## 6 Experimental Results
### Convergence Results  
The convergence curves of GA, PSO and CSA used to accomplish the optimization process presented in this work are presented, using different population sizes to demonstrate the ability of these algorithms to provide accurate control for the quadcopter movements through tuning the PID parameters of the quadcopter. Figures 3-5 show the convergence characteristic curves of the roll control input of the quadcopter obtained by GA, PSO and CSA, respectively, over a different number of population sizes.  
Figure 4: Convergence curves of PSO using various numbers of population sizes.  
Figure 3: Convergence curves of GA with various population sizes.  
It is observed from Figure 3 that there is significant convergence in the evolutionary process of the GA-based approach in optimizing the PID controller of the quadcopter. These convergence curves are promising and affirm the effectiveness of GA in controlling the quadcopter's movements. The convergence curves of controlling the quadcopter movement using the PSO-based approach are rendered for up to 100 iterations. These convergence curves show that PSO lays out fast convergence with very low error. It can be seen from Figure 4 that the convergence curve for the population of size 60 converges more rapidly to the optimal cost than that for the other population sizes, although PSO with population sizes of 40, 60, 80 and 100 roughly reach near fitness cost values. The fitness cost value obtained by PSO for the population of size 20 is not optimal and is significantly higher than the fitness cost values reported by PSO for larger population sizes. Thus, any further increase in the population sizes is likely to reach a very small fitness cost value. It is possible that a very large increase in the size of populations will not significantly reduce the fitness cost value further. The fitness cost value for some population sizes continues to decline slightly with further population sizes. The fitness cost value may reduce further with more than 100 particles in the population of PSO. There is a big difference between the curves of population size 20 and the other curves, which indicates that the controlling process implemented by PSO with 100 particles and 80 particles achieves better control than the controlling process carried out by PSO with 20 particles. The performance of CSA in optimizing the roll input for the quadcopter is shown for up to 100 iterations in Figure 5. These curves represent the fitness cost values obtained using the fitness criterion in Equation (31). The convergence curves in Figure 5 demonstrate that the CSA-based approach provides stable control of the quadcopter with fast convergence and very low error.  
Figure 6 compares the convergence characteristic curves of the roll control input of the quadcopter obtained by GA, PSO and CSA for a population size of 100.  
The convergence curves in Figure 6 confirm that the control method presented based on GA, PSO and CSA reaches a sensible degree of performance in controlling the roll input of the quadcopter for all of these meta-heuristic methods. It is clearly observed that PSO converges to a very small error value, compared to GA and CSA.  
Figure 5: Convergence curves of CSA using various numbers of population sizes.