Assessment and Future Directions of Nonlinear Model Predictive ...

A Two-Time-Scale Control Scheme for Fast Unconstrained Systems 555Objectives++u refuTrajectoryGenerationPlantx refx+slow loop(feedforward)-fast loop(feedback)∆uNE-controller-K(t)∆xFig. 1. Scheme combining trajectory generation (via optimization or system inversion)and NE-controli.e. they are zero along the whole trajectory. Hence, the NE-controller reducesto:∆u(t) =−K(t)∆x(t) (15)K = R −1 Fu T S (16)Ṡ = −Q − SF x − Fx T S + SF uR −1 Fu T S (17)S(t k + T )=P (18)which can be viewed as a time-varying LQR. Note that, if the local systemdynamics are nearly constant, the NE-controller is well approximated by a LQRwith a constant gain matrix K. In contrast, if the system is strongly time-varying,it is necessary to compute the time-varying NE-controller (15)-(18).4 Application to a VTOL Structure4.1 System DynamicsThe simulated example is a VTOL structure.The structure is made of four propellers mounted on the four ends of anorthogonal cross. Each propeller is motorized independently. The propeller rotationalvelocities are opposed as follows (when top viewed, counted counterclockwise):propellers 1 and 3 rotate counterclockwise, while propellers 2 and