# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 6 Simulation Results  
In this section, the estimation accuracy of vehicle states and cornering stiffness coefficients, and the control input performance of the proposed AGFSMC scheme for SbW system road vehicles, are validated over the three different maneuvering tests, in compression with adaptive sliding mode control (ASMC) and adaptive fast sliding mode control (ATSMC). The first test (test 1) is sinusoidal maneuvering with varying tire-road conditions--snowy for the first 30 s and a dry asphalt road for the next 30 s--with the selected coefficient of friction as \(\mu_{t<30}=0.45\), \(\mu_{t\geq 30}=0.85\) and the tire cornering stiffness coefficients for the front and rear wheels as \(C_{f(t<30)}=4000\), \(C_{f(t\geq 30)}=8000\), \(C_{r(t<30)}=5000\), \(C_{r(t\geq 30)}=10,000\), respectively. The second test (test 2) is known as circular maneuvering, conducted over a dry asphalt road. Moreover, a high speed cornering test (test 3) is also introduced to further evaluate the robustness of the proposed scheme. It is worth noting that the first two tests are carried out at longitudinal speed \(V_{x}=10\) m/s and the third test at \(V_{x}=20\) m/s with the same sampling rate of \(\Delta T=0.001\) s. Furthermore, the vehicle and SbW system parameters are listed in Table 2.  
The parameters for the proposed cooperative ASMO and KF estimator with the termination bounds are selected as: \(\hat{L}_{1}(0)=\hat{L}_{2}(0)=8,\rho_{1}=\rho_{2}=10,\sigma=0.001,\epsilon_{1}= \epsilon_{2}=0.005,\epsilon_{3}=0.01,m_{0}=1150\) kg, \(I_{z_{0}}=1430\) kg\(\cdot\)m\({}^{2},\vartheta_{0}=[100\,100]^{T}\), \(P_{0}=10000\times I_{2x2},Q=\left(1\times 10^{-6}\right)I_{2\times 2}\), and \(r_{s}=0.001\). In addition, the parameters for the designed AGFSMC scheme with dead zone bounds are chosen as: \(\lambda_{1}=\lambda_{2}=12\), \(p=7\), \(q=5\), \(\phi=0.8\), \(\beta_{2}=4\), \(t_{p_{o}}=t_{m}=0.016\) m, \(\mu_{o}=0.6\), \(k_{o}=16\), \(\Gamma=I_{4\times 4}\), \(\epsilon_{4}=\epsilon_{5}=0.002\), and the initial conditions are considered as \(\hat{a}(0)=\hat{\beta}_{1}(0)=0\). To compare the performance of the proposed AGFSMC scheme with the adaptive sliding mode control (ASMC), as designed in [25], we used the following equations:  
\[ \begin{split} u=\frac{1}{k}(J_{\epsilon 0}(\lambda\dot{e}+\tilde{ \delta}_{d})+B_{\epsilon 0}\hat{\delta}_{fw}+\tilde{\xi}_{f0}\text{sign}(\hat{\delta}_{fw})+ \alpha s+K\text{sat}(s)+\hat{\rho}_{\tau}\text{tanh}(\delta_{fw}))\\ K=0.1(J_{\epsilon 0}(\lambda|\dot{e}|+|\tilde{\delta}_{d}|)+B_{ \epsilon 0}|\dot{\delta}_{fw}|+\tilde{\xi}_{f0})\\ \dot{\hat{\rho}}_{\tau}=\mu\frac{\phi}{l_{0}}+\mu\text{stanh}( \delta_{fw}),\end{split}  \tag{70}\]  
where the tracking error \(e=\delta_{d}-\delta_{fw}\) and the sliding surface \(s=\dot{e}+\lambda e\) with \(\dot{s}=(s_{k}-s_{k-1})/\Delta t\) are defined in Equation (71). The saturation function \(\text{sat}(\cdot)\) is also taken to be the same as Equation (67) with boundary layer thickness \(\phi=0.8\). Moreover, the nominal SbW system parameters \(J_{\epsilon 0}=3\), \(B_{\epsilon 0k}=12\), \(\tilde{\xi}_{f0}=100\), \(k=18\), and the control parameters \(\lambda=12\), \(\alpha=72\), \(\mu=450\) are selected according to the methodology defined in [25]. For performance comparison with ATSMC, as designed as [27], the calculations are given as follows:  
\[