Assessment and Future Directions of Nonlinear Model Predictive ...

560 S. Gros et al.The choice of the prediction horizon T for the nonlinear MPC scheme is notobvious. Too short a prediction horizon tends to lead to stability problems,while too long an horizon is not desirable from a computational point of view[4]. The prediction horizon chosen for the MPC is T =4[s], with the samplingtime δ =0.1[s].With no parametric uncertainty in C s and C d , the MPC scheme is able tomove the VTOL nicely to the desired setpoints despite the fact that the inputsare computed based on the simplified model. However, MPC struggles whenthe parametric uncertainty on the aerodynamical coefficient exceeds 10 percent.Figure 3 shows the control performance for a 10 percent perturbation, the perturbationbeing applied to the four propellers as indicated in Subsection 4.2.The control is slow, each optimization takes minutes (the exact computationtime depends on the algorithm used) and exhibits a large overshoot.4.4 Two-Time-Scale ControlThe flatness property of the simplified model allows generating the referenceinput and state trajectories algebraically, which reduces the computation timesignificantly. The NE-controller in the fast loop ensures good tracking of thestate references. The cost function is :J = 1 ∫T f[(¯x − ¯x ref ) T Q(¯x − ¯x ref )+(ū − ū ref ) T R(ū − ū ref )]dt2t kwhere the matrices Q and R are the same as for MPC, and (¯x ref , ū ref )arethe reference trajectories generated by the system inversion loop. The choice offinal time for trajectory generation is T f =4[s], and the outputs trajectories arechosen such that the reference velocities and accelerations are zero at T f .This control scheme exhibits a nice behavior as shown in Figure 4. The referenceinput and state trajectories are parametrized using polynomials. Theyare generated once, and no re-calculation is needed. The computation time forthe flatness-based trajectory and feedback generation is fairly low (1.5[s] forthetrajectory considered). Figure 5 displays the gains of the NE-controller. Sincethe gains are strongly time varying, the NE-controller cannot be approximatedby a LQR.4.5 StabilityThe stability analysis of two-time-scale systems is usually treated within thesingular perturbation framework, such as in [5]. However, considering that theNE-controller approximates the optimality objective of the MPC, it is reasonableto seek a stability proof that shows that the slow and the fast loops work towardthe same goal, which does not require a time-scale separation. This work is partof ongoing research.