# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 5 AGFSMC Control Design  
In this section, the estimated dynamics-based adaptive global fast sliding mode control (AGFSMC) is designed in two steps to estimate the uncertain steering parameters and eliminate the effect of varying tire-road disturbance forces, so that the front wheels asymptotically track the driver's reference command in finite time. The tracking error \(e_{\theta}\) between the front wheel angle \(\delta_{fw}\) and the scaled reference hand wheel angle \(\delta_{d}\) is defined as:  
\[e_{\theta}(t)=\delta_{fw}(t)-\frac{\delta_{sw}(t)}{k}=\delta_{fw}(t)-\delta_{ d}(t). \tag{45}\]  
The combination of linear sliding surface and the terminal sliding surface is known as the global fast terminal sliding surface, \(s\), which is defined as [42]:  
\[s=\dot{e}_{\theta}+\lambda_{1}(e_{\theta})^{q/p}+\lambda_{2}e_{\theta}, \tag{46}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) (\(\lambda_{1},\lambda_{2}>0\)), are strictly positive constants, and \(q\) and \(p\), are positive odd numbers, such that \(q<p\). Thus, the time derivative of \(s\) is obtained as:  
\[\dot{s}=\ddot{e}_{\theta}+\lambda_{1}\frac{q}{p}(e_{\theta})^{(\frac{q}{p}-1 )}\dot{e}_{\theta}+\lambda_{2}\dot{e}_{\theta}. \tag{47}\]  
\(\dot{s}\) can be written as:  
\[\dot{s}=\ddot{\delta}_{fw}-\ddot{\delta}_{r}, \tag{48}\]  
where \(\ddot{\delta}_{r}\) is expressed as:  
\[\ddot{\delta}_{r}=\ddot{\delta}_{d}-\bigg{(}\lambda_{1}\frac{q}{p}(e_{\theta} )^{(\frac{q}{p}-1)}+\lambda_{2}\bigg{)}\dot{e}_{\theta}. \tag{49}\]  
Thus, for stabilizing the SbW system (Equation (14)) and exponentially converging the tracking error (Equation (45)) to zero, the two-step closed loop control law \(u\) for the SbW system is designed as:  
\[u=u_{E}+u_{A}, \tag{50}\]  
where, in the first step, the estimated dynamics based control (EDC) \(u_{E}\), is designed to counter the tire-road disturbance acting on the SbW system as follows:  
\[u_{E}=-\text{sign}(s)(|\xi\tau_{ak}|+|\xi\tau_{FK}|), \tag{51}\]  
where \(\xi\tau_{ak}\), is the estimated self-aligning torque, which is computed from the best set of estimated vehicle states and front wheel cornering stiffness provided by the ASMO and KF. \(\xi\tau_{FK}\) is the nominal frictional torque obtained from the nominal set of vehicle parameters, such as mass, nominal coefficient of friction, and the geometry of the vehicle. Both \(|\xi\tau_{ak}|\) and \(|\xi\tau_{Fk}|\) are expressed as:  
\[|\xi\tau_{ak}|=\frac{2\hat{\mathcal{C}}_{f}}{k_{o}}\big{(}t_{p_{o}}+t_{m}\big{)} \Bigg{|}\Bigg{(}\delta_{fw}-\frac{\hat{y}+l_{f}\hat{\psi}}{V_{x}}\Bigg{)}\Bigg{|} \tag{52}\]  
\[|\xi\tau_{Fk}|=\frac{m_{0}g\hat{l}_{r}}{\Big{(}l_{f}+l_{r}\Big{)}k_{0}}\mu_{o}t _{p_{o}}\Big{|}\text{sign}\Big{(}\hat{\delta}_{fw}\Big{)}\Big{|}, \tag{53}\]  
where \(\hat{\dot{y}},\hat{\dot{\psi}}\), and \(\hat{\mathcal{C}}_{f}\) are the observed vehicle states and the front wheel's estimated cornering stiffness, as worked out in the previous section, respectively. \(\mu_{o},t_{po},m_{0}\), and \(k_{o}\) are the nominal system parameters. Second, to tackle the residual disturbance left by the EDC and estimate the uncertain steering parameters, the adaptive global fast sliding mode control (AGFSMC) \(u_{A}\) is designed as follows:  
\[u_{A}=-\text{sign}(s)\left(|y|\hat{a}+\hat{\mathcal{T}}\Bigg{|}\frac{\dot{y}+ l_{f}\dot{\psi}}{V_{x}}\Bigg{|}+\hat{\beta}_{1}|u(t-1)|\right)-\beta_{2}s, \tag{54}\]  
where \(\hat{a}(t)\) is the estimated parameter's vector and \(|y|\) is the signal feedback vector; they are defined as follows:  
\[s=\dot{e}_{\theta}+\lambda_{1}(e_{\theta})^{q/p}+\lambda_{2}e_{\theta}, \tag{46}\]0  
\[s=\dot{e}_{\theta}+\lambda_{1}(e_{\theta})^{q/p}+\lambda_{2}e_{\theta}, \tag{46}\]1  
Moreover, \(\hat{\beta}_{1}\) and \(\beta_{2}\), (\(\hat{\beta}_{1},\beta_{2}>0\)) are the fixed and adaptive gains used to control the convergence speed of AGFSMC, respectively, and \(|u(t-1)|\) is the prior control input obtained at the time step \(t-1\).