A Stochastic Switched Optimal Control Approach to Formation Mission Design for Commercial Aircraft

Abstract

Currently, the main challenge for global aviation is to ensure that the predicted growth in air traffic for the coming decades remains sustainable from an environmental point of view, and that the air traffic management system meets the expected demand for increased capacity. Formation flight offers great promise in terms of improving both the environmental impact of aviation and the capacity of the air traffic management system. This thesis addresses the formation mission design problem for commercial aircraft in the presence of uncertainties. Specifically, it considers uncertainties in the departure times of the aircraft and in the fuel burn savings for the trailing aircraft. Given several commercial flights, the problem consists in arranging them in formation or solo flights and finding the trajectories that minimize the expected value of the direct operating cost of the flights. Since each aircraft can fly solo or in different positions inside a formation, the mission is modeled as a stochastic switched dynamical system, in which flight modes of the aircraft are described by sets of stochastic ordinary differential equations, the discrete states of the system describe the combination of flight modes of the individual aircraft, and the switching logic among the discrete states is defined by logical constraints. The formation mission design problem is formulated as an optimal control problem of a stochastic switched dynamical system and solved using nonintrusive generalized polynomial chaos based stochastic collocation. The stochastic collocation method converts the stochastic switched optimal control problem into an augmented deterministic switched optimal control problem. With this approach, a small number of sample points of the random parameters are used to jointly solve particular instances of the switched optimal control problem. The obtained solutions are then expressed as orthogonal polynomial expansions in terms of the random parameters using these sample points. Depending on the distributions of the random parameters, different types of orthogonal polynomials can be chosen to achieve better precision. This technique allows statistical and global sensitivity analysis of the stochastic solutions to be conducted at a low computational cost. The augmented deterministic switched optimal control problem has been then solved using an embedding approach, which allows switching decision among discrete states to be modeled without relying on binary variables. The resulting problem is a classical optimal control problem which has been solved using a knotting pseudospectral method. The aim of this study is to establish if, in the presence of uncertainties, a formation mission is beneficial with respect to solo flight in terms of the expected value of the direct operating costs. Several numerical experiments have been conducted in which uncertainties regarding the departure times and on the fuel saving during formation flight have been considered. The obtained results demonstrate that benefits can be achieved even in the presence of these uncertainties and that formation flight has great potential to reduce fuel consumption and emissions in commercial aviation.