Optimal Variable-Speed Climb for a Fixed-Wing Aircraft

摘要

In this paper, climb performance optimization is studied to determine control trajectories that minimize the Direct Operating Cost of an air transport. First a model of the vehicle motion is developed. A reduced-order model is then derived, and the calculus of variations is applied to formulate the optimization problem using the reduced-order model. An algorithm is designed to solve the problem, and the efficacy of the solver is validated using the high-order model. Two optimization methods are developed. In the first method, nonlinear programming (NLP) is used to determine the optimal open-loop control. In the second method, the reduced-order model is further simplified-relative to the equations used in the first method-and termed the energy state approximation (ESA) method. The second method is more robust and faster in terms of computing a solution, but the resulting control is marginally sub-optimal relative to the control determined by the first method. Accordingly, the relative benefits of each approach are studied to determine the best design for a real-time embedded application. It is shown that the optimal path determined by a simplified model (i.e., a point mass model and energy state approximation) is indeed a singular arc. It is further shown that if the optimal arrival to the singular arch and the optimal departure from the singular arc are added to the control trajectory, the resulting flight path is a close approximation of the optimal control determined by the NLP method. The optimal controls generated by the first method are tested using the higher-order model and compared to the performance of a typical Flight Management System for a mid-range narrow-body transport. Minimum fuel burn is studied first. The time-related cost to operate the airplane is then included in the cost function to minimize the total Direct Operating Cost. A generic narrow-body transport with modern turbofan engines is the subject of the analyses, but the developed methods are applicable to any fixed-wing aircraft propelled by high-bypass-ratio turbofan engines.