Singular versus boundary arcs for aircraft trajectory optimization in climbing phase

TL;DR

This paper analyzes optimal aircraft climbing trajectories using control theory, comparing boundary and singular arcs under various constraints to improve fuel efficiency and climb time.

Contribution

It introduces a detailed comparison of boundary and singular arcs in aircraft climb optimization considering multiple constraints and cost criteria.

Findings

Boundary and singular arcs are both critical in trajectory optimization.

The study compares common aeronautical procedures with classical control policies.

Results highlight the effectiveness of different strategies under various constraints.

Abstract

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 flight path angle 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.