Resumen de Aircraft trajectory optimization using singular optimal control theory
The optimization of aircraft trajectories using the theory of singular optimal control is studied in this thesis. To describe the aircraft motion, a general nonlinear 3-degree-of-freedom point-mass model is adopted, along with realistic aerodynamic and propulsion models. The controlled motion of an aircraft is modeled as a control system whose performance can be optimized according to some performance index. This control system exhibits different dynamics, constraints and performance indices depending on the flight phase considerd, which leads to a multiphase control system formulation.
An indirect optimization method is applied, in which necessary conditions for optimality are explicitly involved into the problem resolution. The method proposed in this thesis exploits the singular character of the problem in order to provide analytical state-feedback control laws. With this approach, assuming a prescribed solution structure in terms of phase sequence and sequence of singular and bang arcs within each phase, the problem of finding the optimal control is transformed into the problem of finding the values of some unknowns such that the necessary conditions for optimality as well as the initial and final conditions are satisfied, that is, the problem of solving a nonlinear system of equations.
Optimizing global trajectories implies not only addressing each flight phase, but also taking into account the interactions among them as well as looking for a global objective. Therefore, an optimal global trajectory cannot be obtained by simply piecing individually optimized phases together, not even when each phase is optimized with a performance index suitable for a global the optimal control and optimal path structure for a single-phase optimal trajectory also apply at each phase of an optimal multiphase trajectory. As a consequence, prior to applying this approach to the problem of multiphase trajectories of commercial transport aircraft providing minimum fuel consumption, this approach is applied to three auxiliary single-phase problems.
First, the problem of fuel-optimal fixed-rating aircraft climb in the presence of altitude-dependent winds is analyzed. The climb is optimized to give minimum contribution to the global-trajectory fuel consumption. The optimal control is of the bang-singular type, and the optimal trajectories are formed by a singular are and two minimum-path-angle arcs joining the singular are with the given initial and final points. This analysis is used to assess the optimality of a standard climb procedure defined by segments with constant calibrated air speed and Mach number. Linear wind profiles defined by two parameters, the average wind and the wind shear, are considered. The effects of the wind profile and of the initial aircraft weight on the results are studied. Comparison with the optimal results shows that the performance of the optimized standard climb, in terms of global variables such as fuel consumption, flight time and horizontal distance travelled, is very close to optimal.
Second, minimum-fuel cruise at constant altitude with the constraint of a fixed arrival time is analyzed, including the effect of average horizontal winds. Again, the optimal control is of the bang-singular-bang type, and the optimal trajectories are formed by a singular arc and two minimum/maximum-thrust arcs joining the singular arc with the given initial and final points. The effects of average horizontal winds on the optimal results are analyzed, both qualitatively and quantitatively. The influence of the initial aircraft weight and the given cruise altitude is analyzed as well. Two applications are studied: first. The cost of meeting the given arrival time under mismodeled winds, and second, the cost of flight delays imposed on a nominal optimal path. The optimal results are used to assess the optimality of cruising at constant speed; the results show that the standard constant-Mach cruise is very close to optimal.
Third, unpowered descents of commercial transport aircraft are optimized in the presence of altitude-dependent winds, with the objective of maximizing range. The optimal problem and an optimized constant-calibrated-airspeed procedure are analyzed. The optimal control is of the bang-singular-bang type, and the optimal trajectories are formed by a singular arc and two maximum-path-angle arcs joining the singular arc with the given initial and final points. Linear wind profiles defined by two parameters, the average wind and the wind shear, are considered. The effects of both the average, wind and the wind shear on the optimal results, as well as the effects of the aircraft weight, are analyzed. The wind shear is shown to have a clear effect on the maximum range. The comparison between the two sets of results shows that the optimized constant-calibrated-airspeed descent is very close to optimal.
Once the auxiliary single-phase problems are solved, the problem of global trajectories of commercial transport aircraft providing minimum fuel consumption is analyzed. The global trajectories are considered to be composed of three types of phases: climb, cruise, and unpowered descent. The optimal control in every phase is of the bang-singular-bang type, and the optimal climb, cruise and descent trajectories are formed by a singular arc and two minimum/maximum-control arcs joining the singular arc with the given initial and final points. The optimal trajectories and controls, the minimum fuel consumption and some interesting global results are computed for an aircraft performing a climb-cruise-climb-cruise-descent trajectory. Linear wind profiles defined by two parameters, the average wind and the wind shear, are considered. The influence of the aircraft weight and the wind profile on the results is analyzed.
