The open-loop stability of the C-5A operating at sea level conditions was analyzed by examining the static and dynamic stability of the aircraft.

Static stability values: Dynamic stability values:

M_alpha = -0.76 1/s M_q = -0.67 1/s

N_beta = 0.167 1/s^2 M_alpha_dot = - 0.169 1/s^2

L_B = -0.585 1/s^2 N_r = - 0.12 1/s

L_p = -0.329

Based on these values, the aircraft is statically and dynamically stable at sea level.

MATLAB was used to obtain the pole-zero map to evaluate the longitudinal open-loop stability; the poles were obtained from the state-space system (Image 2). Although two poles are stable, the other poles are incredibly close to 0. Thus, this indicates that the aircraft is overall neutrally stable.

The open-loop response was simulated by a Simulink model based on an elevator doublet input based on an elevator doublet input (Image 3). As previously discussed, the pole-zero map indicates that the system response will slowly diverge if not remain neutrally stable. Based on Image 4, the long-term forward velocity response slowly diverges. Images 5 - 7 shows that the responses of the aircraft's AOA, pitch rate and pitch angle remain neutrally stable. Although the divergent response is negligible and that the other responses are neutrally stable, it is still necessary to design an auto-pilot system. Doing so improves the safety of the crewmen and the efficiency of the aircraft.

The lateral-directional stability of the aircraft was evaluated in MATLAB by obtaining the pole-zero map (Image 1) from the state-space system (Image 2). The pole-zero map indicates that the overall open-loop system is unstable due to the presence of unstable poles. 

Aileron and rudder deflections were simulated in Simulink to analyze the open-loop system response. The simulation (Image 5) shows that all system parameters--roll angle, roll rate, sideslip angle, etc--all exponentially diverge. Unlike the results from the longitudinal open-loop analysis, these findings cannot be discarded because the aircraft would catastrophically fail if no auto-pilot control system is implemented. 

Although the C5-A is stable in the longitudinal and lateral-directional axis at 20,000 ft and 40,000 ft, this is not the case at sea level. Before an auto-pilot system was designed, the controllability of the longitudinal and lateral-directional state-space systems were checked. By checking the rank of each state-space system in MATLAB, both systems were found to be controllable in pitch, yaw, and roll.

The auto-pilot system was designed to control one state space variable at a time to simplify the system and to enhance system reliability. The roll, heading, and pitch angles were all controlled by a PID controller. The desired roll and heading angles were achieved by altering the aircraft's ailerons, whereas the desired pitch angle was achieved by altering the aircraft's elevator. To obtain more realistic results, ailerons could not deflect past 15° . Similarly, the elevator could not deflected past 5°.

In the lateral axis, the controller achieved the target roll angle (Image 1). The controller reached the target roll angle within five seconds before steady-state conditions were achieved at 35 seconds. Although there are oscillations during the transient phase, the heading angle remained ±1.2° of its target before finally steadying out. The controller could still be improved by reducing the magnitude of the oscillations and reducing the time to reach steady-state conditions.

The target heading angle was successfully achieved by the controller. Although there were no oscillations once the target heading angle was reached, the time to do so took 40 seconds. To improve the controller the PID gain could be modified to reach the target more quickly, and to smooth out any subsequent oscillations

In the longitudinal axis, the controller achieved the target elevator pitch angle (Image 5). In this axis, an elevator doublet input was used as a disturbance; the target pitch angle is 0°. By six seconds, the pitch angle was within 2.6° of its target, before steady-state conditions were reached at 13 seconds. This shows that the controller is fast-acting and accurate.