12. Practical Stabilization of the Drilling System

When dynamical systems are subject to external perturbations, it is not possible to establish exponential stability; nevertheless, from engineering point of view, the system response may be considered acceptable. This idea gives rise to the notion of ultimate boundedness or practical stability, which allows characterizing the transient behavior of a perturbed system. Under the assumption that the drilling system is subject to external disturbances, and that certain dynamics are frequently disregarded in the models, it is impractical to design control laws aimed at forcing the system response to reach a particular state; instead, we seek to drive the system trajectories into a given domain guaranteeing an acceptable system performance. This chapter concerns the practical stabilization of the drilling system based on two different modeling approaches to describe coupled axial–torsional rod dynamics. The first approach considers a coupled wave–Ordinary Differential Equation (ODE) model. The practical stabilization of the system is addressed via Lyapunov techniques allowing the design of stabilizing controllers to suppress the stick-slip and the bit-bounce. The second approach is based on a coupled neutral-type time-delay equation–ODE model. A pair of feedback controllers is derived from the attractive ellipsoid method which combines Lyapunov strategies and the principle of attractive sets