Regulation of the downside angular velocity of a drilling string with a P-I controller

In this paper, we demonstrate that a simple Proportional-Integral (P-I) controller allows to regulate the angular velocity of a drill-string despite unknown friction torque and by only using measurement of angular velocity at the surface. Our model is an one dimensional damped wave equation coupled at one boundary to dynamics which are subject to uncertainties. The control and the measured output are located at the other side. Writing this balance law system into Riemann coordinates, we design a novel Lyapunov functional in order to prove the exponential stability of the equilibrium in closed loop and show how it implies the regulation of the downside angular velocity.