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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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Abstract. This article deals with the transposition of "fractal robustness" to automatic control. The considered dynamic model which g o verns this phenomenon is a non integer order linear di erential equation where the natural frequency and the damping ratio of the oscillatory mode of the solution are determined. A remarkable result is that the damping ratio is exclusively linked to the non integer order of the di erential equation. This is the corner stone of the CRONE control. The dynamic behaviour of this control is described in tracking and in regulation. The robustness of the damping ratio and of the resonance ratio is demonstrated. An open loop is de ned using closed loop performance speci cations. A C R ONE regulator is approximated by a n i n teger order transmittance. Finally, the principle of the CRONE suspension, the synthesis method and the performance are developed.R esum e. Cet article traite de la transposition en automatique de la "robustesse fractale". Le mod ele dynamique consid er e e s t u n e equation diff erentielle lin eaire d'ordre non entier dont l a f r equence propre et le facteur d'amortissement du mode oscillatoire de la solution sont d etermin es. Un r esultat remarquable tient a ce que le facteur d'amortissement est exclusivement li e a l'ordre non entier de l' equation di erentielle. Un tel r esultat est a l'origine de la commande C R ONE. Le comportement dynamique de cette commande est d ecrit en asservissement e t e n r egulation. La robustesse du facteur de r esonance est d emontr ee. Un transfert en boucle ouverte est d e ni a partir des sp eci cations des performances en boucle ferm ee. Un r egulateur CRONE est synth etis e par une transmittance d'ordre entier. En n, le principe de la suspension CRONE, sa m ethode de synth ese et ses performances s o n t d evelopp es.

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