The CRONE aproach: Theoretical developments and major applications

Abstract This article deals with fractal robustness. Of physical origin, this robustness represents the insensitivity of damping in nature, resulting from the combination of fractality and non integer differentiation. The study case is a natural robust phenomenon: the relaxation of water on a porous dyke whose damping ratio is independent of the mass of moving water. The dynamic model which governs this phenomenon is a non integer order linear differential equation for which the natural frequency and the damping ratio of the oscillatory mode of the solution are determined. A remarkable result is that damping ratio is exclusively linked to differential equation non integer order which is dictated by the fractal dimension of the dyke. Damping robustness is illustrated by two isodamping half-straight lines in the operational plane, and by a frequency template in the Nyquist plane. Then, the principle of the CRONE suspension, the synthesis method and the performances are developed. This suspension system is called the CRONE suspension because of the link with the second-generation CRONE control. The transposition to automatic control of the two isodamping half-straight lines, defines the robust strategy used by the second generation CRONE control. Defining a generalization of the second generation CRONE control, the great principles of the third generation CRONE control are also given. Finally, the problem of time domain identification by non integer model is studied. Among the whole of the methods developed in the CRONE team, a single one is dealt with in this paper. This one is based on the sampling of a non integer differential equation through the Grunwald numeric approximation of the non integer derivative of a time function. An application is carried out within the context of the identification of the dynamic behaviour of a thermal system. As far as the CRONE approach applications are concerned, a first part deals with the first ones showing the performance improvement. A second part is limited to a non exhaustive list of the other applications.

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