Formulação teórica e numérica de problemas de controle ótimo segundo diferentes abordagens matemáticas da dinâmica de aeronaves

Abstract

Optimal control theory is an area of study that has undergone important developments in recent decades. Even today, this is a valuable field of study in the aeronautical sector, where improvements have been sought in the planning of optimal trajectories. Within this sphere, two important branches stand out. One of them deals with the development of numerical techniques for solving trajectory optimization problems in a more general way. In a second branch, it focuses on issues related to modeling the movement of the aircraft and the application of these models in the formulation of the mathematical problem. In both of these fields of study, open discussions are noted, subject to new developments. Concerning numerical methods, direct collocation techniques have been those that have gained wide prominence in research. However, there is the importance of a more in-depth view of its foundations in such a way that they can be properly interpreted and understood. For aircraft motion modeling, there is a need for a more in-depth discussion about the effects related to the choice and application of the different techniques for parameterizing its attitude. Such questions have a strong relationship with the optimal solution and are part of a discussion related to the theoretical formulation of the problem, which predates even the choice of how to solve it, i.e., the numerical formulation. Therefore, this work proposes an alternative form of formulating the problems of trajectory optimization of aircraft, both from a numerical and theoretical point of view. It is shown that the finite element method can be applied in the numerical formulation of optimal control problems, presenting a global architecture in the direct approach by complete parameterization, under which, the collocation method is seen as a particular case. A discussion is also introduced on how the different ways of representing the attitude of the aircraft affect the formulation of the problem. Each form of parameterization of the attitude has specific characteristics, inherent to its mathematical structure, which, if misinterpreted and used, can make the problem poorly formulated, restricting its solution to a certain set of trajectories. Alternative ways are suggested for the treatment of boundary constraints and demonstrate how these issues affect the optimal solution of the problem. The proposed contributions are presented in the solution of a real practical problem in the context of an air race where the optimal trajectories obtained presented a good correlation with experimental data obtained in flight. Finally, it appears that the discussions and proposals carried out throughout this work are of paramount importance for a more coherent approach to the problems of optimizing the trajectory of aircraft. It is therefore suggested that these be taken into account when addressing this class of problems.