Control Design
5.4 Simulation setup
5.5.3 Parameter values in the optimal control problems
The values for the different controller gains are showed in Table 5.1, and their values are discussed in this section.
The PD-controller gains, Kp and Kd, were found through tuning as men-tioned in Section 5.2.4. Experiments with the PD-controller have confirmed that the controller works well, but to make the PD-controller perform even bet-ter, additional tuning of the controller gains could be done. The PD-controller was used to test whether the spacecraft attitude dynamics worked as intended and to produce initial guesses for theL1-controller.
The positive constants,k1,k2, andk3in (5.3a), (5.5a), and (5.7a) were found through tuning and they represent the weights on the different terms in the cost
70 Chapter 5. Control Design functions. The values of k1,k2, andk3 are identical for the three controllers to provide identical conditions for all of them. The results of the optimization are sensitive to changes in the weights, because k1, k2, and k3 specifies how much ’effort’ should be put into optimizing the term related to the respective controller gain. For instance, if there is a large value on the controller gain in front of the term dealing with the final state of ωbob, and a small value on the controller gain in front of the term dealing with the final state of qob, it essentially means that the optimization algorithm considers it more important to reach the desired final state of ωbob. For the path-following cost functions in (5.9a) to (5.9c), the constant a1 was set to the same value as k1 since both constants determine the cost on the term involving ωbob, a2 and a3 were both chosen to equal k2 since all three constants regard terms involving qob, and a4 were chosen to equalk3 since both constants regard theL0- andL1-norm.
The simulation time T is set to 70 s, and the number of control intervals N is set to 50, which yields a step size h = NT = 1.4 s. A larger number of control intervals, i.e., N approaches ∞, would yield a smaller step size. For a smaller step size, the control torques have to become larger at each step in order to obtain the same effect as before. The total amount of torque required to perform a maneuver is the same regardless of the step size. Therefore, when the step size becomes smaller, the control torque will be applied over a shorter time, and the value of the torque needs to be larger to obtain the same total torque as for a larger time step. At one point, the control torques cannot be larger due to saturation on the actuators, which in this case are reaction wheels. If the control torque needed at one time step is larger than the saturation limit, it could be necessary to apply an additional control torque at the following time step to get the same amount of torque. However, if the saturation limit was higher, the reaction wheels may not saturate even though the step size decreases, and it would not be necessary to apply torque at an additional time step. Thus, decreasing the step size while keeping the saturation limit constant could yield a control signal which is less sparse than if the saturation limits were higher.
The spacecraft state variables and the control input that are optimized us-ing CasADi, are of different orders of magnitude and are therefore scaled before the optimization. IPOPT does not scale the optimization variables automatically, even though it does so for the objective function and the constraints (CasADi, 2018). Therefore, the optimization variables had to be scaled before the opti-mization. The scaling variables were found by studying the final values of the state and input before the scaling was applied, followed by some additional tuning. An observation was made during the tuning that the solution to the optimization is very sensitive to the values chosen for the scaling variables.
In Section 5.2.1 it was mentioned that the value , used in the complimen-tary constraints of the controllers (5.3) and (5.5), is a positive constant which imposes slack on the complementarity constraints. (5.3) and (5.5) are relaxed formulations of the general L0-norm minimization problem discussed in Feng
Chapter 5. Control Design 71 et al. (2016) and Section 3.6. (3.31) shows that the complementarity constraint on the minimization problem is equal to zero for the half complementarity for-mulation of the general L0-norm minimization problem, i.e., ξ ◦ x = 0. The relaxed formulation presented in Section 3.6.4, and used in the optimal control problems (5.3) and (5.5), relaxes the complementarity constraints using. The value ofdetermines how strict the complementarity constraint is, i.e. if = 0 it implies that the value of eitherξ orx, or both, has to be exactly zero, and it could be hard to solve an optimization problem having such strict constraints.
On the other hand, if is a nonzero and small value then the values of ξ or x, or both, do not have to exactly zero, they just have to be small. This would provide a less strict constraint on the optimal control problem, and the problem could be easier to solve. During the design process of the maximum hands-off controller and the moving maximum hands-off controller, different values of were tested, resulting in various performance of the controllers. Eventually, the value for was chosen to be 1·10−8 because it provided good results during the tests. Also,=1·10−8 is the value used for the experiments in the work by Feng et al. (2016).
72 Chapter 5. Control Design
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