Assessment and Future Directions of Nonlinear Model Predictive ...

552 S. Gros et al.approaches have been proposed to decrease the computational time required bynonlinear MPC schemes. These approaches rely mostly on cleverly chosen coarseparametrizations [6, 13]. Though very effective, these approaches require that theinput trajectories be sufficiently simple to tolerate low-dimensional representations.Another solution consists in tracking the system trajectories with a fast feedbackloop. If the local dynamics are nearly time invariant, linear control theoryprovides effective tools to design this feedback loop. However, for systems havingstrongly-varying local dynamics, there is no systematic way of designing such afeedback law [1, 15, 16]. This trajectory-tracking problem can be tackled by theneighboring-extremal theory whenever the inputs and states are not constrained.For small deviations from the optimal solution, a linear approximation of the systemand a quadratic approximation of the cost are quite reasonable. In such acase, a neighboring-extremal (NE) controller provides a closed-form solution tothe optimization problem. Hence, the optimal inputs can be approximated usingstate feedback, i.e. without explicit numerical re-optimization.This paper presents two approaches to control a simulated robotic flying structureknown as VTOL (Vertical Take-Off and Landing). The structure has 4 inputsand 16 states. It is a fast and strongly nonlinear system. The control schemesare computed based on a simplified model of the system, while the simulationsuse the original model. The simplified VTOL model is flat [7, 8].The first control approach is based on MPC. The use of repeated optimizationof a cost function describing the control problem provides a control sequence thatsupposedly rejects uncertainties in the system.The second control approach combines a flatness-based feedforward trajectorygeneration in a slow loop and a linear time-varying NE-controller in a faster loop.The slow loop generates the reference input and state trajectories, while the fastloop ensures good tracking of the state trajectories. This control scheme is sufficientlyeffective to make re-generation of the reference trajectories unnecessary.The paper is organized as follows. Section 2 briefly revisits optimization-basedMPC, system inversion for flat systems and NE-control. The proposed two-timescalecontrol scheme is detailed in Section 3. Section 4 presents the simulatedoperation of a VTOL structure. Finally, conclusions are provided in Section 5.2 Preliminaries2.1 Nonlinear MPCConsider the nonlinear dynamic process:ẋ = F (x, u), x(0) = x 0 (1)where the state x and the input u are vectors of dimension n and m, respectively.x 0 represents the initial conditions, and F the process dynamics.