Multiple-Subarc Gradient-Restoration Algorithm, Part 1: Algorithm Structure

Rapid progresses in information and computer technology allow the development of more advanced optimal control algorithms dealing with real-world problems. In this paper, which is Part 1 of a two-part sequence, a multiple-subarc gradient-restoration algorithm (MSGRA) is developed. We note that the original version of the sequential gradient-restoration algorithm (SGRA) was developed by Miele et al. in single-subarc form (SSGRA) during the years 1968–86; it has been applied successfully to solve a large number of optimal control problems of atmospheric and space flight.MSGRA is an extension of SSGRA, the single-subarc gradient-restoration algorithm. The primary reason for MSGRA is to enhance the robustness of gradient-restoration algorithms and also to enlarge the field of applications. Indeed, MSGRA can be applied to optimal control problems involving multiple subsystems as well as discontinuities in the state and control variables at the interface between contiguous subsystems.Two features of MSGRA are increased automation and efficiency. The automation of MSGRA is enhanced via time normalization: the actual time domain is mapped into a normalized time domain such that the normalized time length of each subarc is 1. The efficiency of MSGRA is enhanced by using the method of particular solutions to solve the multipoint boundary-value problems associated with the gradient phase and the restoration phase of the algorithm.In a companion paper [Part 2 (Ref. 2)], MSGRA is applied to compute the optimal trajectory for a multistage launch vehicle design, specifically, a rocket-powered spacecraft ascending from the Earth surface to a low Earth orbit (LEO). Single-stage, double-stage, and triple-stage configurations are considered and compared.     

Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm

The ideal strategy for ship collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between a host ship and an intruder ship. This is a maximin problem or Chebyshev problem of optimal control in which the performance index being maximinimized is the distance between the two ships. Based on the multiple-subarc sequential gradient-restoration algorithm, a new method for solving the maximin problem is developed.Key to the new method is the observation that, at the maximin point, the time derivative of the performance index must vanish. With the zero derivative condition being treated as an inner boundary condition, the maximin problem can be converted into a Bolza problem in which the performance index, evaluated at the inner boundary, is being maximized wrt the controls. In turn, the Bolza problem with an added inner boundary condition can be solved via the multiple-subarc sequential gradient-restoration algorithm (SGRA).The new method is applied to two cases of the collision avoidance problem: collision avoidance between two ships moving along the same rectilinear course and collision avoidance between two ships moving along orthogonal courses. For both cases, we are basically in the presence of a two-subarc problem, the first subarc corresponding to the avoidance phase of the maneuver and the second subarc corresponding to the recovery phase. For stiff systems, the robustness of the multiple-subarc SGRA can be enhanced via increase in the number of subarcs. For the ship collision avoidance problem, a modest increase in the number of subarcs from two to three (one subarc in the avoidance phase, two subarcs in the recovery phase) helps containing error propagation and achieving better convergence results.     

Quasi-steady flight to quasi-steady flight transition in a windshear: Trajectory optimization and guidance

This paper is concerned with the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The take-off 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 the power setting is held at the maximum value and that the aircraft is controlled through 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 absolute path inclination and a reference value, assumed constant. Two types of optimal trajectories are studied: type 1 is concerned with gamma recovery (recovery of the initial value of the relative path inclination); and type 2 is concerned with quasisteady flight recovery (recovery of the initial values of the relative velocity, the relative path inclination, and the relative angle of attack). The numerical results show that the type 1 trajectory and the type 2 trajectory are nearly the same in the shear portion, while they diverge to a considerable degree in the aftershear portion of the optimal trajectory.Next, trajectory guidance is considered. A guidance scheme is developed so as to achieve near-optimum quasi-steady flight recovery in a windshear. The guidance scheme for quasi-steady flight recovery includes three parts in sequence. The first part refers to the shear portion of the trajectory and is based on the result that this 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 instance, variable gamma guidance). The second part (constant gamma guidance) refers to the initial aftershear portion of the trajectory and is designed to achieve almost velocity recovery. The third part (constant rate of climb guidance) refers to the final aftershear portion of the trajectory and is designed to achieve almost complete restoration of the initial quasi-steady state.While the shear guidance and the initial aftershear guidance employ constant gain coefficients, the final aftershear guidance employs a variable gain coefficient. This is done in order to obtain accuracy and prompt response, while avoiding oscillations and overshoots. The numerical results show that the guidance scheme for quasi-steady flight recovery yields a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, ensures the restoration of the initial quasi-steady state, and has good stability properties.This paper is based on Refs. 1 and 2.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, and by Boeing Commercial Aircraft Company. The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, for helpful discussions.     

Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: Trajectory optimization and guidance

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.This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company, and by Air Line Pilots Association.The authors are indebted to Dr. R. L. Bowles (NASA-LRC) and Dr. G. R. Hennig (BCAC) for helpful discussions.     

Wind identification along a flight trajectory,part 2: 2D-kinematic approach

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.     

Wind identification along a flight trajectory,part 3: 2D-dynamic approach

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.     

Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization,Part 2: Application

In this paper, we describe an application of the planar conjugate gradient method introduced in Part 1 (Ref. 1) and aimed at solving indefinite nonsingular sets of linear equations. We prove that it can be used fruitfully within optimization frameworks; in particular, we present a globally convergent truncated Newton scheme, which uses the above planar method for solving the Newton equation. Finally, our approach is tested over several problems from the CUTE collection (Ref. 2).This work was supported by MIUR, FIRB Research Program on Large-Scale Nonlinear Optimization, Rome, Italy.The author acknowledges Luigi Grippo and Stefano Lucidi, who contributed considerably to the elaboration of this paper. The exchange of experiences with Massimo Roma was a constant help in the investigation. The author expresses his gratitude to the Associate Editor and the referees for suggestions and corrections.     

The AR-property for Roberts' example of a compact convex set with no extreme points Part 2: Application to the example

In this second part of our paper, we apply the result of Part 1 to show that the compact convex set with no extreme points, constructed by Roberts (1977), is an AR.

     

Stability of solutions for a class of nonlinear cone constrained optimization problems,part 2: Application to parameter estimation

In this paper we study a problem of parameter estimation in two point boundary value problems. Using a stability theorem for nonlinear cone constrained optimization problems derived in Part 1 of this paper we investigate stability properties of the solutions of the parameter estimation problem in the output-least-squares formulation.