# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 6 Simulation Results
### Circular Maneuvering Test (Test 2)  
The circular maneuvering test is carried out over the dry asphalt road for 25 s with these selected tire-road parameters: \(C_{f}=8000,C_{r}=10000\), and \(u=0.85\).  
Figures 10-12 portray the promising results of the proposed AGFSMC scheme in all aspects during test 2. We can see the fine estimation of vehicle states and cornering coefficients in Figure 10. The estimated cornering coefficients takes less than a second to converge to the sufficient estimation set over the dry asphalt, such as, \(\hat{C}_{f}\cong 7250\), \(\hat{C}_{r}\cong 9050\), and becomes constant after \(e_{3}\) satisfies the  
Figure 9: Control performance of adaptive terminal sliding mode control in test 1: (**a**) Tracking performance; (**b**) Tracking error; (**c**) Control input torque; (**d**) Estimated parameters.  
selected \(\varepsilon_{3}\) bound. Thus, Figure 11 exhibits the excellent tracking response of the front wheels with an observed peak tracking error of 0.008 rad, which eventually converged to the \(\varepsilon_{4}\) bound after the rapid adjustment of all adaptive parameters \(\hat{\int}_{ck}\), \(\hat{B}_{ek}\), \(\hat{\mathcal{F}}\), \(\hat{\mathcal{T}}\), and \(\hat{\beta}_{1}\) to certain constants, as shown in Figure 12.  
Figure 11: Control performance of the proposed AGFSMC scheme in test 2: (**a**) Tracking performance; (**b**) Tracking error; (**c**) Control input torque.  
Figure 10: Estimation results of vehicle states and cornering stiffness coefficients in test 2: (**a**) Estimated lateral velocity; (**b**) Estimated yaw rate; (**c**) Estimated cornering stiffness coefficients.  
In contrast to the proposed scheme, the ASMC shows the worst tracking performance throughout test 2. It can be seen from Figure 13 that the tracking error is unable to obtain any steady state bound and reached a peak value of 0.076 rad, which is almost 9.5 times higher than in the proposed AGFSMC scheme. Moreover, the adaptation law also shows inconsistent behavior in the last 7 s of this test, where the estimated coefficient of self-aligning torque rapidly drops to a highly negative value. As a result, neither tracking error nor sliding surface converged to the steady state boundary at a finite time in the Lyapunov's sense.  
Figure 12: Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 2: (**a–d**) Estimated SbW system parameters; (**e**) Estimated sliding gain.  
Figure 13: Control performance of adaptive sliding mode controller in test 2: (**a**) Tracking performance; (**b**) Tracking error; (**c**) Control input torque; (**d**) Estimated equivalent coefficient of self-aligning torque.  
On the other hand, the ATSMC performed slightly better than the ASMC in terms of tracking response and also managed to converge the tracking error to the steady state bound during test 2. The peak tracking error observed under the ATSMC scheme is 0.067 rad as shown in Figure 14, which is marginally less than the ASMC but almost 8.35 times higher than the proposed scheme. Moreover, the abrupt shift in \(\hat{a}_{1}\) parameter estimation and the multiple control input overshoots are again observed in this test.