TableSat System Identification

As was mentioned previously, one of the main goals of the TableSat project was to develop a model of the TableSat system that can be used to develop and test controllers and state estimators for the system. Once the hardware and software upgrades were complete, a series of tests and experiments were performed to develop the TableSat model. Chapter 5 discusses this model development process in detail. As a starting point, a system of differential equations was assumed as the TableSat equations of motion. These equations model the fan and TableSat dynamics and include nonlinear friction terms for both the fans and TableSat itself.

Including the friction values, there were a total of nine system parameters that needed to be determined to establish the TableSat system model. A com-bination of experimental tests and theoretical equations were used to determine initial values for each of the parameters and to identify and quantify the non-linearities in the system. For example, a torsional pendulum test was used to determine the TableSat moment of inertia, spin up-spin down tests were done to characterize the TableSat friction, and fluid dynamics equations along with the fan specifications were used to estimate the fan speed to voltage constant.

There were two main types of nonlinearities found during these tests. The first was the friction in TableSat itself. From the spin up-spin down tests it was determined the the friction in TableSat, because it spins about a single point, is

constant. It can therefore be compensated for using a simple friction compensa-tion curve that adds addicompensa-tional voltage depending on the direccompensa-tion TableSat is spinning. This method of friction compensation can easily be employed using the friction compensation function discussed above.

The second main nonlinearity in the TableSat system is the friction of the TableSat fans. Through experimentation, it was determined that the fan friction consists of a constant term and a linear term. Compensation for the fan static friction, or dead zone, was mentioned above and is discussed in detail in Chapter 5. There is currently no means to compensate for the linear fan friction. That friction is modelled as a part of the fan time constant. Once the fans have finished accelerating, that term is effectively gone; however, at low voltages there may be additional effects that have not be exactly determined.

Once initial values for the system parameters were determined, a TableSat truth model was developed. The truth model integrates the assumed equations of motion using the parameter values determined through analysis and testing and includes representative sensor noise. After creation of the truth model, it was then used to tune the TableSat parameters by comparing real TableSat data to truth model data and “tweaking” the parameters until the two sets of data matched reasonably well. In this manner, the TableSat truth model was determined to be a reasonable approximation of the real TableSat system.

Once the truth model was developed and the friction nonlinearities

iden-tified, methods for compensating for those nonlinearities were developed and implemented. With friction out of the system, the equations of motion reduced to linear equations of motion that could be solved to obtain a linear model of TableSat, which was also one of the goals of the project. The linear model was verified by comparing both closed-loop and open-loop data from the linear model, truth model, and real TableSat system. The data showed that, when controlling TableSat in closed-loop mode, the linear model does a good job of modelling the system as long as you stay within the saturation level of the actuators. The open-loop data showed that there are still some nonlinearities that are not com-pensated for in the system. The exact nature of these nonlinearities and how to compensate for them still needs to be investigated. Despite these nonlinearities, it is felt that the linear TableSat model is reasonable and can be used to design controllers and state estimators for TableSat.

8.4 Case Studies

With verification of the TableSat system model accomplished, the model was then used to develop and test controllers and state estimators for TableSat. Chapter 7 discusses the development of three different controllers, a simple N-sample averaging state estimator, and a simplified kinematic Kalman Filter. The three controllers are a PD controller, a Model Based Controller/Observer, and a Model Based Controller/Observer with Integral Augmentation. In all three cases, linear

controls methods were employed to develop the controllers. The PD controller was developed using SISO, Root Locus techniques. The other two controllers were developed using MIMO, state space techniques. All three controllers were tested on the linear and truth models as well as the real TableSat system. The results were discussed and explanations for any discrepancies were attempted. The most notable difference between the linear model predictions and truth model and real TableSat responses for all three controllers is that the truth model and real TableSat consistently respond faster than the linear model predicts. Further investigation is needed to determine the cause of this discrepancy.

By combining a simple N-sample averaging state estimator to the PD con-troller, steady state pointing performance could be improved. In addition, noise in the pointing angle was decreased. The degree of improvement in both steady state error and noise depends on the number of samples in the estimator. A 2-sample estimator reduced the steady state error by more than a factor of two and reduced the noise by √

2. A 5-sample estimator eliminated the steady state error and reduced the noise by √

5.

By combining a kinematic Kalman Filter to the PD controller, noise in the state estimates was essentially eliminated. In addition, the smooth estimates revealed interesting oscillations in the transient response of TableSat. Further analysis is needed to determine the cause of these oscillations, but they were also seen in the linear model prediction. The Kalman Filter also showed that there

are still some nonlinearities in the system that have not be fully determined and compensated for. It appears these nonlinearities occur at low speeds and low voltages. Further investigation is needed to determine these nonlinearities and ways to compensated for them.