## 4 Simulation Design and Analysis
### Case 2: Uncertain SBW System with Nonlinearity under Input and Output Time-Varying Delay  
In comparison to Case 1, parameter uncertainties such as inertia, damping, and Coulomb fraction variations are newly considered in Case 2, which are specified in (2). Accordingly, the inertia, damping, and Coulomb fraction variations are formulated by \(\bar{\Delta}_{I_{c}}=0.1)_{e0}\), \(\bar{\Delta}_{B_{e}}=0.1B_{e0}\), and \(\bar{\Delta}_{\xi_{f}}=0.1\xi_{f0}\), respectively, as in (2). In addition, the external disturbance acting on the SBW dynamics modeled in (5) is defined by \(d(t)=\sin(t)\). The time-varying communication delays consist of a fixed component, i.e., \(\tau_{i}=2\times 10^{-2}\) s for  
Figure 5: Tracking 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). the control input and \(\tau_{o}=2\times 10^{-2}\) s for the front wheels' angle feedback, and a varying component with an upper bound of \(5\times 10^{-3}\) for both input and out delays. Please note that the sum of the input and output delays significantly exceeds sampling time \(T_{s}\) in Case 2. In Figure 7a-d, the estimation errors for \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) are depicted. In Case 2, we notice that the performance of the proposed algorithm consistently surpasses the ADRC with respect to estimation, as it exhibits a quicker convergence rate and smaller estimation errors even under large time delays. In Figure 8a-c, it is clear that the tracking error of \(\theta_{s}\) in the proposed control remains smaller than that of the ADRC. Furthermore, the convergence rate of the tracking error and the response rate of the control input in the proposed control are faster than the comparative ADRC. By comparing Figure 7a and Figure 9a, Figure 8b and Figure 9b, it can be observed that adaptive gains \(k_{c}\) and \(k_{o}\) in Case 2 exhibit a broader range than those in Case 1 to counteract the existence of parameter uncertainties, external disturbance, and significant time-varying delays. Thus, the adaptive gains in the proposed AADRC scheme can adaptively update to maintain good performance, whereas the estimation and tracking performance of the ADRC are not as good as the proposed one due to the ADRC's fixed controller and observer gains. Figure 7: Estimation error responses for the uncertain system under the input time delay, \(\tau_{i}=2\times 10^{-2}\) s, the output time delay, \(\tau_{o}=2\times 10^{-2}\) s, and delay variation \(\Delta\tau=5\times 10^{-3}\) s (Case 2). To sum up, it is worth mentioning from Cases 1 and 2 that adaptive mechanisms used in (25) and (30) are, respectively, used to update gains of Observer (24) and Controller (28) elements, which play a focal role in contributing and improving not only the peaking reduction in transient region and the accuracy steady as shown in Figures 5 and 8. The reason for the adaptive mechanisms' contribution is that the adaptive gains of both the proposed observer and controller are, respectively, functions of the estimation and tracking errors, in which those gains increase when the errors become bigger and vice versa. As depicted in Figures 6 and 9, both the proposed observer and controller gains increase largely in the transient region to compensate for the peaking phenomenon in the estimation and tracking trajectories in that region. In addition, those gains continue increasing at different times in the steady-state region, especially when the errors occur largely in that region, leading to the enhancement of observation and tracking accuracy. Furthermore, these adaptively increased gains bring strong robustness as demonstrated in Case 2 compared to the well-known ADRC with fixed gains.