# NESO-Based Path Following Control for
## 4 Case Study
The comparison of along-tracking errors, cross-tracking errors, and heading-tracking errors are shown in Figure 4. It can be shown that the proposed NESO-based backstepping sliding mode control (BSMC) scheme can improve both the transient and steady state tracking errors of the path following for the underactuated hovercraft compared with traditional backstepping control, LESO-based backstepping control, and NESO-based backstepping control schemes. The simulation results of ESO for an underactuated hovercraft are shown in Figure 5. The estimation of velocities for a certain underactuated hovercraft is shown in Figure 5a, which indicates that the  
Figure 6: (**a**) The comparison of heading-tracking errors from different control schemes; (**b**) the comparison of the virtual control and the true control law with the safety limit constraint (SLC). Figure 7: (**a**) The comparison of heading angle from different control schemes; (**b**) the comparison of yaw and surge control forces with different control schemes. proposed NESO has higher accuracy and better transient performance than the LESO proposed in [12]. The approximations of the unknown nonlinear time-varying total uncertainties are plotted in Figure 5b, which show that the time-varying uncertainties can be approximated by NESO but not by LESO. The designed control scheme is robust to the system model. Since the relative wind speed and orientation are time-varying for the hovercraft, the dynamic system of the hovercraft has great time-varying uncertainty. The control simulation results show that the control method does not rely on the accurate model and is robust to the total uncertainties caused by external disturbances and unmodeled factors. Figure 6 shows that the designed control strategy can make the constraint of the nonlinear time-varying turning safety limit possible by introducing an auxiliary variable to achieve the nonlinear time-varying constraint of virtual control. This can significantly improve the safety of the underactuated hovercraft. Compared with the control methods in the literature [25], the yaw controller with the safety limit was designed by control input constraint. However, the safety turning SLC parameter is not directly introduced into the control system, so the constraint effect is not accurate. In our control scheme, the auxiliary limiting variable is introduced into the backstepping process, and the safety limit is directly introduced into the control system. In addition, we consider the nonlinear time-varying turning SLC, which is not taken into the control design in reference [25]. The heading curve of underactuated hovercraft under nonlinear time-varying turning SLC is shown in Figure 7a, which illustrates the heading-tracking error of the underactuated hovercraft converges to zero under the constraint of the nonlinear time-varying turning safety limit. Due to the restriction of the safety limit on the turning velocity, the heading-tracking error slowly approaches zero rather than converges rapidly to zero like that of the control method without SLC. Although the control method without the safety constraint is fast, it does not take the safety factors into account and easily leads to ship capsizing accidents caused by improper control. Figure 7b shows the designed safe yaw and surge control inputs in terms of Theorem 2. It can be known that the proposed disturbances rejection using the NESO-based BSMC law has better stability and smoothness than other control schemes.