Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach

The aim of this paper is to investigate from the numerical point of view the possibility of coupling the Hamilton-Jacobi-Bellman (HJB) equation and Pontryagin's Minimum Principle (PMP) to solve some control problems. A rough approximation of the value function computed by the HJB method is used to obtain an initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered. The CPU time for the proposed method is less than four minutes up to dimension four, without code parallelization

Singular arcs in the generalized Goddard's Problem

We investigate variants of Goddard's problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this report, performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle with the problem of nonsmoothness of the optimal control

Optimal Ecodriving Control: Energy-Efficient Driving of Road Vehicles as an Optimal Control Problem

International audienceTransportation is responsible for a substantial fraction of worldwide energy consumption and greenhouse gas emissions and is the largest sector after energy production. However, while emissions from other sectors are generally decreasing, those from transportation have increased since 1990. Reducing the impact of transportation is a task that is inherently associated with the improvement of energy efficiency, particularly for passenger cars that contribute to almost half of the whole sector

Performance of Mixture-Ratio-Controlled Hybrid Rockets for Nominal Fuel Regression

This paper discusses the impacts of oxidizer-to-fuel mass ratio (O/F) shifts on the flight performance of single-stage sounding rockets using the flight simulations of three scales of O/F-controlled and O/F-uncontrolled hybrid rockets under a nominal fuel regression behavior without uncertainty. The flight simulation code includes three factors dependent on the O/F: thermodynamic states of the burned gas (theoretical Isp), shifts in c∗ efficiency, and nozzle throat erosion. In the flight simulations, a thrust control law was applied to increase the apogee and evaluate the effects of O/F shifts in the thrust curve including throttling. For the best cases in each scale, O/F-controlled hybrid rockets slightly improved the performance by 2.03–2.42% in the averaged specific impulse. However, the performance of the O/F-controlled sounding rockets is essentially the same as the O/F-uncontrolled type under the median regression behavior: especially when considering the slight increases in the mass and complexity of the oxidizer feed system needed for O/F control. Considerable scale effects on the throat erosion and theoretical Isp were observed, but that of the c∗ efficiency was negligible. The improvement of the theoretical Isp was the primary contributor to flight performance, which was responsible for a larger than 70% share in the total Isp increase. The second largest contribution was the improvement of the c∗ efficiency with a share of 21.8–24.3%. The O/F control gave an improvement of throat erosion corresponding to 5.75% in the total Isp increase for the smallest scale; but, with increasing of the scale, the throat area increase ratio became small so that the throat erosion improvement contribution was reduced to 1.21%

Numerical optimal control with applications in aerospace

This thesis explores various computational aspects of solving nonlinear, continuous-time dynamic optimization problems (DOPs) numerically. Firstly, a direct transcription method for solving DOPs is proposed, named the integrated residual method (IRM). Instead of forcing the dynamic constraints to be satisfied only at a selected number of points as in direct collocation, this new approach alternates between minimizing and constraining the squared norm of the dynamic constraint residuals integrated along the whole solution trajectories. The method is capable of obtaining solutions of higher accuracy for the same mesh compared to direct collocation methods, enabling a flexible trade-off between solution accuracy and optimality, and providing reliable solutions for challenging problems, including those with singular arcs and high-index differential-algebraic equations. A number of techniques have also been proposed in this work for efficient numerical solution of large scale and challenging DOPs. A general approach for direct implementation of rate constraints on the discretization mesh is proposed. Unlike conventional approaches that may lead to singular control arcs, the solution of this on-mesh implementation has better numerical properties, while achieving computational speedups. Another development is related to the handling of inactive constraints, which do not contribute to the solution of DOPs, but increase the problem size and burden the numerical computations. A strategy to systematically remove the inactive and redundant constraints under a mesh refinement framework is proposed. The last part of this work focuses on the use of DOPs in aerospace applications, with a number of topics studied. Using example scenarios of intercontinental flights, the benefits of formulating DOPs directly according to problem specifications are demonstrated, with notable savings in fuel usage. The numerical challenges with direct collocation are also identified, with the IRM obtaining solutions of higher accuracy, and at the same time suppressing the singular arc fluctuations.Open Acces

Сингулярные оптимальные управления движением ракет (обзор)

Наведено аналіз сучасного стану і обговорено проблеми удосконалення методів дослідження вироджених варіаційних задач з акцентом на механіку космічного польоту. Увагу приділено дослідженню руху ракет в атмосфері. Включення до складу оптимальних траєкторій дуг сингулярного управління надає змогу в цьому випадку збільшити економічність ракетних двигунів шляхом заміни традиційних двигунів постійної тяги двигунами, що допускають дроселювання величини тяги. Наведені в огляді результати розрахунків для конкретних маневрів можуть слугувати джерелом інформації для прийняття рішень під час конструювання перспективної ракетно-космічної техніки.A review of investigations of the dynamical systems control problems is presented with emphasis on mechanics of space flight. The main attention is drawn to perfecting the methods of solving the degenerate variational problems on motion of rockets in the gravitational fields with allowance for the atmospheric resistance. These problems are immediately associated with the permanently actual problem of the practical astronautics – increasing the mass of useful load that is orbited by the carrier rockets in the circumplanetary orbits. An analysis is given for the modern approaches to solving the problems of control of rockets and spacecrafts motion over the trajectories with singular arcs that are optimal for motion of the variable mass body in the medium with resistance. The presented in this review results of some practical problems enable to estimate an advantage of using the optimal control, realization of which needs the complication of the optimal control system of the rocket engines functioning in comparison with the recent simpler laws of control