Solving Optimal Control Problems for Delayed Control-Affine Systems with Quadratic Cost by Numerical Continuation

- In this paper we introduce a new method to solve fixed-delay optimal control problems which exploits numerical homotopy procedures. It is known that solving this kind of problems via indirect methods is complex and computationally demanding because their implementation is faced with two difficulties: the extremal equations are of mixed type, and besides, the shooting method has to be carefully initialized. Here, starting from the solution of the non-delayed version of the optimal control problem, the delay is introduced by numerical homotopy methods. Convergence results, which ensure the effectiveness of the whole procedure, are provided. The numerical efficiency is illustrated on an example

Continuity of Pontryagin extremals with respect to delays in nonlinear optimal control

Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a new perspective for the numerical implementation of indirect methods, such as the shooting method. We also discuss the sharpness of our assumptions.Comment: arXiv admin note: text overlap with arXiv:1709.0438

On the Accessibility and Controllability of Statistical Linearization for Stochastic Control: Algebraic Rank Conditions and their Genericity

Statistical linearization has recently seen a particular surge of interest as a numerically cheap method for robust control of stochastic differential equations. Although it has already been successfully applied to control complex stochastic systems, accessibility and controllability properties of statistical linearization, which are key to make the robust control problem well-posed, have not been investigated yet. In this paper, we bridge this gap by providing sufficient conditions for the accessibility and controllability of statistical linearization. Specifically, we establish simple sufficient algebraic conditions for the accessibility and controllability of statistical linearization, which involve the rank of the Lie algebra generated by the drift only. In addition, we show these latter algebraic conditions are essentially sharp, by means of a counterexample, and that they are generic with respect to the drift and the initial condition.Comment: 23 page

Prospects on solving an optimal control problem with bounded uncertainties on parameters using interval arithmetic

An interval method based on Pontryagin’s Minimum Principle is proposed to enclose the solutions of an optimal control problem with embedded bounded uncertainties. This method is used to compute an enclosure of all optimal trajectories of the problem, as well as open loop and closed loop enclosures meant to validate an optimal guidance algorithm on a concrete system with inaccurate knowledge of the parameters. The differences in geometry of these enclosures are exposed, and showcased on a simple system. These enclosures can guarantee that a given optimal control problem will yield a satisfactory trajectory for any realization of the uncertainties. Contrarily, the probability of failure may not be eliminated and the problem might need to be adjusted

Prospects on the application of necessary optimality conditions on the resolution of the Goddard problem with unknown bounded parameters using interval arithmetics

International audienc

Missile trajectory shaping using sampling-based path planning

International audienceThis paper presents missile guidance as a complex robotic problem: a hybrid non-linear system moving in a heterogeneous environment. The proposed solution to this problem combines a sampling-based path planner, Dubins' curves and a locally-optimal guidance law. This algorithm aims to find feasible trajectories that anticipate future flight conditions, especially the loss of manoeuverability at high altitude. Simulated results demonstrate the substantial performance improvements over classical midcourse guidance laws and the benefits of using such methods, well-known in robotics, in the missile guidance field of research

Asservissement et Navigation Autonome d'un drone en environnement incertain par flot optique

This thesis deals with navigation of a VTOL Unmanned Aerial Vehicle in unknown or uncertain environment. The use of optical flow is bio-inspired. It provides information on velocity of the vehicle and proximity of obstacles. Two contributions are presented in this work. The first one addresses automatic landing on a static or mobile platform. The maneuver is split into two tasks : stabilization of the linear velocity above the target and vertical landing. The approach shows that regulation of the divergent optical flow around a constant set point enables a smooth landing without collision despite uncertainties on the vehicle and the platform dynamics. The second contribution concerns terrain following with obstacle avoidance. The general approach considered enables us to address different applications such as corner avoidance, steep terrain following, corridor following, etc. Stability analysis assesses robustness and limits of the controller in the presence of various uncertainties such as uncertainties on the geometry of the environment. All the algorithms are simulated and experimented on a quadrotor UAV built at the CEA LIST.Cette thèse porte sur la navigation sans collision d'un véhicule aérien à décollage et atterrissage vertical en environnement inconnu ou incertain. L'utilisation du flot optique, inspirée du monde animal, permet d'obtenir des informations sur la vitesse du véhicule et sur la proximité des obstacles. Deux contributions sont présentées dans ce travail. La première aborde l'atterrissage automatique sur une plateforme statique ou mobile. La manœuvre se décompose en deux tâches : la stabilisation de la vitesse au-dessus de la cible, puis l'atterrissage vertical. L'approche montre que la régulation du flot optique divergent autour d'une consigne constante permet un atterrissage en douceur et sans collision malgré les incertitudes sur la dynamique de la plateforme et du véhicule. La deuxième contribution concerne le suivi de terrain avec évitement d'obstacles. L'approche générale proposée permet d'aborder différentes applications telles que l'évitement d'obstacles frontaux, le suivi de terrain pentu, le suivi de couloir, etc. L'analyse de stabilité évalue la robustesse et les limites du contrôleur en présence de diverses incertitudes telles que les incertitudes sur la géométrie de l'environnement. L'ensemble des algorithmes de commande est simulé et expérimenté sur un mini-drone quadrirotor développé au CEA LIST

Contrôle optimal et planification de trajectoire pour le guidage des systèmes aérospatiaux

In the Guidance-Navigation-Control loop of aeronautical and space systems, guidance specifically concerns the control of the vehicle's center of mass. Due to the complexity of the problems encountered, a classical approach consists first of calculating a reference trajectory by optimization method. A servo-control algorithm is then applied to follow this trajectory. Although simple to implement and computationally inexpensive, this technique does not allow for any unforeseen events (e.g. a change of objective or a deviation of trajectory due to the presence of strong uncertainties). A first part of the work presented in this HDR (accreditation to supervise research) deals with the study of new guidance algorithms ensuring both optimality and robustness in such cases. For this, optimal control methods are used to analyze the structure of the control and numerical shooting methods and continuation techniques are combined to obtain an embedded algorithm. These developments have been applied to target interception problems and to reusable launch vehicle guidance problems.A second part of the work focuses on trajectory planning and control problems in highly constrained environments for which the optimal control techniques used alone are impossible to implement (e.g. in the presence of obstacles or strong hazards). Probabilistic methods are used with optimal control methods for UAV trajectory planning. For terrain-based navigation problem, a robust predictive control method based on a stochastic formalism has been developed to overcome the nonlinear coupling between estimation and control.Dans la boucle de Navigation-Guidage-Pilotage des systèmes aéronautiques et spatiaux, le guidage porte spécifiquement sur la commande du centre de masse du véhicule. En raison de la complexité des problèmes rencontrés, une approche classique consiste tout d'abord à calculer une trajectoire de référence par méthode d'optimisation. Un algorithme d'asservissement est ensuite appliqué pour suivre cette trajectoire. Bien que simple à mettre en œuvre et peu coûteuse en calculs, cette technique n'autorise aucun imprévu (e.g. un changement d'objectif ou une déviation de trajectoire due à la présence de fortes incertitudes). Une première partie des travaux présentés dans cette HDR porte sur l'étude de nouveaux algorithmes de guidage permettant de conserver optimalité et robustesse dans de tels cas. Pour cela, les méthodes du contrôle optimal sont utilisées afin d'analyser la structure de la commande et les méthodes numériques de tir et de continuation sont combinées pour obtenir un algorithme embarqué. Ces développements ont été appliqués à des problèmes d'interception de cible et à des problèmes de retour de lanceurs réutilisables.Une deuxième partie des travaux se concentre sur des problèmes de planification de trajectoire et de commande en environnement fortement contraint pour lesquels les techniques de contrôle optimal utilisées seules sont impossibles à mettre en œuvre (e.g. en présence d'obstacles ou de forts aléas). Des méthodes probabilistes sont utilisées avec des méthodes de contrôle optimal pour la planification de trajectoire d'un drone. Dans le problème de navigation par corrélation de terrain, une méthode de commande prédictive robuste basée sur un formalisme stochastique a été développée pour pallier le problème de couplage entre l'estimation et la commande