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A common problem in the petroleum drilling process is the torsional oscillation generated by the friction present during the cutting process. Torsional oscillations in drill string are particularly difficult to control because the drill string is an underactuated system, it has a very small diameter to length ratio and it is driven at top end with the cutting process at the other end. These factors make the drill string prone to self-excited torsional vibrations caused by the stick-slip of the cutting bit. The system is modeled as a torsional pendulum with two degrees of freedom, where the upper inertia models the top drive and also part of the drilling pipes. The bottom inertia models the bottom hole assembly (BHA). The drill is considered to be a massless torsional spring-damper. The drill string is subjected to friction, which is formulated using a dry friction model. The friction model takes into account Coulomb friction, stiction and Stribeck effect. The latter friction component is the main nonlinear phenomenon that introduces negative damping at the bit; it leads to self-enforcing stick-slip torsional oscillations.In the approach of this work, for the attenuation of these self-excited oscillations a recursive backstepping control strategy is used and it is carried out in continuous time. The main contribution of this work, which is different from the backstepping approaches reported in the literature, is to use a nonlinear/artificial damping as virtual control input. The stability of the system has been proven in the sense of Lyapunov. The goal of the proposed algorithm is to deal the underactuation of the system and to provide a good response for different operating points. The effectiveness and robustness of the controller has been tested in simulations.

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