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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