Approximation of nonlinear dynamic systems by rational series

AbstractGiven an analytic system, we compute a bilinear system of minimal dimension which approximates it up to order k (i.e. the outputs of these two systems have the same Taylor expansion up to order k). The algorithm is based on noncommutative series computation: let s be the generating series of the analytic system; then a rational series g is constructed, whose coefficients are equal to those of s, for all words of length smaller than or equal to k. These words are digitally encoded, in order to simplify the computations of the Hankel matrices of s and g. We then associate with g, a bilinear system, which is a solution to our problem. Another method may be used for computing a bilinear system which approximates a given analytic system (S). We associate with (S) an R-automaton of vector fields and build the truncated automaton by cancelling all the states which have the following property: the length of the shortest successful path labelled by a word that gets through this state is strictly greater than k. Then, the number of states of this truncated automaton yields the dimension (not necessarily minimal) of the state-space

Weighted Petri nets and polynomial dynamical systems

In this article, we show that the generating series of polynomial dynamical systems are exactly the generating series of the subclass of weighted Petri nets where each transition has a single input place with arc weight 1. We propose furthermore an algorithm to check whether a given Petri net corresponds directly to a dynamical system. In many cases, different initial markings correspond to different dynamical systems. We finally prove that the place invariants for the Petri nets correspond to scaling Lie symmetries of the corresponding dynamical system, as well as that the invariants of the symmetry group of the dynamical system corresponds to implicit places in the corresponding Petri net. \\ Dans cet article, nous montrons que les séries génératrices des systèmes dynamiques polynomiaux sont exactement les mêmes que les séries génératrices d'une sous--classe de réseaux de Petri pondérés, dans lesquels chaque transition a une seule place d'entrée avec le poids de l'arc égal à 1. Nous proposons ensuite un algorithme pour vérifier si un réseau de Petri donné correspond directement à un système dynamique. Dans de nombreux cas, des marquages initiaux différents correspondent à des systèmes dynamiques différents. Nous montrons enfin que les invariants de places dans les réseaux de Petri correspondent aux symétries de Lie de changement d'échelle du système dynamique correspondant, ainsi que les invariants du groupe de symétrie du système dynamique correspondent aux places implicites de réseau de Petri correspondant

An algorithm for rule selection on fuzzy rule-based system applied to the treatment of diabetics and detection fraud in electronic payment,

International audienceAbstract—Recently, many papers have proposed automatic techniques for extraction of knowledge from numerical data and for minimization of the number of rules. But few works have been developed for design of experiments and datum plane covering. Most of optimization methods make the assumption that datum plane is sufficiently covered. If this assumption no longer holds, we will see that these methods may not work, since it implies that, before optimization, the fuzzy system gives acceptable results. We present in this paper, an algorithm for selection of fuzzy rules based on datum plane covering. We apply this method to two applications : the problem of treatment of diabetics. Taking the insulin infusion rate as the input and the blood glucose rate as the output, we consider the patient as a black box , whose model has to be obtained from the available measures of inputs–outputs. We dispose of a glycaemia file automatically produced for every person, and an insulin file shared by several persons. the problem of detecting fraud in electronic payment systems. Along with the development of credit cards, fraudulent activities become a major problem in electronic payment systems. Our model is based on specific and usual customer behavior, and deviation from such patterns is suspect

Generating series for drawing output of dynamical systems

We provide the drawing of the output of dynamical system ( ), particularly when the output is rough or near instability points. ( ) being analytical in a neighborhood of the initial state q(0) and de- scribed by its state equations, its output y(t) in a neighborhood of t = 0 is obtained by \evaluating" its generating series. Our algorithm consists in juxtaposing local approximating outputs on successive time intervals [ti; ti+1]0 i n1, to draw y(t) everywhere as far as possible. At every point ti+1 we calculate at order k an approximated value of each component qr of the state; on every interval [ti; ti+1]0 i n1 we calculate an approximated output. These computings are obtained from the symbolic expressions of the generating series of qr and y, truncated at order k, speci ed for t = ti and \evaluated". A Maple package is built, providing a suitable result for oscillating out- puts or near instability points when a Runge-Kutta method is wrong

Drawing Solution Curve of Differential Equation

International audienceWe develop a method for drawing solution curves of dierential equations. This method is based on the juxtaposition of local approximating curves on successive intervals [ti; ti+1]0i

GENERATING SERIES : A COMBINATORIAL COMPUTATION

International audienceThe purpose of this paper is to apply combinatorial techniques for computing coefficients of rational formal series (Gk) in two noncommuting variables and their differences at orders k and k-1. This in turn may help one to study the reliability and the quality of a model for non-linear black-box identification. We investigate the quality of the model throughout the criteria of a measure of convergence. We provide, by a symbolic computation, a valuation relating to the convergence of the family (Bk). This computation is a sum of differ- ential monomials in the input functions and behavior system. We identify each differential monomial with its colored multiplicity and analyse our computation in the light of the free differential calcu- lus. We propose also a combinatorial interpretation of coefficients of (Gk). These coefficients are powers of an operator Θ which is in the monoid generated by two linear differential operators ∆ and Γ. More than a symbolic validation, these computing tools are param- eterized by the input and the system's behavior

Weighted Petri nets and polynomial dynamical systems

In this article, we show that the generating series of polynomial dynamical systems are exactly the generating series of the subclass of weighted Petri nets where each transition has a single input place with arc weight 1. We propose furthermore an algorithm to check whether a given Petri net corresponds directly to a dynamical system. In many cases, different initial markings correspond to different dynamical systems. We finally prove that the place invariants for the Petri nets correspond to scaling Lie symmetries of the corresponding dynamical system, as well as that the invariants of the symmetry group of the dynamical system corresponds to implicit places in the corresponding Petri net. \\ Dans cet article, nous montrons que les séries génératrices des systèmes dynamiques polynomiaux sont exactement les mêmes que les séries génératrices d'une sous--classe de réseaux de Petri pondérés, dans lesquels chaque transition a une seule place d'entrée avec le poids de l'arc égal à 1. Nous proposons ensuite un algorithme pour vérifier si un réseau de Petri donné correspond directement à un système dynamique. Dans de nombreux cas, des marquages initiaux différents correspondent à des systèmes dynamiques différents. Nous montrons enfin que les invariants de places dans les réseaux de Petri correspondent aux symétries de Lie de changement d'échelle du système dynamique correspondant, ainsi que les invariants du groupe de symétrie du système dynamique correspondent aux places implicites de réseau de Petri correspondant

Stabilité glycémique en situation perturbée et adaptation thérapeutique

National audienceDans de précédents travaux, nous avons fourni un modèle du comportement "débit d'insuline/glycémie" du patient diabétique et une régulation de sa glycémie. Ce modèle comportemental est un modèle bilinéaire. Son acquisition consiste à identifier de 3 à 7 paramètres grâce aux données corrélées "débit de perfusion insulinique/glycémie" dont on dispose. Sur les tests initiaux, ce modèle de précision quadratique, présente en moyenne une erreur de 15% sur un intervalle de 15 minutes. Le problème posé est de savoir si ce modèle permet non seulement la prédiction sur les 15 minutes suivantes mais permet en plus de prévoir que le patient entre dans une période d'équilibre stable ou instable de sa glycémie. Il serait alors possible, à l'arrivée d'une perturbation (repas, activité physique, stress) de piloter au plus près les variations de distribution d'insuline, par une adaptation automatique du modèle de correction à la situation nouvelle détectée. En particulier, si l'équilibre est stable, la prédiction sera en principe valable sur un intervalle de temps plus long alors que si l'équilibre est instable, il y aura lieu de diminuer l'intervalle de temps. Partant de ce même modèle, nous proposons ici d'étudier sa stabilité Entrée-Bornée-Sortie-Bornée (EBSB) ce qui signifie en clair que pour une entrée (débit d'insuline) de faible amplitude dans le temps, la sortie (glycémie) a cette même propriété de faible amplitude dans le temps. Selon les équations décrivant le modèle, nous distinguons 3 cas : ou bien la fonction du temps glycémie se calcule explicitement et une conclusion sur la stabilité du modèle s'en déduit, ou bien sans savoir calculer explicitement la fonction glycémie, on est toutefois capable de calculer ses majorant et minorant, s'ils existent, en fonction des majorant et minorant de la fonction du temps débit d'insuline. Enfin, dans le dernier cas, on sait seulement calculer s'il existe des entrées débit d'insuline à valeurs constantes qui soient stabilisantes pour le modèle