Analysis of Successive Loop Closure Autopilots in Simulation

Recall the objective of the designed lateral autopilot was to drive course  to commanded course . Figure 17 below depicts an enlarged view of course  vs commanded course  originally shown in Figure 14 of section 3.1 of this report which gave the simulation results of the successive loop closure control method.

It should be noted that the aircraft was undergoing both altitude and course heading commands during the simulation. Thus, some of the erratic oscillations (every 20 seconds) in Figure 17 below were a result of the aircraft changing altitude at that point in time.

Figure 17: Successive Loop Closure Simulation Results – Course Command Tracking

The lateral aerodynamic coefficients of the Zagi HP Electric Flying Wing are significantly smaller than that of typical, larger UAVs. For example, the aerodynamic coefficients of the primary control derivatives, which describe the moments produced by specific control surface deflections, are multiple orders of magnitude less than that of larger, more stable aircraft. Most notable of these primary control derivatives, with respect to course heading (the commanded variable in Figure 17 above), is the rudder control derivative coefficient. Because the Zagi is a flying wing configured aircraft, it does not have a rudder to assist in the control of course heading.

Other notable aerodynamic coefficients with respect to the Zagi Flying Wing lateral dynamics are the yaw static stability derivative and yaw damping derivative. Both of these aerodynamic coefficients play significant roles in determining a fixed-wing aircraft’s yaw angle  which is equivalent to its course heading. For the Zagi Flying Wing,  and  which are extremely small when compared to the coefficients of a traditional fixed-wing UAV like the Aerosonde UAV (a UAV designed to collect weather data manufactured by Aerosonde Ltd). The aerodynamic coefficients of the Aerosonde UAV and Zagi HP Electric Flying Wing can be found in [2] while a complete table of the Zagi aerodynamic coefficients is provided in Appendix A of the attached report. Additionally, a thorough description of these aerodynamic coefficients can be found in Appendix B of the report.

Recall the objective of the longitudinal autopilot was to drive altitude h to commanded altitude hc and airspeed Va to commanded airspeed Vac. Figure 18 below depicts an enlarged view of altitude h vs commanded altitude hc and airspeed Va vs commanded airspeed Vac originally shown in Figure 14 of section 3.1 of this report which gave the simulation results of the successive loop closure control method.

Figure 18: Successive Loop Closure Simulation Results – Height/Airspeed Command Tracking

It is clear from Figure 18 above that the SLC longitudinal autopilot’s performance is far superior to that of the SLC lateral autopilot; however, as discussed previously, this is primarily due to the fact that the Zagi HP Electric Flying Wing aircraft modeled lacks stability and control with respect to its lateral dynamics. The longitudinal dynamics are influenced by driving the aircraft’s elevons together which is much more similar to that of a traditional fixed wing aircraft than lateral dynamics which can effectively be stabilized with a rudder control surface (which flying wing aircraft’s lack).

Figure 19 below depicts an enlarged view of the control inputs originally shown in Figure 14 of section 3.1 of this report which gave the simulation results of successive loop closure control method.

Figure 19: Successive Loop Closure Simulation Results – Control Variables

Note that due to the Zagi’s flying wing configuration, the rudder control surface is nonexistent and thus  throughout the entirety of the simulation. Additionally, it should be noted that the aircraft was undergoing both altitude and course heading commands during the simulation. Thus, elevator and aileron control actuations were both in effect throughout the entirety of the simulation.

 

Analysis of Linear Quadratic Regulator Autopilots in Simulation

Recall the objective of the designed lateral autopilots was to drive course to commanded course. Figure 20 below depicts an enlarged view of course  vs commanded course  originally shown in Figure 15 of section 3.2 of this report which gave the simulation results of the linear quadratic regulator control method.

Figure 20: Linear Quadratic Regulator Simulation Results – Course Command Tracking

Again, note that the aircraft was undergoing both altitude and course heading commands during the simulation. Thus, some of the erratic oscillations (every 20 seconds) in Figure 20 above were a result of the aircraft changing altitude at that point in time.

The design matrices (Q and R) are used as “tuning knobs” for LQR control systems much like PID gains that are tuned in successive loop closure design. Their values are provided in the attached report.

Recall that  for the lateral LQR autopilot. This means that the most heavily penalized or regulated states of the LQR lateral autopilot were the roll angle and the course heading angle. Additionally, note that the most regulated control input was the aileron control surface action. This intuitively makes sense because, as discussed previously, aileron control surfaces are the primary control inputs used to effect an aircraft’s lateral dynamics. The rudder control surface action  does not effect the Zagi aircraft lateral dynamics because, again, it has no rudder.

Recall the objective of the longitudinal autopilot was to drive altitude, h, to commanded altitude hc, and airspeed Va to commanded airspeed Vac. Figure 21 below gives an enlarged view of altitude h vs commanded altitude hc and airspeed Va vs commanded airspeed Vac using the LQR control design method.

Figure 21: Linear Quadratic Regulator Simulation Results – Height/Airspeed Command Tracking

The Q and R matrices designed for the longitudinal autopilot are provided in the attached report.

Figure 22 below depicts an enlarged view of the control inputs originally shown in Figure 15 of section 3.2 of the report which gave the complete simulation results of the LQR control method.

Figure 22: Linear Quadratic Regulator Simulation Results – Control Variables

It is visually easy to see that the LQR autopilot responses given in Figures 20 and 21 are superior to those of the SLC control design in Figures 17 and 18. Additionally, when comparing the control inputs of the SLC and LQR controller designs given in Figures 19 and 22, respectively, it is clear that the LQR controller is much steadier in applying control surface actuations. This is particularly visible in the comparative simulation data for the aileron control surface.