Interval numerical observer: Application to a discrete time nonlinear fish model

International audienceThe aim of this work is to reconstitute the state of a discrete-time nonlinear system representing a dynamical model of a harvested fish population. For this end, we are going to use a numerical method of building an interval observer for the consider discrete-time model fish population. We adapt to this model an algorithm called "Interval Moving Horizon State Estimation" (IMHSE) which gives an estimated interval of the system states. This algorithm is carried out in [8] and work well for a general class of discrete-time systems.Le but de ce travail est de reconstruire les états d’un système discret non linéaire représentant la dynamique d’une population de poissons soumise à l’action de la pêche. Pour cela nous allons utiliser une méthode numérique de synthèse d’un observateur intervalle du modèle discret de la population de poissons considéré. Nous adaptons à ce modèle un algorithme appelé "Interval Moving Horizon State Estimation" (IMHSE) qui permet d’estimer les états du système par des intervalles. Cet algorithme est développé dans [8] et marche bien pour une classe générale de systèmes discrets

A simple adaptive observer for a class of continuous linear time varying system with discrete output

International audienceIn this preliminary work an adaptive observer for continuous-discrete systems is proposed. It is well known that adaptive observer is very important for state and parameters estimation; most of known results concern continuous systems with continuous-output or discrete-time systems with discrete outputs (see for instance [1,3,7,11]). In this paper we consider a linear system with discrete outputs and we propose an algorithm to estimate both the state and the parameter without any persistence condition. The proposed adaptive observer is shown to be quite promising due to the exponential error convergence

An observer for a nonlinear age-structured model of a harvested fish population.

International audienceWe consider an age-structured model of a harvested population. This model is a discrete-time system that includes a nonlinear stock-recruitment relationship. Our purpose is to estimate the stock state. To achieve this goal, we built an observer, which is an auxiliary system that uses the total number of fish caught over each season and gives a dynamical estimation of the number of fish by age class. We analyse the convergence of the observer and we show that the error estimation tends to zero with exponential speed if a condition on the fishing effort is satisfied. Moreover the constructed observer (dynamical estimator) does not depend on the poorly understood stock-recruitment relationship. This study shows how some tools from nonlinear control theory can help to deal with the state estimation problem in the field of renewable resource management

On the stock estimation for some fishery systems

International audienceIn this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called "observer" can be helpful to deal with the resource stock estimation in the field of renewable resource management. It is often difficult or expensive to measure all the state variables characterising the evolution of a given population system, therefore the question arises whether from the observation of certain indicators of the considered system, the whole state of the population system can be recovered or at least estimated. The goal of this paper is to show how some techniques of control theory can be applied for the approximate estimation of the unmeasurable state variables using only the observed data together with the dynamical model describing the evolution of the system. More precisely we shall consider two fishery models and we shall show how to built for each model an auxiliary dynamical system (the observer) that uses the available data (the total of caught fish) and which produces a dynamical estimation $\hat x(t)$ of the unmeasurable stock state $x(t)$. Moreover the convergence speed of $\hat x(t)$ towards $x(t)$ can be chosen

Weak heteroclinic solutions and competition phenomena to anisotropic difference equations with variable exponents

Tyt. z nagłówka.Bibliogr. s. 744.In this paper, we prove the existence of weak heteroclinic solutions for a family of anisotropic difference equations under competition phenomena between parameters.Dostępny również w formie drukowanej.KEYWORDS: anisotropic difference equations, heteroclinic solutions, discrete Hölder type inequality, competition phenomena

Sur quelques problèmes d'observateurs, Applications à certains modèles d'écosystème aquatique

The goal of this thesis is twofold. First we show that some tools from control theory are helpful to adress the stock estimation problem for an exploited sh population. The stock estimation is one of the most important problem in shery science. To make a policy decision about the exploitation of renewable ressources, it is necessary to take into account the state of the resource stocks. This implies the need of a good estimate of the available resource. Mathematical models are more and more used to describe the evolution of biological systems. In the rst part of this work, we consider two mathematical models for shery resources. The rst one is a " stage structured " model that describes the dynamics of a population divided in stage-classes (according to age, length or weight) and submitted to the shing action. The second model is a " global " model that describes the evolution of a sh population that can move between an area where it can be harvested and a reserve area where no shing is allowed. Both models are given by systems of ordinary di erential equations. For our work, we show that a tool from nonlinear control theory called " observer " can be helpul to deal with the resource stock estimation in the eld of renewable resource management. More precisely we built a dynamical systems that uses the available data ( shing efort and the total or partial of caught sh) and which produces a dynamical estimation of the stock state. In the second part of this thesis, an adaptive observer for continuous-discrete systems is proposed. It is well known that adaptive observer is very important for join state and parameters estimation ; most of known results concern continuous systems with continuous-output or discrete-time systems with discrete output. In our work, we consider a linear system (obtained after some output injection transformation) with discrete outputs and we propose an algorithm to estimate both the state and the parameter.We use rst the persistence conditions to show the convergence of the proposed algorithm, after, we propose an algorithm for the join state and parameter estimation without any persistance condition. The proposed adaptive observer are shown to quite promising due to the exponentieal error convergence.Cette thèse a un double objectif ; en effet, dans un premier temps, le but est d'utiliser les outils de l'Automatique moderne(La théorie du contrôle en mathématique) pour résoudre des problèmes liés à la gestion des ressources de la bio-diversité en général, et de manière particulière à la gestion des ressources halieutiques pour ce qui concerne l'estimation des stocks. Nous avons considéré deux modèles de pêches, l'une structurée en stades et l'autre issue des modèles globaux. Nous avons utilisé sur ces deux modèles des techniques mathématiques pour construire des estimateurs d'état appelés Observateurs pour estimer l'état réel des systèmes que nous considérons. Ces techniques sont bien connues dans l'étude des systèmes modélisant les phénomènes physiques, mais rares sont les documents qui font état de l'utilisation des observateurs dans les systèmes modélisant certains écosystèmes telles les populations de poissons. Nous avons ici tenté d'associer la rigueur mathématique dans l'estimation des états des ces systèmes qui sont construits très souvent de manière empirique et complexe. La seconde partie de cette thèse traite des observateurs adaptatifs d'une classe de systèmes dont la sortie est à temps discret. En effet, nous considérons une classe de systèmes pouvant se mettre sous une forme particulière appelée forme adaptative, après une transformation via l'injection de sortie. Nous développons un algorithme récursif permettant d'estimer aussi bien l'état que quelques paramètres du système. La particularité de cette étude des observateurs adaptatifs est liée au fait qu'ici nous avons une sortie discrète, contrairement à ce que nous pourrions rencontrer dans la littérature pour les observateurs adaptatifs. Dans notre approche de ces observateurs adaptatifs continus discrets, nous avons d'abord construit l'estimateur d'état et de paramètres en trouvant un gain constant qui permet la convergence de l'algorithme ; ensuite nous considérons le même problème d'estimations d'état et de paramètres, mais sous un autre angle. Nous avons donc utilisé des techniques de l'excitation persistante pour permettre la convergence de l'algorithme

Interval numerical observer: Application to a discrete time nonlinear fish model

The aim of this work is to reconstitute the state of a discrete-time nonlinear system representing a dynamical model of a harvested fish population. For this end, we are going to use a numerical method of building an interval observer for the consider discrete-time model fish population. We adapt to this model an algorithm called "Interval Moving Horizon State Estimation" (IMHSE) which gives an estimated interval of the system states. This algorithm is carried out in [8] and work well for a general class of discrete-time systems.Le but de ce travail est de reconstruire les états d’un système discret non linéaire représentant la dynamique d’une population de poissons soumise à l’action de la pêche. Pour cela nous allons utiliser une méthode numérique de synthèse d’un observateur intervalle du modèle discret de la population de poissons considéré. Nous adaptons à ce modèle un algorithme appelé "Interval Moving Horizon State Estimation" (IMHSE) qui permet d’estimer les états du système par des intervalles. Cet algorithme est développé dans [8] et marche bien pour une classe générale de systèmes discrets

Stability analysis for a class of discrete schistosomiasis models with general incidence

Abstract In this paper, we propose to study a class of discrete schistosomiasis models with general incidence function. This model is derived from a continuous schistosomiasis model (in Appl. Math. 4:1682-1693, 2003) by using the backward Euler discretization method with step size h = 1 $h=1$ . We visit some basic properties of this discrete model and we study the stabilities of the equilibria by constructing some appropriate Lyapunov functional for the endemic equilibrium

Weak heteroclinic solutions of anisotropic difference equations with variable exponent

In this article, we prove the existence of heteroclinic solutions for a family of anisotropic difference equations. The proof of the main result is based on a minimization method, a change of variables and a discrete Holder type inequality