## 4 Simulation Results and Discussion
### Results and Discussion
Interestingly, the PID control system follows the  
Figure 10: Control inputs in tracking the step-type trajectory. (**a**) Without disturbances; (**b**) With disturbances. trajectory in real-time, while the LQI-GA is lagging behind, as seen in their time responses in Figure 12. However, Figures 13 and 14 clarify that the PID control system requires a lot of rotational corrections and control efforts to achieve this result. They are important factors in evaluating the feasibility of the system in a real application. The integral of square values (ISVs) presented in Table 3 shows the consumed kinematic energies of the water and the weight-shifting actuators, in which the proposed LQI-GA is significantly more efficient than the other. Figure 11: Tracking performances with the continuous trajectory. **(a)** Without disturbances; **(b)** With disturbances. Figure 12: Time responses of the robot position with the continuous trajectory. **(a)** Without disturbances; **(b)** With disturbances. Mathematical analyses of the stability and performance of the proposed system have been given in the previous Section. In particular, the robustness of the closed-loop system has been proved via the Lyapunov function candidate (21), and the boundedness of the control error has been given in (23). The GA-based tuning controller gains minimize the cost function (25), which is equivalent to minimizing the control error, even in the presence of disturbances, and minimizing the variation of the weight-shifting actuation system. The proper performances in all simulation scenarios verify the theoretical analyses. The water flows smoothly, especially in comparison with the turbulent flow, due to the control of the system, are also shown in Fig. 14.  
Figure 14: Control inputs in tracking the continuous trajectory. (**a**) Without disturbances; (**b**) With disturbances. Figure 13: Time responses of the robot orientation with the continuous trajectory. (**a**) Without disturbances; (**b**) With disturbances. the nozzles' flow in the study [24]. This is a great feature since it helps prevent the flutter instability of the water-conveying hose and enhances the total stability of the system. Meanwhile, PID is a popularly standard control technique, and the cascade control structure has been adopted for aerial systems in many studies--for example, in the references [19; 24; 30] of the manuscript. The controlled quality depends on the PID controller parameters, which are not easily tuned in cascade control. The result is that the PID-controlled system seems to be slow in tracking the step-type reference but fast and overshot in tracking the sinusoid trajectory. Factors such as the saturation of the inputs and the objective of the reduction of the non-minimum phase phenomenon are difficult to incorporate into the design of the PID controllers. Thus, the PID control system requires a lot of rotational corrections to maintain the tracking performance, and noticeable chattering appears in the PID control inputs. That leads to significant control efforts and consumed water in the PID control system. On average, the LQI-GA consumes 2.28% less water and uses 83.85% less kinetic energy of the actuator than the PID control system.