Abstract

In the present day, the aviation sector is one of the largest contributor of carbon dioxide emissions in the world. As air traffic growth is expected to outweigh the industry's efforts to reduce air pollution, the problem of minimizing fuel consumption in commercial flight becomes of utmost importance. This thesis proposes an optimal control framework for the optimization of aircraft trajectories in Flight Management Systems (FMS), focusing on the problem known as the Economy Mode. This problem consists of minimizing the direct operating cost of the flight in compliance with a crew-supplied cost index.
The objective of the FMS is to obtain optimal true airspeed references that will then be followed by the pilot or the autopilot. The optimal top-of-climb and top-of-descent must be computed as well. A novel approach is proposed based on solving the problem analytically using a combination of Pontryagin's maximum principle and the Hamilton-Jacobi-Bellman equation. For the cruise phase, a sub-optimal algebraic solution for the true airspeed is obtained in a state-feedback form, which reduces to the well-known maximum range case when the cost index vanishes. For the climb and the descent, the sub-optimal speed is the positive root inside the aircraft's flight envelope of a 5\ts{th} degree polynomial whose coefficients involve only the state variables and the aircraft-specific coefficients, which can be found easily with fast-converging algorithms such as Newton's method. The exact optimal trajectories are computed numerically using the shooting method, and simulations show that the sub-optimal trajectories are close enough for all practical purposes. Moreover, the trajectories exhibit the expected behavior regarding the locations of the top-of-climb and top-of-descent. Having attained an analytic solution for the cruise and a computationally inexpensive formulation for the climb and the descent, the need to have a performance database in the system is eliminated thus making its implementation faster in real-time.
Overall, the developments presented in this work not only provide a very efficient means of implementing the optimal speed schedules in an on-board FMS, but also extend the theory of aircraft performance to the more general minimum-cost case based on the cost index.