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Aircraft Performance Optimization

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... Cost-optimal control of fixed-wing aircraft has been the subject of research for decades ([1]- [8]). The approach described here builds on this previous work. ...
... Since the vehicle operates at subsonic speeds, a flat, non-rotating earth is assumed. To make the problem tractable, the longitudinal and lateral motion are assumed decoupled, and a reduced-order model of the aircraft motion in the vertical plane is derived [8] using the so-called energy state approximation [1]. ...
... The states of Problem 8 are h, E, and m and V is an explicit function of the energy and altitude. To solve Problem 8, an approach similar to the one used in [8] is adopted. Since the dynamic equations are affine with respect to the control input γ and the cost is independent of the input, the velocity set is convex and the system does have an optimal solution. ...
... ATM has gained considerable interest among the research community in the last 20 years. References [4][5][6] present trail-blazing research on aircraft performance optimization. Reference [4] introduces various fixed end-point flight path optimization problems, such as fuel minimization with a fixed range, time minimization with a fixed range, and fuel minimization with fixed range and fixed time conditions. ...
... References [4][5][6] present trail-blazing research on aircraft performance optimization. Reference [4] introduces various fixed end-point flight path optimization problems, such as fuel minimization with a fixed range, time minimization with a fixed range, and fuel minimization with fixed range and fixed time conditions. References [5,6] demonstrate trajectory optimization in a vertical plane considering the direct operating cost for subsonic transport aircraft. ...
... This introduces drag following (9) and (8); therefore, thrust can be estimated from (4). Fuel flow is generally expressed as shown in (11) with thrust-specific fuel consumption . ...
Article
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The increase in air traffic worldwide requires improvement of flight operational efficiency. This study aims to reveal the potential benefits, namely, savings on fuel consumption and flight time, which are expected for Japanese airspace, by statistically evaluating the operational efficiency defined by average differences of fuel consumption, flight time, and flight distance between the original and the optimized flight of domestic flights in Japan. The aircraft position and time data used in this study were obtained from Collaborative Actions for Renovation of Air Traffic Systems Open Data—the radar data released by the Japan Civil Aviation Bureau. Flight information, such as air data and fuel flow, is estimated by applying meteorological data and aircraft performance model to the position information of radar data. Each reconstructed trajectory is optimized in terms of flight fuel consumption and flight time with an assumed cost index (CI). Dynamic programming is used as the trajectory optimization method. The flight fuel consumption and flight time of the optimized flight are compared with the original values to evaluate the operational efficiency. Herein, approximately one-third of 1-day data, i.e., 1087 cases of four aircraft types, are analyzed with reasonable CI settings. Our research findings suggest that flight fuel consumption and flight distance can be saved by 312 kg and 19.7 km, respectively, on average for the object flights. Following a statistical comparison between the original and the optimized flights, it was observed that two types of features, namely, flying on a detoured path and flying with nonoptimal altitude and speed in the cruise phase, are major factors which deteriorate the total operational efficiency in terms of fuel consumption, flight time, and flight distance.
... Climb trajectory optimization is an old research problem mainly studied in 1970s to 1980s. Most researchers used the maximum principle, and it was concluded that maximum thrust is approximately optimal [7][8][9][10]. Burrows [11] stated that reducing thrust near TOC can save fuel, but he did not clarify the reason. ...
... Small TSFC means that little fuel is required to produce a unit thrust, and so the fuel is efficiently used. In most previous studies, TSFC is assumed to be constant [7,8], which might affect the optimal climb trajectory. Burrows's paper does not explicitly mention TSFC in detail but seems to use more realistic fuel flow data with B767. Figure 3 shows the normalized TSFC of Large Aircraft 1 under typical flight condition according to the BADA4 model. ...
Article
This paper proposes a new fuel-saving climb procedure by reducing thrust near top of climb. Aircraft usually climb at maximum thrust during the climb phase. However, as revealed in this paper, the maximum thrust during climb is not always optimal considering the fact that the engine fuel efficiency of the maximum thrust is worse than that of the cruise thrust. Potential fuel savings of 40–80 lbs by a large jet airliner can be the result of this different engine thrust, but such savings cannot be fully achieved because of aircraft capability and air traffic control constraints. Therefore, the author proposes a practical fuel-saving climb procedure to satisfy the constraints of current aircraft and air traffic control operation. It is shown that a simple rule (800 ft/min climb from 6000 ft below the optimal cruise altitude) can save more than half of the potential saving (20–50 lbs) regardless of the initial weight with the same aircraft type. It is also confirmed that the proposed procedure works for other aircraft types.
... For the cruise (minimum-fuel) problem, the energy model is known to exhibit non-convexity of the velocityset, leading to unrealizable chattering-cruise [14,19,24] in certain cases. Schultz and Zagalsky [19] showed that the first-order necessary conditions are satisfied for the steady-state cruise arc for the intermediate model where the thrust and the flight-path angle are the con-trol variables. ...
... For the cruise (minimum-fuel) problem, the energy model is known to exhibit non-convexity of the velocityset, leading to unrealizable chattering-cruise [14,19,24] in certain cases. Schultz and Zagalsky [19] showed that the first-order necessary conditions are satisfied for the steady-state cruise arc for the intermediate model where the thrust and the flight-path angle are the con-trol variables. However, Speyer [22] showed that the solution obtained by Schultz and Zagalsky fails the Generalized Legendre-Clebsch condition in the vector control form [11,15]. Schultz [20], using the full pointmass model, showed that the steady-state cruise arc satisfies both the state-Euler equations and the Generalized Legendre-Clebsch condition; hence, it is a candidate solution for the minimum fuel, fixed-range problem. ...
... Due to the nonlinearity and complexity in aircraft dynamics, OTC problems are most often solved using numerical methods. A number of successful mathematical algorithms have been developed over the last decades [11][12][13][14][15]. These techniques can be generally classified into direct and indirect methods. ...
Article
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We present a novel stochastic optimal control framework that accounts for various types of uncertainties, with application to reentry trajectory planning. The formulation of the optimal trajectory control problem is presented in the context of an indirect method where a functional objective associated with the terminal vehicle speed is to be minimized. Uncertain input parameters in the optimal trajectory control model, including aerodynamic parameters and initial and terminal conditions, are modeled as aleatory random variables, while the statistical parameters of these aleatory distributions are themselves random variables. The parametric and model uncertainties are simultaneously propagated through an extended polynomial chaos expansion (EPCE) formalism. Several metrics are described to evaluate response statistics and presented as insightful tools for robust decision making. Specifically, the response probability density function (PDF) reflecting influence of both epistemic and aleatory uncertainties is obtained. By sampling over the random variables representing model error, an ensemble of response PDFs is generated and the associated failure probability is estimated as a random variable with its own polynomial chaos expansion. Besides, the sensitivity index functions of response PDF with respect to the statistical parameters are evaluated. Coupling parametric and model uncertainties within the EPCE framework leads to a robust and efficient paradigm for multilevel uncertainty propagation and PDF characterization in general optimal control problems.
... Long range cruise trajectories minimising fuel for a fixed mass aircraft have previously been researched by others using control variables of thrust and flight path angle (Schultz and Zagalsky 1972;Speyer 1973) or lift and thrust (Schultz 1974;Menon 1989), but these have not accounted for the wind field. In these papers the fuel flow is minimised for a set distance of flight. ...
Article
Full-text available
With the advent of improved aircraft situational awareness and the need for airlines to reduce their fuel consumption and environmental impact whilst adhering to strict timetables, fixed-time, fuel-optimal routing is vital. Here, the aircraft trajectory planning problem is addressed using optimal control theory. Two variants of a finite horizon optimal control formulation for fuel burn minimization are developed, subject to arrival constraints, an aerodynamic fuel-burn model, and a data-driven wind field. In the first variant, the control variable is expressed as a set of position-dependent aircraft headings, with the optimal control problem solved through a reduced gradient approach at a range of fixed airspeeds. The fuel optimal result is taken as the lowest fuel use recorded. In the second variant, both heading angle and airspeed are controlled. Results from three months of simulated flight routes between London and New York show that permitting optimised en-route airspeed variations leads to fuel savings of 0.5% on an average day (and up to 4% on certain days), compared with fixed airspeed flights. We conclude that significant fuel savings are possible if airspeeds are allowed to vary en route to take optimal advantage of the wind field.
... Therefore, the cruise point of design, where the aircraft components are optimized for this particular situation, is vital for the aircraft performance (Schultz and Zagalsky 1972). Consequently, the proposed wing and HTP design optimization will be performed under the previously mentioned cruise point of design, given the importance of this configuration. ...
Article
The efficient design of the aerodynamic surfaces in the aircraft allows the optimal performance of the vehicle and the reduction of the fuel consumption. Among these surfaces, the wing is the main contributor to the force which lifts the aircraft enabling the flight. However, the application of this force out of the center of gravity generates a moment that must be balanced through a force applied in the Horizontal-Tail-Plane (HTP) reaching the longitudinal trim of the aircraft. Traditionally, the design of the wing and HTP have been performed iteratively attaining suboptimal or time ineffective results. In this paper, we solve this problem through the application of a Genetic Algorithm for the combined optimization of the wing and the HTP by adjusting the aspect ratio, the taper ratio, the twist angle and the incidence angle of both surfaces to produce the optimal balance of lift adjusting its distribution to an elliptical configuration and enabling the longitudinal trim. Results show the automatic adjustment of the parameters of the aerodynamic surfaces thus fulfilling the objectives of this paper.
... Aircraft vertical motion is usually represented through one of these three different models: rigid-body, point-mass or energy-state [89]. The rigid-body model is the more representative but requires large computation times with respect to energy-state models, since more variables are involved in the calculations. ...
Thesis
The continued increase of air traffic, which doubles every 15 years, produces large economic benefits but poses environmental issues that put at risk the sustainable development of air transport. Other factors such as jet fuel prices volatility, the introduction of new environmental regulations and intense competition in the airline industry, have stimulated in the last years research on trajectory optimization and flight efficiency topics. The Flight Management System (FMS) is an onboard avionic system, standard in all transport aircraft, which is used by flight crews to manage the lateral and vertical flight-plan. Since current avionic systems are limited in terms of computational capacity, the computations performed by their algorithms are usually done on the basis of conservative hypotheses. Thus, notorious deviations may occur between FMS computations and the actual flight profile flown by the aircraft. The goal of this thesis is to develop an onboard function, which could be integrated in future Airbus cockpits, that computes optimal trajectories, readjusts the flight strategy according to the dynamic aircraft condition and minimizes operating costs. Flight energy management principles has been used for optimizing aircraft trajectories in descent and approach phases with respect to fuel consumption, greenhouse gas and noise emissions. The proposed function has been developed on the basis of dynamic programming techniques, in particular the A* algorithm. The algorithm minimizes a certain objective function by generating incrementally the search space. The exploration of the search space gives the optimal profile that links the aircraft current position to the runway threshold, independently of the current flight mode and aircraft energy condition. Results show 13% fuel savings and a decrease of 12% in gas emissions compared with a best-in-class FMS. Furthermore, the algorithm proposes the flight strategy to dissipate the excess of energy in situations where aircraft fly too high and/or too fast close to the destination runway. A preliminary operational evaluation of the computed trajectories has been conducted in the flight simulators. These tests demonstrate that the computed trajectories can be tracked with current guidance modes, although new modes should be required to decrease the workload of flight crews. In conclusion, this paper constitutes a solid background for the generation of real-time optimal trajectories in light of the automation of descent and approach flight phases.
... The aircraft motion in the vertical plane is represented by a point-mass model that provides a sufficient level of representativeness [25] from a performance perspective. The generation of the trajectory focuses on slow dynamics variables and disregards fast dynamics such as flight path angle rate (γ). ...
... The aircraft motion in the vertical plane is represented by a point-mass model that provides a sufficient level of representativeness [31] from a performance perspective. The generation of the trajectory focuses on slow dynamics variables and disregards fast dynamics such as flight-path angle rate (γ). ...
... The aircraft motion in the vertical plane is represented by a point-mass model, which provides a fair compromise between representativeness [23] and computing performance. The energy-limit trajectory stabilizes the aircraft in the minimum ground distance for any wind condition, and then provides the maximum energy dissipation in the shortest distance. ...
Article
The increased number of daily flights in the last decades has led to the densification of major airports and the proliferation of air traffic delays. The approach phase is inherently complex and labor extensive for flight crews, whose mission is to follow a stabilized approach. Current flight management systems do not provide much help in situations of high energy, where crews apply challenging energy management techniques to land in the destination runway. Go-around procedures shall be initiated as soon as the aircraft is not at the correct energy condition at the stabilization gate, despite the increase of workload this type of operation induces for crews and air traffic controllers. The continuation of a nonstabilized approach may put at peril the safety of flight. This paper presents an algorithm that computes the trajectory that stabilizes the aircraft in the minimum distance, which is also known as the high-energy-limit trajectory. The provision of this trajectory contributes to raising flight crew awareness of the aircraft energy state, and it informs that stabilization is not possible by giving the minimum distance to follow a stabilized approach. Flight simulator results helped to assess the operational concept and improve the representativeness of real flight operations. The calculation of the trajectory on a real-time basis provides flight crews with useful information of the aircraft energy condition, which improves flight safety and may ultimately reduce the number of nonstabilized approaches and go-around procedures.
... Aircraft TO models have gained much popularity over the last decades. For instance, Schultz and Zagalsky (1972) present solutions for several fixed endpoint aircraft TOPs using calculus of variations. In Raivio et al. (1996), a nonlinear programming-based method is proposed to compute optimal trajectories for a descending aircraft. ...
Article
Over the past few years, Unmanned Aerial Vehicles (UAVs) have become more and more popular. The complexity of routing UAVs has not been fully investigated in the literature. In this paper, we provide a formal definition of the UAV Routing and Trajectory Optimisation Problem (UAVRTOP). Next, we introduce a taxonomy and review recent contributions in UAV trajectory optimisation, UAV routing and articles addressing these problems, and their variants, simultaneously. We conclude with the identification of future research opportunities.
... Aircraft TO models have also gained much popularity over the last decades. For instance, Schultz & Zagalsky (1972) present solutions for several fixed end point aircraft TOPs using calculus of variations. In Raivio et al. (1996), a nonlinear programming-based method is proposed to compute optimal trajectories for a descending aircraft. ...
Article
Unmanned Aerial Vehicles (UAVs) are becoming increasingly popular over the past few years. The complexity of routing UAVs has not been fully investigated in the literature. In this survey, we aim to review recent contributions in UAV trajectory optimisation, UAV routing and contributions addressing these two problems simultaneously. A unified framework is introduced to describe UAV routing and trajectory optimisation problems. We conclude with the identification of future research opportunities.<br/
... The process of deriving the equations of motion, analytical solutions and the properties of optimal trajectories are available in many texts (Houghton and Brock 1960;Vinh 1981;Shevell 1989;Asselin 1997). (Betts 1998) (Schultz and Zagalsky 1972). Research on optimum horizontal guidance techniques and on aircraft trajectory optimisation in the horizontal plane was also carried out by (Erzberger and Lee 1971;Kishi and Pfeffer 1971;Heymann and Ben-Asher 1997). ...
Thesis
Full-text available
Flight simulators are becoming more sophisticated in replicating actual flying manoeuvres and conditions. Despite the advancement of technology, a flight simulator cannot perfectly represent a particular aircraft in all aspects. For example, the mathematical model of the aircraft is never fully accurate, the motion and visual systems have physical limitations that make the full representation of the sensation of flying less than perfect. Regulatory authorities around the world are beginning to approve—or are considering the approval of—single engine gas turbine (SEGT) aircraft for regular public transport (RPT) operations. This will require the flight simulator industry to consider exploring the use of flight simulators for SEGT aircraft in RPT operations. The application of flight simulators for initial pilot training for both civil and military pilots is still relatively unexploited. For example, training of ab-initio pilots in emergencies such as forced landings is still carried out in aircraft. Similarly, almost all training of combat manoeuvres for military pilots is also carried out in aircraft. The issues involved in doing such training in simulators are not well developed in the literature. This study raises some issues for training pilots to fly forced landings and examines the impact that these issues may have on the design of simulators for such training. In particular, it focuses on the trajectory that a pilot must fly after an engine failure and how pilots could be trained for this manoeuvre in a simulator. A sensitivity study of the effects of errors in the aerodynamic parameters was carried out and the requirements for determining these parameters for simulators were examined. This study also investigated the effect that the tolerances prescribed in the Manual of Criteria for the Qualification of Flight Simulators have on the performance of flight simulators used for pilot training. A simplified analytical model for the Beech Bonanza model E33A aircraft with retractable undercarriage was used to determine the effect of the aforementioned to the tolerances on forced landings. It was found that the effect of the tolerances is highly sensitive on the nature of the manoeuvre flown and that in some cases, negative transfer of training may be induced by the tolerances. For an engine failure height at 650 ft AGL, the results show that, following a turn around manoeuvre for landing, the touchdown points vary significantly from the reference model. This issue is of concern to flight simulator manufacturers in determining their tolerance standards. An investigation on the effect vertical atmospheric turbulence, based on the MIL-F-8785C specifications, has on a forced landing manoeuvre was also carried out. 100 vertical turbulence profiles were randomly generated and were applied to a simplified forced landing analytical model. The results show that the nature of the manoeuvre flown is highly sensitive to the vertical gust and is therefore important to flight simulation. The straight glide to touchdown manoeuvre shows possible touchdown locations that vary from approximately 637 ft to 828 ft from the engine failure location, while for the continuous 360° attempt, it ranges from 0 ft to 396 ft. The vertical gust has the most effect during the initial phase of the forced landing flight manoeuvre since a small deviation in the glide angle due to vertical gust will have a nonlinear effect on the horizontal distance. The effect for a 360° turn to touchdown is least affected since the turn radius is less susceptible to vertical gust. Therefore, it can be concluded that vertical turbulence has the most effect on the straight glide to touchdown manoeuvre and lesser effect on the turning manoeuvre. This analysis demonstrates the importance of analyses of vertical gust to flight simulation and the consideration of such requirements within the context of particular manoeuvres to be flown, as in some cases, negative transfer of training may be induced by the vertical gust. The vertical gust sensitivity analysis shows that a simulator may incur significant errors in the task of handling an engine failure after take-off for a single engine aircraft. This raises the question of the ability to use simulators to train pilots aptly for engine failure after take-off using the tolerances as specified in current regulations since the resultant errors are manoeuvre dependent. A forced landing trajectory optimisation was carried out using Genetic Algorithm (GA). The selection of GA control parameters can have a significant impact on the effectiveness of this optimisation algorithm. Therefore, the selection for a suitable set of control parameters in the GA analyses for both direct-value and real-value encoding was carried out to determine the best selection of population size, crossover rate, mutation rate and coefficients for non-uniform. The results show that there is minimal or no improvement in the fitness value when the population size exceeds four times the chromosome length and the computational cost increases linearly with the population size. The forced landing manoeuvre analyses with pre-selected touchdown locations and pre-selected final headings were carried out for an engine failure at 650 ft AGL for bank angles varying from banking left at 45° to banking right at 45°, and with an aircraft’s speed varying from 75.6 mph to 208 mph corresponding to 5% above airplane’s stall speed and airplane’s maximum speed respectively. Simulations were carried out for a time averaged atmospheric turbulence model and for a simplified thermal model for various crosswinds at three test locations. The results show that certain pre-selected touchdown locations are more susceptible to horizontal wind. The results for the forced landing manoeuvre with a pre-selected location show minimal distance error while the quality of the results for the forced landing manoeuvre with a pre-selected location and a final heading show that the results depended on the pre-selected location and on the final heading. For certain pre-selected touchdown locations and final headings, the airplane may either touchdown very close to the pre-selected touchdown location but with greater final heading error from the pre-selected final heading or touchdown with minimal final heading error from the pre-selected final heading but further away from the pre-selected touchdown location. Analyses for an obstacle avoidance forced landing manoeuvre were also carried out where an obstacle was intentionally placed in the flight path as found by the GA program developed for without obstacle. The program developed successfully found flight paths that will avoid the obstacle and touchdown near the pre-selected location. In some cases, there exist more than one ensemble grouping of flight paths. The distance error depends on both the pre-selected touchdown location and where the obstacle was placed. The distance error tends to increase with the addition of a specific final heading requirement for an obstacle avoidance forced landing manoeuvre. Again, as with the case without specific final heading requirement, there is a trade off between touching down nearer to the pre-selected location and touching down with a smaller final heading error. Although GA is capable of locating some of the best values for optimisation, it does not guarantee that it is a minimum value. Therefore, another optimisation technique, Sequential Quadratic Programming (SQP) was used to find more accurate and precise solutions. The results obtained agree well with the results obtained using GA. This research concludes with a practical forced landing manoeuvre strategy in the presence of unknown wind conditions. The main limitations of the research are highlighted and several areas for further research are suggested.
... Other studies followed (e.g., refs. 2,3,4,5), including the gradient-based computer solutions of Bryson,et al. (refs. 6,7). ...
... Вопросы оптимизации полета пассажирских самолетов не теряют актуальности уже несколько десятилетий. Этой проблеме посвящены многие работы отечественных и зарубежных авторов [1][2][3][4][5][6][7][8][9][10][11][12]. ...
Article
This paper presents a novel approach for generating the best possible climb trajectory that ensures minimum fuel and time efficient climb. The problem is first formulated using standard steady climb equations, which generate a unique combination of flight velocity and flight path angle at each altitude. A possible scenario, such as air density, mass, available power, and required powered variation with altitude, is taken into account when defining the problem. Thereafter, sliding-mode-based trajectory tracking control is formulated with its design procedures, system stability with applied control inputs, finite-time convergence analysis, and complete architecture. A Hansa-3 research aircraft is considered as an example model to demonstrate the work. The findings of generated trajectory are then produced and discussed. In order to follow the design trajectory and achieve the same, the sliding-mode-based control command is supplied. The novelty of the present work lies in proposed strategy of trajectory generation, wherein the aircraft path and velocity are found out to make the fuel and time efficient climb possible. Subsequently, robust control law is developed which shows the applicability of the proposed work on autopilot. The results show that the proposed controller not only controls the aircraft but is also able to follow the design trajectory with minimal errors. To further explore the impact of aircraft mass on climb performance, repeated set simulation is carried out. The outcome is compared with conventional climb, which promises its practical implementation since the proposed solution is simple and compatible to integrate with the existing aircraft autopilot.
Conference Paper
View Video Presentation: https://doi.org/10.2514/6.2022-3838.vid The trajectory optimization of single aircraft is an old problem, and there are already established optimization methods, indirect method and direct method. However, the indirect method requires analytical calculations and it is difficult to automate it in incorporating the constraints. On the other hand, the direct method requires computational costs for iterative calculations and does not assure the conversion to the global optimal solution. This paper proposes a new numerical optimization method to find the optimal descent trajectory, which hardly requires computational cost. The obtained results are compared to the ones obtained by the existing direct methods with various conditions such as different cost index and wind condition. The result shows that the trajectory calculated by the proposed method agrees well with the one calculated by the direct method.
Article
A study was performed to characterize an optimum return trajectory for a sailplane after a rope-break failure during an aerotow launch procedure. The performance of an SGS 1-26E sailplane was simulated using the equations of motion for quasi-steady flight in a time-stepping routine and published aerodynamic polar data. A gradient-based optimization algorithm was implemented using the simulated trajectory considering the glide velocity, bank angles, and runway offset angle to determine the minimum rope-break altitude from which a successful return could be theoretically produced. The SGS 1-26E sailplane was used in a flight-test campaign for empirical turn modeling and validation of the performance simulation. The minimum altitude where a return trajectory and downwind runway landing could be successfully completed after a rope-break event was observed to be 76.4 ft above ground level. A headwind and crosswind presence was observed to decrease the minimum rope-break altitude with increasing wind velocity up to a critical value, after which rope-break altitudes began to increase. These minimum rope-break altitudes serve as a theoretical indication that safe trajectories can be performed at lower failure altitudes than the commonly practiced decision altitude of 200 ft for certain wind conditions.
Article
It is a challenge to achieve fast altitude control for hypersonic flight due to the inherently insufficient sensitivity of air density to the near-space altitudes. A dominant open-loop feedforward term and a minor altitude error correction term are combined together to obtain the elevation angle command in this paper. A cascade control strategy is used to calculate the elevation command while ensuring the attitude stability. Moreover, an extended state observer is employed to achieve precise and robust attitude control simultaneously, and a stability margin tester is used to tune the controller explicitly. Extensive simulations are conducted to validate the effectiveness, robustness and measurement noise sensitivity of the proposed method.
Chapter
Periodical cruise can raise the fuel efficiency of hypersonic vehicle. Therefore, it is crucial to seek optimal characteristic parameters of periodical trajectory to establish a feasible guidance law in practice. In this paper, we employ the popular, mature and efficient nonlinear optimization scheme, pseudo-spectral method (PSM), to optimize the hypersonic periodic cruise in an innovative way by utilizing the symmetric property of periodic curve. Two optimization strategies, combined and entire ones, with artificial constraints for the sake of symmetry are provided and embedded within the PSM to obtain the optimal numerical solution. Then the parameters of specific periodic functions are fitted through nonlinear square method according to these numerical solutions. The numerical results of these two proposed methods and the optimal steady-state method are compared and investigated, demonstrating the effectiveness of the proposed methods.
Article
Full-text available
The paper presents the results of optimization of the flight program for a passenger aircraft. In mathematical modeling, real operational limitations and possible changes in atmospheric parameters were taken into account. The mathematical model is characterized by the description of the flight of an airplane as a material point and the description of a gas turbine engine by a model of the first level of accuracy. The following operational limitations were considered: the possibility of cruising at certain levels, the restriction on the vertical speed with a decrease, the possibility of transition to the next echelon with an available stock of thrust of 20%. The optimization of the flight program is based on the criterion of the minimum amount of fuel expended for the flight, at a given range. Calculations were conducted for several standards of air temperature change, (depending on the climatic zone). A comparative analysis of the obtained optimization results is carried out, the degree of influence of the change in atmospheric conditions is estimated. When comparing flights in different climatic zones, the fuel costs were compared for a flight program optimized for the given atmospheric conditions and for a flight program optimized for use in the mathematical model of the International Standard Atmosphere (ISA).
Thesis
In recent years, employing Unmanned Aerial Vehicles (UAV) to collect data and making measurements has gained popularity. Often, the use of UAVs allows for a reduction in costs and improvements of other performance criteria. The academic routing community has acknowledged the interest of companies and organisations in adopting UAVs in their operations. However, constraints due to the flight dynamics of UAVs have often been neglected. Finding feasible trajectories for UAVs in a routing problem is a complex task, but it is necessary to ensure the feasibility of the routes. In this thesis we introduce the Unmanned Aerial Vehicle Routing and Trajectory Optimisation Problem (UAVRTOP), the problem of optimising the routes and trajectories of a fleet of UAVs subject to flight dynamics constraints. Motivated by a disaster assessment application, we propose a variant of the UAVRTOP, in which a fleet of autonomous aerial gliders is required to photograph a set of points of interest in the aftermath of a disaster. This problem is referred to as the Glider Routing and Trajectory Optimisation Problem (GRTOP). In this work, we propose a single-phase Mixed-Integer Non-linear Programming (MINLP) formulation for the GRTOP. Our formulation simultaneously optimises routes and the flight trajectories along these routes while the flight dynamics of the gliders are modelled as ordinary differential equations. We avoid dealing with non-convex dynamical constraints by linearising the gliders’ Equations of Motion (EOMs), reducing the proposed MINLP into a Mixed-Integer Second-Order Cone Programming (MISOCP) problem. Another contribution of this work consists of proposing a multi-phase MINLP formulation for a modified version of the GRTOP. We do not attempt to solve this formulation directly, instead we propose a hybrid heuristic method that is composed of two main building blocks: (i) a Sequential Trajectory Optimisation (STO) heuristic, designed to cope with the challenging task of finding feasible (flyable) trajectories for a given route; and (ii) a routing matheuristic, capable of generating routes that can be evaluated by STO. We perform computational experiments with real-life instances based on flood risk maps of cities in the UK as well as in a large number of randomly generated instances.
Article
The paper deals with the optimal flight of an aircraft in a fuel constrained intercept maneuver. After a preliminary study of the problem itself an algorithm for real time optimization is presented. It is based on a reduced system of the state equations, where the direction of the velocity vector is treated as control. The integration of the system is done with a specially designed integration formula. The parameterized optimal control is approximated by a nonlinear programming method. State constraints are directly incorporated by elimination of parameters. The algorithm constructed in this way is taken as an element of a feedback guidance scheme. For some boundary conditions the flight path controlled by the feedback guidance is compared to the open-loop solution of the optimal control problem (“validationThe paper deals with the optimal flight of an aircraft in a fuel constrained intercept maneuver. After a preliminary study of the problem itself an algorithm for real time optimization is presented. It is based on a reduced system of the state equations, where the direction of the velocity vector is treated as control. The integration of the system is done with a specially designed integration formula. The parameterized optimal control is approximated by a nonlinear programming method. State constraints are directly incorporated by elimination of parameters. The algorithm constructed in this way is taken as an element of a feedback guidance scheme. For some boundary condi tions the flight path controlled by the feedback guidance is compared to the open-loop solution of the optimal control problem ("validation" of the guidance algorithm). A close agreement of both types of trajectories is observed. of the guidance algorithm). A close agreement of both types of trajectories is observed.
Article
Aircraft Cruise Performance is examined using energy state approximation and chattering controls to investigate fuel range optimal trajectories. Classical steady-state cruise, simple chattering-cruise and the extended chattering cruise models are studied as constrained parameter-optimization problems. The term "extended chattering" refers to vehicle system modeling extended to maintain vertical equilibrium only on the average. Computational results are presented for three different aircraft. This study shows that simple chattering cruise for certain specific energies can result in substantial fuel savings over classical steady state cruise. However extended chattering cruise results in only marginal fuel savings when compared to simple chattering cruise.
Chapter
Ein Flug (eine Flugmission) setzt sich aus verschiedenen Flugabschnitten zusammen, Bild 3.1.1. Wichtige Flugabschnitte sind:
Chapter
Die Grundgleichungen (1.6.17) beschreiben in allgemeiner Form die Flugbahn eines Luftfahrzeugs unter der Einwirkung von aerodynamischen, Antriebs- und Gewichtskräften. Sind diese Kräfte unausgeglichen, so werden zusätzliche Trägheitskräfte wirksam; Betrag und/oder Richtung des Geschwindigkeitsvektors sind dann zeitlich veränderliche Größen. Zur Bestimmung der Flugbahn in konkreten Fällen müssen die Grundgleichungen — ein System von Differentialgleichungen — integriert werden. Diese Aufgabe ist nur in einfachen Sonderfällen analytisch lösbar, da einige der in den Grundgleichungen enthaltenen Größen nichtlinear von den Zustandsgrößen V, γ und H abhängen und unter Umständen nur in der Form von Kurvenoder Tabellen bekannt sind.
Chapter
The object of the design synthesis process dealt with in the previous chapters is to achieve the goals laid down in the design specification. The first cycle of the iterative design process will be concluded with an analysis of the operational characteristics for the purpose of investigating to what extent the design requirements have been met. Some general comments on the prediction of aerodynamic characteristics are made in this chapter. Definitions and subdivisions of the drag according to several schemes are discussed. The choice of operational limit speeds and the determination of n-V diagrams are then briefly reviewed. A procedure to analyze the flight profile, reserve fuel quantity and payload-range characteristics is given, followed by some general aspects of climb and field performance. The chapter concludes with certain aspects of operating economy and some critical notes on the use of standard formulas for estimating direct operating costs.
Article
Alternative numerical and analytical approaches for trajectories calculation are presented. Unlike previous works which were pointed at optimized trajectories only, thus limited to certain definitions of climb-cruise-decent maneuvers, for fixed wing airframes only, the current treatment permits the usage of a full detailed aerodynamic and engine models, being implemented in the exact flight mechanics equations, generalized enough for both fixed and rotary wing airframes. The current analysis defines practical approaches to solve any type of maneuver asked by the operational squadrons. In the paper we give examples of maneuvers subjected to unique constraints such as constant ground airspeed, rate of climb, etc. The new maneuvers' trajectories were found to be pretty close to known trajectories of similar maneuvers such as the constant true air speed or constant path angle maneuvers. The approach shown here suggests that for fixed wing airframes, according to the constraint subjected in the maneuver, a delicate two dimensional scheme should be used in order to find the throttle settings and the angle of attack. The rotary wing schemes are found to be much easier to be solved numerically in each time step, as the induced velocity could be found analytically. Furthermore, we can always turn the equation for the path angle to an algebraic one. However, due to the small value of the absolute velocity of the rotary wing airframes, the integration process restrict us to a much smaller integration time step compared to a typical jet fighter simulation. We have proven that implementing such maneuvers in a global mission performance simulation may give the flight squadrons a way to obtain more accurate results based on their realistic way maneuvering the airframe.
Article
Algorithms for airliner trajectory control in accordance with 4-D navigation principles are considered. Vertical component of control is designed with the suboptimal algorithms. These algorithms are based on the results of the solution of problem about minimum fuel consumption in a flight of given range within specified time. Energy approximation and singular perturbation method are used in solving. The algorithms make available to take into account probable wind on a route. Particular attention is given to arrival of an aircraft in the required time. In connection with this, special control trajectory algorithm of the continuous descent is examined. This algorithm relies on the approximation of optimal continuous descent by the profile "CAS - constant number M - constant CAS" (here CAS is calibrated air speed). The algorithm provides some time window of arrival at the cost of the profile parameter variation. It may be used also for compensation of probable wind impact. Horizontal component of control is constructed according to a preplanned route. A route can be replanned at flight in connection with the necessity of fly-around of weather dangerous or prohibited air areas that had not been taken into account at flight planning. The method of trajectory generation is proposed to provide safe flyaround of such areas. This method relies on using so called "safety map".
Article
Modeling and digital simulation are important phases in the development cycle of the civil flight management system. The distributed real-time simulation structure of the flight management system was proposed. The core function modules of the flight management system including flight planning, flight guidance, trajectory prediction, performance calculation, navigation, navigation database and display were modeled and the simulation algorithms were developed. Based on the avionics full-duplex switched ethernet (AFDX) protocol, the simulation module of the flight management computer, integrated navigation system and automatic flight control system were designed and integrated into the distributed system. A generalized AFDX-based distributed simulation system framework of the flight management system was presented. Simulation results demonstrate that the distributed AFDX-based framework can be implemented in real time applications with improvements on the performance of the network.
Article
Analytical and numerical approaches for identification of the general structure of terminal cost functions are presented. Unlike previous works which were pointed at numerical and analytical methodologies that allowed an accurate mission performance optimization, the current treatment handles the "Inverse Problem", and identifies the rationale of a given unfamiliar yet optimized controller. The current general mathematical problem was raised from a given climb schedule of fighter aircraft, hard coded in core avionics computers, that was not decipherable by any means by the operational pilots nor in the classical flight mechanics literature. The mathematical treatment addressed a feasible simple numerical way to identify the cost function defined for the specific controller. The goal was to allow the most accurate calculation in all conditions of flight regimes, timely efficient, for all fixed wing aircrafts, using a detailed aerodynamics and engine databases to acquire the unknown cost function. The analytical solution for the problem, suggested a simple integral equation on the "Rationale Function", that had been proven to be correct for known optimal controllers such as const equivalent air speed while descending without operating engine. Furthermore, the numerical approach of the methodology enables the operational pilots to understand the rationale of the controller via the usage of scalar weight factors using structure of terminal cost function of time, distance and fuel consumed. The algorithm was applied to solve the operational problem of an unknown schedule which was a linear function of Mach number versus the altitude. Using our methodology, we have found out that the linear mach schedule might be used as a replacement for the classical constant calibrated air speed descent, which is generally used by pilots. Such schedule might be programmed in the aircraft computers, and increase the classical descent performance of the aircraft. Moreover, we may use the methodology to improve already programmed linear mach schedules, when the aerodynamic properties of the airframe are changed. The result, that was found as a by product, gives us a practical usage of the terminal cost function structure procedure as a simple numerical tool, to reveal even more challenging optimal trajectories applied to any aircraft with it own unique aerodynamic properties.
Article
The long-range aircraft cruise problem is very important for airlines which desire to minimize fuel consumption. Many papers have been appeared including the case which the criteria involve cruising time. Most of them, however, solve the problems under condition of a steady-state flight. In other word, they optimize the constrained problem. In the present operation, the altitude and velocity are chosen the optimal ones that are derived from the solution of a minimum fuel problem for a steady-state level flight. This solution satisfies the neccessary conditions of optimality. However, it is shown that the steady cruise is non-convex. The non optimality of a steady cruise is also reported, and the optimal solutions are solved using a periodic control that repeats the same sequence of unsteady controls and trajectories. In this paper, the long range minimum fuel problem is solved numerically under the assumption that the optimal thrust control becomes a bang-bang type. The aircraft weight change during one period is ignored. Under the boundary condition that the initial states are the same as the terminal ones, the optimal problem is solved. The equations of motion for an aircraft is the point-mass model, the atomospheric density is a funcion of the altitude. It is shown that the optimal arc consists of two quasi-steady subarcs. It is also shown that the same subarcs are derived by using a reduced order model. The optimal controls for this model become the chattering type. The control and state variables of this solution are characteristic of the unsteady optimal cruise.
Article
An approach to optimize multiphase trajectories of commercial transport aircraft is presented. The approach is based on the theory of hybrid optimal control, and it is applied to the case of minimum-fuel trajectories. The multiphase trajectories are composed of three types of phases, climb, cruise, and descent, in a given sequence. In each phase, the optimal control is scalar and of the bang–singular–bang type, and the optimal path is formed by a singular arc and two minimum/maximum-control arcs joining the singular arc with the initial and final switching points. An indirect numerical method is developed, which takes into account the structure of the solution directly in the algorithm and exploits the singular character of the problem. In the analysis, the effects of horizontal winds are taken into account; general along-track wind and crosswind profiles are considered, dependent both on altitude and along-track position. The optimal trajectories are computed for a model of a Boeing 767-300ER performing a climb–cruise–descent trajectory with a one-step cruise climb, and for different wind profiles. The influence of the wind profiles on the optimal results is analyzed and shown to be important.
Thesis
Full-text available
In this thesis, a study of the optimization of aircraft trajectories using parametric optimization theory is presented. To that end, an approach based on the use of predefined trajectory patterns and parametric optimization is proposed. The trajectory patterns are in fact flight intents, formed flight segments which model procedures commonly flown by airlines, following air-traffic-control rules. The patterns allow to describe the trajectory by a small number of parameters, whose values, continuous or discrete, can be chosen to optimize a given property of the trajectory. A mixed-integer nonlinear programming (MINLP) problem is formulated to obtain the optimum values. Firstly, the approach is used to solve the general problem of minimum direct-operating-cost cruise (unsteady, with variable mass, and without any constraint on speed or altitude) with given range. The trajectory pattern considered in this application is formed by segments at constant Mach number and constant altitude, restricted to take discrete values: Mach numbers multiple of 0.01, and altitudes defined by flight levels. The unrestricted problem in which the Mach numbers and the altitudes are continuous variables is also considered. The optimized procedures define not only the optimum values of speed and altitude for the different cruise segments, but also the optimum lengths of each segment. The main objective of this application is to analyze how the optimized procedures change when the Mach numbers and the altitudes are restricted to take discrete values. The effects of the cost index, of the initial aircraft weight and of an average horizontal wind in the optimized procedures are also analyzed. Next, the problem of minimum-fuel cruise at constant altitude with fixed range and fixed arrival time is solved and the optimized procedures obtained using the proposed approach are compared with known optimal laws obtained using singular optimal control theory. The trajectory pattern considered in this application is similar to the previous one, formed by segments at constant Mach number restricted to take discrete values. The comparison shows that the optimized procedures approximate very well the optimal laws and give results that are very close to the optimal values. Following, the approach is applied to a set of aircraft, taking into account the losses of separation that may arise among them. An algorithm for conflict resolution (CR) is presented, in which conflict-free trajectories are optimized. The optimality criterium is defined so that the deviation from the intended (preferred) trajectories in the lateral profile is minimized. This problem is solved in two phases: one in which a first valid solution is found by means of a random search, and another one in which this first valid solution is optimized. The resolution trajectory patterns take into account changes of the nominal waypoints (vectoring) and changes of the aircraft speeds. The algorithm is applied to the case of multiple conflicts among commercial transport aircraft in converging traffic in the terminal area. Different scenarios are considered, which include locked aircraft, that is, aircraft whose trajectories are known and fixed. The cost of the global resolution process is assessed, in terms of extra distance travelled, extra flight time and extra fuel consumed for each aircraft. Finally, the previous CR algorithm is extended to solve the problem of optimizing conflictfree trajectories that meet scheduled times of arrival (STA). In this application, the resolution process has three steps: avoidance, which generates conflict-free trajectories that meet the given sequence of arrival (which is a hard constraint of the problem); recovery, in which the resolution trajectories are modified to meet the STA (which is a primary objective); and, optimization, to minimize a combination of costs (secondary objective). Two algorithms are presented: one in which the optimization step is applied globally (to all aircraft) after the other two steps are performed for all aircraft, and another one in which the optimization step is applied locally to each aircraft after the other two steps are performed for the given aircraft; this second algorithm is efficient when the scenario is very demanding (in which the global optimization is not effective). Results are presented for two scenarios, one with a traffic of 30 aircraft in an hour, all of the same wake-turbulence category, and another one with 35 aircraft in an hour and with aircraft of different categories. In all cases, a kinetic trajectory predictor (nonlinear point-mass model with variable mass) is used, which is accurate, flexible and transparent, and provides the high-fidelity prediction required in all the applications.
Article
During the last several years, performance computers have established their claim as cost-effective avionics devices by demonstrating significant fuel savings. Many of these devices command different flight control strategies, yet all claim to yield (near) optimal performance. This paper addresses an in-depth study performed to determine the effectiveness of different flight control strategies on performance. Specifically, the issues of constant IAS/Mach climbs and descents versus more general speed schedules and constant maximum climb thrust versus modulated thrust schedules are addressed. Finally, the benefits of utilization of navigation data for descent guidance is presented. While a high bypass ratio business jet was used as the design example, analogous performance has been found for low bypass jet transport aircraft.
Conference Paper
Raising the fuel efficiency per mile is a crucial criterion of trajectory design for a hypersonic vehicle. This paper presents several numerical methods to such problem based on HL-20 model. This is a typical Optimal Periodic Control (OPC) problem. By simplifying the HL-20 model, the new high-fidelity guess is given to obtain a more reliable optimal solution. Furthermore, the optimal solutions are obtained by applying the software-SNOPT and IMSL to solve the nonlinear optimization problems with enough constraints and then the contrast between the obtained results is shown by using different auxiliary calculation tools. Actually, the cruising mode with periodic altitude and velocity is more fuel-efficient than the previous results also by the methods of OPC, thus, better than the traditional one with constant altitude and velocity. Therefore, an improved trajectory can be realized based on the constant optimal mode. The simulation results verify the effectiveness of the method.
Book
Full-text available
This book deals with aircraft flight performance. It focuses on commercial aircraft but also considers examples of high-performance military aircraft. The framework is a multidisciplinary engineering analysis, fully supported by flight simulation, with software validation at several levels. The book covers topics such as geometrical configurations, configuration aerodynamics and determination of aerodynamic derivatives, weight engineering, propulsion systems (gas turbine engines and propellers), aircraft trim, flight envelopes, mission analysis, trajectory optimization, aircraft noise, noise trajectories, and analysis of environmental performance. A unique feature of this book is the discussion and analysis of the environmental performance of the aircraft, focusing on topics such as aircraft noise and carbon dioxide emissions.
Article
Full-text available
It is shown that the methodology of optimal periodic control is useful in analyzing the dynamics of aircraft cruise. Relaxed steady-state analysis suggests that oscillatory aircraft motion may reduce fuel consumption, particularly when altitude constraints are imposed. Aircraft with high thrust to drag ratios and low wing loading are favored.
Conference Paper
Optimal aircraft cruise performance occurs along particular trajectories traversed through Mach number and altitude space as the mission fuel mass is expended. Ideally, these optimal trajectories depend primarily on the interplay between aerodynamic performance and engine fuel burn rate and on the lapse of these metrics with Mach and altitude. In many situations, however, the ideal operating conditions are constrained by point performance capabilities of the aircraft. For example, stall may limit the efficiency of low speed loiter operations, and maximum speed and altitude capability may reduce the achievable specific range. In this paper, we investigate the implications of thrust limits on aircraft range performance in comparison to unconstrained operations at the cruise conditions corresponding to optimal aerodynamic efficiency. We derive the conditions of optimality for both a two-parameter quadratic drag polar and a drag polar with wave drag, and we demonstrate the concepts by exploring Mach-altitude "sky maps" for three subsonic turbojet aircraft examples. Finally, we explore the implications of changes in the aircraft drag polar, thrust loading, and wing loading on optimal aerodynamic performance. Copyright © 2012 by Brian J. German, Michael D. Patterson, and Timothy T. Takahashi.
Article
We develop here the theory of operating an airplane so as to minimize an arbitrary function of the end-values of the generalized coordinates. A propeller-driven airplane is treated as a particle in equilibrium, subject to the forces of drag, lift, thrust, and gravity. We assume that the specific fuel consumption is a function of the power only, and that the available power is independent of the altitude. The problem is shown to be of the Bolza type in the Calculus of Variations, with the complications arising from the presence of inequalities, discontinuities, and variables whose derivatives do not enter the problem explicitly. The Euler-Lagrange equations are derived and discussed.
Conference Paper
Trajectory generation for flight planning and prediction in Air Traffic Control (ATC) is formulated as a parameter optimal control problem in consistence with ATC procedures. A gradient algorithm is devised for accurate numerical solutions. The climb and descent phases are first studied separately, and are then combined, together with the cruise phase, to study complete trajectories from liftoff to touchdown. For climb, decent, and the entire flight, optimal trajectories are calculated that respectively minimize flight time, fuel consumption, and when applicable, distance traveled, and these trajectories are compared. To evaluate qualities of generated trajectories, two sensitivity concepts are used. Open-loop sensitivities measure changes of generated trajectories due to modeling errors, and reflect the reliability of the trajectory generation process. In contrast, closed-loop sensitivities measure deviations of actual trajectories from generated trajectories caused by modeling errors and/or flight conditions, where actual trajectories are obtained with the pilots or autopilots actively tracking flight objectives extracted from the generated trajectories. They reflect trajectory predictability. Closed-loop sensitivities are used to compare the effectiveness of different trajectory tracking logics in reducing trajectory prediction errors. In this paper, vertical flights in ATC are considered and defined through a series of flight segments that model the current ATC procedures, where each segment is specified with a set of flight objectives. Aircraft flights are described by a dynamic point-mass. Trajectory tracking is modeled via the method of feedback linearization. Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Conference Paper
A generalized approach for Bingo fuel and trajectory algorithm calculation is presented. Unlike previous works which were based on look up tables for the different portions of the flight profile, the current treatment permits the usage of a full detailed aerodynamic and engine models, being implemented in the exact flight mechanics equations of motion. The current analysis defines a practical approach for bingo fuel optimization technique, which can be applied to any turbo jet engine propelled aircraft. A higher accuracy model was developed to be capable to solve the "Non-Classic Bingo" problem: for a given final weight and final distance, a flight profile should be found to give the minimum fuel, for an a-priori unknown initial weight. The novel approach is found to be much more practical than the tedious way of solving the well known non-linear optimal control problem: The calculation time is reduced dramatically, and moreover, the simplicity for the pilot's commands during the flight is preserved. The approach was implemented and analyzed on various aircraft models in service in the Israeli Air Force. A perfect match (less than 4% deviation) to the flight manual calculations was found. With this timely efficient algorithm in hand, being implemented in a computer code, the Bingo problem may be solved almost immediately, and may give the flight squadrons a way to calculate their own long range missions, for unique drag polar stores, without the supervision of the engineers. Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Article
A systematic and rapid steepest-ascent numerical procedure is described for solving two-point boundary-value problems in the calculus of variations for systems governed by a set of nonlinear ordinary differential equations. Numerical examples are presented for minimum time-to-climb and maximum altitude paths for a supersonic interceptor and maximum-range paths for an orbital glider.
Article
For a fairly general aircraft model and a large class of drag models, steady-state cruise for a long time span is nonoptimal with respect to fuel economy. This is proved by a second-order variational analysis, using a frequency-domain version of the classical Jacobi (conjugate point) optimality condition. The variational analysis suggests a sinusoidal perturbation away from steady-state cruise which improves fuel economy (as confirmed numerically), but is still not optimal. The form of the optimal trajectory for long duration cruise is unknown. However, two intuitive reasons for improved fuel economy using cycle cruise paths are given.
Article
This paper considers energy-optimal path planning and perpetual endurance for unmanned aerial vehicles equipped with solar cells on the wings, which collect energy used to drive a propeller. Perpetual endurance is the ability to collect more energy than is lost during a day. This paper considers two unmanned aerial vehicle missions: 1) to travel between given positions within an allowed duration while maximizing the final value of energy and 2) to loiter perpetually from a given position, which requires perpetual endurance. For the first mission, the subsequent problem of energy-optimal path planning features the coupling of the aircraft kinematics and energetics models through the bank angle. The problem is then formulated as an optimal control problem, with the bank angle and speed as inputs. Necessary conditions for optimality are formulated and used to study the optimal paths. The power ratio, a nondimensional number, is shown to predict the qualitative features of the optimal paths. This ratio also quantifies a design requirement for the second mission. Specifically, perpetual endurance is possible if and only if the power ratio exceeds a certain threshold. Comparisons are made of this threshold between Earth and Mars. Implications of the power ratio for unmanned aerial vehicle design are also discussed. Several illustrations are given.
Article
Acceleration-constrained hypersonic flight is examined in the context of optimal periodic control and regulation. Fuel-optimal periodic trajectories are found that yield an 11% improvement over the best static cruise solution, with a maximum vehicle acceleration of 5g. These optimal periodic trajectories were implemented using a periodic regulator in feedback form, which minimized the second variation of the cost. This regulator was extended to account for a slowly varying system parameter (such as vehicle mass) and was shown to perform strikingly well in the constrained hypersonic cruise problem.
Article
For a host of dynamic systems, periodic motion is more efficient than steady-state operation. This work focuses on atmospheric fuel-efficient periodic flight. To understand some of the generic properties of periodic optimal flight, the theoretical aspects are presented, which form the basis for the numerical computation of periodic optimal paths. Furthermore, the simplest problem for which periodic optimal control can be induced is formulated and various solutions based on asymptotic analysis are given. Intuitive physical mechanisms that contribute to producing periodic optimal flight are discussed on the basis of various levels of approximation of the aircraft dynamic models and upon perturbation analysis about the steady-state cruise path. The performance improvement and the periodic optimal flight path of a hypersonic vehicle are computed. To mechanize this flight path, a neighboring optimum perturbation guidance law is developed and its performance is presented.
Article
A solution method for determining three-dimensional minimum-time aircraft trajectories for aircraft is described. The method is based on Euler-Lagrange optimization theory and energy state approximations. The optimal controls are found by either maximizing or minimizing a modified Hamiltonian. The starting extremum operation and the switching conditions from one extremum operation to the other are determined. The solution is computed by interation on the two constant adjoint variables. The solution convergence, however, is sensitive to these parameters. A method based on the characteristics of the heading adjoint variable is used to reduce this sensitivity. A roll angle chattering solution, which occurs during the minimum power energy loss segment of the trajectory and causes computational problems, is removed by including penalty on the roll angle magnitude in the performance criterion. A number of specific problems are then solved.
Article
Singular perturbation theory is used to derive a near-optimal mid-course guidance law incorporating a linear combination of flight time and terminal specific energy as the performance index. State variables in the slow time-scale model consist of down range, cross range, specific energy, and heading angle. A near-optimal approach using feedback linearization is developed to account for altitude and flight-path-angle dynamics neglected in the slow time-scale solution. Implementation logic and simulation results for an engagement scenario are given.
Article
Applying the minimum principle, maximum endurance flight is considered as an optimal cyclic control problem with a state variable constraint. It is shown that a significant increase in the maximum endurance can be achieved by dynamic flight having an optimal cyclic control when compared to the best steady-state flight. The optimal altitude range within the flight envelope is determined. Furthermore, it is shown that the powerplant type represents a key factor when compared to other aircraft characteristics. From this, it follows that turbojet-type engines have properties that can increase endurance via cyclic control.
Article
In this paper, a discussion is presented of the approach maneuvers of an aircraft in the vertical plane. A derivation is then given of the basic optimum solutions for minimizing the time and noise during the landing approach. The direct use of the maximum principle is made so as to obtain preliminary optimum solutions for the simplified dynamics of the aircraft.
Article
Airplane performance sensitivity to the choice of lateral and vertical profiles is studied based on data recorded on in-service flights of a 727-200 airplane equipped with JT9D-7 engines. Some of the practical considerations in implementing the real-time optimum profile algorithms and some of the difficulties encountered with the optimum profile computations are discussed. Predicted fuel and time savings are related to the air traffic control (ATC) environment experienced by the airlines to obtain net savings predictions with varying degrees of ATC restriction.
Article
Since the steady-state cruise path of an idealized point mass model of an atmospheric vehicle operating in the hypersonic flight regime is dynamically not fuel minimizing, closed periodic paths are numerically determined. By application of second-order conditions for local optimality, a periodic extremal path for a flat Earth is shown to be locally minimizing and produces an improvement in fuel usage of 4. 2% over the steady-state cruise path. Application of these second variational conditions to extremal paths for the spherical Earth failed. Nevertheless, these paths produce improved fuel performance over the associated steady-state cruise path.
Article
Advances in hypersonic vehicle technologies, including high-temperature materials, lightweight structures, and high thrust-to-weight ratio engines, indicate that a strategic vehicle at higher Mach numbers (approximately 6) is feasible. An efficient performance analysis method is developed along with a control strategy to evaluate potential hydrogen-fueled turbojet/ramjet propulsion systems for advanced technology hypersonic cruise vehicles. A conceptual strategic reconnaissance mission is evaluated for tradeoffs between throttle and angle of attack control in minimizing fuel consumption/maximizing range. Many mission, flight, and vehicle-related requirements and constraints are satisfied in the design process. In addition, powered hypersonic flight produces unique performance characteristics not encountered at subsonic speeds.
Article
Roughly a quarter of the total fuel savings of the new generation of large jet transports will come from the capability of computing fuel optimal flight trajectories between departure point and destination. The shape of the trajectory in the vertical plane is treated here. A simplified mathematical model is described including spline fits to the drag and fuel flow functions. A suboptimal trajectory is found using the maximum principle of optimal control and singular perturbation theory. The inner or boundary-layer solutions are identified as the climb or descent segments of the flight, while the outer solution corresponds to cruise. The inner solutions are expanded to second order in the vicinity of the outer solution to develop cruise control laws for cost-effective response to altered in-flight conditions.
Article
The energy-state approximation, properly interpreted, is quite adequate for performance optimization of supersonic aircraft. In addition to its use for determining flight profiles for minimum time and minimum fuel to climb to a given altitude and speed, this paper shows that the energy-state approximation may also be used to determine flight profiles for maximum-range glide from a given altitude and speed to another altitude and speed; maximum range at a fixed throttle setting for a given amount of fuel (or, what is the equivalent problem, minimum fuel for a givenange at a fixed throttle setting; maximum total range for a given amount of fuel (or minimum fuel for a given total range); maximum range in a given time (or minimum time for a given range).
Article
The sensitivity of minimum time and minimum fuel flight paths to variations in aircraft parameters in different atmospheric conditions was investigated using the energy state approximation. Numerical results are presented for a typical supersonic aircraft in Standard Day, Hot Day, and Cold Day atmospheres. Minimum time and minimum fuel flight paths in the vertical plane are well documented in the literature. This paper shows how flight time and fuel consumption are affected by changes in thrust, weight, drag coefficients (CDo and Cm), and specific fuel consumption (SFC) in each of three different atmospheric conditions (Standard Day, Hot Day, and Cold Day). For each variation, the effect on performance (flight time or fuel consumption) is determined for the nominal paths. Then for each variation, the flight path is adjusted to be either time optimal or fuel optimal. As a result of this analysis, it was found that flight time and fuel consumption are sensitive to variations in CDO, SFC, and aircraft weight. There is only a slight sensitivity to variations in Cot and no flight time sensitivity to variations in SFC. It was found that a Hot Day atmosphere tended to degrade aircraft performance by increasing flight time and fuel consumption along the flight path, while a Cold Day atmosphere tended to enhance performance. Adjusting the nominal flight paths to be time optimal or fuel optimal for the conditions being considered was found to be desirable for only a limited number of conditions. In a Standard Day atmosphere, the only conditions for which path adjustment significantly improved performance are large thrust reductions and large increases in CDO. In a Hot Day atmosphere, path adjustment improved performance for every variation with the exception of thrust increases and CDO decreases. In a Cold Day atmosphere, path adjustment failed to significantly improve performance for any variation. © 1974, American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Article
Singular controls have appeared rather frequently in dynamic optimization problems during the last two decades. Although special methods have been effective in analysing these singular problems, it is only recently that a unifying theory has been developed. This survey paper traces the early developments in the subject through to the most recent work of Jacobson and others in the establishment of necessary, sufficient, and necessary and sufficient conditions for optimality of singular extremals.
Article
This paper advances prior results on the problem of time-free, Mayer-optimal control of a blunt space vehicle's flight in an atmosphere. The extremally steered, first order, singular arc is partially singular and the Lie-bracket formalism employed previously for the totally singular case is not quite applicable. However, by differentiating the switching function, a relationship is obtained between the totally and partially singular arcs. For the special case of coplanar flight, the singular thrust control is expressed in a nonlinear state feedback form. This form indicates that the singular thrust magnitude is separable as the sum of five types of forces: one due to gravity alone, another due to centrifugal effects, a third due to drag alone, a fourth due to atmospheric density gradient, and finally a fifth due to a coupling between these forces. The extremal steering angle along the singular arc is obtained as an implicit function of the states. An explicit solution is obtained for the special case of a low-density atmosphere. For proper combinations of steering and rocket exhaust speed, the singular arc satisfies the generalized Legendre-Clebsch condition and the Goh-Robbins condition making it a strong candidate for Mayer optimality.
Article
A method of identifying suboptimum flight profiles for short-haul propeller-driven aircraft, using multivariate optimization techniques, is outlined. These profiles are defined in terms of simple parameters which can be readily monitored and controlled by the crew without the assistance of an autopilot. A sequence of minimum-fuel studies, based on a typical modern commuter aircraft, is presented. The effects of some operational constraints are examined. It is found that a 1. 5% fuel saving can be achieved relative to the current flight-manual operating techniques. The climb and descent segments are most significant in this type of operation.
Article
A maximum principle application shows that a fuel-optimal aircraft trajectory which arrives at an assigned time is also a direct operating cost-optimal trajectory for certain time cost. The time cost is minimum for maximum endurance cruise. Only for minimum time cost can an optimal trajectory contain a hold or path stretching segment. These concepts are illustrated for three aircraft: the Boeing 737, 747, and 767. Complete optimal trajectories are shown for the 747. Cruise control implications are investigated using the 767 model. Finally, optimal descents are compared to Mach/CAS descents for the 737.
Article
A near-optimal periodic solution to the maximum-endurance cruise problem is investigated. Point-mass models are developed for different types of aircraft. Energy-state methods are used to determine minimum-fuel climb and maximum-endurance descent schedules in the altitude-airspeed plane, which are then pieced together with transition arcs to form a periodic cruise solution. A trajectory tracking controller is designed to make the point-mass models track the periodic cruise trajectories. The tracking controller is designed using the feedback linearization methodology. Closed-loop simulations are then used to compute the fuel consumption resulting from the use of periodic trajectories. These values are compared to the steady-state, optimal-endurance cruise fuel consumption values. For an F/A-18 aircraft model, it was found that savings of about 17% could be realized if the engines can be turned off when the aircraft is not on the climb schedule. However, if the throttle cannot be set below flight idle, the periodic cruise trajectory is found to produce worse performance than the steady-state cruise, primarily due to poor specific fuel consumption at idle throttle setting. Simulations with a model of an F-4 in periodic cruise with idealized engine characteristics non-zero minimum throttle did show a modest improvement over the steady-state cruise performance, but only by 2.7%.
Article
A simulation of a projected 1980-era Navy attack aircraft suitable for man-in-the-loop investigations of energy management is developed. A set of energy management systems and display subsystems are defined based on an optimum climb and cruise concept proposed by Rutowski and Bryson which uses the energy-state approximation. The potential value of the technique is demonstrated by digital flight simulation. The technique is easily mechanized and closely approximates optimum flight paths previously computed only in extensive land-based facilities. It is within the present state of the art in airborne systems and could greatly alleviate pilot workload while simplifying cockpit instrumentation. (Author)
Article
A simple and efficient performance analysis and optimization method is developed for finding optimum flight-path trajectories between fixed endpoints. A methodology based on the energy method is derived for optimizing transitions between climb/cruise and cruise/descent for a hypersonic reconnaissance vehicle (Mach 6). Powered hypersonic flight produces some unique performance characteristics for which the methodology satisfies mission, flight, and vehicle-related design requirements and constraints. The transitions are optimized for minimum fuel burned between end points. Results from the optimization study on transitional trajectories are presented to illustrate the methodology
Article
Minimum-time interception of a target flying at a constant speed and altitude by a missle-carrying fighter aeroplane is analysed using the method of forced singular perturbations (FSP). The engagement starts beyond visual range and terminates when the firing envelope of the missile is reached. Since in a medium-range scenario the interceptor does not reach its maximum specific energy, the solution consists of matched initial and terminal boundary layers. The paper concentrates on interception of high flying targets for which the terminal phase is also a climbing subarc. The solution of this problem is approximated in a uniformly valid explicit feedback form using a range-dependent weighting parameter. Application to an interceptor with realistic aerodynamic and propulsion models serves as an illustrative example.
Article
An efficient performance analysis method is developed to evaluate potential hydrogen fueled turbojet, ramjet and scramjet propulsion for advanced technology hypersonic aircraft. An algorithm, based on energy management and calculus of variations, is developed for determining optimal trajectories. The methodology applies both techniques simultaneously in obtaining the optimal trajectory. Many mission, flight, and vehicle-related requirements and constraints are satisfied in the process. In addition, powered hypersonic flight produces unique performance characteristics not encountered at subsonic speeds. With the analysis method, two potential hypersonic vehicles are evaluated, both for maximizing payload and minimizing time to climb, and the resulting optimal trajectories are presented.
Article
Sequential quadratic programming is used to solve several minimum-fuel flight path optimization problems for a point-mass aircraft flying in a vertical plane. Each optimal control problem involves two control functions: a “steering” control and a “thrust” control. Range is the independent variable.Four dynamic models are examined. The usual point-mass model requires four states (speed, flight path angle, altitude, and mass) and angle of attack as the steering control. An intermediate model with three states (speed, altitude, and mass) neglects the flight path angle dynamics and uses flight path angle as the steering control. An energy-state approximation and a point-mass model including engine dynamics are also treated. Explicit function representations are used to model the aircraft's thrust, lift, and drag characteristics.Numerical solutions are presented for several fuel-optimal transition flight paths between specified states. A comparison of results for the various dynamic models is also given.
Article
An algorithm, based on the energy-state method, is derived for calculating optimum trajectories with a range constraint. The basis of the algorithm is the assumption that optimum trajectories consist of, at most, three segments: an increasing energy segment (climb); a constant energy segment (cruise); and a decreasing energy segment (descent). This assumption allows energy to be used as the independent variable in the increasing and decreasing energy segments, thereby eliminating the integration of a separate adjoint differential equation and simplifying the calculus of variations problem to one requiring only pointwise extremization of algebraic functions. The algorithm is used to compute minimum fuel, minimum time, and minimum direct-operating-cost trajectories, with range as a parameter, for an in-service CTOL aircraft and for an advanced STOL aircraft. For the CTOL aircraft and the minimum-fuel performance function, the optimum controls, consisting of air-speed and engine power setting, are continuous functions of the energy in both climb and descent as well as near the maximum or cruise energy. This is also true for the STOL aircraft except in the descent where at one energy level a nearly constant energy dive segment occurs, yielding a discontinuity in the airspeed at that energy. The reason for this segment appears to be the relatively high fuel flow at idle power of the engines used by this STOL aircraft. Use of a simplified trajectory which eliminates the dive increases the fuel consumption of the total descent trajectory by about 10 percent and the time to fly the descent by about 19 percent compared to the optimum.
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Results are presented for wind tunnel tests of an elastic wing model to verify the theoretical predictions for the aeroelastic instability of an oblique wing. The model wing has an elliptic planform of 10 to 1 axis ratio and a symmetrical airfoil section of 7-1/2% thickness/chord ratio. The wing is of wood and as may be seen in the photographs presented, slack wires are used to limit the amplitude of unstable motions. The fuselage is mounted on bearings permitting freedom of roll, but provision is made to clamp the fuselage for some of the tests. It is found that freedom in roll increases the dynamic pressure at which aeroelastic instability first appears. With the model free in roll, the effectiveness of the ailerons in maintaining trim is not noticeably affected by passage through the speed at which the wing would become unstable if clamped.
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A method for generating a minimum-fuel, fixed-range, fixed-time-of-arrival flight path in an on-board flight management system computer for commercial aircraft is described. It is shown that up to 6% of the fuel otherwise used can be saved by means of this capability, despite time-of-arrival delays of up to 30 min, by a medium-range, tri-jet transport aircraft.
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A number of studies dealing with fuel minimization are concerned with three-dimensional flight. However, only Neuman and Kreindler (1982) consider cases involving commercial jet transports. In the latter study, only the climb-out and descent portions of complete long-range flight paths below 10,000 ft altitude have been investigated. The present investigation is concerned with the computation of minimum-fuel nonturning and turning flight paths for climb-outs from 2000 to 10,000 ft for long-range flights (greater than 50 n mi), and for complete flight paths of lengths between 5 and 50 n mi.
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A real-time computer algorithm to control and optimize aircraft flight profiles is described and applied to a three-dimensional minimum-time intercept mission. The proposed scheme has roots in two well known techniques: singular perturbations and neighboring-optimal guidance. Use of singular-perturbation ideas is made in terms of the assumed trajectory-family structure. A heading/energy family of prestored point-mass-model state-Euler solutions is used as the baseline in this scheme. The next step is to generate a near-optimal guidance law that will transfer the aircraft to the vicinity of this reference family. The control commands fed to the autopilot (bank angle and load factor) consist of the reference controls plus correction terms which are linear combinations of the altitude and path-angle deviations from reference values, weighted by a set of precalculated gains. In this respect the proposed scheme resembles neighboring-optimal guidance. However, in contrast to the neighboring-optimal guidance scheme, the reference control and state variables as well as the feedback gains are stored as functions of energy and heading in the present approach. Some numerical results comparing open-loop optimal and approximate feedback solutions are presented.