ALBERT

Notes: Abstract This paper is concerned with optimal flight trajectories in the presence of windshear. The penetration landing problem is considered with reference to flight in a vertical plane, governed by either one control (the angle of attack, if the power setting is predetermined) or two controls (the angle of attack and the power setting). Inequality constraints are imposed on the angle of attack, the power setting, and their time derivatives. The performance index being minimized measures the deviation of the flight trajectory from a nominal trajectory. In turn, the nominal trajectory includes two parts: the approach part, in which the slope is constant; and the flare part, in which the slope is a linear function of the horizontal distance. In the optimization process, the time is free; the absolute path inclination at touchdown is specified; the touchdown velocity is subject to upper and lower bounds; and the touchdown distance is subject to upper and lower bounds. Three power setting schemes are investigated: (S1) maximum power setting; (S2) constant power setting; and (S3) control power setting. In Scheme (S1), it is assumed that, immediately after the windshear onset, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant; in this scheme, the only control is the angle of attack. In Scheme (S2), it is assumed that the power setting is held at a constant value, equal to the prewindshear value; in this scheme, the only control is the angle of attack. In Scheme (S3), the power setting is regarded as a control, just as the angle of attack. Under the above conditions, the optimal control problem is solved by means of the primal sequential gradient-restoration algorithm (PSGRA). Numerical results are obtained for several combinations of windshear intensities and initial altitudes. The main conclusions are given below with reference to strong-to-severe windshears. In Scheme (S1), the touchdown requirements can be satisfied for relatively low initial altitudes, while they cannot be satisfied for relatively high initial altitudes; the major inconvenient is excess of velocity at touchdown. In Scheme (S2), the touchdown requirements cannot be satisfied, regardless of the initial altitude; the major inconvenient is defect of horizontal distance at touchdown. In Scheme (S3), the touchdown requirements can be satisfied, and the optimal trajectories exhibit the following characteristics: (i) the angle of attack has an initial decrease, which is followed by a gradual, sustained increase; the largest value of the angle of attack is attained near the end of the shear; in the aftershear region, the angle of attack decreases gradually; (ii) initially, the power setting increases rapidly until maximum power setting is reached; then, maximum power setting is maintained in the shear region; in the aftershear region, the power setting decreases gradually; (iii) the relative velocity decreases in the shear region and increases in the aftershear region; the point of minimum velocity occurs at the end of the shear; and (iv) depending on the windshear intensity and the initial altitude, the deviations of the flight trajectory from the nominal trajectory can be considerable in the shear region; however, these deviations become small in the aftershear region, and the optimal flight trajectory recovers the nominal trajectory. A comparison is shown between the optimal trajectories of Scheme (S3) and the trajectories arising from alternative guidance schemes, such as fixed controls (fixed angle of attack, coupled with fixed power setting) and autoland (angle of attack controlled via path inclination signals, coupled with power setting controlled via velocity signals). The superiority of the optimal trajectories of Scheme (S3) is shown in terms of the ability to meet the path inclination, velocity, and distance requirements at touchdown. Therefore, it is felt that guidance schemes based on the properties of the optimal trajectories of Scheme (S3) should prove to be superior to alternative guidance schemes, such as the fixed control guidance scheme and the autoland guidance scheme.

Notes: Abstract This paper deals with the identification of the wind profile along a flight trajectory by means of a three-dimensional kinematic approach. The approach is then applied to a recent aircraft accident, that of Flight Delta 191, which took place at Dallas-Fort Worth International Airport on August 2, 1985. In the 3D-kinematic approach, the wind velocity components are computed as the difference between the inertial velocity components and the airspeed components. The airspeed profile is obtained from flight measurements. The inertial velocity profile is obtained by integration of the measured inertial acceleration. The accelerometer biases and the impact values of the inertial velocity components are determined by matching the computed flight trajectory with the measured flight trajectory, available from the digital flight data recorder (DFDR) and air traffic control radar (ATCR). This leads to a least-square problem, which is solved analytically. Key to the precision of the identified wind profile is the correct identification of the accelerometer biases and the impact velocity components. In turn, this depends on the proper selection of the integration time. Because the measured data are noise-corrupted, unstable identification occurs if the integration time is too short. On the other hand, stable identification takes place if the integration time is properly chosen. Application of the method developed to the case of Flight Delta 191 shows that the identification problem has a stable solution if the integration time is larger than 180 sec. Numerical computation shows that, for Flight Delta 191, the maximum wind velocity difference determined with the 3D-kinematic approach was ΔW x =124 fps in the longitudinal direction, ΔW y =66 fps in the lateral direction, and ΔW h =71 fps in the vertical direction.

Notes: Abstract This paper deals with the identification of the wind profile along a flight trajectory by means of a two-dimensional kinematic approach. In this approach, the wind velocity components are computed as the difference between the inertial velocity components and the airspeed components. The airspeed profile is obtained from flight measurements. The inertial velocity profile is obtained by integration of the measured inertial acceleration. The accelerometer biases and the impact values of the inertial velocity components are determined by matching the computed flight trajectory with the measured flight trajectory, available from the digital flight data recorder and air traffic control radar. This leads to a least-square problem, which is solved analytically for both the continuous formulation and the discrete formulation. Key to the precision of the identification process is the proper selection of the integration time. Because the measured data are noise-corrupted, unstable identification occurs if the integration time is too short. On the other hand, if the integration time is too long, the hypothesis of two-dimensional motion (flight trajectory nearly contained in a vertical plane) breaks down. Application of the 2D-kinematic approach to the case of Flight Delta 191 shows that stable identification takes place for integration times in the range τ = 120 to 180 sec before impact. The results of the 2D-kinematic approach are close to those of the 3D-kinematic approach (Ref. 1), particularly in terms of the inertial velocity components at impact (within 1 fps) and the maximum wind velocity differences (within 2 fps). The 2D-kinematic approach is applicable to the analysis of wind-shear accidents in take-off or landing, especially for the case of older-generation, shorter-range aircraft which do not carry the extensive instrumentation of newer-generation, longer-range aircraft.

Notes: Abstract This paper deals with the identification of the wind profile along a flight trajectory by means of a two-dimensional dynamic approach. In this approach, the wind velocity components are computed as the difference between the inertial velocity components and the airspeed components. The airspeed profile as well as the nominal thrust, drag, and lift profiles are obtained from the available DFDR measurements. The actual values of the thrust, drag, and lift are assumed to be proportional to the respective nominal values via multiplicative parameters, called the thrust, drag, and lift factors. The thrust, drag, and lift factors plus the inertial velocity components at impact are determined by matching the flight trajectory computed from DFDR data with the flight trajectory available from ATCR data. This leads to a least-square problem which is solved analytically under the additional requirement of closeness of the multiplicative factors to unity. Application of the 2D-dynamic approach to the case of Flight Delta 191 shows that, with reference to the last 180 sec before impact, the values of the multiplicative factors were 1.09, 0.84, and 0.89; this implies that the actual values of the thrust, drag, and lift were 9% above, 16% below, and 11% below their respective nominal values. For the last 60 sec before impact, the aircraft was subject to severe windshear, characterized by a horizontal wind velocity difference of 123 fps and a vertical wind velocity difference of 80 fps. The 2D-dynamic approach is applicable to the analysis of windshear accidents in take-off or landing, especially for the case of older-generation, shorter-range aircraft which do not carry the extensive instrumentation of newer-generation, longer-range aircraft. The same methodology can be extended to the investigation of aircraft accidents originating from causes other than windshear (e.g., icing, incorrect flap position, engine malfunction), above all if its precision is further increased by combining the 2D-dynamic approach and the 2D-kinematic approach.

Notes: Abstract Standard wind identification techniques employed in the analysis of aircraft accidents are post-facto techniques; they are processed after the event has taken place and are based on the complete time histories of the DFDR/ATCR data along the entire trajectory. By contrast, real-time wind identification techniques are processed while the event is taking place; they are based solely on the knowledge of the preceding time histories of the DFDR/ATCR data. In this paper, a real-time wind identification technique is developed. First, a 3D-kinematic approach is employed in connection with the DFDR/ATCR data covering the time interval τ preceding the present time instant. The aircraft position, inertial velocity, and accelerometer bias are determined by matching the flight trajectory computed from the DFDR data with the flight trajectory available from the ATCR data. This leads to a least-square problem, which is solved analytically every β seconds, with β/τ small. With the inertial velocity and accelerometer bias known, an extrapolation process takes place so as to predict the inertial velocity profile over the subsequent β-subinterval. At the end of this subinterval, the extrapolated inertial velocity and the newly identified inertial velocity are statistically reconciled and smoothed. Then, the process of identification, extrapolation, reconciliation, and smoothing is repeated. Subsequently, the wind is computed as the difference between the inertial velocity and the airspeed, which is available from the DFDR data. With the wind identified, windshear detection can take place (Ref. 1). As an example, the real-time wind identification technique is applied to Flight Delta 191, which crashed at Dallas-Fort Worth International Airport on August 2, 1985. The numerical results show that the wind obtained via real-time identification is qualitatively and quantitatively close to the wind obtained via standard identification. This being the case, it is felt that real-time wind identification can be useful in windhsear detection and guidance, above all if the shear/downdraft factor signal is replaced by the wind difference signal (Ref. 1).

Notes: Abstract This paper is concerned with windshear detection in connection with real-time wind identification (Ref. 1). It presents a comparative evaluation of two techniques, one based on the shear/downdraft factor and one based on the wind difference index. The comparison is done with reference to a particular microburst, that which caused the 1985 crash of Flight Delta 191 at Dallas-Fort Worth International Airport. The shear/downdraft factor has the merit of combining the effects of the shear and the downdraft into a single entity. However, its effectiveness is hampered by the fact that, in a real situation, the windshear is accompanied by free-stream turbulence, which tends to blur the resulting signal. In turn, this results in undesirable nuisance warnings if the magnitude of the shear factor due to free-stream turbulence is temporarily larger than that due to true windshear. Therefore, proper filtering is necessary prior to using the shear/downdraft factor in detection and guidance. One effective way for achieving this goal is to average the shear/downdraft factor over a specified time interval τ. The effect of τ on the average shear/downdraft factor is studied. Another effective way of offsetting the effects due to free-stream turbulence is to employ the wind difference index for detection and guidance. The wind difference index is computed over a specified time interval τ along the trajectory of the aircraft and is more stable than the shear/downdraft factor signal with respect to interference effects due to free-stream turbulence. Yet, the wind difference index can be determined using the same aerodynamics and inertial instrumentation necessary for determining the shear/downdraft factor. The effect of τ over the wind difference index is studied; it is found that, for τ relatively large, the wind difference index tends to a stable value, while the average shear/downdraft factor tends to vanish. It must be noted that any unfavorable shear (inner core of a downburst) is both preceded and followed by a favorable shear. The total wind velocity difference associated with the regions of favorable shear is exactly the same as that associated with the region of unfavorable shear. On the other hand, the shear/downdraft factor averaged over the regions of favorable shear is much smaller than that averaged over the region of unfavorable shear. As a consequence, the wind difference index is more useful than the average shear/downdraft factor in detecting a potentially dangerous windshear situation while the aircraft is still flying in the region of favorable shear.

Notes: Abstract This paper is concerned with optimal flight trajectories in the presence of windshear. The abort landing problem is considered with reference to flight in a vertical plane. It is assumed that, upon sensing that the airplane is in a windshear, the pilot increases the power setting at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative. The performance index being minimized is the peak value of the altitude drop. The resulting optimization problem is a minimax problem or Chebyshev problem of optimal control, which can be converted into a Bolza problem through suitable transformations. The Bolza problem is then solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. Numerical results are obtained for several combinations of windshear intensities, initial altitudes, and power setting rates. For strong-to-severe windshears, the following conclusions are reached: (i) the optimal trajectory includes three branches: a descending flight branch, followed by a nearly horizontal flight branch, followed by an ascending flight branch after the aircraft has passed through the shear region; (ii) along an optimal trajectory, the point of minimum velocity is reached at about the time when the shear ends; (iii) the peak altitude drop depends on the windshear intensity, the initial altitude, and the power setting rate; it increases as the windshear intensity increases and the initial altitude increases; and it decreases as the power setting rate increases; (iv) the peak altitude drop of the optimal abort landing trajectory is less than the peak altitude drop of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory; (v) the survival capability of the optimal abort landing trajectory in a severe windshear is superior to that of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory.

Notes: Abstract This paper is concerned with the optimal transition and the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The abort landing problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered. It is assumed that a transition from descending flight to ascending flight is desired; that the initial state corresponds to quasi-steady flight with absolute path inclination of −3.0 deg; and that the final path inclination corresponds to quasi-steady steepest climb. Also, it is assumed that, as soon as the shear is detected, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative. First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the instantaneous altitude and a reference value, assumed constant. By suitable transformations, the Chebyshev problem is converted into a Bolza problem. Then, the Bolza problem is solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. Two types of optimal trajectories are studied, depending on the conditions desired at the final point. Type 1 is concerned with gamma recovery (recovery of the value of the relative path inclination corresponding to quasi-steady steepest climb). Type 2 is concerned with quasi-steady flight recovery (recovery of the values of the relative path inclination, the relative velocity, and the relative angle of attack corresponding to quasi-steady steepest climb). Both the Type 1 trajectory and the Type 2 trajectory include three branches: descending flight, nearly horizontal flight, and ascending flight. Also, for both the Type 1 trajectory and the Type 2 trajectory, descending flight takes place in the shear portion of the trajectory; horizontal flight takes place partly in the shear portion and partly in the aftershear portion of the trajectory; and ascending flight takes place in the aftershear portion of the trajectory. While the Type 1 trajectory and the Type 2 trajectory are nearly the same in the shear portion, they diverge to a considerable degree in the aftershear portion of the trajectory. Next, trajectory guidance is considered. Two guidance schemes are developed so as to achieve near-optimum transition from quasi-steady descending flight to quasi-steady ascending flight: acceleration guidance (based on the relative acceleration) and gamma guidance (based on the absolute path inclination). The guidance schemes for quasi-steady flight recovery in abort landing include two parts in sequence: shear guidance and aftershear guidance. The shear guidance is based on the result that the shear portion of the trajectory depends only mildly on the boundary conditions. Therefore, any of the guidance schemes already developed for Type 1 trajectories can be employed for Type 2 trajectories (descent guidance followed by recovery guidance). The aftershear guidance is based on the result that the aftershear portion of the trajectory depends strongly on the boundary conditions; therefore, the guidance schemes developed for Type 1 trajectories cannot be employed for Type 2 trajectories. For Type 2 trajectories, the aftershear guidance includes level flight guidance followed by ascent guidance. The level flight guidance is designed to achieve almost complete velocity recovery; the ascent guidance is designed to achieve the desired final quasi-steady state. The numerical results show that the guidance schemes for quasi-steady flight recovery yield a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, allows the aircraft to achieve the final quasi-steady state, and has good stability properties.

Notes: Abstract This paper is concerned with optimal flight trajectories in the presence of windshear. With particular reference to take-off, eight fundamental optimization problems [Problems (P1)–(P8)] are formulated under the assumptions that the power setting is held at the maximum value and that the airplane is controlled through the angle of attack. Problems (P1)–(P3) are least-square problems of the Bolza type. Problems (P4)–(P8) are minimax problems of the Chebyshev type, which can be converted into Bolza problems through suitable transformations. These problems are solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. Numerical results are obtained for a large number of combinations of performance indexes, boundary conditions, windshear models, and windshear intensities. However, for the sake of brevity, the presentation of this paper is restricted to Problem (P6), minimax ∣Δh∣, and Problem (P7), minimax ∣Δγ∣. Inequality constraints are imposed on the angle of attack and the time derivative of the angle of attack. The following conclusions are reached: (i) optimal trajectories are considerably superior to constant-angle-of-attack trajectories; (ii) optimal trajectories achieve minimum velocity at about the time when the windshear ends; (iii) optimal trajectories can be found which transfer an aircraft from a quasi-steady condition to a quasi-steady condition through a windshear; (iv) as the boundary conditions are relaxed, a higher final altitude can be achieved, albeit at the expense of a considerable velocity loss; (v) among the optimal trajectories investigated, those solving Problem (P7) are to be preferred, because the altitude distribution exhibits a monotonic behavior; in addition, for boundary conditions BC2 and BC3, the peak angle of attack is below the maximum permissible value; (vi) moderate windshears and relatively severe windshears are survivable employing an optimized flight strategy; however, extremely severe windshears are not survivable, even employing an optimized flight strategy; and (vii) the sequential gradient-restoration algorithm (SGRA), employed in its dual form (DSGRA), has proven to be a powerful algorithm for solving the problem of the optimal flight trajectories in a windshear.

Notes: Abstract This paper is concerned with guidance strategies for near-optimum performance in a windshear. This is a wind characterized by sharp change in intensity and direction over a relatively small region of space. The take-off problem is considered with reference to flight in a vertical plane. First, trajectories for optimum performance in a windshear are determined for different windshear models and different windshear intensities. Use is made of the methods of optimal control theory in conjunction with the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. In this approach, global information on the wind flow field is needed. Then, guidance strategies for near-optimum performance in a wind-shear are developed, starting from the optimal trajectories. Specifically, three guidance schemes are presented: (A) gamma guidance, based on the relative path inclination; (B) theta guidance, based on the pitch attitude angle; and (C) acceleration guidance, based on the relative acceleration. In this approach, local information on the wind flow field is needed. Next, several alternative schemes are investigated for the sake of completeness, more specifically: (D) constant alpha guidance; (E) constant velocity guidance; (F) constant theta guidance; (G) constant relative path inclination guidance; (H) constant absolute path inclination guidance; and (I) linear altitude distribution guidance. Numerical experiments show that guidance schemes (A)–(C) produce trajectories which are quite close to the optimum trajectories. In addition, the near-optimum trajectories associated with guidance schemes (A)–(C) are considerably superior to the trajectories arising from the alternative guidance schemes (D)–(I). An important characteristic of guidance schemes (A)–(C) is their simplicity. Indeed, these guidance schemes are implementable using available instrumentation and/or modification of available instrumentation.