Considering the attractor structure of chaotic maps for observer-based synchronization problems

Abstract

To cite this version:Gilles Millérioux, Floriane Anstett, Gérard Bloch. Considering the attractor structure of chaotic maps for observer-based synchronization problems. AbstractThe main purpose of this paper is to state some sufficient conditions for global synchronization of chaotic maps. The synchronization is viewed as a state reconstruction problem which is tackled by polytopic observers. Unlike most standard observers, polytopic observers can account for a special property of chaotic dynamics. Indeed, it is shown that many chaotic maps can be described in a so-called convexified form, involving a time-varying parameter which depends on the chaotic state vector. Such a form makes it possible to incorporate knowledge on the structure of the compact set wherein the parameter lies. This set depends implicitly on the structure of the chaotic attractor. It is proved that the conservatism of the polyquadratic stability conditions for the state reconstruction, stated in a companion paper, can be reduced when the corresponding Linear Matrix Inequalities involve the vertices of the minimal convex hull of this set. Theoretical developments along with special emphasis on computational aspects are provided and illustrated in the context of adaptive synchronization.

An observer looks at the cell temperature in automotive battery packs

“…Then, as the state matrix A(ρ k ) for LPV systems depends linearly on ρ k , it can be rewritten in a polytopic form (Millérioux et al, 2005;Halimi et al, 2013) as…”

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“…, θ N are the N vertices of the convex polytope D ρ . Since A d depends linearly on ρ k , it is shown in (Millérioux et al, 2005) that A d (ρ k ) can always be decomposed in a polytopic form as:…”

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“…TheĀ i 's are constant matrices and are named vertices. In (Millerioux et al, 2005), the conditions under which such a rewriting is possible are provided along with the computational aspects for finding out thē A i 's. Therein, it is explained why (11) includes piecewise linear maps, Lur'e systems and most of chaotic systems with polynomial nonlinearities such as the Logistic map, the Henon map, the Mandelbrot map.…”