## 3 Results and Discussion  
The control system is applied in the pedestal for tracking the satellite so that the antenna always faces the satellite. This ensures maximum energy absorption from the satellite signal. Hence, the satellite trajectory should be given to the system as the set point for the tracking process. This data in the form of elevation (\(El\)) and azimuth (\(Az\)) angles (which are calculated by NOAA software for P6 satellite, manufactured in India) is applied to the model. The P6 satellite trajectory, based on elevation and azimuth anglesversus time, is illustrated in Figure 2 during one pass over the earth station. Using the mapping in (1), the satellite trajectory, in terms of \(\theta_{1}\) and \(\theta_{2}\), is shown in Figure 3, which is utilized as the reference input for tracking in the simulations of this section. The pedestal parameters are presented in Table A1. Two independent controllers are used for controlling the \(X\) and \(Y\) axes. The free parameters of APFC are selected as \(\lambda=0.995,N=3,T=5,p=8,H_{1}=21,H_{2}=46\) for both axes. In addition, The PID coefficients for both axes are set as \(K_{P}=50\), \(K_{I}=20\), and \(K_{d}=500\). The tuning of the PID coefficients is carried out by trial and error. However, \(K_{P}\) and \(K_{I}\) have been selected so as to achieve a short response time for the closed-loop system and to reduce the steady state tracking error as much as possible. It should be noted that the large value of \(K_{d}\) in the PID controller results in the predictive behaviour of the controller. The reference input and the response of the closed-loop system with APFC and PID controllers are illustrated in Figure 4. According to the figure, it seems that both the PID and APFC controllers can track the satellite. For more clarity, the tracking errors for \(X\) and \(Y\) axes are shown in Figure 5. As seen, the tracking error is significantly smaller with the APFC in comparison to the PID controller and the closed-loop system with the APFC algorithm based on Laguerre functions is capable of tracking the desirable trajectory more accurately. This occurs to a greater extent during the moments when the satellite trajectory passes the degree of zero along the \(Y\) axis. Figure 3: Satellite trajectory in terms of \(\theta_{1}\) and \(\theta_{2}\) as the reference inputs. Figure 2: Satellite trajectory presented in the form of elevation (\(EI\)) and azimuth (\(Az\)) angles as the reference inputs. The control efforts of the APFC and PID controllers are illustrated in Figure 6. It is obvious that the ranges of the control efforts for APFC and PID controllers are significantly different. An interesting feature of APFC is its desirable performance against the interference effects of the two channels. This is due the consideration of these effects in the system identification at every moment, which affects the system model in the next moment. If the satellite trajectory approaches zero along the \(Y\) axis (this occurs when the satellite is located in horizon), maximum acceleration is achieved along the inferior axis of the pedestal. This case is the same as when passing a satellite over an El/Az pedestal (zenith pass). In other words, the singular point for the X-Y pedestal is zero degrees along the inferior axis, namely, the \(Y\) axis. Hence, the control effort generated by the PID controller is higher at the moment, corresponding to zero degrees. However, in comparison to the APFC, the tracking error for the PID controller is significantly larger. On the other hand, the interference effect for the \(X\) and \(Y\) channels at these moments (corresponding to near zero degrees) is higher. Generally, the control effort generated by the PID controller is greater than that of the APFC, which may result in the saturation of actuators in the control system. Figure 4: The reference input, the response of the closed–loop control system using APFC scheme and PID controller for \(X\) and \(Y\) axes.