SummaryIn this article, the minimum time and fuel consumption of an aircraft in its climbing phase are studied. The controls are the thrust and the lift coefficient and state constraints are taken into account: air slope and speed limitations. The application of the maximum principle leads to parameterize the optimal control and the multipliers associated with the state constraints with the state and the costate and leads to describe a multipoint boundary value problem, which is solved by multiple shooting. This indirect method is the numerical implementation of the maximum principle with state constraints and it is initialized by the direct method, both to determine the optimal structure and to obtain a satisfying initial guess. The solutions of the boundary value problems we define give extremals, which satisfy necessary conditions of optimality with at most 2 boundary arcs. Note that the aircraft dynamics has a singular perturbation but no reduction is performed.
In this article, we are interested in optimal aircraft trajectories in climbing phase. We consider the cost index criterion which is a convex combination of the time-to-climb and the fuel consumption. We assume that the thrust is constant and we control the air slope of the aircraft. This optimization problem is modeled as a Mayer optimal control problem with a single-input affine dynamics in the control and with two pure state constraints, limiting the Calibrated AirSpeed (CAS) and the Mach speed. The candidates as minimizers are selected among a set of extremals given by the maximum principle. We first analyze the minimum time-to-climb problem with respect to the bounds of the state constraints, combining small time analysis, indirect multiple shooting and homotopy methods with monitoring. This investigation emphasizes two strategies: the common CAS/Mach procedure in aeronautics and the classical Bang-Singular-Bang policy in control theory. We then compare these two procedures for the cost index criterion.
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