International audienceAfter an overview of the results dedicated to the stability of systems described by differential equations involving fractional derivatives also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate orders hypothesis, it is shown that a direct extension of the second Lyapunov's method is a tedious task. If the fractional order ν is such that 0 < v < 1, the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI introduction. These conditions are applied to the gain margin computation of a CRONE suspension

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International audienceThis paper highlights several misinterpretations that arise in the field of fractional systems analysis using a representation known in the literature as "state space description". Given these misinterpretations, some results already published and based on this description are questionable. Thus alternative descriptions are proposed

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This paper proposes a new proof of the Matignon's stability theorem. This theorem is the starting point of numerous results in the field of fractional order systems. However, in the original work, its proof is limited to a fractional order ν such that 0 < ν < 1. Moreover, it relies on Caputo's definition for fractional differentiation and the study of system trajectories for non-null initial conditions which is now questionable in regard of recent works. The new proof proposed here is based on a closed loop realization and the application of the Nyquist theorem. It does not rely on a peculiar definition of fractional differentiation and is valid for orders ν such that 1 < ν < 2.

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback law is tackled here. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on numerical examples from the literature.