Case Studies
7.4 MBCO with Integral Augmentation
The results of the MBCO show a steady state error in the desired pointing angle.
One possible cause of this steady state error is the imperfect friction compen-sation in the TableSat system. According to the truth model, if the friction compensation is adequate, the MBCO should be able to point TableSat in the desired direction with little or no steady state error. However, as was observed previously, we have not exactly modelled or compensated for all of the friction in the system, especially friction at low speeds and low voltage commands. In addition, the MBCO is essentially a Proportional-Derivative controller, meaning that there is no guarantee that it can remove all steady state error for a step input. To ensure zero steady state error for a step input, integral augmentation can be added to the MBCO.
7.4.1 Design Methodology
From single-input, single-output linear controller design theory, we know that adding an integrator to the forward path of a system will allow the closed loop system to perfectly track any step input. When extended to input, multi-output systems, this implies that there must be an integrator in each input (or output) channel. Consider the augmented TableSat plant dynamics:
˙
x = Ax + bu
˙u = v y = Cx
(7.38)
which can be written in state space form as follows:
x˙ TableSat does not have full state feedback, a state estimator is needed to estimate the missing state. The same state estimator used in the MBCO, the Kalman Filter, can be used with the augmented plant, yielding the following closed loop dynamics:
where Ka and La are the controller and estimator gain matrices, respectively, for the augmented plant. Ka and La can be found in the same manner as K and L in Section 7.3. For finding Ka, r and Q are defined as in Equation 7.33, with Ca replacing C. For finding La, W2 is defined as in Equation 7.35 and W1 is modified to include the additional plant state, u:
W1 =
After finding Ka and La, the MBCO with integral augmentation can be translated into TableSat form:
where, as before, the extra zeros in B1care needed to give the matrix the correct dimension. As with the MBCO, the above controller is defined in continuous time and will need to be converted to discrete time before implementing it on TableSat. The predicted closed-loop poles for the MBCO with integral augmen-tation (MBCOi) are -0.32, -5.2±10.6, and -15.0. After discretization, the MBCOi controller definition is:
nc = 4
0.9935 −1.06e−4 −9.85e−10 −2.02e−8
−2.16e−4 0.7752 1.57e−5 −4.79e−4
−31.69 −1060 1.0096 56.42
−0.9143 −1.303 −6.43e−4 0.7171
Like the MBCO, the MBCO with integral augmentation (MBCOi) was tested on the linear model, truth model, and real TableSat system with a simple pass through state estimator. Figure 7.15 shows the predicted output from the linear model with a step input of 50 degrees. As can be seen, the linear model predicts an overdamped response, with a slightly longer settling time that the MBCO. The MBCOi reaches its steady state in about 25 seconds. But, as with the MBCO, the MBCOi predicts no steady state error.
The linear model prediction can now be compared to the predicted results
0 10 20 30 40 50 60
Linear Model Predictions for MBCOi
Time (sec)
Position (deg)
Figure 7.15: Predicted results from the linear model for a step input of 50 degrees using the MBCO with integral augmentation.
from the truth model and real TableSat system. Figure 7.16 shows the predicted and actual TableSat response for a desired target of 50 degrees. As can be seen in the figure, the truth model prediction and real TableSat results are similar to the linear prediction, but like the MBCO, the truth model and real TableSat system respond quicker, reaching steady state in about 10 seconds. From the figure, it can be seen that the truth model predicts no steady state error, which is an improvement over the MBCO, which showed a steady state error of about 2.0 degrees. This improvement in steady state pointing implies that the integral augmentation did remove the steady state error in the truth model as desired.
On the other hand, the real TableSat results still show a steady state error of
about 2.0 degrees, which implies the integral augmentation did not act exactly as hoped.
0 5 10 15 20 25 30 35 40 45
0 10 20 30 40 50 60
Time (sec)
Position (deg)
MBCO with Integral Augmentation
Real TSat Truth Model
Figure 7.16: Predicted results from the truth model plotted with actual TableSat results using the MBCOi with a desired target of 50 degrees.
Chapter 8
Conclusions
The main goal throughout the TableSat project has been to develop and model a system that can be used to demonstrate the often abstract concepts of control systems engineering and show how those concepts can be applied to a real system.
More specifically, TableSat represents a complete, single degree of freedom space-craft that uses sensors and actuators to estimate and control the attitude and rate of the system. While the focus has been on using TableSat to demonstrate SISO linear controls techniques, those taught to undergraduate engineering students, it was also desirable to show how TableSat could also be used to demonstrate more complicated multivariable and state space controls techniques. Originally, Table-Sat started as a highly nonlinear system. To effectively function as a teaching tool for linear controls theory, however, TableSat needs to act as a linear system;
and it must be possible to accurately model the system. Through hardware and software upgrades it was possible to reduce some of the nonlinearities in the Ta-bleSat system. Once the hardware and initial software upgrades were complete,
system modelling was used to identify and sometimes quantify additional nonlin-earities in the system. Once identified, in many cases it was possible to eliminate or reduce these nonlinearities, allowing for the development of a linear system model and a truth model. Both models were then verified and used to develop example controllers and state estimators for the new TableSat system. Finally, these controllers and estimators were tested on the real TableSat system and the results compared to predictions based on the TableSat models.