# Optimal Learning and Self-Awareness Versus PDI
## 3 Results
### Control Implementation  
The performance of the three control system implementations is depicted in Figure 6. Comparing the three cases allowed further analysis on the differences between feedforward, feedback, and the combined deterministic artificial intelligence system. The feedforward and the deterministic artificial intelligence systems are more precise than the feedback method. This is because they are based on exact control Equations (9) and (10) respectively. Conversely, the feedback controller is based off a PDI controller that suffers phase lag [53], and is therefore less precise. Additionally, the gains in a PDI or PID controller must be finely tuned with pre-determined gain values, which can be an iterative and time-consuming task because controller performance varies greatly depending on the values. Lastly, the deterministic artificial intelligence configuration represents an error combination of both the feedforward and feedback plots. This allows the analytical accuracy of the feedforward equation to be updated with the responsiveness of the feedback correction by learning. Plot scaling precludes easy visualization of the error, so tabulated numerical results are provided in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\phi_{actual}\)–\(\phi_{desired}\) & \(\theta_{actual}\)–\(\theta_{desired}\) & \(\psi_{actual}\)–\(\psi_{desired}\) & **Computational** \\  & **(degrees)** & **(degrees)** & **(degrees)** & **Burden (s)** \\ \hline \(u_{ff}\) & \(-3.93\times 10^{-4}\) & \(4.18\times 10^{-4}\) & \(-1.48\times 10^{-2}\) & 24.7 \\ \(u_{fb}\) & \(2.30\times 10^{-8}\) & \(-7.31\times 10^{-8}\) & \(2.97\times 10^{-8}\) & 36.3 \\ D.A.I. & \(1.10\times 10^{-8}\) & \(-7.55\times 10^{-9}\) & \(4.85\times 10^{-9}\) & 24.1 \\ \hline \hline \end{tabular}  
* using a 0.001 time-step. Reminder: \(\psi\) is the axis of maneuver.  
\end{table} 
Table 2: Actual desired Euler angle errors and associated run times for the three cases 1. Figure 5: Time-step analysis comparing \(\theta_{actual}-\theta_{desired}\) and \(\omega_{actual}-\omega_{desired}\) errors. Figure 6: Control (Newton-meters) versus time in seconds for the three configurations. Figure 7 shows the Euler angle tracking error over time. Figure 7,b shows that the error is different for each controller. The feedback controller fluctuates initially as it corrects to reduce error over time. The feedforward controller is excellent initially, but slowly deviates as error accrues without correction. Lastly, the deterministic artificial intelligence system is the best, starting with minimal error, and furthermore correcting that error over time. Figure 7 shows the position error in the yaw \(\psi\) channel, which is the channel in which a 30-degree maneuver was commanded. The feedforward controller again starts off with minimal error before deviating over time, while the feedback controller starts with the greatest amount of error that is quickly damped, and the deterministic artificial intelligence is the best of both. Table 2 compares the boundary condition satisfaction at the final time of the maneuver. The results show that the deterministic artificial intelligence system (D.A.I.) has both the least amount of error and the shortest computational runtime. The feedforward controller is the worst in accuracy due to an inability to correct for error, while the feedback controller can correct, but takes longer to do so. The feedback controller is hypothesized to perform worse because Equation (2) tries to model Equation (1), but can only poorly approximate it, yielding inaccuracies. Figure 8 is a revisualization of the data presented in Figure 7 for clarity of conclusions. This depiction is more intuitive and illustrates the change in angular yaw position over time for each controller, as well as magnifying the post maneuver oscillations and damping (or lack of such, in the case of Figure 8,b). Commanding \(\{