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.