# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 5 AGFSMC Control Design
### Convergence Proof
Substituting Equations (61), (63) and AGFSMC \(u_{A}\) (Equation (54)) into (Equation (60)), then the inequality is written as:  
\[\begin{array}{rl}\dot{V}_{3}\leq&s[-J_{ek}\dot{\widetilde{\xi}}_{r}-B_{ek} \dot{\xi}_{fw}-sign(s)(\hat{\hat{\gamma}}_{k}|\ddot{\hat{\gamma}}_{r}|+\hat{ B}_{ek}|\dot{\hat{\delta}}_{fw}|\\ &\quad+\hat{\mathcal{T}}|\delta_{fw}|+\hat{\mathcal{T}}\Bigg{|}\frac{\dot{ \vartheta}+l_{f}\dot{\Psi}}{V_{x}}\Bigg{|}+\hat{\beta}_{1}|u(t-1)|-\beta_{2}s ]\\ &\quad+|s|\mathcal{F}|\mbox{sign}(\dot{\hat{\gamma}}_{fw})|+|s|\mathcal{T}| \delta_{fw}|+|s|\mathcal{T}\Bigg{|}\frac{\dot{y}+l_{f}\dot{\Psi}}{V_{x}} \Bigg{|}\\ &\quad+\dot{\hat{a}}^{T}\Gamma^{-1}\widetilde{a}+\widetilde{\beta}_{1}\dot{ \hat{\beta}}_{1}\\ &=-(|s|\hat{J}_{ek}|\dot{\hat{\delta}}_{r}|+sJ_{ek}\dot{\widetilde{\xi}}_{r})- (|s|\hat{\beta}_{ek}|\dot{\hat{\delta}}_{fw}|+sB_{ek}\dot{\hat{\delta}}_{fw}) \\ &\quad-(|s|\mathcal{F}|sign(\dot{\hat{\delta}}_{fw})|-s|\mathcal{F}|\mbox{sign }(\dot{\hat{\delta}}_{fw})|)\\ &\quad-(|s|\mathcal{F}|\delta_{fw}|-|s|\mathcal{F}|\delta_{fw}|)+|s|\mathcal{T} \Bigg{|}\frac{\dot{y}+l_{f}\dot{\Psi}}{V_{x}}\Bigg{|}-|s|\hat{\mathcal{T}} \Bigg{|}\frac{\dot{\vartheta}+l_{f}\dot{\Psi}}{V_{x}}\Bigg{|}\\ &\quad-|s|\hat{\beta}_{1}|u(t-1)|-\beta_{2}s^{2}+\dot{\hat{a}}^{T}\Gamma^{-1} \widetilde{a}+\widetilde{\beta}_{1}\dot{\hat{\beta}}_{1}\\ &=-|s||\dot{\widetilde{\xi}}_{r}|\widetilde{J}_{ek}-|s||\dot{\delta}_{fw}| \widetilde{B}_{ek}-|s||sign(\dot{\hat{\delta}}_{fw})|\widetilde{\mathcal{F}} -|s||\delta_{fw}|\widetilde{\mathcal{T}}\\ &\quad-|s|\left(\hat{\mathcal{T}}\Bigg{|}\frac{\dot{\vartheta}+l_{f}\dot{ \Psi}}{V_{x}}\Bigg{|}-\mathcal{T}\Bigg{|}\frac{\dot{y}+l_{f}\dot{\Psi}}{V_{x }}\Bigg{|}\right)-|s|\hat{\beta}_{1}|u(t-1)|-\beta_{2}s^{2}\\ &\quad+\dot{\hat{a}}^{T}\Gamma^{-1}\widetilde{a}+\widetilde{\beta}_{1}\dot{\hat{ \beta}}_{1}\\ &=-|s||y|\,\dot{\widetilde{a}}+\dot{\hat{a}}^{T}\Gamma^{-1}\widetilde{a}-|s| \left(\hat{\mathcal{T}}\Bigg{|}\frac{\dot{\vartheta}+l_{f}\dot{\Psi}}{V_{x}} \Bigg{|}-\mathcal{T}\BiggWith the adaptation laws of \(\hat{\hat{a}}\) (Equation (57)) and \(\hat{\hat{\beta}}_{1}\) (Equation (58)), substituting into Equation (63) satisfies:  
\[\dot{V}_{3}\leq-|s|\left(\mathcal{T}\Bigg{|}\frac{\dot{\hat{y}}+l_{f}\dot{\hat{y }}}{V_{x}}\Bigg{|}-\mathcal{T}\Bigg{|}\frac{\dot{y}+l_{f}\dot{\hat{y}}}{V_{x}} \Bigg{|}\right)-|s|\beta_{1}|u(t-1)|-\beta_{2}s^{2} \tag{64}\]  
The convergence proof shows that the proposed AGFSMC is stable and the inequality (Equation (64)) ensures that the global fast terminal sliding surface variable exponentially converges to zero (\(s=0\)) in the finite time. **Remark 3**. : _The signum function \(\mathrm{sign(s)}\) incorporates the chattering and discontinuity in the proposed controller. Therefore, to eliminate the chattering phenomenon the signum function is replaced by the boundary layer saturation function \(\mathrm{sat}(\cdot)\) such that Equations (51) and (53) are re-written as:_  
\[u_{E}=-\mathrm{sat}(s)(|\xi\tau_{\mathrm{ak}}|+|\xi\tau_{\mathrm{FK}}|) \tag{65}\]  
\[u_{A}=-\mathrm{sat}(s)\left(|y|\hat{a}+\mathcal{T}\Bigg{|}\frac{\dot{\hat{y}} +l_{f}\dot{\hat{y}}}{V_{x}}\Bigg{|}+\hat{\beta}_{1}|u(t-1)|\right)-\beta_{2}s. \tag{66}\]  
_The boundary layer saturation function is defined as:_  
\[\mathrm{sat}(s)= \begin{cases}\frac{s}{\hat{\phi}}&|s|<\phi\\ \mathrm{sign}(s)&\mathrm{otherwise}\end{cases} , \tag{67}\]  
_where \(\phi>0\) represents the boundary layer thickness. Due to the boundary layer, the closed-loop error cannot converge to zero. However, a carefully selected value of \(\phi\) would lead the error to a user-specified bounded region._  
**Remark 4**. : _In order to avoid overestimation of \(\hat{a}\) and \(\hat{\beta}_{1}\), which can lead the control input \(u(t)\) to saturation, Equations (57) and (58) can be re-written for the permissible bounds of \(e_{\theta}\) using the discontinuous projection mapping [36] as follows:_  
\[\hat{a}= \begin{cases}0&\mathrm{if}\ |e_{\theta}|\leq\varepsilon_{4}\\ \Gamma|y|^{\top}|s|&\mathrm{otherwise}\end{cases}