5.5.2 PD + Adaptive Nonlinear Compensation Tracking Performance

This section examines the performance of the PD plus adaptive nonlinear compensation controller when tracking a multi axis sinusoidal tumble trajectory. The test started out using only the PD controller with the gains tuned to the nominal PD+NL values ([lambda]=18, Kd=1).
The specified angular velocity trajectory about each axis was formed by a summation of two sine waves of different magnitudes and periods. The angular magnitude and period of each of the individual elements of the trajectory are listed in Table 5-1.

Table 5-1 Angular magnitude and period of each of the individual trajectory elements

After the PD controller tracked for several cycles, the adaptive controller was activated. The adaptation gain was set to ([Gamma]=100). The results are shown in Figure 5-27.

Figure 5-27 Multi-axis sinusoidal tumble trajectory tracking results for PD and PD+Adaptive NL using G to reduce noise sensitivity (after 245 sec). Adaptation gain ([Gamma]=100).

To help illustrate the overall trends for both the angular error and commanded moment, the data from the Figure 5-27 was averaged, and divided into two charts. On each chart, the data stream is divided such that the test starts at t=0 and continues until the point at which adaptation was activated. The adaptive data then also starts at t=0 so that the relative performance of the adaptive controller may be more easily observed. The average angular error data is shown in Figure 5-28, and the average moment data is shown in Figure 5-29.

Figure 5-28 Results from Figure 5-27 showing average angular error before (upper line), and after (lower line) adaptation was activated.

Notice in Figure 5-28 that once the PD controller is started, the error continues to increase with time. The average PD tracking error is 13.8°. Once the adaptation algorithm is activated, the error is asymptotically reduced to approximately 1.8°.

Figure 5-29 Results from Figure 5-27 showing average commanded moment before (upper line), and after (lower line) adaptation was activated.

Comparison of Figure 5-28 with Figure 5-29 shows that for the PD controller, as the angular error increases, the moment required to track the desired trajectory also increases. The average commanded moment using the PD controller is approximately 35% of the maximum moment commandable by the thruster system. Once the adaptation algorithm is activated, the commanded moment is steadily reduced to an average of 29% of maximum. So while the adaptive controller reduces the average angular error by a factor of 7.7 over the PD controller (using the same gains), it also reduces the average required moment by a factor of 1.2.
The adaptive controller can also compensate for certain dynamic effects that have not been included in the system model by "lumping" them in with the dynamic system parameters. For example, thruster asymmetries may be compensated for by the inertia parameters. If B is a matrix that describes the modification of the nominal thruster system performance (such as would be produced by a failed thruster), then the modified rotational dynamics can be given by


.

(5.3)

An excellent example of this capability was given by way of an unplanned anomaly. While conducting initial testing of adaptive controller tracking with the right dexterous manipulator extended, one of the x-thrusters failed. This failure produced significant asymmetries in pitch and yaw moment capability, while leaving roll unchanged. With the manipulator extended, and a significantly modified moment profile, the vehicle dynamics were very different from those for which the controllers were tuned.
In interest of examining controller performance in the presence of anomalies, testing was continued this degraded state. The tuned PD controller ([lambda]=68, Kd=1) was tested first. The specified trajectory was the same as the one described in Table 5-1. The results are shown in Figure 5-30.

Figure 5-30 Multi-axis sinusoidal tumble trajectory tracking results for tuned ([lambda]=68, Kd=1) PD controller. Vehicle had right dexterous manipulator extended, and one failed thruster.

The average angular tracking error was approximately 13°, with a maximum angular error of about 30°. The adaptive controller was then activated, using ([lambda]=68, Kd=1) with adaptation gain [Gamma] =100. Even in this degraded state, the adaptive controller can improve the vehicle performance. The average angular tracking error was asymptotically reduced to approximately 4°, with a maximum angular error of about 8° (Figure 5-31).

Figure 5-31 Multi-axis sinusoidal tumble trajectory tracking results for tuned ([lambda]=18, Kd=1, [Gamma]=100) PD+Adaptive NL controller. Vehicle had right dexterous manipulator extended, and one failed thruster.