Control Design Method: Successive Loop Closure

In this section, the basics of control system design are described for the successive loop closure method. The primary goal of the autopilot is to control the inertial position and attitude of the UAV. Longitudinal dynamics, which include forward speed, pitching, climbing/descending motions, are assumed to be decoupled from the lateral dynamics, which include rolling, yawing motions and the side-to-side or turning motions. The decoupling of lateral and longitudinal dynamics is used often, if not unanimously, in design of aircraft autopilots because it yields good performance while simultaneously simplifying development.

The block diagram in Figure 7 below shows the lateral autopilot using the successive loop closure control method. The derivative gain provides roll rate damping for the innermost loop. Roll attitude is regulated with proportional and integral gains. Course heading is regulated with the proportional and integral gains. With successive loop closure, the gain values are designed in succession beginning with the innermost loop and working outward. The inner loops of the lateral autopilot are used to control roll angle and roll rate while the outer loop commands course hold.

Figure 7: Successive Loop Closure Lateral Autopilot Block Diagram from [2

Note that the innermost transfer function  in Figure 7 above describes the aircraft’s linearized loateral dynamics and is derived in detail in [2], and additionally, in Appendix C of the attached report. Outputs and their respective transfer functions found in the right-half of Figure 7 above are also derived in detail in [2], and in Appendix C of the attached report.

A key design requirement in tuning the PID gains of both lateral and longitudinal SLC autopilots is that each successive loop (innermost working outward) must have a lower bandwidth than the previous – ideally by a factor of 5 to 10. This ensures inner loops run at higher frequencies than outer loops. This is highly intuitive as inner loop variables are included in the feedback of outer loops.

Longitudinal autopilot design is typically more complicated than lateral autopilot design because of the influence of airspeed on aircraft longitudinal dynamics. The objective in designing the longitudinal autopilot is to regulate airspeed and altitude using the throttle and elevators as actuators. The block diagram in Figure 8 below shows the longitudinal autopilot using the successive loop closure control method. The innermost loop’s objective is to regulate aircraft pitch attitude. Design of this loop will follow methods very similar to that of the roll attitude loop in the lateral autopilot. The outer loop controlling altitude must have gains chosen such that bandwidth of altitude-from-pitch loop is less than bandwidth of pitch-attitude-hold loop.

Figure 8: Successive Loop Closure Longitudinal Autopilot Block Diagram from [2]

Note that the innermost transfer function in Figure 8 above describes the aircraft’s linearized longitudinal dynamics and is derived in detail in [2], and additionally, in Appendix C of the report. Outputs and their respective transfer functions found in the right-half of Figure 8 above are also derived in detail in [2], and additionally, in Appendix C of the attached report.

 

Control Design Method: Linear Quadratic Regulator

The derivation of the equations above for LQR controller design was adapted from [3] for this project and can be found in the attached report. Similar derivations can be found in both [4] and [5].