# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 6 Simulation Results
### Sinusoidal Maneuvering Test (Test 1)  
The reference steering wheel angle is generated by:  
\[\delta_{d}=0.4\sin(0.5\pi t)\text{rad}. \tag{72}\]  
Figure 5 shows the simultaneously estimated lateral velocity, yaw rate, and cornering stiffness coefficients. It is observed that the cooperative ASMO and KF scheme intelligently cope with the tire-road variations and estimate the vehicle states and cornering stiffness coefficients by self-tuning the gains according to the driving environment. Figure 5c shows that the estimated \(\hat{C}_{f},\hat{C}_{r}\) have not only converged to the neighborhood of the actual values in both dry and snowy conditions, but also become constant after the condition \(e_{3}\) reached a specified termination bound. Figure 6 represents the tracking response and the control input performance of the AGFSMC scheme against the varying tire-road disturbance forces. We can see from Figure 6b that the proposed methodology effectively eliminates the impact of self-aligning torque (Equation (20)) and Coulomb frictional torque (Equation (21)) from the SbW system and ensures that the front wheels are precisely tracking the reference steering angle with a steady state tracking error of 0.002 rad. It is noted that at the beginning of sinusoidal maneuvering, after 3 s, the tracking error reached the peak value of 0.01 rad. This is because we started all the parameter estimations from very low values, such as \(\hat{w}_{0}=[100\;100]^{T}\), \(\hat{a}(0)=\hat{\beta}_{1}(0)=0\). Therefore, right after the peak error, all the estimated parameters converged to the sufficient estimation set. As a result, the peak tracking error also converged to the steady-state dead zone region. Moreover, Figure 7 shows the estimated SbW system parameters and the sliding gain adaptation profile. It is observed that the estimated SbW system parameters did not converge to the listed actual constants, but due to the adaptive capability of the proposed control scheme, all parameters as well as the sliding gain are adaptively adjusted in time for both driving conditions, which ensure the closed-loop stability of the SbW system. Hence, the outstanding steering performance of the SbW  
Figure 5: Estimation results of vehicle states and cornering stiffness coefficients in test 1: (**a**) Estimated lateral velocity; (**b**) Estimated yaw rate; (**c**) Estimated cornering stiffness coefficients. system vehicle is achieved against the nonlinear tire-road disturbance forces and the uncertain SbW system parameters. Figure 8 demonstrates that the steering performance of the ASMC scheme is not as good as that of the proposed AGFSMC scheme. This is because the hyperbolic tangent function used in ASMC is unable to replicate the actual self-aligning torque acting on the SbW system. Also, the adaptation law  
Figure 6: Control performance of the proposed AGFSMC scheme in test 1: (**a**) Tracking performance; (**b**) Tracking error; (**c**) Control input torque. Figure 7: Estimated SbW system parameters and sliding gain with AGFSMC scheme in test 1: (**a–d**) Estimated SbW system parameters; (**e**) Estimated sliding gain. cannot estimate the appropriate equivalent coefficient of self-aligning torque to compensate for the varying tire-road conditions. Consequently, the overall tracking error is much higher, particularly in the dry asphalt road condition: the tracking error peaks to the steady-state value of 0.06 rad, which is almost 30 times higher than in the proposed scheme. Although the ASMC scheme has the information of nominal parameters and utilized the saturation function, it incorporates high-frequency chattering during the first 3 s of the simulation, where the reference angle is set to zero. Figure 8: Control performance of adaptive sliding mode controller in test 1: (**a**) Tracking performance; (**b**) Tracking error; (**c**) Control input torque; (**d**) Estimated equivalent coefficient of self-aligning torque.