Problèmes de commande optimale stochastique généralisés

Cette thèse est divisée en deux grands chapitres, dont le premier porte sur des problèmes de commande optimale en dimension un et le deuxième sur des problèmes en dimension deux ou plus. Notons bien que, dans cette thèse, nous avons supposé que le facteur temps n'intervient pas. Dans le premier chapitre, nous calculons, au début, l'équation de programmation dynamique pour la valeur minimale F de l'espérance mathématique de la fonction de coût considérée. Ensuite, nous utilisons le théorème de Whittle qui est applicable seulement si une condition entre le bruit blanc v et les termes b et q associés à la commande est satisfaite. Sinon, nous procédons autrement. En effet, un changement de variable transforme notre équation en une équation de Riccati en G= F', mais sans conditions initiales. Dans certains cas, à partir de la symétrie des paramètres infinitésimaux et de q, nous pouvons en déduire le point x' où G(x')=0. Si ce n'est pas le cas, nous nous limitons à des bonnes approximations. Cette même démarche est toujours possible si nous sommes dans des situations particulières, par exemple, lorsque nous avons une seule barrière. Dans le deuxième chapitre, nous traitons les problèmes en dimension deux ou plus. Puisque la condition de Whittle est difficile à satisfaire dans ce cas, nous essayons de généraliser les résultats du premier chapitre. Nous utilisons alors dans quelques exemples la méthode des similitudes, qui permet de transformer le problème en dimension un. Ensuite, nous proposons une nouvelle méthode de résolution. Cette dernière linéarise l'équation de programmation dynamique qui est une équation aux dérivées partielles non linéaire. Il reste à la fin à trouver les conditions initiales pour la nouvelle fonction et aussi à vérifier que les n expressions obtenues pour F sont équivalentes.This thesis is divided into two chapters: the first one deals with some optimal control problems in one dimension and the second one with these problems in two or more dimensions. Note that, in this thesis, the time variable is not taken into account. In Chapter 1, at first we compute the dynamic programming equation for the minimal expected value F of the cost function considered. Next, we apply Whittle's theorem if the condition between the noise v and the functions b and q associated with the control variable is satisfied. Otherwise, we proceed differently. Indeed, if we make a change of variable, we obtain a Riccati equation for G= F', but without initial conditions. In some cases, from the symmetry of the infinitesimal parameters and of the function q, we can deduce the point x' where G(x')=0. If this is not possible, we limit ourselves to good approximations. The same approach is still possible if we are in specific situations, for example, when we have only one barrier. In Chapter 2, we discuss problems in dimension two or more. Since the condition in Whittle's theorem is difficult to satisfy in this case, we try to generalize the results obtained in Chapter 1. We then use in some examples the method of similarity solutions, which enables us to transform the problem into a one-dimensional one. Next, we propose a new resolution method. This method linearises the dynamic programming equation, which is a non-linear partial differential equation. Finally, we must find initial conditions for the new function, and also verify that the n expressions for F are equivalent

Application of genetic algorithm to a forced landing manoeuvre on transfer of training analysis

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. It focuses on flight trajectories that a pilot of a single-engine general aviation aircraft should fly after engine failure and how pilots can be better simulator trained for this forced landing manoeuvre. A sensitivity study on the effects of errors and an investigation on the effect of tolerances in the aerodynamic parameters as prescribed in the Manual of Criteria for the Qualification of Flight Simulators have on the performance of flight simulators used for pilot training was carried out. It uses a simplified analytical model for the Beech Bonanza model E33A aircraft and a vertical atmospheric turbulence based on the MIL-F-8785C specifications. It was found that the effect of the tolerances is highly s ensitive on the nature of the manoeuvre flown and that in some cases, negative transfer of training may be induced by the tolerances. A forced landing trajectory optimisation was carried out using Genetic Algorithm. 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. 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 depend on the end constraints. 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 h eading 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 methodology 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. 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