## 5 Results of Simulation
In contrast, the proposed control system adjusts the acceleration input to account for the acceleration limit, which allows the vehicle to reach its target speed. Therefore, unlike the standard MRAC, the proposed control system made it possible to reach the target speed. As shown in the bottom graph of Figure 9b, the standard MRAC exhibits consistently higher peak jerk values compared to the proposed control system throughout the acceleration phase--with a maximum jerk of approximately 0.8 m/s\({}^{3}\). In contrast, the proposed control system achieved a significantly lower maximum jerk (approximately 0.5 m/s\({}^{3}\)). Furthermore, noticeable oscillations are observed in the standard MRAC during the 50 to 60 s period, while the proposed control system effectively suppressed these oscillations, which confirms improved stability. Figures 8 and 9 clearly demonstrate that as the operating environment becomes increasingly severe, the tracking performance of conventional control methods gradually deteriorates. In particular, the standard MRAC struggles to handle increased vehicle mass,  
Figure 9: (a) Simulation results for Scenario 3; (b) simulation results for Scenario 4.  
which results in hitting acceleration limits. Consequently, the actual plant fails to follow the reference model, which increases the tracking error. This, in turn, affects the adaptive gain and leads to instability in the control input. As a result, not only does the plant fail to track the desired speed, but the control input also suffers from high-frequency noise, which further degrades control performance. In contrast, the proposed control system actively modified the reference model when acceleration limits were detected, which ensured that the plant could feasibly follow the reference model under the given physical constraints. This modified reference model not only helped the plant track the desired speed more effectively but also mitigated instability in the adaptive gain, ultimately suppressing high-frequency noise in the control input. As shown in Figure 9, these benefits become even more pronounced under severe operating conditions, which highlights the superior robustness and adaptability of the proposed method. To quantitatively evaluate the performance of the proposed adaptive control method, a comparative analysis was conducted using two conventional control strategies: standard MRAC and proposed MRAC. Three key performance indices were selected to assess the responsiveness of the system and the impact of the control inputs on actuator load:  
* Raising Time: The time required for the vehicle to reach 90% of the reference speed from 10%, representing the speed of the control response. * Speed Error Area (Integrated Absolute Error, IAE): The integral of the absolute difference between the actual vehicle speed and the reference speed over time used to evaluate tracking accuracy. * Maximum Jerk: The peak rate of change in acceleration over time, indirectly reflecting the smoothness of control inputs. Lower jerk values correspond to reduced actuator stress and improved ride comfort. These indices serve as comprehensive metrics for evaluating speed-tracking ability, disturbance reflection, and control input stability--factors closely related to the physical acceleration limitations and high-gain noise sensitivity considered in this study. The results of the evaluation for Scenarios 1 through 4 are summarized in Table 4.  
Based on overall comparison results, the proposed longitudinal adaptive controller demonstrates faster raising time even under physical input constraints, thanks to its ability to reshape the reference model to accommodate acceleration limits. It also achieves the lowest cumulative speed error, indicating improved disturbance rejection and more precise tracking of the reference speed. Moreover, the proposed controller consistently produces lower maximum jerk values compared to standard MRAC, confirming its ability to generate smoother control inputs and reduce mechanical stress on actuators.