Sequential Gradient-restoration Algorithm Research Articles

Properties of the sequential gradient-restoration algorithm (SGRA), part 1: Introduction and comparison with related methods

The sequential gradient-restoration algorithm (SGRA) was developed in the late 1960s for the solution of equality-constrained nonlinear programs and has been successfully implemented by Miele and coworkers on many large-scale problems. The algorithm consists of two major sequentially applied phases. The first is a gradient-type minimization in a subspace tangent to the constraint surface, and the second is a feasibility restoration procedure. In Part 1, the original SGRA algorithm is described and is compared with two other related methods: the gradient projection and the generalized reduced gradient methods. Next, the special case of linear equalities is analyzed. It is shown that, in this case, only the gradient-type minimization phase is needed, and the SGRA becomes identical to the steepest-descent method. Convergence proofs for the nonlinearly constrained case are given in Part 2.

Optimal penetration landing trajectories in the presence of windshear

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.

Marangoni instability with non-uniform volumetric energy sources due to incident radiation

The effects of non-uniform volumetric energy sources, temperature-dependent viscosity and surface-tension on Marangoni convection in an incompressible fluid layer are studied. The nonlinear temperature profile arises from external incident radiation. The critical conditions for the onset of Marangoni instability in a microgravity environment ( 10 −6–10 −3 g ) are determined numerically after a normal mode-type linear stability analysis. The perturbation equations are solved as an optimal control problem in the calculus of variations using Miele's sequential gradient-restoration algorithm (SGRA). The results indicate that viscosity plays a significant role in Marangoni convection relative to surface-tension. External radiation is a destabilizing factor while the degree of heat transfer (larger values of Bi) from the fluid to the adjacent gas is a stabilizing factor in Marangoni instability.

Optimal trajectories for LEO-to-LEO aeroassisted orbital transfer

This paper considers both classical and minimax problems of optimal control arising in the study of noncoplanar, aeroassisted orbital transfer. The maneuver considered involves the transfer from one planetary orbit to another having different orbital inclination, but the same radius. An example is the LEO-to-LEO transfer of a spacecraft with a prescribed plane change, where LEO denotes low Earth orbit. The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by modulating the lift coefficient (hence, the angle of attack) and the angle of bank. The presence of upper and lower bounds on the lift coefficient is considered. Three different transfer maneuvers are studied. Type 1 involves four impulses and four space plane changes; Type 2 involves three impulses and three space plane changes; and Type 3 involves three impulses and no space plane change. In Type 1, the initial impulse directs the spacecraft away from Earth, and then is followed by an apogee impulse propelling the spacecraft toward Earth; in Types 2 and 3, the initial impulse directs the spacecraft toward Earth. A common element of these maneuvers is that they all include an atmospheric pass, with velocity depletion coupled with plane change. Within the framework of classical optimal control, the following problems are studied: (P1) minimize the energy required for orbital transfer; (P4) maximize the time of flight during the atmospheric portion of the trajectory; (P5) minimize the time integral of the square of the path inclination. Within the framework of minimax optimal control, the following problem is studied: (Q1) minimize the peak heating rate. Numerical solutions for Problems (P1), (P4), (P5), (Q1) are obtained by means of the sequential gradient-restoration algorithm for optimal control problems under following conditions: for Problem (P1), the plane change components and the apogee radius are optimized; for Problems (P4), (P5), (Q1), the plane change components and the apogee radius are kept at the same levels determined for Problem (P1). This is done in order to impart good energy characteristics to the solutions of Problems (P4), (P5), (Q1). The engineering implications of the solutions are discussed, in order to determine the most useful solutions in the light of energy requirements and heat transfer requirements. In particular, the merits of the nearly-grazing solution (P5) are discussed. For all maneuver types, it appears that the nearly-grazing solution (P5) is an excellent compromise between energy requirements and heating requirements.

Optimal abort landing trajectories in the presence of windshear

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.

Transformation Techniques for Minimax Optimal Control Problems and their Application to Optimal Flight Trajectories in a Windshear: Optimal abort Landing Trajectories

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 employinq 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, these conclusions are reached: (i) the optimal trajectory includes three branches: a descendinq 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 settinq rate increases; (iv) the peak altitude.

Optimal trajectories for aeroassisted, noncoplanar orbital transfer

This paper considers both classical and minimax problems of optimal control which arise in the study of noncoplanar, aeroassisted orbital transfer. The maneuver considered involves the transfer from a high planetary orbit to a low planetary orbit with a prescribed atmospheric plane change. An example is the HEO-to-LEO transfer of a spacecraft with a prescribed plane change, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit. The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by modulating the lift coefficient (hence, the angle of attack) and the angle of bank. The presence of upper and lower bounds on the lift coefficient is considered. Within the framework of classical optimal control, the following problems are studied: (P1) minimize the energy required for orbital transfer; (P2) minimize the time integral of the heating rate; (P3) minimize the time of flight during the atmospheric portion of the trajectory; (P4) maximize the time of flight during the atmospheric portion of the trajectory; (P5) minimize the time integral of the square of the path inclination; and (P6) minimize the sum of the squares of the entry and exit path inclinations. Problems (P1)–(P6) are Bolza problems of optimal control. Within the framework of minimax optimal control, the following problems are studied: (Q1) minimize the peak heating rate; (Q2) minimize the peak dynamic pressure; and (Q3) minimize the peak altitude drop. Problems (Q1)–(Q3) are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. Numerical solutions for Problems (P1)–(P6) and Problems (Q1)–(Q3) are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. The engineering implications of these solutions are discussed, and it appears that the energy solution (P1) and the nearly-grazing solution (P5) are superior to the remaining solutions. While the nearly-grazing solution (P5) requires more energy than the energy solution (P1), at the same time it involves less integrated heating rate, less peak heating rate, less peak dynamic pressure, and less peak total acceleration. Therefore, it is felt that both solutions (P1) and (P5) should be considered as candidates for flight operations, and hence candidates for spacecraft design.

EFFECTS OF INTERNAL HEAT GENERATION AND VARIABLE VISCOSITY ON MARANGONI CONVECTION

The problem of Marangoni convection in a horizontal fluid layer with internal heat generation is solved using the sequential gradient-restoration algorithm (SGRA) developed for optimal control problems. A general discussion of the method is given. The effects of the temperature-dependent surface tension and viscosity are studied by examining the critical Marangoni numbers and wave numbers. The validity of the method is established by comparison with the work in the literature for the case of a linear temperature profile across the fluid layer.

Dynamic optimization problems with bounded terminal conditions

Bounded terminal conditions of nonlinear optimization problems are converted to equality terminal conditions via the Valentine's device. In so doing, additional unknown parameters are introduced into the problem. The transformed problems can still be easily solved using the sequential gradient-restoration algorithm (SGRA) via a simple augmentation of the unknown parameter vector π. Three example problems with bounded terminal conditions are solved to verify this technique.

Effects of Internal Heat Generation and Variable Viscosity on Marangoni Convection

The problem of Marangoni convection in a horizontal fluid layer with internal heat generation is solved using the sequential gradient-restoration algorithm (SGRA) developed for optimal control problems. A general discussion of the method is given. The effects of the temperature-dependent surface tension and viscosity are studied by examining the critical Marangoni numbers and wave numbers. The validity of the method is established by comparison with the work in the literature for the case of a linear temperature profile across the fluid layer.

Solution to the orr–sommerfeld equation for liquid film flowing down an inclined plane: An optimal approach

AbstractThe Orr–Sommerfeld equation is solved numerically for a layer of liquid film flowing down an inclined plane under the action of gravity using the sequential gradient‐restoration algorithm (SGRA) The method consists of solving the governing equation as it is a Bolza problem in the calculus of variations. The neutral stability curves, eigenvalues and eigenfunctions to the stability problem can be determined simultaneously during the process.

Primal-dual properties of sequential gradient-restoration algorithms for optimal control problems 2. General problem

The problem of minimizing a functional, subject to differential constraints, nondifferential constraints, initial constraints, and final constraints, is considered in connection with sequential gradient-restoration algorithms (SGRA) for optimal control problems. Both the primal formulation and the dual formulation are presented. Depending on whether the primal formulation is used or the dual formulation is used, one obtains a primal sequential gradient-restoration algorithm (PSGRA) or a dual sequential gradient-restoration algorithm (DSGRA). For the problem under consideration, it is found convenient to split the control vector into an independent control vector and a dependent control vector, the latter having the same dimension as the nondifferential constraint vector. This modification enhances the computational efficiency of both the primal formulation and the dual formulation. The system of Lagrange multipliers associated with (i) the gradient phase of SGRA and (ii) the restoration phase of SGRA is examined. For each phase, it is shown that the Lagrange multipliers are endowed with a duality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter. These duality properties have considerable computational implications: they allow one to reduce the auxiliary optimal control problems associated with (i) and (ii) to mathematical programming problems involving a finite number of parameters as unknowns.

Primal and dual formulations of sequential gradient-restoration algorithms for trajectory optimization problems

One of the most effective first-order algorithms for solving trajectory optimization problems is the sequential gradient-restoration algorithm (SGRA). Originally developed in the primal formulation (PSGRA), this algorithm is extended to incorporate a dual formulation (DSGRA). Both the primal formulation and the dual formulation involve a sequence of two-phase cycles, each cycle including a gradient phase and a restoration phase. In turn, each iteration of the gradient phase and the restoration phase requires the solution of an auxiliary minimization problem (AMP). In the primal formulation, the AMP is solved with respect to the variations of the state, the control and the parameter. In the dual formulation, the AMP is solved with respect to the Lagrange multipliers. A characteristic of the dual formulation is that the AMP's associated with the gradient phase and the restoration phase of SGRA can be reduced to mathematical programming problems involving a finite number of parameters as unknowns. A comparison of the primal formulation and the dual formulation is presented. The comparison is done in terms of several trajectory optimization problems having current aerospace interest.

Application of the sequential gradient-restoration algorithm to thermal convective instability problems

The problem of the thermal stability of a horizontal incompressible fluid layer with linear and nonlinear temperature distributions is solved by using the sequential gradient-restoration algorithm developed for optimal control problems. The hydrodynamic boundary conditions for the layer include a rigid or free upper surface and a rigid lower surface. The resulting disturbing equations are solved as a Bolza problem in the calculus of variations. The results of the study are compared with the existing works in the literature.

Approximate solutions to minimax optimal control problems for aeroassisted orbital transfer

This paper considers minimax problems of optimal control arising in the study of aeroassisted orbital transfer. The maneuver considered involves the coplanar transfer from a high planetary orbit to a low planetary orbit. An example is the HEO-to-LEO transfer of a spacecraft, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit. The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by means of lift modulation. The presence of upper and lower bounds on the lift coefficient is considered. The following minimax problems of optimal control are investigated: (i) minimize the peak heating rate, problem P1; and (ii) minimize the peak dynamic pressure, problem P2. It is shown that problems P1 and P2 are approximately equivalent to the following minimax problem of optimal control: (iii) minimize the peak altitude drop occurring in the atmospheric portion of the trajectory, problem P3. Problems P1–P3 are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. However, the need for these transformations can be bypassed if one reformulates problem P3 as a two-subarc problem of optimal control, in which the first subarc connects the initial point and the point where the path inclination is zero, and the second subarc connects the point where the path inclination is zero and the final point: (iv) minimize the altitude drop achieved at the point of junction between the first subarc and the second subarc, problem P4. Note that problem P4 is a Bolza problem of optimal control. Numerical solutions for problems P1–P4 are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. Numerical examples are presented, and their engineering implications are discussed. In particular, it is shown that, from an engineering point of view, it is desirable to solve problem P3 or P4, rather than problems P1 and P2.

Analytical Determination of Optimal Control and Optimal Location for Underground Railways

This paper considers the problem of determining both optimal control and optimal location for an underground railway in order to minimize the energy consumption of round trip between two consecutive stations. If this partial optimization is conducted for every station pair through the line, the optimization of whole network can be achieved. A simple dynamical second order model of a train is used in the study, considering parameter values obtained from rolling equipment employed in the Santiago Underground Railways System (METRO). In addition, a set of restrictions over the state and control variables is considered for train operation, such as velocity restriction, acceleration restriction, trackage slope restrictions in different points of the way, etc. Only the station situations are known and correspond to actual stations of Santiago METRO. The technique utilized to solve the problem is the sequential Gradient- Restoration Algorithm developed by A. Miele et als. Applying this technique, both optimal control applied over train and optimal location on which the train must move are obtained. The actual location of Santiago METRO and the new trackage derived from this technique are compared, obtaining an important energy saving. This comparison is performed on the basis that in both cases (actual and new locations) optimal control applied over train is derived from Sequential Gradient-Restoration Algorithm. In the former case (actual) the location is given and only the optimal is generated, but in the later case (new) location and optimal control are generated from optimization method used.

Optimal trajectories for aeroassisted orbital transfer

This paper considers both classical and minimax problems of optimal control arising in the study of aeroassisted orbital transfer. The maneuver considered involves the coplanar transfer from a high planetary orbit to a low planetary orbit. An example is the HEO-to-LEO transfer of a spacecraft, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. Trajectory control is achieved by means of lift modulation. An appropriate performance index is minimized. Specifically, within the frame of classical optimal control, the performance index is the energy required for orbital transfer; within the frame of minimax optimal control, the performance index is the peak heating rate. The presence of upper and lower bounds on the lift coefficient is considered. Numerical solutions are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. Numerical examples are presented, and their engineering implications are discussed.

Supplementary optimality properties of the restoration phase of sequential gradient-restoration algorithms for optimal control problems

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the restoration phase. It is shown that the Lagrange multipliers associated with the restoration phase not only solve the auxiliary minimization problem of the restoration phase, but are also endowed with a supplementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.

Supplementary Optimality Properties of the Family of Gradient-Restoration Algorithms for Optimal Control Problems

The problem of minimizing a functional subject to differential constraints, nondifferential constraints, initial constraints, and final constraints is considered within the frame of the family of gradient-restoration algorithms for optimal control problems. This family includes sequential gradient-restoration algorithms (SGRA) and combined gradient-restoration algorithms (CGRA).The system of Lagrange multipliers associated with (i) the gradient phase of SGRA, (ii) the restoration phase of SGRA, and (iii) the combined gradient-restoration phase of CGRA is examined. It is shown that, in each case, the Lagrange multipliers are endowed with a suppiementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.These supplementary optimality properties have considerable computational implications: they allow one to reduce the study of an iteration of (i), (ii), (iii) to a mathematical programming problem involving a finite number of parameters as unknowns.

Supplementary optimality properties of gradient-restoration algorithms for optimal control problems

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the gradient phase. It is shown that the Lagrange multipliers associated with the gradient phase not only solve the auxiliary minimization problem of the gradient phase, but are also endowed with a supplementary optimality property: they minimize the error in the optimality conditions, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.