# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 4 ASMO and KF  
In this section, we will first design the ASMO for yaw rate \(\dot{\psi}\) and lateral velocity \(\dot{y}\), and then use the KF parameter estimator to estimate the tire cornering stiffness coefficients under varying road conditions. In order to design the observer, a few assumptions are made, such as: the yaw rate \(\dot{\psi}\) is directly measurable from yaw rate sensor; and the vehicle's longitudinal velocity is obtained from \(V_{x}=r_{e}\omega\), where \(r_{e}\) is the effective tire radius and \(\omega\) is the averaged free wheels angular speed measured from the wheel encoders. Moreover, it is considered that there is no effect of gravitation acceleration \(g\) on the lateral acceleration \(a_{y}\), such that the \(a_{y}\) measurement model is defined as [29]:  
\[a_{y,sensor}=\ddot{y}+V_{x}\dot{\psi}. \tag{23}\]  
Therefore, the lateral velocity can be obtained from a strapdown algorithm [35] as follows:  
\[\dot{y}(t)=\dot{y}(t-1)+\int(a_{y,sensor}-V_{x}\dot{\psi})dt, \tag{24}\]  
where \(\dot{y}(t-1)\) is the prior lateral velocity. The conventional sliding mode observer (SMO) for vehicle states (Equation (11)) can be designed as:  
\[\ddot{\ddot{y}}=-A_{11}\dot{\dot{y}}-A_{12}\dot{\dot{\psi}}+B_{1}\delta_{fw}+ L_{1}\text{sign}\Big{(}\dot{y}-\dot{\dot{y}}\Big{)} \tag{25}\]  
\[\ddot{\ddot{\psi}}=-A_{21}\dot{\dot{y}}-A_{22}\dot{\dot{\psi}}+B_{2}\delta_{fw }+L_{2}\text{sign}\Big{(}\dot{\psi}-\dot{\dot{\psi}}\Big{)}, \tag{26}\]  
where \(A_{11},A_{12},B_{1}\), and \(B_{2}\) are the elements of Equation (12) with nominal \(m_{0}\) and \(I_{z0}\); \(L_{1}\) and \(L_{2}\) are the observer gains, which must satisfy the following conditions, such that:  
\[L_{1}>\text{max}(|A_{11}e_{1}|+|A_{12}e_{2}|) \tag{27}\]  
\[L_{2}>\text{max}(|A_{12}e_{1}|+|A_{22}e_{2}|), \tag{28}\]  
where \(e_{1}=\dot{y}-\dot{\dot{y}}\) and \(e_{2}=\dot{\psi}-\dot{\dot{\psi}}\). The road surface variation is a critical factor for tuning the observer gains \(L_{1}\) and \(L_{2}\) during the design process. Any inappropriate selection of \(L_{1}\) and \(L_{2}\) will significantly reduce the SMO performance, resulting in a possible deviation of state estimation from the original trajectory. Due to the aforementioned fact, an adaptive gain based sliding mode observer [34] is proposed, which improves the estimation performance by adapting the observer gains according to tire road conditions. Therefore, Equations (25) and (26) are changed to new forms, as follows:  
\[\ddot{\hat{y}} =-A_{11}\dot{\hat{y}}-A_{12}\dot{\hat{y}}+B_{1}\delta_{fw}+\hat{L} _{1}(t)\text{sign}(e_{1}) \tag{29}\] \[\ddot{\hat{\hat{y}}} =-A_{21}\dot{\hat{y}}-A_{22}\dot{\hat{y}}+B_{2}\delta_{fw}+\hat{L} _{2}(t)\text{sign}(e_{2}), \tag{30}\]  
where the ASMO gain adaptation law for \(i=1,2\) is expressed as:  
\[\dot{\hat{L}}_{i}(t)= \begin{cases}\rho_{i}|e_{i}|,&|e_{i}|>\varepsilon_{i}\\ 0,&\text{otherwise}\end{cases}  \tag{31}\]  
where \(\hat{L}_{i}(t)>0\), is strictly positive time varying adaptive ASMO gain. \(\rho_{i}\) is a positive scalar used to adjust the adaption speed. \(\varepsilon_{i}\ll 1\) are small positive constants used to activate the adaptation mechanism with the condition defined in Equation (31); therefore, as the error converges to the bound \(|e_{i}|\leq\varepsilon_{i}\) in finite time, \(\hat{L}_{i}(t)\) will stop increasing. For convergence proof, the Lyapunov function of ASMO for lateral velocity is defined as:  
\[V_{1}=\frac{1}{2}\varepsilon_{1}^{2}+\frac{1}{2\rho_{1}}\widetilde{L}_{1}, \tag{32}\]  
where \(\widetilde{L}_{1}=\hat{L}_{1}-L_{1}\) is the adaptive gain convergence error. The derivative of \(V_{1}\), with the consideration that \(\dot{L}_{1}=0\), is obtained as follows:  
\[