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Multiphase mixed-integer optimal control applied to 4D trajectory planning in air traffic management

In this paper an approach to aircraft trajectory optimization is presented in which integer variables and continuous variables are considered. Integer variables model decision making processes, and continuous variables describe the state of the aircraft which evolves according to differential-algebraic equations. The problem is formulated as a multiphase mixed-integer optimal control problem. It is transcribed into a mixed integer nonlinear programming problem by applying a 5th degree Gauss-Lobatto direct collocation method and then solved using a nonlinear programming based branch-and-bound algorithm. The approach is applied to the following en-route flight planning problem: given an aircraft point mass model, a wind forecast, a 3D airspace structure, and the relevant flying information regions with their associated overflying costs, find the control inputs that steer the aircraft from the initial fix to the final fix following a route of waypoints and performing step climbs, while minimizing certain performance indexes in which fuel based, environmental based, time based, and overflying based costs are considered during the flight. The decision making process arises in determining the optimal sequence of waypoints and the optimal sequence of flight levels. The optimal times at which the step climbs are performed and the waypoints are to be overflown are also to be determined. Numerical results are presented and discussed, showing the effectiveness of the approach.