Aplicación de técnicas analíticas y numéricas de control óptimo en la determinación de actuaciones integrales de vehículos aeroespaciales

Abstract

The analysis of the integral performance of aerospace vehicles is of great relevance in the design and evaluation of the mission, as it will define fundamental aspects of their operation, such as the range or the autonomy. This research seeks to determine the optimal trajectories of these platforms. The approach to this type of scenario requires defining the dynamics of the vehicles; specifying their characteristics, including the corresponding propulsion systems, aerodynamics, etc.; modeling the operational conditions and establishing the constraints imposed by the mission. Therefore, in general, the problem is posed as an Optimal Control one. In particular, the integral performance of three types of platforms are considered. On the one hand, an analysis of the cruise phase is carried out to optimize the range of a reciprocating engine and propeller light aircraft, as well as a commercial aircraft with a turbofan engine. The optimal trajectories of a HAPS (High Altitude Pseudo Satellites) in station-keeping are also determined, trying to minimize the cost of the energy consumption. The systematic resolution of such variety of problems requires a numerical method. Within this context, this work proposes an algorithm based on a direct transcription numerical method as a robust and versatile tool to solve the resulting Optimal Control problems. Different discretization schemes are implemented, as well as mesh refinement techniques and different solvers that provide the algorithm with the necessary flexibility to adapt to different types of scenarios. In addition, due to the singular nature of some of these problems, a regularization term that allows these situations to be addressed is defined. The method is validated with several examples from the literature, including characteristic scenarios such as bang-bang controls or singular problems, so that its effectiveness is tested in a wide variety of circumstances. In all cases, we recovered the solutions of the references, with relative differences of less than 1 %, validating the solutions obtained by the method. Next, the problem of maximizing the range of a propeller-driven aircraft in a level flight cruise is analyzed within the framework of Optimal Control. A simple model of aircraft with a parabolic drag polar is assumed. The modeling of its propulsive system, in contrast to usual studies, considers that the specific fuel consumption and the propeller efficiency are functions of the velocity and the engine power. To conduct the study, we select a notional Piper Cherokee PA-28 as representative of light aircraft. With the existing data, the airplane and mission features, a cruise at 7000 ft, are characterized. Moreover, from the engine and the propeller performance charts, the model of the propulsive system is constructed (Full model). We also derive two simplified models: the Von Mises model, with constant specific fuel consumption and propeller efficiency, and the PA one, defined by a constant specific fuel consumption and propeller efficiency depending on the velocity. The problem is solved numerically by means of a direct transcription method. The regularization term incorporated makes it possible to obtain a solution for the Von Mises and PA models, since the lead to singular problems. In addition, an indirect method is implemented to determine the singular arc through the Kelley condition, which allows to find the contribution of inertial terms, neglected in the Breguet equations, whose impact is minimal —of the order of 100 meters in the range of the aircraft. Subsequently, the numerical algorithm is validated again with this analytical solution. In this context, the velocity and mass (state variables), the power throttle (control), and the best range are determined. The Full model gives a maximum range of 1492 km. The differences with the Von Mises and PA models are about 24 km and 1 km, respectively. A non-optimal steady cruise is also analyzed, providing a significant reduction of the flight time, with 27 km decrease in the range, about 2 %; a quite different situation from the turbojet case, which presents reductions of the order of 10 %. The evolution of the state variables and control in the steady cruise, however, separates from the Full model. That is not the case for the PA model, which almost reproduces the Full model results. Additionally, it leads to a clear image of the physics involved in the flight: the best range comes from maximizing the product of the propeller and aerodynamic efficiencies with respect to the velocity. That determines the optimal arc. Next, we present a study equivalent to the previous one, but for a turfan aircraft with a turbofan engine. In this case, a Boeing 757-200, representative of narrow-body commercial airplanes is taken for the analysis. In the modeling of its propulsive system, once again, some simplifications usually employed in this type of works are removed. Hence, the dependences of the thrust and the specific fuel consumption on the velocity are considered derived from the available performance charts. Similarly to the propeller-driven aircraft, we derive a simplified model based on the Von Mises airplane model, i.e., we assume constant maximum thrust and specific fuel consumption for the given flight altitude. The implemented algorithm based on a direct transcription numerical method —with regularization term for resulting singular cases— is employed to solve the problem. Additionally, by using an indirect method, the analytical solution of the Von Mises scenario can be obtained, as well as the Breguet formulation. In this way, the impact of the inertial terms and the differences with the case of reciprocating-engined aircrafts can be determined. The results show that they have a small contribution, although compared to the reciprocating-engine propeller aircraft, it is significantly higher —in the order of about 30 km for the aircraft of the study. After the optimization process, the solution of the problem is obtained, including the state variables, the control and the best range. The Full model provides a maximum range of 3411.80 km, about 63 km longer than the Von Mises model. In addition, in this case, the resulting flight time is also lower, about 36 minutes, which represents a variation of about 12 %. On the other hand, for the constant speed program the range is significantly reduced, around 11% compared to the Full model, which are consistent with other results presented in the literature. The evolution of the state variables shows similar trends between the Full model and the Von Mises one, with mean differences of slightly less than 10 %. Finally, an analysis is conducted showing the influence of the altitude in the determination of optimal trajectories. Due to the high speeds associated with turbojet, or turbofan, aircrafts, an increase in flight level means that, in certain situations of the Von Mises model, the limit of the maximum velocity is reached, leading to the optimal trajectory having to be modified reducing the range with respect to the scenario in which this restriction is not effective. On the other hand, for the Full model, this circumstance does not occur since having lower associated velocities does not reach the maximum limit established. Finally, a new type of aerospace vehicles is analyzed. Due to the latest technological breakthroughs, High-Altitude Pseudo Satellites (HAPS) have recently become a feasible solution with great potential in the aerospace industry for Earth observation and communications, among other applications. Minimizing the energy consumption of these solar powered platforms is critical and, in the case of lighter than air vehicles, leads to smaller and more manageable platforms. When stratospheric airships perform a station-keeping mission, a certain displacement from reference point on the Earth’s surface is usually admissible. This flexibility makes it possible to define an optimal control law for the airship that minimizes the energy required to fly in a 24-hour cycle, leading to a sprint and drift trajectory. This research analyzes the impact on the energy balance of the mission that stems from the changes in the allowed station-keeping radius. It also considers the effects of the daylight hours, the wind intensity, and the characteristics of the on-board energy system. Once the associated Optimal Control problems have been formulated, they are rigorously solved numerically by means of the direct transcription method with regularization implemented. The results define the optimal sprint and drift trajectories adapted to every scenario, providing the time evolution of the available power that controls the flight. The effect of the different environmental, operational, and platform design parameters on the mission is studied. Thus, the relevance of the battery-solar panel cost ratio is shown, as it determines the change in the value of energy between day and night. In addition, the effect of daylight hours on the cost of the energy is calculated, as well as the influence of the wind, which has a greater impact at night —as energy is more expensive in this period. The analysis indicates that following the optimal trajectory leads to weight savings in the energy system of about 5.4 kilograms per kilometer of the station-keeping radius. It entails that, for example, if a 20 kilometer radius is allowed, the energy required decreases more than 6% and the payload capacity increases about a 43% when compared to the fixed-point flight.