## 2 SBW Plant Dynamical Modeling
### Dynamical Modeling of the Nominal SBW Plant with Uncertainties  
The differential equation of motion for the SBW plant can be expressed as follows using the simplified model of the SBW system described in [29]:  
\[ \begin{split} J_{e}\ddot{\theta_{s}}+B_{e}\dot{\theta_{s}}& =\kappa u-\tau_{c}-\tau_{\mathrm{sel}}\\ \kappa&=\kappa_{1}\cdot\kappa_{2}\cdot\kappa_{3} \cdot\kappa_{4}\\ \tau_{c}&=\xi_{f}\text{sign}(\dot{\theta_{s}}) \end{split}  \tag{1}\]  
Figure 1: Schematic SBW system with communication time delays. where \(\theta_{s}\), \(\theta_{s}\), and \(\ddot{\theta}_{s}\), are, respectively, the angular position, velocity, and acceleration of the steering motion of the front wheels; \(u\) indicates the steering motor unit's voltage torque that is designed by the proposed controller; \(\tau_{sel}\) is the self-aligning torque applied to the steering system; \(\tau_{c}\) is the frictional torque caused by the Coulomb friction; and \(\text{sign}(\cdot)\) is the standard signum function. Table 1 lists all of the other system parameters and their corresponding descriptions, values, and units. Additionally, Table 1 provides the scaling factors related to \(\kappa\) along with their respective descriptions and values. It should be noted that, for the purposes of this study, \(\kappa\) can be regarded as a constant because, at least for the duration of our investigation, its value does not change noticeably over time. Nonetheless, the nonlinearities and external disturbances of the model may lead to unpredictable and fluctuating system parameters [2]. The system temperature fluctuations, erratic external road loads that result in tiny deformations in the steering system, and other factors can all contribute to these parameter variations. The parametric uncertainties can be regarded as bounds, according to [2], and are given here with corresponding bounds. \[ \begin{split}|\Delta_{I_{e}}|&=|I_{e}-J_{\text{ \(\ell\)}0}|\leq\bar{\Delta}_{I_{e}};\quad(\bar{\Delta}_{I_{e}}=0.1J_{\text{ \(\ell\)}0})\\ |\Delta_{B_{e}}|&=|B_{e}-B_{\text{\(\ell\)}0}|\leq \bar{\Delta}_{B_{e}};\quad(\bar{\Delta}_{B_{e}}=0.1B_{\text{\(\ell\)}0})\\ |\Delta_{\bar{\zeta}_{f}}|&=|\bar{\zeta}_{f}-\bar{ \zeta}_{f0}|\leq\bar{\Delta}_{\bar{\zeta}_{f}};\quad(\bar{\Delta}_{\bar{\zeta}_ {f}}=0.1\bar{\zeta}_{f0})\end{split}  \tag{2}\]  
where \(J_{\text{\(\ell\)}0}\), \(B_{\text{\(\ell\)}0}\), and \(\bar{\zeta}_{f0}\) represent the nominal components of the SBW model parameters; \(\bar{\Delta}_{I_{e}}\), \(\Delta_{B_{e}}\), and \(\Delta_{\bar{\zeta}_{f}}\) denote the uncertain parts of the SBW model parameters; \(\bar{\Delta}_{I_{e}}\), \(\bar{\Delta}_{B_{e}}\), and \(\bar{\Delta}_{\bar{\zeta}_{f}}\) refer to the upper bounds of the corresponding model parameters. The velocities of the front wheels cannot always be measured due to practical constraints of the SBW system. Moreover, a front-wheel camber angle that is too inadequate may arise from the absence of weight distribution on the front wheels [30]. Consequently, during the steering process, the tires might not sense exact self-aligning torques. In our investigation, we assume that the tire slip angles are small. Therefore, to approximate the self-aligning torque, a hyperbolic tangent function can be used [31; 32] as  
\[\tau_{\text{sel}}=\rho_{\tau}\text{tanh}(\theta_{s}) \tag{3}\]  
where \(\rho_{\tau}\) is the time-varying coefficient related to various road statuses; it is given elaborately in Simulation Outcomes section. In addition, we define function \(\text{tanh}(\cdot)\) as in the following expression:  
\[\text{tanh}(z)=\frac{e^{2z}-1}{e^{2z}+1}. \tag{4}\]