5.6 Discussion

This section will summarize some of the major points from this chapter.
Initial tuning of the PD controller produced an average regulation error of 0.7°. In Section 5.2.1, a gain modification strategy was introduced that "softened" the response of the controller as the angular increased. This modification allowed the proportional gain to be increased beyond the original limit cycle boundary. The resulting new average regulation error was 0.3°. Thus the gain modification strategy reduced regulation errors by a factor of 2.3 (Figure 5-32).

Figure 5-32 Comparison of average regulation error before (left) and after (right) the gain modification strategy from Section 5.2.1 was added.

Initial testing of the PD plus nonlinear compensation controller in Section 5.3.1 initially produced poor tracking performance due to problems with thruster saturation caused by propagation of high frequency sensor noise through the controller. Initial tracking performance for the PD plus nonlinear compensation controller using the PD gain set ([lambda]=68 and Kd=1) produced an average angular tracking error of 7.8° It was found that reduction of [lambda] reduced the effect of the noise, thus initially reducing the tracking error. Reducing [lambda] arbitrarily low reduces the effective proportional gain of the closed loop system, thus increasing tracking errors. A minimum average angular tracking error of 5.9° was found at [lambda]=18. Thus gain tuning reduced the angular error by a factor of 1.3 (Figure 5-33).

Figure 5-33 Comparison of average angular tracking error for the PD+NL controller before (left) and after (right) the "proportional" gain [lambda] was tuned to reduce average angular tracking error (Figure 5-20)

Once the gains had been tuned for the nonlinear compensation controller, a strategy for reducing the effect of noise was implemented (Section 5.3.1). The constant G from (5.2), reduces the gain [lambda] within the noisiest term of the nonlinear controller. With G=100, the average angular tracking error was reduced by a factor of 3.3 to 1.8° (Figure 5-34).

Figure 5-34 Comparison of average angular tracking error for the tuned PD+NL controller before (left) and after (right) the noise reduction strategy from Section 5.3.1 was introduced.

Now that the nonlinear controller was tuned, its tracking performance could be compared with that of the tuned PD controller. The average angular tracking error for the tuned PD controller ([lambda]=68 and Kd=1) was 4.1°. The tuned PD plus nonlinear compensation controller ([lambda]=18, Kd=1 with G=100) produced an average angular tracking error of 1.8° for the same trajectory, thus reducing the average error by a factor of 2.3 (Figure 5-35).
The average commanded moment required for producing the results in Figure 5-35 was then examined. The average commanded moment for the tuned PD controller was 55% of the maximum commandable by the thruster system, while the tuned PD plus nonlinear compensation controller required an average commanded moment of 47% for the same trajectory, thus reducing the average commanded moment by a factor of 1.2 (Figure 5-36). Not only does the nonlinear compensation controller produce lower average tracking errors than the tuned PD controller, but it does so using a lower average commanded moment.

Figure 5-35 Comparison of average angular error for the tuned PD controller (left) and the tuned PD+NL controller (right) while tracking a sinusoidal yaw trajectory (Section 5.3.1).

Figure 5-36 Comparison of average commanded moment for the tuned PD controller (left) and the tuned PD+NL controller (right) while tracking a sinusoidal yaw trajectory (Section 5.3.1).

In Section 5.5.2, the PD plus adaptive nonlinear compensation controller was tested. The test started with the PD controller using the nominal PD+NL gains ([lambda]=18, Kd=1 with G=100) tracking a multi-axis sinusoidal tumble trajectory. This produced an average angular tracking error of 13.9°. When the adaptive controller was activated (starting with the parameter estimates initialized to zero and the adaptation gain [Gamma] set to 100), the average angular tracking error was asymptotically reduced toward 1.8°. Thus the adaptive nonlinear compensation controller produced a factor of 7.7 reduction of average angular tracking error over a quaternion based PD controller using the same gains (Figure 5-37).
Additionally, the average commanded moment is reduced from 35% of the maximum level commandable by the thrusters to 29% (Figure 5-38). The nonlinear adaptive controller significantly improves tracking performance, while also reducing the commanded moment by a factor of 1.2.

Figure 5-37 Comparison of average angular error for the PD controller (left) and the PD+Adaptive NL controller (right) while tracking a sinusoidal multi-axis tumble trajectory (Section 5.5.2). Nominal PD+NL gains were used for both controllers.

Figure 5-38 Comparison of average commanded moment for the PD controller (left) and the PD+Adaptive NL controller (right) while tracking a sinusoidal multi-axis tumble trajectory (Section 5.5.2). Nominal PD+NL gains were used for both controllers.