Grazing Analysis for Synchronization of chaotic hybrid systems

International audienceIn this paper, a Grazing bifurcation analysis is proposed and a way to chaos is presented. Moreover, based on this analysis an observer design for the synchronization of chaotic hybrid system is given

Control design of an HIV-1 model

In this paper, we formulate a dynamic mathematical model that describes the interaction of the immune system with the human immunodeficiency virus (HIV), combined with nonlinear continuous feedback control. The detailed computations of two linearizing inputs is presented. It results in the design of a first fully linearizable system and a second partially linearizable one. The proposed controllers have the ability to drive the system close to the healthy equilibrium state. Numerical simulations demonstrate them effectiveness by maintaining virus concentration in very low levels and healthy cells in satisfactory levels

Observability forms for switched systems with Zeno phenomenon or high switching frequency

International audienceThis paper deals with the observability of a class of switched systems with Zeno phenomenon or high switching frequency. Particularly, three observability forms are proposed and the observability for each form with knowledge of filtered switching signal is analyzed. Meanwhile, sufficient and necessary conditions for the existence of a diffeomorphism to transform a class of switched systems into one of such forms are presented. Examples and simulations are given at the end to highlight the theoretical results

Mathematical analysis of a model for chronic myeloid leukemia

In this paper, a mathematical analysis of a model describing the evolution of chronic myeloid leukemic with effect of growth factors is considered. The corresponding dynamics are represented by a system of ordinary differential equations of dimension 5. This system described the interactions between hematopoietic stem cells (H.S.C), hematopoietic mature cells (M.C), cancer hematopoietic stem cells, cancer hematopoietic mature cells and the associated growth factor concentration. Our research is, henceforth, carried out on the existence and the uniqueness of the solution of this system. The next substantive concern will be a discussion on the local and global stability of the corresponding steady states. Three scenarios, however, corresponding to different actions of hematopoiesis on stem cells (differentiate cells or both cells) are considered

Observability Normal Forms for a class of switched systems with zeno phenomena

International audienceThis paper deals with necessary and sufficient conditions to transform a class of switched systems to a particular form dedicated to observer design with and without zeno phenomena. Meanwhile, sufficient observability conditions for switched system with or without zeno phenomena are given. In the last section, some observer structures are proposed upon two academical examples

Symmetries impact in chaotification of piecewise smooth systems

International audienceThis paper is devoted to a mathematical analysis of a route to chaos for bounded piecewise smooth systems of dimension three subjected to symmetric non-smooth bifurcations. This study is based on period doubling method applied to the associated Poincaré maps. Those Poincaré maps are characterized taking into account the symmetry of the transient manifolds. The corresponding Poincaré sections are chosen to be transverse to these transient manifolds, this particular choice takes into account the fact that the system dynamics crosses the intersection of both manifolds. In this case, the dimension of the Poincaré map (defined as discrete map of dimension two) is reduced to dimension one in this particular neighborhood of transient points. This dimension reduction allows us to deal with the famous result "period three implies chaos". The approach is also highlighted by simulations results applied particularly to Chua circuit subjected to symmetric grazing bifurcations

Un nouveau type d'observateur impulsionnel

International audienceDans cet article, dans un premier temps, un nouveau type d'observateur hybride pour une classe de systèmes linéaires continus avec des mesures discrètes est présenté. L'approche utilisée est basée sur des conditions suffisantes de stabilité des systèmes dynamiques linéaires impulsifs. Celles-ci seront rappelées. Puis, des exemples seront donnés afin de mettre en évidence la conception de ces observateurs et montrer que ceux-ci sont généralisables aux systèmes non linéaires et aux mesures de type clairsemé (c-'a-d en dessous de la fréquence de Shannon-Nyquist)

Discussion about sliding mode algorithms, Zeno phenomena and observability

International audienceThis chapter is devoted to a discussion about the relations between first and high order slidingmode algorithms and both types of Zeno (Chattering and Genuinely) behaviors of switched dynamical systems. Firstly, the Henstock-Kurzweil integral is recalled in order to set up the problem of switched systems with Zeno phenomena, which enables to include Filippov solution and take into account some singularities. Then, observer designs based on the well-known super twisting algorithm are proposed. For this kind of problems, the importance of finite time convergence of the observation error is studied, and some simulations are given to highlight the discussion. Lastly, the two tanks example is given in order to point out the differences between both Zeno phenomena types, to show that there is life after Zeno and that a higher order sliding mode observer can be efficient before, during and after both Zeno phenomena types

Symmetries impact in chaotification of piecewise smooth systems

International audienceThis paper is devoted to a mathematical analysis of a route to chaos for bounded piecewise smooth systems of dimension three subjected to symmetric non-smooth bifurcations. This study is based on period doubling method applied to the associated Poincaré maps. Those Poincaré maps are characterized taking into account the symmetry of the transient manifolds. The corresponding Poincaré sections are chosen to be transverse to these transient manifolds, this particular choice takes into account the fact that the system dynamics crosses the intersection of both manifolds. In this case, the dimension of the Poincaré map (defined as discrete map of dimension two) is reduced to dimension one in this particular neighborhood of transient points. This dimension reduction allows us to deal with the famous result "period three implies chaos". The approach is also highlighted by simulations results applied particularly to Chua circuit subjected to symmetric grazing bifurcations

Chaotic behavior analysis based on corner bifurcations

International audienceIn this paper, a mathematical analysis in order to generate a chaotic behavior for piecewise smooth systems submitted to one of it's specific bifurcations, namely the corner one, is proposed. This study is based on period doubling method