Multi-control commercial aircraft trajectory optimization in a vertical plane with state-inequality constraints via singular control theory

Abstract

We present a computational framework based on singular control theory to optimize commercial aircraft flights in a vertical plane. The optimization problem involves the lift coefficient (a regular control), and throttle setting (a singular control). However, given a small aerodynamic path angle typically associated with commercial aircraft flights and the significant time-scale separation in flight dynamics, it is customary to treat the aerodynamic path angle as a singular control and replace the lift coefficient as a function of states. This results in a model that is singular on both controls. This research sheds light on several key challenges that have either been unexplored, such as the possibility of switches in both controls even during a specified flight phase, or underexplored, such as state-inequality constraints and wind shear, in relation to the pure singular model. Adhering to the Pontryagin’s maximum principle, we develop a switching-point algorithm to solve the pure singular model for the direct operative cost. Extensive analysis is conducted to understand the impact of various relaxed parameters. The simulation results obtained from the pure singular model are validated through first and second order optimality conditions. Finally, we compare the solutions obtained from the pure singular model with those of the original model, ultimately concluding that the pure singular model yields exceptionally accurate results. In conclusion, the paper highlights the computational time and accuracy advantages of the proposed optimization framework. Furthermore, it underscores the noteworthy observation that the sensitivity of the associated Hamiltonian system with respect to aircraft mass is marginal.