Assessment and Future Directions of Nonlinear Model Predictive ...

634 J.J. Arrieta–Camacho, L.T. Biegler, and D. SubramanianRADARWP FSM { S : } NMPC u∗ AIRCRAFT zFig. 1. Schematic view of the coupling between the FSM and the NMPC block. Thecurrent state together with the set-point and radar readings cause the FSM to updatethe set S which, in return, alters the structure of the nominal OCP within the NMPCblock. The optimal control action u ∗ obtained in the NMPC block is implemented inthe system.In order for each aircraft to reach a given target in an efficient manner, wedefine the following objective functional for each prediction horizon k:⎡⎤J[z(k), ū, N] = ∑ ⎢P i ⎣ 1 ∫ tk F(ū22 1,i +ū 2 2,i)dt + ηi Φ(¯z i , z spi )| ⎥t k ⎦ (6)Fit k Iwhere u 1,i and u 2,i are the forward and vertical load factors for aircraft i, respectively.In this application it suffices to consider the load factor as the accelerationexperienced by the aircraft. We choose to minimize the load factor terms becausethere is a direct relation between the acceleration of an aircraft and fuel consumption(higher forward or upward accelerations require more fuel) and pilotsafety and comfort.In (6), the contributions of each aircraft are added up, weighted by a factorP i ≥ 0representingthepriority of each aircraft. Each contribution includesan integral term, that measures the control effort, and an exact penalty termΦ(¯z i , z spi )| t k = ‖¯z i(t k F F ) − zsp i ‖ 1, weighted by a factor η i ≫ 0, that enforces thetarget.The target is imposed with an exact penalty term and not with a hard constraint,because it is not possible to know aprioriwhen the aircraft will reachthe target. If the aircraft are far from their targets, the exact penalty formulationencourages a closer distance to the target, without necessarily reaching it. Onthe other hand, if the target can be reached within the time horizon of the NMPCcontroller, the exact penalty is equivalent to a hard constraint, provided that theweighting factor η i is sufficiently large (see [9]; in this work we use η i =10 5 ).If the target can be reached within the time horizon k, the FSM transitions tothe lock mode, which reduces the interval t F by one unit at k +1, until thetarget is reached. The penalty term has important implications on the stabilityproperties, as discussed in the next section. The objective functional (6) togetherwith the above constraints and appropriate initial conditions specify the OCPgiven to the NMPC block.The FSM was implemented in Matlab as a collection of switch statements.The optimization step of the NMPC block is implemented in AMPL using