## 4 Simulation Design and Analysis
### Case 1: Nominal SBW System with Nonlinearity under the Fixed Input and Output Time Delay  
In the first case, the SBW system's tracking performance is assessed under the nominal nonlinear SBW dynamics in the presence of fixed communication time delays. To investigate this case, both the parametric uncertainties as expressed in (2) and external disturbance \(d(t)\) as given in (5) are set to zero. Furthermore, the input and output time delays as illustrated in Figure 1 are assigned values of \(\tau_{i}=5\times 10^{-3}\) s and \(\tau_{o}=5\times 10^{-3}\) s, respectively. Thus, the sum of \(\tau_{i}\) and \(\tau_{o}\) matches the sampling time \(T_{s}\). In Figure 3(a)-d, the estimation errors for \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) are presented, highlighting the superior estimation performance of the proposed algorithm when compared to the benchmark ADRC method. It is evident that estimation errors \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) of the proposed algorithm are lower than those of the comparison approach. In addition, the estimation error in the proposed approach converges faster. Moreover, it is observed that some spikes occur in the estimation errors in both methods due to the observer gains' dependency on the discrepancy between \(x_{1}\) and its estimation. Nevertheless, the AESO exhibits a faster convergence rate compared to the ESO in the comparative method, with estimation errors being smaller in the proposed approach. In Figure 4(a)-c, the trajectories of steering angle \(\theta_{s}\), the tracking error of \(\theta_{s}\), and control input \(u(t)\) are displayed under both the proposed algorithm and the comparative method. In Figure 4(a),b, it is evident that the convergence rate achieved under the proposed control method outperforms that of the ADRC. Furthermore, the tracking error in the proposed algorithm is reduced, as the impact of time delays is effectively mitigated by the AESO in the proposed control approach. In addition, Figure 4(c) clearly illustrates that the proposed control approach exhibits a more expeditious response in the control input  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Control** & **Proposed** & **ADRC** \\
**Parameters** & **(24)–(28)** & **(48) and (49)** \\ \hline \(\omega_{c}\) & 25 & 25 \\ \(k_{c}\) & 700 & - \\ \(\omega_{o}\) & 125 & 125 \\ \(k_{o}\) & \(1\times 10^{9}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of Control Schemes in All Cases.  
when compared to the ADRC. To achieve a comprehensive understanding of the proposed control scheme, Figure 6a,b presents adaptive gain \(k_{c}\) of the AGSEFBC and adaptive gain \(k_{o}\) of the AESO, respectively, along with the corresponding fixed gains of ESO in the ADRC. By comparing Figure 4a and Figure 6a, Figure 5b and Figure 6b, it can be observed that adaptive gains \(k_{c}\) and \(k_{o}\) vary in response to the estimation error of \(x_{1}\) and the tracking error of \(\theta_{s}\), respectively. In contrast, it is evident that the gains in the ADRC remain fixed, as illustrated in Figure 6a,b.  
Figure 4: Estimation error responses for the nominal system with nonlinearity under the input time delay, \(\tau_{i}=5\times 10^{-3}\) s, and the output time delay, \(\tau_{o}=5\times 10^{-3}\) s (Case 1).