Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximization for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimization problems with the help of convex semidefinite programming (optimization over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach

A New Mixed Iterative Algorithm to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem

International audienceThe optimal fuel impulsive time-fixed rendezvous problem is reviewed. In a linear setting, it may be reformulated as a non convex polynomial optimization problem for a pre-specified fixed number of velocity increments. Relying on variational results previously published in the literature, an improved mixed iterative algorithm is defined to address the issue of optimization over the number of impulses. Revisiting the primer vector theory, it combines variational tests with sophisticated numerical tools from algebraic geometry to solve polynomial necessary and sufficient conditions of optimality. Numerical examples under circular and elliptic assumptions show that this algorithm is efficient and can be integrated into a rendezvous planning tool

Characterization and computation of control invariant sets within target regionsfor linear impulsive control systems

Linear impulsively controlled systems are suitable to describe a venue of real-life problems, going from disease treatment to aerospace guidance. The main characteristic of such systems is that they remain uncontrolled for certain periods of time. As a consequence, punctual equilibria characterizations outside the origin are no longer useful, and the whole concept of equilibrium and its natural extension, the controlled invariant sets, needs to be redefined. Also, an exact characterization of the admissible states, i.e., states such that their uncontrolled evolution between impulse times remain within a predefined set, is required. An approach to such tasks -- based on the Markov-Lukasz theorem -- is presented, providing a tractable and non-conservative characterization, emerging from polynomial positivity that has application to systems with rational eigenvalues. This is in turn the basis for obtaining a tractable approximation to the maximal admissible invariant sets. In this work, it is also demonstrated that, in order for the problem to have a solution, an invariant set (and moreover, an equilibrium set) must be contained within the target zone. To assess the proposal, the so-obtained impulsive invariant set is explicitly used in the formulation of a set-based model predictive controller, with application to zone tracking. In this context, specific MPC theory needs to be considered, as the target is not necessarily stable in the sense of Lyapunov. A zone MPC formulation is proposed, which is able to i) track an invariant set such that the uncontrolled propagation fulfills the zone constraint at all times and ii) converge asymptotically to the set of periodic orbits completely contained within the target zone.Fil: Sánchez, Ignacio Julián Rodolfo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Louembet, Christophe. Centre National de la Recherche Scientifique; Francia. Universite de Toulose - Le Mirail; FranciaFil: Actis, Marcelo Jesús. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe; Argentina. Universidad Nacional del Litoral. Facultad de Ingeniería Química; ArgentinaFil: González, Alejandro Hernán. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentin

Contributions au guidage pour le rendez-vous spatial par résolution du problème de commande optimale impulsionnelle

National audienceThis manuscript of Habilitation à Diriger des Recherches deals with the guidance for spacecraft rendezvous. The orbital rendezvous consists in docking two spacecraft originating from their initial relative position and speed in free or fixed time. More generally, rendezvous guidance means computing all the maneuvers performed in the relative motion context. The relative motion describes the motion of a "chaser" spacecraft with thrust abilities in reference frame attached to the inert spacecraft in its orbit and called "target" spacecraft. The propulsion mode of the chaser being chemical, it is supposed to provide strong acceleration on small period of time with the respect to orbital period. These high thrusts allow a rapid evolution of the speed during these intervals. The difference of scale between the orbital dynamics and the speed evolution leads us to consider that the chemical propulsion causes instantaneous jumps in the speed history from the orbital motion point of view. The rendezvous guidance problem is described in this manuscript as an optimal control problem addressed from different perspectives. It is, at first, considered unconstrained and treated by solving the Neustadt problem. Then technological restrictions on the relative movement are accounted such as visibility conditions, collision avoidance, thusters limitations and uncertainties on the navigation data and the control misexecution. In this context, the so-called direct methods for solving the optimal control problem are put in place while providing certificates of constraint satisfaction.Ce manuscrit d'Habilitation à Diriger des Recherches aborde le problème du guidage en rendez-vous orbital. Le rendez-vous orbital consiste à réaliser la jonction entre deux véhicules orbitaux à partir de leurs position et vitesse initiales en temps libre ou fixé. De manière plus générale, le rendez-vous concerne toutes les manoeuvres exécutés dans un contexte de mouvement relatif. Le mouvement relatif décrit le mouvement d'un satellite muni de moyens de propulsion, appelé chasseur, dans le référentiel lié à un satellite considéré inerte évoluant naturellement sur son orbite et appelé cible. Le mode propulsion du chasseur est chimique et procure de forte accélération sur des temps d'activation très petits devant la période de révolution orbitale. Ces fortes poussées permettent une évolution significative de la vitesse durant ces intervalles. La grande différence d'échelle entre la dynamique orbitale et celle de la vitesse nous amène à considérer que la propulsion chimique provoque des sauts de vitesse lorsque l'on s'intéresse au mouvement orbital. Le problème de guidage en rendez-vous orbital est décrit dans ce manuscrit comme un problème de commande optimale abordé depuis différentes perspectives. Il est, dans un premier temps, considéré non contraint et traité en résolvant le problème de Neustadt. Puis, des restrictions d'ordre technologique sur le mouvement relatif sont envisagées telles que des conditions de visibilité, d'évitement de collision, les limites physiques des propulseurs ou encore des incertitudes sur les données de navigation et l'exécution de la commande. Dans ce contexte, des méthodes de résolution du problème de commande optimale dites directes sont mis en place tout en procurant des certificats de satisfaction des contraintes

Génération de trajectoires optimales pour systèmes différentiellement plats - Application aux manoeuvres d'attitude sur orbite

Cette thèse aborde la problématique de la génération de trajectoires d’attitude pour les satellites d’observation d’orbite basse. Les techniques développées sont basées sur le concept de la platitude différentielle dont les propriétés permettent de résoudre le problème de commande optimale sans intégration des équations différentielles représentant la dynamique du véhicule. Nous avons proposé deux démarches de résolution différentes. La première démarche est basée sur la collocation par les B-splines. Cette méthode permet de transformer le problème de commande optimale en un problème de programmation non linéaire. Afin d’améliorer les performances numériques de cette méthode, nous avons défini un problème convexe à partir de l’approximation convexe de l’espace admissible aux contraintes. La deuxième méthode consiste à convertir le problème, initialement formulé en termes de programmation semi-infinie, en un problème de programmation semi-définie positive. Cette approche a pour finalité de vérifier les contraintes imposées sur le continuum temporel et non plus en un nombre d’instants fixés pour les méthodes par collocation directe. Les techniques développées dans le cadre de cette thèse ont été testées et validées sur un simulateur industriel du CNES. Les résultats montre alors que les méthodes par platitude permettent un gain important par rapport aux méthodes classiques du CNES notamment du point de vue de l’écart à la trajectoire et de l’excitation de modes souples.This thesis discusses the design of slew manoeuvers for Earth observation satellite. First, We show that the non linear model of reaction wheels actuated satellite satisfies the flatness property. Second, it is demonstrated that the main interest of path planning by flatness approach allows to solve the optimal control problem without the need of dynamics integration. By parameterizing the flat outputs by B-spline, we developed an non linear parametric problem that is solved with NLP solvers. To improve the computational quality of this methodology, we describe a convex subset where all constraints are satisfied. Using this subset, we can design a faster collocation-based path planner. The main drawback of this methodology is that constraints are only checked at collocation points. To overcome this drawback, we have developed a new trajectory generation scheme based on a B-spline positivity theorem that leads to efficient and tractable programming scheme using LMI techniques. Finally, we tested trajectories calculated by our flatness methodology in a CNES’ Demeter simulator. We compared our results with the results from CNES trajectories generation tools. Our methodology improve the the optimal maneuvering time, the used control torque by diminishing the deflect to the initial trajectory and excitation of flexible structures

Collision avoidance in low thrust rendezvous guidance using flatness and positive B-splines

International audienceCollision avoidance issue in close proximity orbital rendezvous is addressed in this paper. The arising optimal control problem is solved by means of differential flatness in order to apply algebraic geometry tools. These tools based on B-splines parametrization and positive piecewise polynomial concept provide a certification of constraints satifaction ontinuously in time contrary to classical collocation techniques. The non convexity derived from the avoidance constraint is overcome by using time-varying convex approximation

Using Differential Flatness for solving the Minimum-Fuel Low-Thrust Geostationary Station-Keeping Problem

International audienceThe minimum fuel station keeping of a geostationary satellite equipped with electric thrusters undergoing the Earth non-spherical disturbing gravitational potential up to degree and order 2 is first recast by transforming the linear time varying relative dynamics in a time invariant one with a Floquet-Lyapunov transformation. In a second step, the flatness property of the system is used to convert the genuine optimal control problem to a linear programming problem. This problem is solved for an ideal geostationary satellite with one thruster mounted on each face