \(z\) can be written in [PERSON]'s discretized form as:  
\[w(k) =w(k-1)+v(k) \tag{41}\] \[V_{2}=\frac{1}{2}\varepsilon_{2}^{2}+\frac{1}{2\rho_{2}}\widetilde{L}_{2}. \tag{35}\]0  
where \(v\) and \(r\) are the zero mean process noise and measurement noise, respectively. In order to improve the estimation performance and the convergence accuracy of KF, the difference \(e_{3}=z_{t}-H\dot{w}_{t}^{-}\), known as residual, is utilized to switch off the KF estimator. Therefore, on the basis of