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Abstract

This paper deals with the stabilization of an anti-stable string equation with Dirichlet actuation where the instability appears because of the uncontrolled boundary condition. Then, infinitely many unstable poles are generated and an infinite dimensional control law is therefore proposed to exponentially stabilize the system. The idea behind the choice of the controller is to extend the domain of the PDE so that the anti-damping term is compensated by a damping at the other boundary condition. Additionally, notice that the system can then be exponentially stabilized with a chosen decay-rate and is robust to uncertainties on the wave speed and the anti-damped coefficient of the wave equation, with the only use of a point-wise boundary measurement. The efficiency of this new control strategy is then compared to the backstepping approach.

Abstract

This article deals with the stability analysis of a drilling system which is modelled as a coupled ordinary differential equation / string equation. The string is damped at the two boundaries but leading to a stable open-loop system. The aim is to derive a linear matrix inequality ensuring the exponential stability with a guaranteed decay-rate of this interconnected system. A strictly proper dynamic controller based on boundary measurements is proposed to accelerate the system dynamics and its effects are investigated through the stability theorem and simulations. It results in an efficient finite dimension controller which subsequently improves the system performances.

Abstract

This paper presents the design of a nonlinear control law for a typical electromagnetic actuator system. Electromagnetic actuators are widely implemented in industrial applications, and especially as linear positioning system. In this work, we aim at taking into account a magnetic phenomenon that is usually neglected: flux fringing. This issue is addressed with an uncertain modeling approach. The proposed control law consists of two steps, a backstepping control regulates the mechanical part and a sliding mode approach controls the coil current and the magnetic force implicitly. An illustrative example shows the effectiveness of the presented approach.

Abstract

With the increasing power of computers, real-time algorithms tends to become more complex and therefore require better guarantees of safety. Among algorithms sustaining autonomous embedded systems, model predictive control (MPC) is now used to compute online trajec-tories, for example in the SpaceX rocket landing. The core components of these algorithms, such as the convex optimization function, will then have to be certified at some point. This paper focuses specifically on that problem and presents a method to formally prove a primal linear programming implementation. We explain how to write and annotate the code with Hoare triples in a way that eases their automatic proof. The proof process itself is performed with the WP-plugin of Frama-C and only relies on SMT solvers. Combined with a framework producing all together both the embedded code and its annotations, this work would permit to certify advanced autonomous functions relying on online optimization.

Abstract

This paper presents a control law based on Hybrid Dynamical Systems (HDS) theory for a dc-ac converter. This theory is very suited for analysis of power electronic converters, since they combine continuous (voltages and currents) and discrete (on-off state of switches) signals avoiding, in this way, the use of averaged models. Here, practical stability results are induced for this tracking problem, ensuring a minimum dwell-time associated with an LQR performance level during the transient response and an admissible chattering around the operating point. The effectiveness of the resultant control law is validated by means of simulations and experiments.

Abstract

The ROADEF/EURO challenge is a contest jointly organized by the French Operational Research and Decision Aid society (ROADEF) and the European Operational Research society (EURO). The contest has appeared on a regular basis since 1999 and always concerns an applied optimization problem proposed by an industrial partner. The 2014 edition of the ROADEF/EURO challenge was led by the Innovation & Research department of SNCF, a global leader in passenger and freight transport services, and infrastructure manager of the French railway network. The objective of the challenge was to find the best way to store and move trains on large railway sites, between their arrivals and departures. Since trains never vanish and traffic continues to increase, in recent years some stations have been having real congestion issues. Train management in large railway sites is of high interest for SNCF, which is why it was submitted to the operations research community as the industrial problem for the 2014 edition of the ROADEF/EURO challenge. This paper introduces the special section of the Annals of Operations Research volume devoted to the ROADEF/EURO challenge 2014, as well as the methods of the finalist teams and their results.

Abstract

This paper focusses on the design of a robust switching control law for an uncertain discrete-time switched affine system. In order to cope with model uncertainties, a novel control law is introduced and its parameters result from an optimization problem, aiming at reducing the volume of the attractive and invariant set, where the solutions of the closed-loop systems converge to. The design is based on a quadratic Lyapunov function and guarantees global practical stability and robustness with respect to parameter variations. Our method and the associated relaxed control law are then compared with existing conditions from the literature and are validated through numerical examples.

Abstract

The problem of fixed-time fuel-optimal trajectories with high-thrust propulsion in the vicinity of a Lagrange point is tackled via the linear version of the primer vector theory. More precisely, the proximity to a Lagrange point i.e. any equilibrium point-stable or not-in the circular restricted three-body problem allows for a linearization of the dynamics. Furthermore, it is assumed that the spacecraft has ungimbaled thrusters, leading to a formulation of the cost function with the 1-norm for space coordinates, even though a generalization exists for steerable thrust and the 2-norm. In this context, the primer vector theory gives necessary and sufficient optimality conditions for admissible solutions to two-value boundary problems. Similarly to the case of rendezvous in the restricted two-body problem, the in-plane and out-of-plane trajectories being uncoupled, they can be treated independently. As a matter of fact, the out-of-plane dynamics is simple enough for the optimal control problem to be solved analytically via this indirect approach. As for the in-plane dynamics, the primer vector solution of the so-called primal problem is derived by solving a hierarchy of linear programs, as proposed recently for the aforementioned rendezvous. The optimal thrusting strategy is then numerically obtained from the necessary and sufficient conditions. Finally, in-plane and out-of-plane control laws are combined to form the complete 3-D fuel-optimal solution. Results are compared to the direct approach that consists in working on a discrete set of times in order to perform optimization in finite dimension. Examples are provided near various Lagrange points in the Sun-Earth and Earth-Moon systems, hinting at the extensive span of possible applications of this technique in station-keeping as well as mission analysis, for instance when connecting manifolds to achieve escape or capture.

Résumé

Le problème généralisé des moments (PGM) est un problème d’optimisation linéaire sur des espaces de mesures. Il permet de modéliser simplement un grand nombre d’applications. En toute généralité il est impossible à résoudre mais si ses données sont des polynômes et des ensembles semi-algébriques alors on peut définir une hiérarchie de relaxations semidéfinies (SDP) – la hiérarchie moments-sommes-de-carrés (moments-SOS) – qui permet en principe d’approcher la valeur optimale avec une précision arbitraire. Le travail contenu dans cette thèse adresse deux facettes concernants le PGM et la hiérarchie moments-SOS: Une première facette concerne l’évolution des relaxations SDP pour le PGM. Le degré des poids SOS dans la hiérarchie moments-SOS augmente avec l’ordre de relaxation. Lorsque le nombre de variables n’est pas modeste, on obtient rapidement des programmes SDP de taille trop grande pour les logiciels de programmation SDP actuels, sauf si l’on peut utiliser des symétries ou une parcimonie structurée souvent présente dans beaucoup d’applications de grande taille. On présente donc un nouveau certificat de positivité sur un compact semi-algébrique qui (i) exploite la parcimonie présente dans sa description, et (ii) dont les polynômes SOS ont un degré borné à l’avance. Grâce à ce nouveau certificat on peut définir une nouvelle hiérarchie de relaxations SDP pour le PGM qui exploite la parcimonie et évite l’explosion de la taille des matrices semidéfinies positives liée au degré des poids SOS dans la hiérarchie standard. Une deuxième facette concerne (i) la modélisation de nouvelles applications comme une instance particulière du PGM, et (ii) l’application de la méthodologie moments-SOS pour leur résolution. En particulier on propose des approximations déterministes de contraintes probabilistes, un problème difficile car le domaine des solutions admissibles associées est souvent non-convexe et même parfois non connecté. Dans notre approche moments-SOS le domaine admissible est remplacé par un ensemble plus petit qui est le sous-niveau d’un polynôme dont le vecteur des coefficients est une solution optimale d’un certain SDP. La qualité de l’approximation (interne) croît avec le degré du polynôme et la taille du SDP. On illustre cette approche dans le problème du calcul du flux de puissance optimal dans les réseaux d’énergie, une application stratégique où la prise en compte des contraintes probabilistes devient de plus en plus cruciale (e.g., pour modéliser l’incertitude liée á l’énergie éolienne et solaire). En outre on propose une extension des cette procedure qui est robuste à l’incertitude sur la distribution sous-jacente. Des garanties de convergence sont fournies. Une deuxième contribution concerne l’application de la méthodologie moments-SOS pour l’approximation de solutions généralisés en commande optimale. Elle permet de capturer le comportement limite d’une suite minimisante de commandes et de la suite de trajectoires associée. On peut traiter ainsi le cas de phénomènes simultanés de concentrations de la commande et de discontinuités de la trajectoire. Une troisième contribution concerne le calcul de solutions mesures pour les lois de conservation hyperboliques scalaires dont l’exemple typique est l’équation de Burgers. Cette classe d’EDP non linéaire peut avoir des solutions discontinues difficiles à approximer numériquement avec précision. Sous certaines hypothèses, la solution mesurepeut être identifiée avec la solution classique (faible) à la loi de conservation. Notre approche moment-SOS fournit alors une méthode alternative pour approcher des solutions qui contrairement aux méthodes existantes évite une discrétisation du domaine.

Abstract

The generalized moment problem (GMP) is a linear optimization problem over spaces of measures. It allows to model many challenging mathematical problems. While in general it is impossible to solve the GMP, in the case where all data are polynomial and semialgebraic sets, one can define a hierarchy of semidefinite relaxations – the moment-sums-of-squares (moment-SOS) hierachy – which in principle allows to approximate the optimal value of the GMP to arbitrary precision. The work presented in this thesis addresses two facets concerning the GMP and the moment-SOS hierarchy: One facet is concerned with the scalability of relaxations for the GMP. The degree of the SOS weights in the moment-SOS hierarchy grows when augmenting the relaxation order. When the number of variables is not small, this leads quickly to semidefinite programs (SDPs) that are out of range for state of the art SDP solvers, unless one can use symmetries or some structured sparsity which is typically present in large scale applications. We provide a new certificate of positivity which (i) is able to exploit the structured sparsity and (ii) only involves SOS polynomials of fixed degree. From this, one can define a new hierarchy of SDP relaxations for the GMP which can take into account sparsity and at the same time prevents from explosion of the size of SDP variables related to the increasing degree of the SOS weights in the standard hierarchy. The second facet focusses on (i) modelling challenging problems as a particular instance of the GMP and (ii) solving these problems by applying the moment-SOS hierarchy. In particular we propose deterministic approximations of chance constraints a difficult problem as the associated set of feasible solutions is typically non-convex and sometimes not even connected. In our approach we replace this set by a (smaller) sub-level-set of a polynomial whose vector of coefficients is a by-product of the moment-SOS hierarchy when modeling the problem as an instance of the GMP. The quality of this inner approximation improves when increasing the degree of the SDP relaxation and asymptotic convergence is guaranteed. The procedure is illustrated by approximating the feasible set of an instance of the chance-constrained AC Optimal Power Flow problem (a nonlinear problem in the management of energy networks) which nowadays becomes more and more important as we rely increasingly on uncertain energy sources such as wind and solar power. Furthermore, we propose an extension of this framework to the case where the underlying distribution itself is uncertain and provide guarantees of convergence. Another application of the moment-SOS methodology discussed in this thesis consider measure valued solutions to optimal control problems. We show how this procedure can capture the limit behavior of an optimizing sequence of control and its corresponding sequence of trajectories. In particular we address the case of concentrations of control and discontinuities of the trajectory may occur simultaneously. In a final contribution, we compute measure valued solutions to scalar hyperbolic conservation laws, such as Burgers equation. It is known that this class of nonlinear partial differential equations has potentially discontinuous solutions which are difficult to approximate numerically with accuracy. Under some conditions the measure valued solution can be identified with the classical (weak) solution to the conservation law. In this case our moment-SOS approach provides an alternative numerical scheme to compute solutions which in contrast to existing methods, does not rely on discretization of the domain.

Mots-Clés / Keywords

Moments; Positive polynomials; Sparsity; Chance constraints; Measure valued solutions; Semidefinite relaxations; Polynômes positifs; Parcimonie; Contraintes probabilistes; Solutions mesures; Relaxations semidéfinies;

Abstract

This paper deals with the stability of discrete‐time networked systems with multiple sensor nodes under dynamic scheduling protocols. Access to the communication medium is orchestrated by a weighted try‐once‐discard or by an independent and identically‐distributed stochastic protocol that determines which sensor node can access the network at each sampling instant and transmit its corresponding data. Through a time‐delay approach, a unified discrete‐time hybrid system with time‐varying delays in the dynamics and in the reset conditions is formulated under both scheduling protocols. Then, a new stability criterion for discrete‐time systems with time‐varying delays is proposed by the discrete counterpart of the second‐order Bessel‐Legendre integral inequality. The developed approach is applied to guarantee the stability of the resulting discrete‐time hybrid system model with respect to the full state under try‐once‐discard or independent and identically‐distributed scheduling protocol. The communication delays can be larger than the sampling intervals. Finally, the efficiency of the presented approach is illustrated by a cart‐pendulum system.