Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples

Revisión de literatura de jerarquía volúmenes acotantes enfocados en detección de colisiones

(Eng) A bounding volume is a common method to simplify object representation by using the composition of geometrical shapes that enclose the object; it encapsulates complex objects by means of simple volumes and it is widely useful in collision detection applications and ray tracing for rendering algorithms. They are popular in computer graphics and computational geometry. Most popular bounding volumes are spheres, Oriented-Bounding Boxe s (OBB’ s), Axis-Align ed Bound ing Boxes (AABB’ s); moreover , the literature review includes ellipsoids, cylinders, sphere packing, sphere shells , k-DOP’ s, convex hulls, cloud of points, and minimal bounding boxe s, among others. A Bounding Volume Hierarchy is ussualy a tree in which the complete object is represented thigter fitting every level of the hierarchy. Additionally, each bounding volume has a cost associated to construction, update, and interference te ts. For instance, spheres are invariant to rotation and translations, then they do not require being updated ; their constructions and interference tests are more straightforward then OBB’ s; however, their tightness is lower than other bounding volumes. Finally , three comparisons between two polyhedra; seven different algorithms were used, of which five are public libraries for collision detection.(Spa) Un volumen acotante es un método común para simplificar la representación de los objetos por medio de composición de formas geométricas que encierran el objeto; estos encapsulan objetos complejos por medio de volúmenes simples y son ampliamente usados en aplicaciones de detección de colisiones y trazador de rayos para algoritmos de renderización. Los volúmenes acotantes son populares en computación gráfica y en geometría computacional; los más populares son las esferas, las cajas acotantes orientadas (OBB’s) y las cajas acotantes alineadas a los ejes (AABB’s); no obstante, la literatura incluye elipses, cilindros empaquetamiento de esferas, conchas de esferas, k-DOP’s, convex hulls, nubes de puntos y cajas acotantes mínimas, entre otras. Una jerarquía de volúmenes acotantes es usualmente un árbol, en el cual la representación de los objetos es más ajustada en cada uno de los niveles de la jerarquía. Adicionalmente, cada volumen acotante tiene asociado costos de construcción, actualización, pruebas de interferencia. Por ejemplo, las esferas so invariantes a rotación y translación, por lo tanto no requieren ser actualizadas en comparación con los AABB no son invariantes a la rotación. Por otro lado la construcción y las pruebas de solapamiento de las esferas son más simples que los OBB’s; sin embargo, el ajuste de las esferas es menor que otros volúmenes acotantes. Finalmente, se comparan dos poliedros con siete algoritmos diferentes de los cuales cinco son librerías públicas para detección de colisiones

Reachability of Uncertain Linear Systems Using Zonotopes

International audienceWe present a method for the computation of reachable sets of uncertain linear systems. The main innovation of the method consists in the use of zonotopes for reachable set representation. Zonotopes are special polytopes with several interesting properties : they can be encoded efficiently, they are closed under linear transformations and Minkowski sum. The resulting method has been used to treat several examples and has shown great performances for high dimensional systems. An extension of the method for the verification of piecewise linear hybrid systems is proposed

Privatized distributed anomaly detection for large-scale nonlinear uncertain systems

In this article two limitations in current distributed model based approaches for anomaly detection in large-scale uncertain nonlinear systems are addressed. The first limitation regards the high conservativeness of deterministic detection thresholds, against which a novel family of set-based thresholds is proposed. Such set-based thresholds are defined in a way to guarantee robustness in a user-defined probabilistic sense, rather than a deterministic sense. They are obtained by solving a chance-constrained optimization problem, thanks to a randomization technique based on the Scenario Approach. The second limitation regards the requirement, in distributed anomaly detection architectures, for different parties to regularly communicate local measurements. In settings where these parties want to preserve their privacy, communication may be undesirable. In order to preserve privacy and still allow for distributed detection to be implemented, a novel privacy-preserving mechanism is proposed and a so-called privatized communication protocol is introduced. Theoretical guarantees on the achievable level of privacy, along with a characterization of the robustness properties of the proposed distributed threshold set design, taking into account the privatized communication scheme, are provided. Finally, simulation studies are included to illustrate our theoretical developments

Distributed Fault-Tolerant Control of Large-Scale Systems: an Active Fault Diagnosis Approach

The paper proposes a methodology to effectively address the increasingly important problem of distributed faulttolerant control for large-scale interconnected systems. The approach dealt with combines, in a holistic way, a distributed fault detection and isolation algorithm with a specific tube-based model predictive control scheme. A distributed fault-tolerant control strategy is illustrated to guarantee overall stability and constraint satisfaction even after the occurrence of a fault. In particular, each subsystem is controlled and monitored by a local unit. The fault diagnosis component consists of a passive set-based fault detection algorithm and an active fault isolation one, yielding fault-isolability subject to local input and state constraints. The distributed active fault isolation module - thanks to a modification of the local inputs - allows to isolate the fault that has occurred avoiding the usual drawback of controllers that possibly hide the effect of the faults. The Active Fault Isolation method is used as a decision support tool for the fault tolerant control strategy after fault detection. The distributed design of the tube-based model predictive control allows the possible disconnection of faulty subsystems or the reconfiguration of local controllers after fault isolation. Simulation results on a well-known power network benchmark show the effectiveness of the proposed methodology

Détection d’intersection via l’application de Gauss; revue et nouvelles techniques

This paper delves into the problem of detecting the intersection of two convexpolyhedra.It does so through the lens of Minkowski sums and Gauss maps, and with a biastowards applications in computer graphics and robotics.In the first part, we show how Minkowski sums and Gauss maps come into play,give a brief survey of techniques for pairs of simple shapes and describe alow-level optimization of a naive algorithm for convex polyhedra, which isapplied to tetrahedra.Novel applications to the ray casting problem are also given.In the second part, we take a more abstract approach to the problem anddescribe a new and very efficient and robust algorithm for detecting theintersection of two convex shapes.The new technique works directly on the unit sphere, interpreted as the sphereof directions.In particular, it is compared favourably to the ubiquitous algorithm ofGilbert, Johnson and Keerthi.Cet article discute du problème (décisionnel) de la détection de l'intersectionde deux polyèdres convexes. Il porte particulièrement sur les applications dece problème en informatique graphique et en robotique. La discussion s'y faitdu point de vue des sommes de Minkoswki et de l'application de Gauss.Dans la première partie, nous rappellons le rôle de ce point de vue dans lacompréhension de la géométrie du problème. Nous donnons un bref aperçu destechniques conçues pour certaines paires de formes simples, et nous proposonsun algorithme naïf mais optimisé, traitant des polyèdres convexes quelconques.Nous traitons en exemple une application aux paires de tétraèdres et uneapplication au problème du lancer de rayons.En deuxième partie, nous approchons le problème de manière plus abstraite etdécrivons un nouvel algorithme robuste et rapide pour la détection del'intersection de deux objets convexes (non nécessairement polyédrique).Ce nouvel algorithme travaille directement sur la sphère unité que nousinterprétons comme l'espace des directions. En particulier, notre nouvelletechnique est comparée favorablement à celle, fort répandue, de Gilbert,Johnson et Keerthi